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Modelling of Pyroelectric Detectors
and Detection by Digital Signal
Processing Algorithms
A thesis submitted to The University of Manchester for the degree of
Doctor of Philosophy in the Faculty of Engineering and Physical
Sciences
2012
Spyros Efthymiou
School of Electrical and Electronic Engineering
1
TABLE OF CONTENTS ABBREVIATIONS .......................................................................................................... 6
ABSTRACT ...................................................................................................................... 9
DECLARATION ............................................................................................................ 10
COPYRIGHT STATEMENT ......................................................................................... 11
ACKNOWLEDGMENT ................................................................................................. 12
1 Introduction ............................................................................................................. 13
2 Background ............................................................................................................. 17
2.1 Pyroelectric detectors ...................................................................................... 17
2.1.1 Theory of operation ................................................................................... 18
2.1.1.1 Radiation to thermal conversion ........................................................ 19
2.1.1.2 Temperature to electric charge conversion ........................................ 23
2.1.1.3 Pyroelectric current to voltage conversion ........................................ 28
2.1.1.4 Voltage responsitivity of CM-PED ................................................... 30
2.1.2 Modelling of pyroelectric detectors .......................................................... 31
2.1.2.1 Modelling by numerical methods (FEM) .......................................... 31
2.1.2.2 Modeling in the Laplace domain ....................................................... 32
2.2 Pulsed signal detection .................................................................................... 33
2.3 Pulsed signal theory ........................................................................................ 34
2.3.1 Synchronous demodulation ....................................................................... 37
2.3.1.1 Principle of operation ........................................................................ 37
2.3.1.2 Quadrature synchronous demodulation ............................................. 39
2.3.1.3 Noise and errors of QSD .................................................................... 40
2.3.1.3.1 Error analysis ................................................................................. 41
2.3.1.3.2 Noise performance......................................................................... 43
2.3.2 Gated integration ....................................................................................... 46
2.3.2.1 Principle of Operation ........................................................................ 46
2
2.3.2.2 Noise analysis of gated integration .................................................... 50
3 Tools and methods .................................................................................................. 52
3.1 Modelling pyroelectric detectors..................................................................... 52
3.1.1 Modelling with finite element methods .................................................... 53
3.1.1.1 Geometric modelling ......................................................................... 53
3.1.1.2 Subdomains and material properties .................................................. 54
3.1.1.3 Boundary conditions .......................................................................... 56
3.1.1.3.1 Specified temperature .................................................................... 56
3.1.1.3.2 Specified heat flux ......................................................................... 56
3.1.1.3.3 Highly conductive layer ................................................................ 57
3.1.1.3.4 Convective cooling ........................................................................ 57
3.1.1.3.5 Surface to surface or surface to ambient radiation ........................ 58
3.1.1.4 Meshing ............................................................................................. 59
3.1.1.5 Simulation parameters and setup ....................................................... 59
3.1.1.6 Modeling a pyroelectric detector array .............................................. 59
3.1.2 Modelling in Laplace domain ................................................................... 61
3.1.2.1 The transfer function of PED thermal model..................................... 61
3.1.2.2 Voltage responsitivity of a current-mode pyroelectric detector ........ 63
3.1.2.3 Transfer function of a TIA ................................................................. 64
3.1.2.3.1 Stability and compensation ........................................................... 65
3.1.2.3.2 Overall TF and voltage responsitivity of a PED ........................... 66
3.1.2.4 Noise considerations .......................................................................... 67
3.1.2.4.1 Johnson noise ................................................................................ 67
3.1.2.4.2 Op-amp input current noise (shot noise) ....................................... 69
3.1.2.4.3 Op-amp input voltage noise........................................................... 69
3.1.2.4.4 Total RMS output noise of the TIA ............................................... 71
3.1.3 Modelling of a PED in NI LabVIEW ....................................................... 72
3.1.3.1 Transient and frequency modelling of a PED .................................... 72
3
3.1.3.2 Noise simulation of a PED in LV ...................................................... 75
3.1.4 Pulsed performance of PEDs .................................................................... 76
3.2 Pulsed detection and SP with PEDs ................................................................ 78
3.2.1 Measuring the pulsed response of a PED with QSD and SMGI ............... 78
3.3 Gated quadrature synchronous demodulation ................................................. 81
3.3.1 Theory of operation ................................................................................... 81
3.3.1.1 Generation of the gating signal .......................................................... 82
3.3.1.2 Gating function .................................................................................. 83
3.3.1.3 Quadrature synchronous detection..................................................... 84
3.3.1.4 Determining M for an arbitrary pulse shape ...................................... 85
3.4 Signal processing software .............................................................................. 88
3.4.1 Overview of SPS ....................................................................................... 88
3.4.2 Signal generation/acquisition .................................................................... 90
3.4.3 Implementation of signal processing methods .......................................... 91
3.4.3.1 Estimation of M and D for GQSD and SMGI ................................... 92
3.4.3.2 Gating function .................................................................................. 93
3.4.3.3 Statistical analysis of the SP outputs ................................................. 94
3.4.4 Modeling of PEDs and measurements in SPS .......................................... 95
4 Evaluation of methods............................................................................................. 97
4.1 Evaluation of PED models .............................................................................. 97
4.1.1 Validating the FEM model ........................................................................ 98
4.1.1.1 Extracting the experimental thermal response of SPH-43 ................. 98
4.1.1.2 FEM results of the thermal response of SPH-43 ............................. 100
4.1.1.3 Comparing results within the LV-based Simulator ......................... 100
4.1.1.4 Summary of the objectives .............................................................. 102
4.2 Evaluation of SP methods ............................................................................. 103
4.2.1 Simulated data ......................................................................................... 103
4.2.1.1 Cases for evaluation ......................................................................... 103
4
4.2.1.2 Input signal ...................................................................................... 104
4.2.1.3 Sampling and processing settings .................................................... 106
4.2.1.4 Estimation of the output SNR and relative error ............................. 107
4.2.2 Experiments ............................................................................................ 108
4.3 Simulated measurements with SPH-43 ......................................................... 109
4.3.1 Simulation settings .................................................................................. 109
4.3.1.1 Excitation signal settings ................................................................. 109
4.3.1.2 PED settings ..................................................................................... 110
4.3.2 Digital signal processing ......................................................................... 111
4.3.2.1 GQSD settings ................................................................................. 112
4.3.2.2 SMGI settings .................................................................................. 113
4.3.2.3 CQSD settings ................................................................................. 114
4.3.2.4 Simulations summary ...................................................................... 115
5 Results and discussion .......................................................................................... 116
5.1 PED modelling .............................................................................................. 116
5.1.1 Verification of the FEM model ............................................................... 116
5.1.2 Simulation results of PED arrays ............................................................ 118
5.2 DSP Evaluation: GQSD, CQSD & SMGI .................................................... 121
5.2.1 Simulation results .................................................................................... 121
5.2.1.1 White noise ...................................................................................... 121
5.2.1.2 1/f Noise ........................................................................................... 122
5.2.2 Performance comparison on acquired and simulated data ...................... 124
5.3 Simulated measurements with SPH-43 ......................................................... 126
5.3.1 Case I: High voltage responsitivity (RF = 100GΩ) ................................. 126
5.3.2 Case II: Low voltage responsitivity (RF = 1GΩ) .................................... 129
5.3.2.1 Scenario 1: SNR performance for constant OTC and varying feff ... 129
5.3.2.2 Scenario 2: SNR performance for a constant feff and varying OTC 131
6 Conclusions and future work ................................................................................ 133
5
6.1 Conclusions ................................................................................................... 133
6.2 Future work ................................................................................................... 136
REFERENCES .............................................................................................................. 138
A APPENDIX ........................................................................................................... 143
A.1 Solution of the heat transfer ODE for a time invariant input Φ(t) ................ 143
A.2 Solution of the heat transfer ODE to a sinusoidal input Φ(t) ........................ 144
A.3 Solution of the heat transfer ODE in Laplace domain .................................. 145
B APPENDIX ........................................................................................................... 146
B.1 Fourier series representation of a rectangular pulse train f(t) ....................... 146
B.2 Fourier transform of a rectangular pulse ....................................................... 148
C APPENDIX ........................................................................................................... 149
C.1 Layout of a geometric model of pyroelectric detector SPH-43 .................... 149
C.2 Geometric model of an array of multiple pyroelectric elements based on SPH-
43 150
D APPENDIX ........................................................................................................... 151
D.1 Derivation of a non-ideal transfer function of transimpedance amplifier ..... 151
6
ABBREVIATIONS
AC Alternating current
ADC Analogue to digital conversion
BA Boxcar averaging
BC Boundary conditions
BPF Band-pass filter
CAD Computer aided design
CM Current mode
CQSD Conventional quadrature synchronous demodulation
DAC Digital to analogue converter
DC Direct current
DFG Difference frequency generation
DSP Digital signal processing
EGI Exponential gate integrator
EM Electromagnetic
EMS Electromagnetic spectrum
ETC Effective time constant
FDM Finite difference method
FEA Finite element analysis
FEM Finite element method
FET Filed-effect transistor
FFT Fast Fourier transform
FIR Finite Impulse Response
FS Fourier series
FT Fourier transform
GI Gated integration
GQSD Gated quadrature synchronous demodulation
GUI Graphical user interface
GW Gate window
HBE Heat balance equation
HPF High pass filter
IIR Infinite impulse response
IR Infrared
LHS Left hand side
7
LIA Lock-in amplifier
LMA Lumped mass approach
LV LabVIEW
MBT Maximum bandwidth threshold
MEMS Micromachining and microelectromechanical systems
NI National instruments
NMR Nuclear magnetic resonance
OLG Open loop gain
Op-amp Operational amplifier
OTC Observed time constant
PCB Printed circuit board
PDE Partial differential equation
PED Pyroelectric detector
PSD Phase sensitive demodulation
QSD Quadrature synchronous demodulation
RC Resistor-capacitor
RHS Right hand side
RMGI Recovery-mode gated integration
RMS Root mean square
RTC Radiation to thermal conversion
SAR Surface-to-ambient radiation
SD Synchronous demodulation
SMGI Static-mode gated integrator
SNIR Signal-to-noise improvement ratio
SNR Signal-to-noise ratio
SP Signal processing
SPS Signal Processing Software
SSR Surface-to surface radiation
TCC Thermal to Current Conversion
TDS Time domain spectroscopy
TemTeT Temperature Terahertz Tomography
TF Transfer function
THz Terahertz
TIA Transimpedance amplifier
8
VI Virtual instrument
VM Voltage mode
9
ABSTRACT
Pyroelectric Detector (PED) models are developed considering the classical heat
balance equation to simulate the detector’s response under specified radiation
conditions. Studies on the behaviour of a PED are presented under the conditions of step
function and a pulsed load. Finite Element Methods (FEMs) have been used to obtained
3D models of the resulting temperature field in a Lithium Tantalate (LiTaO3)
pyroelectric crystal, incorporated in a complete commercial detector, taking into
account details of its geometry and thermal connectivity. The novelty is the achieved
facility to predict the response to pulsed radiation, which is valuable for the engineering
of pulsed-source sensor systems requiring detection at room temperature.
In this thesis, we present a signal processing (SP) algorithm, which combines the
principle of Quadrature Synchronous Demodulation (QSD) and Gated Integration (GI),
to achieve achieve an improved signal-to-noise ratio (SNR) in pulsed signal
measurements. As a first step, the pulse is bracketed by a gating window and the
samples outside the window are discarded. The gate duration is calculated to ensure that
the periodic signal at the output has an “apparent” duty factor close to 0.5. This signal
is then fed continuously for QSD to extract the magnitude and phase of its fundamental
component, referenced to a sinusoidal signal with period defined by the gate length. An
improved SNR performance results not only from the increase of the average signal
energy, but also from the noise suppression inherent to the QSD principle. We introduce
this method as Gated Quadrature Synchronous Demodulation (GQSD), emphasizing the
synergy between GΙ and QSD.
10
DECLARATION
No portion of the work referred to in the thesis has been submitted in support of an
application for another degree or qualification of this or any other university or other
institute of learning.
11
COPYRIGHT STATEMENT
Copyright in text of this thesis rests with the author. Copies (by any process) either in
full, or of extracts, may be made only in accordance with instructions given by the
author and lodged in the John Rylands University Library of Manchester. Details may
be obtained from the Librarian. This page must form part of any such copies made.
Further copies (by any process) of copies made in accordance with such instructions
may not be made without the permission (in writing) of the author.
The ownership of any intellectual property rights, which may be described in this thesis,
is vested in The University of Manchester, subject to any prior agreement to the
contrary, and may not be made available for use by third parties without the written
permission of the University, which will prescribe the terms and conditions of any such
agreement.
Further information on the conditions under which disclosures and exploitation may
take place is available from the Head of School of Electrical and Electronic Engineering
or the Vice-President and Dean of the Faculty of Life Sciences, for Faculty of Life
Sciences’ candidates.
12
ACKNOWLEDGMENT
First, I would like to thank my family who has always supported me to study hard and
find my own paths in life.
I would like to sincerely thank my supervisor Prof. Krikor B. Ozanyan for his guidance
throughout my university studies. I am grateful to all colleges of SISP group for
making my time in Manchester unforgettable.
I would like to acknowledge Dr. Paul Wright and Prof. Patrick Gaydecki for assisting
me throughout the project and also for providing me the tools I needed to complete
experiments and obtain results.
Finally, I would like to thank the TempTeT group for the financial support to complete
my PhD.
13
CHAPTER
1 Introduction
Terahertz (THz) waves are non-ionizing electromagnetic (EM) waves in the frequency
range of 0.3THz to 3THz and lying between the infrared (IR) and microwave parts of
the EM spectrum. An attractive feature at this wavelength range is the ability to
penetrate materials, which are usually opaque to both visible and IR radiation.
Examples of these materials are paper, textiles and plastics, even a few centimetres of
brick. However, T-rays can be blocked by a metallic object or a thin layer of water. THz
wave technology is growing fast, creating the opportunity of imaging applications that
have not been possible previously. Another powerful potential of the THz spectral range
rises from the fact that it can be implemented as a coherent technique for making both
amplitude and phase measurements. Time-domain detection of THz pulses interacting
(reflecting or penetrating) with a sample can be Fourier-transformed into the frequency
domain, revealing a wealth of spectral information about the sample in terms of
amplitude and phase. Moreover, THz technology allows precise measurements of
parameters such as refractive index or absorption coefficients of a sample. However,
THz systems are still not widely accepted in practice because of their complexity and
limitations in efficiency, size and cost.
Optical absorption tomography imaging [1] is based on reconstruction from a
sufficiently large number of carefully chosen line-of-sight measurements, yielding path-
integrals of the measured quantity. In conventional absorption tomography, this quantity
is usually referred to as absorptivity. In desirable refinements, such as temperature
tomography, the main challenge in is that measurements equivalent to taking path-
integrals of temperature are not possible. However, the relative population of two
molecular vibrational-rotational states is temperature dependent and can be quantified
by absorption measurements [2], enabling temperature mapping from two tomographic
images.
14
Figure 1.1 Block diagram of a proposed setup for Fast T-ray Tomography with four THz
projections (“sheets”), between emitter and detection blocks at 0o, 45o, 90o, 135o across the object
Indeed, if two absorptivity images, A1 and A2, are generated by reconstruction of line-
of-sight data, the temperature and concentration maps can be calculated from A1 and
A2 pixel-wise by a robust algorithm or with the help of pre-calculated look-up tables.
The development of work done in this thesis was greatly influenced from the overall
aims and objectives of the Temperature Terahertz Tomography (TempTeT) project to
demonstrate a THz Tomographic imaging system operating at millisecond image frame
rates. The diagram in Fig. 1.1 indicates the proposed setup for Fast T-ray Tomography
indicating four THz projections at 0, 45, 90, 135 degrees. The final goal to achieve a
millisecond image frame rate is possible only by building a true multi-channel,
simultaneous measurement system with a large number of independent measurement
channels (32) and no moving parts (hence Fast Tomography).
The focus of TempTeT was on the THz emitting targets and the THz detectors
incorporated into the emitter and detector blocks, respectively (see top right corner of
Fig. 1.1). Tools and methods described in the following chapters are developed
according to the requirements for a single emitter-detector block which is then
multiplied times the number of projections and deployed in the final system. Properties
of the emitted radiation such as type, power, frequency etc. were essential in defining
the addressed problem. Additionally, the predefined detector of choice has created a
multipath planning arrangement, where detailed studies of the selected sensors were
necessary to suggest optimal configuration of the front-end electronics as well as
adequate selection of a SP processing scheme that will maximise the SNR of the
measurement. The system was targeting water vapour absorption suitable for
temperature imaging, which were found to lie in the range 1.0 THz to 2.5 THz.
15
Pyroelectric detectors (PED) are inexpensive room temperature sensors that are often
found in IR based applications. These sensors are AC coupled devices and thus they are
responsive only to modulated radiation. This feature suggests conventional detection
methods, such as synchronous demodulation (SD). SD techniques allow selective
amplification and extraction of the low frequency envelope, which quantifies the energy
of the transmitted signal. Pulsed narrowband THz radiation is generated by Difference
Frequency Generation (DFG) and it is usually employed by a master laser and
frequency-shifted optical parametric oscillator. The nature of the lasers used for
tuneable DFG dictates fairly low duty cycles, of the order of 10-3 and smaller. Even
though PEDs have attractive advantages in comparison to other sensors, they do have
some characteristics that make them less desirable in pulsed regime applications.
Firstly, they are naturally slow detectors implying long integration times. However,
speed, and hence the operational bandwidth, is limited by the electrical cut-off
frequency, which is usually traded with sensitivity. Lastly, the AC mode of operation
raises the question in terms of what is the optimal method to be utilised in order to
achieve a high quality measurement.
From the principal motivation of the current research outlined above, we can derive the
main aim of this work:
to develop methodology for the exploitation of inexpensive room-temperature THz
detector arrays in multichannel tomography systems with pulsed sources
To achieve this aim, extensive studies of PEDs have been undertaken to understand in
detail their principles of operation under various conditions, involving their pulse
performance at low duty cycle values. Subsequently, existing digital signal processing
(DSP) methods were considered in conjunction with the analysis of generic PEDs
leading to unique modelling approaches of commercially available devices. As will be
detailed further, this modelling led to the development of an original signal processing
method suitable for PEDs and PED arrays working with pulsed THz sources.
The content of this thesis is organised as follows. The first part of chapter 2 describes
the background theory and finite element method modeling of PEDs, while the second
16
part introduces the principles of operation of SD and Gated Integration (GI). The
developed modelling tools and methods are discussed in chapter 3. Additionally, the
novel Signal Processing (SP) algorithm, allowing efficient pulsed measurements, is
explained in detail. Evaluation of methods is addressed in chapter 4 where all methods
and simulations are verified and compared either to experimentally derived data or
against their performances. Moreover, this chapter introduces the reader to a unified
approach of modelling the performance of a PED. This approach allows the
implementation of SP algorithms to investigate their suitability with the detector of
choice. Chapter 5 demonstrates results of various experiments and simulations,
highlighting the advantages of the considered SP methods when used to measure the
pulsed data trains. Finally, conclusions and future developments are presented in
chapter 6.
17
CHAPTER
2 Background
The processing and analysis of electrical signals is a fundamental problem for many
engineers and scientists. During the last part of the twentieth century, rapid growth in
new industrial technology has been encouraged by the development of electronics and
particularly computers resulting to a parallel growth in new instruments and measuring
signal processing units [3]. The selection of a particular DSP algorithm depends on the
nature of the raw output signal of the incorporated transducers, in our particular case,
PED. In this chapter, PEDs are reviewed with respect to principle of operation and
modelling as intended to be used in the TempTeT project. Furthermore, adequate signal
processing techniques are discussed emphasizing their advantages and disadvantages in
pulsed detection with PEDs. These studies led to the development of a novel DSP
algorithm to measure efficiently the PED response to pulsed emitted radiation.
2.1 Pyroelectric detectors
Thermal detectors are often utilised in radiometric applications to convert the absorbed
radiation to a proportional electrical signal. In our particular case, PEDs have been the
detectors of choice due to their advantage to measure efficiently the intensity of THz
radiation [4]. Electronic devices like broadband antennas and Schottky diodes are often
used in THz systems. However, recent improvements in the category of thermal
detectors, such as Gollay cells, micro-bolometers and PEDs, have encouraged their use
in THz applications. Even though Golay cells are devices traditionally designed to
measure THz radiation, they do have major disadvantages. Compared to PEDs, Golay
cells are extremely fragile, large and cumbersome. Moreover, they tend to be slow in
response, very sensitive to mechanical vibration and very expensive.
18
Figure 2.1 Examples of commercial PEDs. (a) and (b) PEDs of different element size (b) 3D
assembly structure of a multi-channel PED
On the other hand, the compactness and small design of PEDs provides a good prospect
in the manufacturing of detector arrays of different size (see Appendix A). In addition to
that, PEDs are room temperature devices with competitive sensitivity, faster response
times and are relatively inexpensive. Bolometers have also been considered but their
limitation to operate efficiently at room temperature had led to the conclusion that the
choice of PEDs was the most effective and low cost solution for the requirement of the
TempTeT project. The following subsections describe in detail the physics and
modelling of PEDs according to the existing literature. Lastly, the important features of
PEDs are summarised, highlighting their effect to an optimal design and calibration of a
measuring system.
2.1.1 Theory of operation
The advances in micromachining and microelectromechanical systems (MEMS) have
been of great importance in the design and fabrication of micro (μ) scale PEDs. The
sensitive element of a PED is made out of a ferroelectric crystal and is usually located in
the middle of the device allowing direct exposure to incident radiation. Examples of
commercially available PEDs are shown in Fig. 2.1 where (a) and (b) illustrate two
PEDs of the same manufacturer but with different crystal dimensions and (c) depicts the
subassembly of a complex PED structure that contains two pyroelectric crystals within a
TO-39 housing can. Nowadays, dedicated design and simulation tools are used to model
various PED types, extending their use in a wide range of applications. PEDs belong to
the class of thermal detectors, in which absorbed radiation is first converted to heat and
subsequently to a measurable quantity.
19
Figure 2.2 Energy conversion stages of a PED
The overall conversion process breaks down to three stages as depicted by the red
arrows in Fig. 2.2. The rectangular blocks indicate the resulting quantity from each
conversion stage. The remaining text of this section describes each conversion stage
along with the existing literature on PEDs.
2.1.1.1 Radiation to thermal conversion
In thermal conversion processes, basic laws of thermodynamics are applied to form the
framework for heat transfer. An object exposed to varying radiative flux Φ(t) will
experience a rise (denoted by the Greek letter “Δ”) in temperature ΔTc that is
proportional to the heat flow within its volume. The distributed temperature within the
thermally sensitive element, in this case the pyroelectric crystal, is considered as the
measurable quantity whereas the incident radiant flux acts as the stimulus. The resulting
rise of the temperature distribution ΔTc(r,t) in an object, is obtained from the solution of
the general heat transfer, partial differential equation (PDE) shown in (2.1), derived in
most of heat transfer textbooks [5, 6].
The first three partial derivatives on the left hand side (LHS) of (2.1) represent the rate
of temperature change with respect to position. The term k denotes the thermal
conductivity, given by (2.2) where q is the heat flow, d is the thickness of the material
and ΔTc is the variation of temperature within the pyroelectric crystal.
The right hand side (RHS) term, term denotes the rate of temperature change within the
crystal under varying radiative boundary conditions.
tCρ
zk
zyk
yxk
xcccc
ΔΤΔΤΔΤΔΤ
(2.1)
ΔΤ
dqk (2.2)
20
According to literature [7-17], it is often convenient to model the thermal transient
response only as a function of time by neglecting the space dependent term on the LHS
of (2.1). This approximation is based on the assumption that for a non-complex sensor
structure, a pyroelectric crystal with micro-scale thickness (<60μm) and relatively small
area will not significantly change the temperature in all three directions. However,
according to [18-22], more complicated structures have been approximated to 1D
models to analytically obtain the thermal transient response of a PED both as a function
of time and space. Extending the latter approach, computer aided design (CAD) tools
apply numerical and iterative methods (i.e. finite element methods) on 3D models of
PEDs, or any other, improving further the accuracy of the overall simulation process
[23-25]. For reasons that are mentioned in the following sections, this review describes
the thermal transient response of the PED only as a function of time. By vanishing the
space dependent term and taking into account radiation boundary conditions, (2.1)
yields a first order ordinary differential equation (ODE) as shown in (2.3) where η
denotes the percentage of radiation converted to head.
The total thermal conductance (G) and capacitance (C) of a multilayer PED model are
calculated with the expressions (2.4) and (2.5) respectively where k is the specific
thermal conductance, ρ is the material density, C is the heat capacity and d is the layer
thickness. The thermal conductance G defines the time rate of steady state heat transfer
through a material of thickness d and area A, induced by a unit temperature difference
between the body surfaces. Therefore, an expression for G can be deduced by
rearranging (2.2) as shown in (2.6) where R is called thermal resistance and is the
reciprocal of the thermal conductance.
The corresponding layers are denoted by the subscripts of each quantity. The solution
of (2.3) for a particular moment of time yields an average temperature transient
response of the PED.
tΦη
G
t
dt
tdC
Tth
ddTth
ΔΤΔΤ(2.3)
m
mmT
th
Tth d
Ak
d
Ak
d
Ak
RG
2
22
1
11
1(2.4)
mmmmTth dACρdACρdACρC 22221111
(2.5)
d
kq
RG
ΔΤ
1(2.6)
21
This method is often addressed as the “lumped” capacitance method or Lumped Mass
Approach (LMA) [26, 27]. Assuming that the incident radiant flux is represented as
function of time Φ(t), the analytical solution of (2.3) is given by (2.7) (see Appendix
A.1).
where τth denotes the thermal time constant of the PED and is defined by the ratio of the
total thermal capacitance and total thermal conductance. The thermal response of the
PED to a sinusoidal input is described by (2.9) where the time dependent variable Φ(t)
is replaced by a sinusoidal signal expressed in polar form [21].
The lumped system shown in Fig. 2.3a, implies that the electric current within the
consisting components (resistor, capacitor) propagates instantaneously through the
system. This however, assumes that the size of the components is small compared to the
wavelength of the propagated signal implying a constant current throughout their
physical dimensions. Based on the same principle, a simplified “lumped” thermal model
of the PED is often associated to an equivalent electrical resistor-capacitor (RC)
network [8, 22, 26] to simulate its thermal transient response where electrical
components corresponds to the thermal properties of the PED (radiation, total thermal
capacitance and resistance).
Figure 2.3 PED equivalent electrical RC network of the PED thermal model
dttΦeC
eηt
tτ
Tth
τ
dth
th11
ΔΤ (2.7)
tωjeΦωtjΦ 0(2.8)
tωj
Tth
Tth
0d e
CωjG
Φηtωj
ΔΤ (2.9)
22
The equivalent Laplace transformed circuit of the PED thermal “lumped” model is
shown in Fig. 2.3b, in which all signals are represented by their Laplace transforms, all
components by their impedances, and initial conditions by their equivalent sources.
Laplace transforms provide a convenient and systematic approach in the analysis of
linear systems, where integro-differential equations are replaced by algebraic functions
in the complex plane. The three conversion stages of the PED can be conveniently
represented by suitably interconnected linear subsystems, each of which can be easily
analysed. Each subsystem is characterised by a transfer function, defined by the ratio of
the Laplace transforms of the output and input, when all initial conditions are zero.
As shown in (2.10), the ratio between the Laplace transform of the temperature change
ΔTd(s) and the Laplace transform of the incident radiation Φ(s), yields to the transfer
function of the equivalent thermal model of the PED, where RthT denotes the thermal
resistance, τth the thermal time constant and s is the Laplace operator (s = jω) (see
Appendix A.3). In terms of signal analysis, the PED lumped model behaves like a single
pole (1st order) low pass filter (LPF) with a time constant defined by the thermal
properties of the constituent materials.
Assuming sinusoidal input conditions, (2.11) and (2.12) (next page) corresponds to the
magnitude and phase response of the PED thermal model as a function of frequency,
respectively. Consequently, the bode plots of the magnitude and phase of the Radiation
to Thermal Conversion (RTC) transfer function (TF) are illustrated in Fig. 2.4a and Fig.
2.4b respectively.
Figure 2.4 Magnitude (a) and phase (b) response of the equivalent lumped circuit of a PED.
th
Tthd
τs
Rη
sΦ
ssH
1
ΔΤ1
(2.10)
23
The bandwidth of the PED thermal response is defined by the thermal cut-off frequency
fth. Beyond that point, the magnitude drops 20dB per decade (6dB/octave). The relation
between the fth and the thermal time constant is shown in (2.13).
2.1.1.2 Temperature to electric charge conversion
Pyroelectricity is an old phenomenon, which was investigated by the Curies et al before
the 1900s. The possibility of using the pyroelectric effect for IR detection was explored
after an important study in 1956, where a dynamic method was introduced to study the
pyroelectric effect [11]. The pyroelectric effect can be described by the
thermodynamically reversible interactions that may occur among the thermal,
mechanical and electric properties of a crystal. These interactions are shown in Fig. 2.5a
where the line joining pairs of circles signify that a small change in one of the variables
produces a corresponding change in the other. The physical properties of heat capacity,
elasticity, and electrical permittivity are defined by the three short bold lines that
connect pairs of the thermal, elastic and electric variables. Coupled effects are also
illustrated in the figure, denoted by lines joining pairs of circles at difference corners of
the diagram.
Figure 2.5 (a) Graphical illustration of the thermal, mechanical and electrical interactions within a
pyroelectric crystal where. (b) Planar projection of BaTiO3 lattice model illustrating the displacement of the atoms from the equilibrium.
221
1 th
Tth
τω
RηωjH
(2.11)
thωτωjH 11 tan
(2.12)
thth πτ
f2
1 (2.13)
24
For example, a small temperature increases θ produces an increase in entropy σ
proportional to the heat capacity divided by temperature. Pyroelectricity is a coupled
effect that relates a change in temperature to a change in macroscopic displacement D in
units of C/m2. The pyroelectric effect is often quantified a pyroelectric coefficient p,
which is simply the rate of change of spontaneous polarisation (Ps) with temperature.
According to Fig. 2.5a, a pyroelectric crystal has a primary and a secondary pyroelectric
effect. In the first, the crystal is rigidly clamped under a constant strain S to prevent
expansion or contraction. A change in temperature causes a change in electric
displacement as shown by the green line. The secondary effect is a result of crystal
deformation. Thermal expansion causes strain that alters the electric displacement via a
piezoelectric process, as signified by the dashed red lines. It is important to mention
here that pyroelectric crystals fall in the category of ferroelectric materials. The
polarization of these crystals can be reversed by sufficiently strong electric field. They
are also characterised by the Curie temperature, above which the material is paraelectric
(non-polar). Below this temperature, ferroelectrics are polar and can exhibit
pyroelectricity Thus, all ferroelectric materials are pyroelectric but only some
pyroelectric materials (those in which polarization may be switched by external field)
are ferroelectric. Clearly, all ferroelectric materials are piezoelectric.
The existence of the pyroelectric effect in any solid material requires that three
conditions be satisfied. Firstly, the molecular structure must have a nonzero dipole
moment. Secondly, the material must have non-centre of symmetry and thirdly the
material must have either no axis of rotational symmetry or a single axis of rotational
symmetry that is not included in an inversion axis. Of the 32 crystal point-group
symmetries, only 10 permit the existence of pyroelectricity. The lattice model in Fig.
2.5b shows a projection of the unit cell of barium titanate (BaTiO3) on the (100) plane
at a temperature of 291 K. The displacement of the atoms from their equilibrium
positions on a cubic lattice gives rise to the spontaneous polarization; its variation with
temperature is the pyroelectric effect [17].
25
The pyroelectric current is derived from the basic Maxwell’s equations when applied to
phenomena that involve electric fields in a matter [28]. Under these conditions and
according to Gauss’s law, a macroscopic displacements of positive and negative charges
of a dielectric material is formed and is described by an electric dipole moments given
by (2.14)
where s is a vector directed from the negative to the positive charge with a magnitude
equal to the distance between the charges. The electric polarization of a material is
defined by the product of the dipole moment m and the number of molecules per unit
volume. This is shown in (2.15) where N enumerates the molecules of the pyroelectric
crystal per unit volume.
The volumetric charge density of the accumulated charge caused by a three-
dimensional variation of the electric polarisation Ps is given by (2.16)
where qb represents the bound charge that is displaced due to the applied electric fields
and does not move freely through the material. The expression shown in (2.16) relates
the polarization process to the differential form of Guass’s law for electric fields. The
latter states that, the electric field produced by the electric charge within a material,
diverges from positive charge and converges upon negative charge. This is
mathematically, expressed by (2.17) where the dot product on the LHS describes the
tendency of the field to “flow” away from a specified location while and RHS denotes
the total electric charge density divided by the permittivity of free space ε0.
Thus, if q is equal to the sum of the bound charge density qb and free charge density qf
within the crystal, then (2.17) can be rewritten as shown in (2.18).
sm Q (2.14)
mP Ns (2.15)
sbq P (2.16)
0E
ε
q (2.17)
0E
ε
qq fb (2.18)
26
Figure 2.6 Pyroelectric crystal with its surfaces attached to conductive electrodes allowing current
flow due to change in temperature
Substituting (2.16) in (2.18) and solving for qf, the total free charge density is obtained.
The term in the parenthesis is written as a vector and is often called the “displacement
current” given by
Assuming that the electric field within the material is very small, the divergence of the
displacement current D is equal to the divergence of the spontaneous polarization Ps
resulting a charge density qf as shown in (2.21).
Integrating over a cylindrical-shaped volume that covers just the area of the surfaces
that are normally oriented to the polarization vector and applying Stokes’ theorem the
pyroelectric charge density J is obtained as shown in (2.22).
As shown in Fig. 2.6, the top and bottom electrodes are attached to the pyroelectric
crystal to collect the charge accumulated on the two surfaces. The derivative of (2.22)
yields the pyroelectric current Ip(t) where p and A are the pyroelectric coefficient and
area of the crystal respectively.
sf εq PE0 (2.19)
sε PED 0 (2.20)
fq sPD (2.21)
AdAdVdVqJSVV
f sss PPP (2.22)
dt
td
d
d
dt
Ad
dt
dJtI d
dp
ΔΤ
ΔΤss PP
(2.23)
dt
tdpAtI d
pΔΤ
(2.24)
27
Figure 2.7 PED current-magnitude (a) and current-phase (b) response
Performing the Laplace transform of (2.24) and rearranging, the transfer function H2(s)
of the Thermal to Current Conversion (TCC) stage is obtained as shown in (2.25).
To relate the incident radiation Φ(s) with the pyroelectric current Ip(s) equations (2.25)
and (2.10) are combined, yielding a transfer function (2.26) of which the magnitude
denotes the current responsitivity Ri. The magnitude and phase of Ri is shown in (2.27)
and (2.28).
From (2.26), one may observe that the current response of the PED acts like a high pass
filter (HPF). PEDs are AC coupled devices, and therefore only responsive to any input
energy flux that causes the temperature of the crystal to change. Evaluating (2.27) and
(2.28) for a specified frequency range, the current magnitude and phase response of the
PED is obtained as shown in Fig. 2.7a and Fig. 2.7b, respectively.
In addition to pyroelectricity, some ferroelectric crystals display piezoelectric
properties. Such crystals induce an electric change when they are under mechanical
stress. Under the same principle, an electric field across the piezoelectric crystal may
develop mechanical strain.
pAss
sIsH
d
p ΔΤ2
(2.25)
th
Tthp
i τs
pARηsspAsH
sΦ
sIsR
11 (2.26)
221 th
Tth
i
τω
ωpARηsR
(2.27)
thi ωτπ
sR 1tan2
(2.28)
28
This phenomenon is due to the non-centrosymetric structure of these crystals (21 of the
32 classes) allowing a more flexible motion of ions along one axis than others. By
applying pressure on the crystal, the charge separation between the individual atoms of
the crystal changes, creating an electric potential.
2.1.1.3 Pyroelectric current to voltage conversion
The voltage response of the detector is obtained by converting the pyroelectric current
to a voltage. This can be achieved in two modes; the Current Mode (CM) and Voltage
Mode (VM), depicted in Fig. 2.8 (a) and (b) respectively. At this stage, the PED can be
considered as a very high-impedance, low current-source (~nA to μA) with Rd
representing the output electrical resistance and Cd the output electrical capacitance of
the pyroelectric device. In both modes, the component Cc defines the equivalent
capacitance of the connecting cables to the input of the current to voltage (I-to-V)
converter. The ideal transfer functions of the VM and CM readout circuits are given by
(2.29) and (2.30) respectively. In (2.29), Re is the total resistance of the parallel
configuration of Rd and the load resistance Rin (Rd || Rin) and Ce is the capacitance of the
connecting cable and detector in parallel (Cd + Cc). Therefore, the electrical time
constants for the VM (τvm) and CM (τcm) circuits are given by the product ReCe and RfCf
respectively [29].
Figure 2.8 PED readout circuits (a) Voltage Mode (VM) and (b) Current Mode (CM)
vm
e
p
VMpVM
τs
R
sI
sVsH
13 (2.29)
cm
f
p
CMpCM
τs
R
sI
sVsH
13 (2.30)
29
Despite the low pass frequency behaviour of both pre-amplifier modes, , important
differences between the two must not be neglected, to ensure that the selected mode
satisfies the requirements of the intended application. According to [30], the VM read-
out circuit fits best in low frequency modulation schemes due to the high capacitance of
the sensor. On the contrary, since the input of the a CM is “virtually grounded” the
output signal is independent of the sensors capacitance and as a result is much faster.
In VM, the operational amplifier (op-amp) buffer shown in Fig. 2.8a, is often a field-
effect-transistor (FET) in common drain configuration where a shunt resistance Rs is
usually connected to the drain pin to provide thermal stabilization. Even though
integrating an FET within the detector minimises the complexity of electronics needed,
this approach restrains the bandwidth of the circuit at very low frequencies (0.01 – 0.1
Hz). Consequently, the separation of the modulated signal and the ambient temperature
drift becomes not trivial. This problem can be mitigated by adding a second stage
amplifier or a compensating crystal, acting as a notch filter at the frequency of
disturbance. As these solutions may suffer from other disadvantages such as high circuit
complexity and sensitivity reduction, the CM configuration provides a more robust
solution. In contrast to the VM, the feedback action of the op-amp in CM
(transimpedance stage), forces the two amplifier inputs to the same voltage, presenting a
virtual ground to the PED output current.
Comparing the two modes, the CM offers the advantage of higher sensitivity with
relatively smaller electrical time constants. Additionally, proper op-amp selection
ensures low output offsets and temperature drifts. The advantages of CM PED along
with the requirements of the TempTeT project had led to its selection and purchase from
Spectrum Detectors1 (SPH-43). Even though the ideal transfer function of CM
preamplifier is identical to a simple 1st order LPF, in practice, the finite open-loop gain
of the op-amp leaves a residual signal across the PED that in turn yielding a new
bandwidth limit and potentially oscillatory behaviour [31]. Throughout this work,
various CAD tools have been used to study the behaviour of this particular detector in
detail. Rigorous analysis of the CM preamplifier is discussed in the following chapters
including the noise performance of SPH-43.
1 Spectrum Detector has been bought by Gentec-EO since June 7, 2010.
30
Figure 2.9 Magnitude (a) and phase (b) voltage response of the PED as a function of frequency.
2.1.1.4 Voltage responsitivity of CM-PED
From basic control theory, the overall transfer function of a CM-PED is the product of
its constituent transfer functions. This product is also equal to the ratio of the Laplace
transform of the input radiant flux Φ(s) and the Laplace transform of the
transimpedance output voltage Vp(s) as.
Thus, with a voltage responsitivity given by (2.32) the output magnitude and phase of a
CM-PED is shown in Fig. 2.9 as a function of frequency. The thermal and electrical cut-
off frequencies of the PED are denoted by fth and fel respectively.
In summary, a PED, operating in either CM or VM, behaves like a band-pass filter
(BPF) with a bandwidth, defined by its thermal and electrical cut-off frequencies. In the
design process of a front-end detection scheme, it is often desired to maximise the SNR.
Although the overall magnitude of the detector may increase by enlarging the overall
thermal resistance RthT , it is however possible that higher values of Rth
T will cause both,
the magnitude of denominator and numerator of (2.32), to increase keeping intact the
voltage response of the PED while its bandwidth is reduced. On the other hand, the
voltage responsitivity and bandwidth of the PED is increased as the total heat
capacitance of the PED thermal model is reduced. It can also be speculated that the SNR
cm
f
th
Tthp
T τs
RpAs
τs
Rη
sΦ
sVsH
11
)( (2.31)
2222 11 cmth
fTth
vT
τωτω
pARRηRsH
(2.32)
31
improves for large values of feedback resistor Rf.. However, this may result to a
significant increase of noise and consequently deteriorating the output SNR.
Adequate and accurate modelling of a PED is therefore considered necessary to
investigate possible solutions to maximise the output SNR not only by increasing the
output magnitude but also by reducing the output noise.
2.1.2 Modelling of pyroelectric detectors
2.1.2.1 Modelling by numerical methods (FEM)
FEM analysis is used in cases where the spatial temperature distribution of complicated
structures cannot be neglected. Two-dimensional (2D) FEM modeling is reported in
[24], to simulate the temperature field of a multilayer pyroelectric thin film detector.
With the advances in technology, supercomputers are now more than capable to
perform transient analysis of 3D models. In [32] and [33] the obtained temperature
transient response of a 3D model led to novel designs, enhancing significantly the rate
of temperature variation and voltage responsitivity of a PED. Nonetheless, FEM
analysis allows further studies on modelling the thermal behaviour of an array of
multiple pyroelectric elements [23, 25]. In this research, we focus on 3D PED models
where the general 3D heat transfer equation is solved according to the material
properties and geometry dimensions of the model. A simplified model of a PED, is used
in Fig. 2.10 to elucidate briefly a FEM analysis performed in COMSOL Multiphysics
[26]. The materials used in the PED model can be selected from existing material
libraries, to link their physical properties to the corresponding subdomains; if a
particular material is not available, these properties are provided manually. A simple
structure of a multilayer PED model consists of five main layers; the top electrode, the
pyroelectric crystal, the bottom electrode, air and electrically isolated substrate. The top
and bottom electrodes are often neglected due to their very small thickness (~20nm).
Figure 2.10 Simplified multi-layer structure of PED 3D model
32
Subsequently, the model is meshed, dividing the whole structure in finite size elements
[34, 35]. Initial conditions and boundary constraints must be set for each surface and
subdomain to obtain a solution of the governing partial differential equation. FEM
modelling is a good choice for solving partial differential equations over complicated
structures. Moreover, it can handle a variety of engineering problems, individually or
combined, such as solid mechanics, dynamics, fluids, electrostatic etc.
2.1.2.2 Modeling in the Laplace domain
The main disadvantage of FEM modelling is that, a general closed-form solution, which
would permit one to examine the system’s response to changes in various parameters,
cannot be produced. On the other hand, the simplicity of an equivalent “lumped”
network of a thermal model becomes attractive with the sacrifice of some degree of
accuracy. In addition to this, the thermal equivalent model of a PED, along with the
concomitant electronics, can be modelled individually, or as a complete system, to
obtain the response of the PED. Computer-simulation tools such as SPICE can be a
powerful aid to obtain the response of a PED as described in [15]. A complete
equivalent electrical circuit is used in [9] where the equivalent electrical components of
the PED thermal model and its consisting electronics are varied to observe their impact
on performance. Extending this idea in Laplace domain, the transfer functions
describing each conversion stage, are used in [13] where Matlab/Simulink is utilised to
simulate the thermal, current and voltage response of the PED.
33
2.2 Pulsed signal detection
Pulsed regime applications are often encountered in various fields such as spectroscopy,
imaging, radar systems and many others, where the rich spectral content of pulsed
signals is advantageous in the transmission of information. Pulse radar systems utilise
short pulse transmission at the required energy while adequate signal processing
techniques are used to efficiently measure the received pulsed signals [36]. These
systems are often used in air-traffic control systems, ballistic missile and air defence
systems, targeting systems etc.
The discovery of ultrashort pulses has been also a breakthrough within the THz region
of the electromagnetic spectrum (EMS) where time domain spectroscopy (TDS)
techniques (range 30GHz to 3THz) are used for the characterization of various materials
[37]. Subsequently, THz-TDS provides time information that allows the development of
various three-dimensional THz tomographic imaging modalities [38]. Optimal
performance is ultimately dependent on the quality of the measured physical quantity.
Adequate SP techniques are used to measure the signal parameter, which corresponds to
the quantity of interest. However, appropriate selection of a SP technique is strongly
related to the voltage response of the front-end sensors and therefore the study of the
detected signal must not be considered trivial.
In our particular case, the detected signal is obtained by a pulsed excited PED. This is
due to the specific THz source developed and used in the project, based on Difference
frequency generation from two ns pulses emitted from a fibre laser [39]. For the
purposes of this research, the transmitted pulsed signals are considered synchronous and
therefore the first obvious SP method is quadrature synchronous demodulation2 (QSD).
However, QSD is not always an optimal choice in pulsed systems and therefore other
techniques such as GIs are deployed, due to their ability to recover fast waveforms and
resolve features down to a nano-second (ns) level [40]. The rest of this section is
organised as follows. First, pulsed signals are examined in the time and frequency
domains. Then QSD and GI are described separately, focussing on their signal
processing performance with emphasis on noise.
2 In literature, Quadrature Synchronous Demodulation is often referred to as Dual-Phase Lock-In Amplification or Phase Sensitive Detection.
34
2.3 Pulsed signal theory
In signal theory, an ideal pulse pattern has the meaning of an instantaneous amplitude
change, from the baseline to a higher value and back to the baseline (Vbase). Periodical
repetition of this pattern yields an ideal pulse train with a period and pulse duration
(denoted here by T and χ, sec). The duty cycle of a pulse train is given by (2.33).
The energy of a pulse train signal is often characterized by the average voltage (Vavg)
equal to the product of the peak-to-peak voltage (Vpp) and the duty cycle δ as shown in
(2.34).
The duty cycle δ is a dimensionless variable and is often described in percentage (%),
defining the amount of signal period occupied by the pulse length. An example of an
ideal rectangular pulse train is shown in Fig. 2.11 where Vpp and Vavg is the peak-to-peak
and average voltage of the pulse train respectively. In electronic engineering, it is often
convenient to perform analysis of analogue and/or digital signals in the frequency
domain. The mathematical intuition of frequency domain analysis has first been
announced in 1807 by Joseph Fourier claiming that, an arbitrary function (with or
without discontinuities) which is defined within a finite interval can always be
approximated by a sum of sinusoids of different magnitudes and frequencies [41].
Figure 2.11 Example of an ideal pulse train
T
χδ (2.33)
pkpkavg VδVT
χV
0(2.34)
35
Studying signals in the frequency domain often reveals information that cannot be
obtained in the time domain, especially when the signal is mixed with background
noise. The compact trigonometric Fourier series (FS) of a rectangular pulse train is
shown in (2.35) (see Appendix B)
where θh and h are the phase and absolute magnitude of each harmonic enumerated by
the letter h, δ is the duty cycle and f0 is the pulse repetition rate. Transformation of a
time dependent function to the frequency domain, and vice versa, is possible by the
Fourier transform (FT). Applying the FT on an aperiodic signal, a continuous function
is formed with respect to frequency. If however the signal is periodic then the output of
the FT is sampled at the frequency harmonics specified by the FS harmonics. This is
shown in Fig. 2.12 where the dotted line represents the FT of an aperiodic rectangular
pulse and the bullet-spikes depict the FT of a periodic extension of the same rectangular
pulse, with period of T0.
Figure 2.12 Fourier transforms of a rectangular pulse (dotted line) and its periodic extension
(bullet-spikes)
0
02cos
h
hhpp θthfπVδts
(2.35)
πδhhπ
Vpph 2cos12 (2.36)
δhπ
δhπθh 2sin
12costan 1 (2.37)
36
The frequency spectrum of a rectangular pulse train has three important features. First,
the fundamental harmonic has always the highest magnitude. Second, the spectrum of a
rectangular pulse signal with a duty cycle of 0.5 consists of odd multiples of the
fundamental harmonic. Lastly, the magnitude of the fundamental component is
maximised when the δ is 0.5. One of the objectives of a signal recovery process is to
extract a parameter that corresponds to the transmitted energy of the pulse train, or any
arbitrary pulsed signal. For example, a SP technique may be used to measure the
average energy of the time domain pulsed signal which in the frequency domain,
corresponds to the magnitude of the zero-frequency component. Other techniques may
use the demodulation principle to extract the magnitude and phase of any of the
consisting harmonics of interest. Naturally, optimal performance of a SP method must
ensure high output SNR, either by maximising the amplitude of the output signal or by
reducing the output noise. The following sections describe the principles of operation of
a static-mode gated integration and a quadrature synchronous demodulation. Their noise
performance is discussed under pulsed input conditions.
37
2.3.1 Synchronous demodulation
Synchronous demodulation has many applications in engineering and it is considered as
one of the most effective and widely used techniques for recovering signals that are
buried in noise. This technique is often called lock-in detection or lock-in amplification
(LIA) with the term “lock” defining the synchronous relationship between the input and
a reference signal. It is worth mentioning that even though the principle of operation of
LIAs is based on SD, this technique has the tendency to be described as an instrumental
lock-in technique capable of recovering signals dominated by noise. However, in
literature [42-45] the two methods are considered equal, and therefore in this thesis the
general terms SD and quadrature SD (QSD) are used. Due to the large literature and
bibliography on SD, this review addresses only their principles of operation and some
other fundamental aspects that are related to the content of the following chapters.
2.3.1.1 Principle of operation
As shown in Fig. 2.13, the block diagram of a simple SD consists of a phase shift unit, a
multiplier and a LPF. Assume that the input signal s(t) and reference signal r(t) are both
pure, noiseless sinusoids given by (2.38) and (2.39) respectively
where φ0 and φr, f0 and fr, and S0 and R denote the phase, frequency and amplitude of
s(t) and r(t) respectively. With the assumption that f0 is equal to fr, the multiplication of
the two signals yields to a sinusoid of double frequency with an additional DC term
proportional to the phase difference between the two.
Figure 2.13 Block diagram of a simple synchronous demodulator
000 2sin φtfπSts (2.38)
rr φtfπRtr 2sin (2.39)
38
The resulting signal is mathematically expressed in (2.40) where Δφ denotes the phase
difference between the reference and the input signal.
Under ideal filtering conditions, the frequency dependent term on the RHS of (2.40) is
rejected while the DC term appears at the output of the LPF. Therefore, the output of
SD is a DC signal proportional to the amplitude and phase difference of the reference
and input signals assuming that the amplitude of r(t) is one (2.41). Due to this phase
dependency, SD may also be addressed as phase sensitive demodulation (PSD).
Maximum sensitivity is achieved, ensuring that the phase difference Δφ is zero or as
close to zero as possible. Phase shifter circuits are often provided at the input of the
PSD to make this possible.
Figure 2.14 Principle of operation of SD in Fourier domain. The input and reference spectra are
shown in (a) and (b) respectively. Their product (c) is multiplied with the magnitude response (d) of the LPF of which the output is shown in Fourier (e) and time domain (f).
rφφtfπRS
φRS
trtstI 0000 2cos2
Δcos2
(2.40)
φStVI Δcos
20
(2.41)
39
With the assumption that Δφ is ideally zero, a more concrete understanding of this
technique is obtained when described in the Fourier domain. The Fourier spectra of a
sinusoidal input signal and a sinusoidal reference signal are shown in Fig. 2.14a and
Fig. 2.14b respectively. Subsequently, the spectrum of their product, depicted in Fig
2.14c, is multiplied by the magnitude response of the LPF (Fig. 2.14d) to reject the
frequency dependent term. Thus, the remaining component is the phase sensitive term
as illustrated in Fig. 2.14e. Assuming a 1st order LPF, the output transient response with
zero initial conditions is shown in Fig. 2.14f. The transfer function of a 1st order LPF is
given by (2.42) where τsd denotes the time needed for the output to reach ~63% of its
maximum value. Following basic signal processing theory [41] the time constant τsd
and the cut-off frequency fsd of a 1st order LPF is given by (2.43) and (2.44)
respectively.
2.3.1.2 Quadrature synchronous demodulation
In radiometric applications, it is often desirable to measure both the magnitude and
phase of the detected signal with respect to the provided reference. This is achieved by
multiplying the input signal s(t) with a pair of reference sinusoids which one of them is
shifted by π/2, as shown in (2.45) and (2.46).
Figure 2.15 shows the block diagram of a QSD, which is essentially a duplicated
version of SD where the second PSD is fed with the same input signal as the first but
driven by a reference signal that is shifted 90o. Subsequently, each of the two products
in (2.45) are low pass filtered to obtain the “in-phase” and “quadrature” components,
VI(t) and VQ(t) respectively.
sdLPD τs
sH
1
1
(2.42)
sdsdsd fπωτ
2
11
(2.43)
sdsd πτ
f2
1 (2.44)
trtstQ
trtstI
q
i
(2.45)
where o
rqq
rii
fπRtr
fπRtrtr
902sin
2sin
(2.46)
40
Figure 2.15 Basic block diagram of a QSD
The vector magnitude VM(t) and the phase θ of the frequency component of s(t)
correlating with r(t) are given by (2.47) and (2.48).
where ΔΘ(t) is the phase difference between the harmonic of interest and the reference
signal. This configuration is often referred to as a dual-phase LIA, and is capable of
displaying the output signal in rectangular of polar form. The main advantage over the
single-phase configuration is that, QSD provides a continuous magnitude reading
regardless of possible phase variations, avoiding the necessity of persistent phase
alignment by the phase-shifter circuit.
2.3.1.3 Noise and errors of QSD
According to [40, 43, 46], the expected signal-to-noise improvement ratio (SNIR) of a
SD depends on the the input and output noise bandwidths, Bni and Bno respectively.
Their relation to the SNIR is shown in (2.49) where the SNRi and SNRo are the input
and output SNR respectively.
tVtVtV QIM22
(2.47)
ttV
tVθ
I
QΔΘtan 1
(2.48)
no
ni
i
o
B
B
SNR
SNRSNIR (2.49)
41
Figure 2.16 Common application of QSD used to measure the attenuation of the intensity of a
chopped laser beam after passing through a physical substance
In this review, the error and noise performance of QSD is discussed in two parts. The
first part, considers the case where the input signal is not a pure sinusoid but instead, is
a square wave with an additional DC component. With a reference signal correlating
with the fundamental component of the square wave, this analysis describes how the
DC component along with odd harmonics comprising the square wave, may influence
the output of QSD. In the second part, the same square wave signal is mixed with
Gaussian noise to address the noise performance of the QSD.
2.3.1.3.1 Error analysis
The block diagram shown in Fig. 2.16 illustrates a common application where a QSD is
used to measure intensity attenuation of transmitted radiation due to interaction with a
physical substance. The raw output of the emitter is a continuous wave (CW) laser beam
mechanically chopped (modulated) at a frequency specified by a sinusoidal reference.
According to the theory discussed earlier, the output of QSD will relate to the
magnitude and phase of the harmonic referenced by the modulation frequency. For the
purposes of this analysis, we assume that the sample is completely transparent to the
incident radiation, resulting in maximum QSD output voltage. It is also assumed that the
output of the sensor s(t) is a noiseless square wave with peak-to-peak voltage AV and
DC offset (A/2)V. According to the Fourier theorem, any arbitrary signal is composed
by a sum of an infinite number of sinusoids.
42
Figure 2.17 Time domain square wave (a) with its Fourier spectrum (b). The fundamental
frequency for this example is 1kHz
In this particular case, the Fourier series representation of a square wave shown in Fig.
2.17a is given by (2.50) and is graphically expressed in Fig. 2.17b.
where h enumerates the harmonics of s(t). The Fourier spectrum of each PSD output
consists of the phase sensitive (correlated harmonic) and the sum of higher frequencies
(non-correlated harmonics). This is clearly illustrated in Fig. 2.18a and 2.18b where the
Fourier spectra of the resulting in-phase I(t) and quadrature Q(t) components are
computed using the fast Fourier transform (FFT) and plotted respectively. The phase
sensitive terms of each PSD product, are depicted by the blue dotted lines while the
remaining harmonics are un-correlated. An additional harmonic appears at the
modulation frequency due to the existence of the DC input offset, and in this particular
case has a frequency of 1 kHz. The filtering stage of QSD weights the output signals of
the PSDs according to the frequency response of the LPFs.
Figure 2.18 Fourier spectra of the in-phase (a) and quadrature (b) components resulting from the
product between the reference signal and the input square wave
1
0
12
212sin
2h
h
tfπh
π
AAts (2.50)
43
Figure 2.19 Illustration of the magnitude frequency response of the LPF along with the spectra of
the in-phase and quadrature components
The magnitude frequency response of an nth order RC LPF is given by (2.51) where fsd
is the cut-off frequency of the LPF and n is the order of the filter.
Ideally, the purpose of the LPFs is to extract only the DC terms of I(t) and Q(t) and
subsequently to obtain the vector magnitude VM(t) of the reference harmonic. However,
due to the non-ideal behaviour of practical filters, the energy of some of the un-
correlated harmonics will contribute to the output of the LPF, increasing the magnitude
of VM(t). According to (2.51) minimum errors occur at low cut-off frequencies and high
order filters. This is shown in Fig. 2.19 where the frequency responses of the LPFs are
plotted for various values of n at a cut-off frequency of 500 Hz.
2.3.1.3.2 Noise performance
The Fourier and time domain representation of a noisy square wave is shown in
Fig. 2.20.
Figure 2.20 A noisy square wave represented in the time (a) and Fourier (b) domains
n
sd
LPF
f
f
ωjH2
1
1
(2.51)
44
Figure 2.21 Exemplified frequency spectrum of a detected modulated signal signifying the
importance of modulating signals at higher rates
The difference between the error and noise performance of QSD is that output errors are
stationary while output noise causes the output signal to fluctuate around the mean
value. QSD has several key features that could substantially improve the output SNR. In
principle, the cut-off frequency and order of the LPF define the amount of noise that is
transferred to the output due to the random fluctuation of noise harmonics. Naturally,
the noise performance of QSD improves as the order of the LPF increases and its cut-off
frequency decreases. However, narrower bandwidths yield to long time constants, and
hence slow transient response. With the advances in DSP, the design of finite and
infinite impulse response filters (FIR and IIR) allow the realisation of versatile and
dynamic filters mimicking almost ideal conditions.
An attractive feature of QSD, is the ability to reject low frequency drifts and noise
signals with power spectral densities proportional to 1/f (β) (0 < β < 3). The QSD
principle is based on the idea of shifting (modulating) a low frequency component, to a
higher frequency where white noise dominates 1/f noise. The experimental
configuration illustrated in 2.16 elucidates a common setup to apply QSD on a
modulated CW laser beam. Without modulation (no chopping), the spectrum of the
detected signal would consist of a single DC harmonic plus the additional noise and
interference inherited from integrated electronics, external vibrations, interfering
electromagnetic fields etc. Figure 2.21 illustrates the Fourier spectrum of pink noise and
two sinusoidally modulated signals; signal A at 2 Hz and signal B at 1 kHz. Assuming
equal magnitudes of the two signals, demodulation at 1 kHz results in SNR
improvement of approximately 30 dB, due to a significant reduction of noise.
45
Figure 2.22 Magnitude of the first three harmonics of a pulse train against a varying duty cycle δ
According to (2.36) the magnitude of each individual harmonic varies sinusoidally with
respect to duty cycle δ. To illustrate this effect, the magnitude of the first three
harmonics of a pulsed signal is plotted for δ taking values from 0 to 1, as shown in Fig.
2.22. When locked on the fundamental harmonic of the pulsed signal, the sensitivity of
QSD significantly decreases as the duty cycle of the input signal deviates from 0.5.
This is illustrated in Fig. 2.23 where the output of a QSD is plotted against the duty
cycle δ of the input pulse train. Utilising a QSD to measure narrow pulsed signals that
suffer from severe noise conditions is highly inefficient, especially if the noise
dominates over the whole spectrum of the input signal. Under this extreme condition,
the QSD extracts the integrated amount of noise within a bandwidth specified by the
cut-off frequency of the demodulation filters. Consequently, cases in which the pulse
peak power is held fixed, reduction of δ to very small values (<1%) may drive the
magnitude of the pulsed input spectrum below the noise floor making it inseparable
from unwanted signals.
Figure 2.23 Experimental performance of a commercial LIA3 for various values of duty cycle δ
3 Model 7265 – DSP Lock-In Amplifier
46
2.3.2 Gated integration
Devices that are based on gated integration or else referred to as Boxcar averaging
(BA), have been extremely useful in applications such as pulsed nuclear magnetic
resonance (NMR) spectrometry where the SNR is poor [47]. These devices perform
continuous integration of a certain portion of a repetitive input signal, even if pulsed
with a very low duty cycle (δ < 0.1) [48]. The length of the signal selection is often
tagged by the term gate window (GW), and thus the name of “gated” integration.
Simple and inexpensive GI systems with a wide frequency range have been proposed
and successfully demonstrated in [49, 50]. Advances in the development of
microcomputers, had led to the implementation of algorithms with auto-zeroing
capabilities, used to minimise the effects of source fluctuations during average [51]. In
this review, the GI basic principle of operation is discussed, exploiting two modes of
operation. In addition to that, a brief noise analysis of this technique reveals the
advantages and disadvantages with a view to input noise and signal conditions.
2.3.2.1 Principle of Operation
GIs operate in two modes, the static-mode GI (SMGI) and a recovery-mode GI (RMGI).
In the former, a fixed time segment within the period of the input signal is continually
integrated by applying an adjustable triggering delay. On the other hand, RMGI uses a
variable triggering delay on a narrow GW, thus scanning it across the input signal. Their
main difference is that, a SMGI yields a low-frequency output (the signal envelope) that
corresponds to the mean of the gated time segment of the input signal. Instead, RMGI
recovers its temporal shape in non-real time. This review only takes into account the
SMGI, since waveform recovery has not been useful for the purposes of these research.
However, details on both modes are available in existing literature [40]. The block
diagram of a general GI is shown in Fig. 2.24. The input signal s(t) is applied at the
input of a gate, which in analogue implementation, can be simply a switch that is
triggered by binary gating signal g(t).
Figure 2.24 Block diagram of a gated integrator
47
Figure 2.25 Transient output response of a linear GI (A) and exponential GI (B)
Figure 2.26 Gated integration (Boxcar Averager [40]) with linear output averaging
Hence, integration of s(t) occurs only when g(t) is “high” (gate open). For low values of
g(t) (gate closed) the input signal is disconnected from the integrator. With the
assumption that the integrator is simply a an analogue RC LPF network, the capacitor
stores the energy accumulated while the g(t) was high. After sufficient number of gates,
the output voltage across the capacitor will approximate the average value of the gated
segment of the input signal. GIs often incorporate two types of averaging; exponential
and linear. In exponential GI (EGI), the output rises asymptotically towards the
averaged value of the gated segment. Linear GI (LGI) is achieved by adding each
consecutive output sample and subsequently dividing the result by the number of gates.
Commercial GI units4 often incorporate additional filtering stages to provide further
noise reduction. A comprehensive illustration of this technique is shown in Fig. 2.26
whereas Fig. 2.25 illustrates an exemplified transient response depicting the two
filtering approaches.
4 C.f. EG&G 162 mainframe boxcar averager, where gated integration is performed in separate processor modules (163 and/or 164)
48
Figure 2.27 Transient and steady state response of SMGI with gate size less than (a) and equal (b)
to the pulse duration
The gating function of a GI has been an important starting point to the development of
the novel SP algorithm (explained later on) and therefore, it is necessary to describe its
functionality in more rigorous manner. For this analysis, we assume that a noiseless
pulsed rectangular train of 20% duty cycle is applied at the input of the gating block
(see Fig. 2.24). As long as the duration of the gate (tGW) remains smaller than the pulsed
duration (χ), the output of the integrator eventually saturates to the peak amplitude of
the rectangular pulsed train. This is illustrated in Fig. 2.27 where the input (blue line)
and output (red line) time domain signals are plotted for for tGW less than χ (a), and tGW
equal to χ (b) .
On the other hand, at the time instant where tGW becomes greater than χ the capacitor of
the RC network begins to discharge through the resistor. Naturally, the longer the gate
length, the larger the oscillations at the output signal. In addition to this, the amplitude
of the ripples depends also on the time constant of LPF (τGI). Small value of τGI implies
that the integrator will be fast enough to charge the capacitor to a higher voltage and
thus increasing the ripple effect. This is shown in Fig. 2.27 where (a) illustrates the
transient (left graph segment) and steady state (right graph segment) response for tGW
greater than χ and (b) signifies the effect of a smaller τGI to the output signal. It is
important to note here, that EGIs performance must be examined in terms of an
effective and observed time constant; ETC and OTC respectively.
49
Figure 2.28 Transient and steady state response of SMGI for a gate length twice the duration of the
pulse (a). The same response is shown in (b) but with an ETC must smaller that in (a)
The ETC is equivalent to τGI whereas the OTC is proportional to the gate size tGW, the
ETC and repetition period T0. The OTC is defined by the product between the period of
the input signal and the amount of repetitive periods needed until the output of the LPF
reaches 63% of its steady-state response. The latter is simply the ratio between the ETC
and gate size tGW. Finally, an expression for the OTC is deduced as shown in (2.52). If
τGI is less than tGW the OTC is defined by the ETC of the RC integrating network.
One characteristic of a perfect GI is that its output level does not change between gate
openings. This kind of ideal GI is described as having infinite hold time. The benefit of
ideal GI is that, signal detection capabilities would be independent of the gate duty
cycle. Thus, for any gate size, the trigger delay could be made low as possible without a
limit and still not have adverse effect on the signal recovery capabilities. However, in
reality GI suffer from leakage currents, which cause the voltage at the output to change
between gating openings. Assuming that the gate is set narrow enough to average
continuously the peak voltage of a pulsed signal, the output changes that occur between
the gate windows due to leakage current will tend to bring the output level away from
the expected value.
Depending on the relative amplitude of the desired value and effect of leakage, there
will always be some degree of error. Nevertheless, proper configuration of the GI
GWGIGW
GIS tτT
t
τOTC for 0 (2.52)
50
instruments can usually make the error negligibly small. In many applications, leakage
effect degradation can be greatly reduced by the use of digital memory. This requires
both analogues to digital (ADC) and digital to analogue converters (DAC). Under this
configuration, the capacitor serves as a sample and hold circuit which temporarily stores
the value upon a gate trigger. Subsequently, the digital stored value is converted to
analogue and while the gate is close is applied at the input of integrator. Essentially the
DAC loads ideal initial condition to the integrator preventing it from the potential
droop.
2.3.2.2 Noise analysis of gated integration
Detail noise analysis of static and recovery mode GIs, is reported in literature [40, 45].
Assume that, a SMGI is used in the scheme described in Fig. 2.16. The reference signal
g(t) is a pulsed binary signal, used to sample the input as shown in Fig. 2.29a. Since the
integration is constrained within the boundaries of the gate length, it is assumed that in
an ideal integrator (no signal leakage with time) the signal at the input of the LPF is
equivalent, to the concatenation of successive sampling intervals.
Figure 2.29 Signal analysis of the gating function of a SMGI where (a) noisy input signal (solid
line) with the gating signal (dotted line), (b) merge of the selected portions (blocks) and (c) Fourier spectrum of the merged signal
This forms a non-real time gated signal sG(t), which in this case is spectrally equivalent
to a square wave with a fundamental harmonic defined by the reciprocal of tGW.
51
Naturally, if the gate length is exactly equal to the length of the input pulse, then the
resulting gated signal is defined only by a single DC term. In this case, the time domain
gated signal sG(t) and its Fourier spectrum are shown in Fig. 2.29b and 2.29c
respectively.
Therefore, it can be concluded that the output SNR of a static-mode EGI, is directly
related not only to the cut-off frequency of the RC filter but also to the gate length tGW.
Moreover, small values of tGW, maximises the output by measuring the peak amplitude of
the input signal, and hence high probability of SNR improvement. The conclusion of
this analysis agrees with the SNIR (2.53) given by a commercial GI unit5 (2.53).
The SNIR of a linear averager (SNIRLA) is simply the square root of the number of
repetitive gated samples [47]. It is worth noting that, even though the SNIRLA is
independent of τGI and tGW, these values may affect the rate at which the SMGI output
increases. In practice, the SNIRLA is superior to SNIREA for large number of repetitions.
However, their difference becomes negligible if the duration of the exponential
averaging lasts for at least 5τGI.
GI may be considered as repetitive averaging and is often used to minimize the effect of
broadband (white) noise [52] by bandwidth narrowing. It also maximizes the detected
signal amplitude. However, in the presence of flicker noise, the SNR performance of a
SMGI is unsatisfactory; this becomes obvious when it is analysed in the frequency
domain. Gated integration is equivalent to the filtering of a continuous signal which is
assembled from consecutive signal blocks. These blocks are made adjacent by the
gating signal g(t), since signal integration takes place only when the gate is open. For
GWs of any size, the bandwidth of SMGI is located at at the baseband region. Thus, the
output of an SMGI is a low-frequency envelope. Even with sharp and narrow bandwidth
LPFs, the most substantial part of Flicker noise will still appear at the output,
significantly degrading the SNIR.
5 C.f. EG&G 162 mainframe boxcar averager, where gated integration is performed in separate processor modules (163 and/or 164)
GW
GIEA t
τSNIR
2
(2.53)
52
CHAPTER
3 Tools and methods
PEDs are predominantly presented in literature, in terms of their response to a
sinusoidal or a square wave excitation with 0.5 duty cycle [26]. In this work, a
commercially available PED (SPH-43) is studied in a versatile environment where a
three-dimensional (3D) finite element model is employed to simulate its thermal
transient response. The geometric model of SPH-43 is drawn to match the actual
dimensions and material properties of the device. With respect to experimental results,
the thermal model is verified and subsequently extended to create a reliable model to
study the thermal behaviour of possible array geometries of multiple pyroelectric
elements. The temperature transient response obtained from the finite element analysis
(FEA), is subsequently processed within a novel, LabVIEW (LV) based Signal
Processing Software (SPS), to derive the voltage response of SPH-43. SPS performs
noise and stability analysis of the associated electronics integrated within the detector.
Finally, a novel SP method that combines the principle of QSD and GI is implemented
in a DSP block, within SPS, to achieve better signal-to-noise ratio (SNR) in pulsed
signal measurements.
3.1 Modelling pyroelectric detectors
As already discussed in chapter 3, a PED transforms incident radiation to an electrical
signal in three conversion stages, denoted as CS1, CS2 and CS3. In CS1, the radiation
flux Φ(t) is absorbed causing a temperature change ΔTd(t) in the pyroelectric crystal. In
CS2, the rate of temperature change yields a pyroelectric current Ip(t) due to the
pyroelectric effect and in CS3, a readout circuit is employed to convert the resulting
current to a voltage. This section, reports on various approaches to model a commercial
pyroelectric device (SPH-43), allowing reliable prediction of its obtainable voltage
responsitivity.
53
3.1.1 Modelling with finite element methods
All thermal detectors behave according to the heat transfer principles. When the
sensitive element of a PED is exposed to radiation, heat is transferred through its
structure while some energy is lost, due to the molecular motion of the surrounding air.
Conduction, convection and radiation are the three basic mechanisms of heat transfer.
The governing equation of a heat transfer problem is described by the partial differential
equation (PDE) shown in (2.1). In terms of practical convenience, crude boundary
conditions regarding the three mechanisms of heat conduction, are often assumed to
simplify (2.1) to a one-dimensional (1D) PDE, which is then used to describe the
thermal behaviour of the PED. LMA is often used to simplify (2.1) to a single variable,
time dependent, ordinary differential equation (ODE), assuming that heat transferred
occurs instantaneously. However, when complicated boundaries and conditions are
required, the best alternative approach in terms of flexibility and accuracy is provided
by numerical methods such as finite element methods (FEM), finite difference methods
(FDM) etc. Standard software packages, such as COMSOL Multiphysics, ANSYS, etc.,
incorporate heat transfer modules, capable of solving complex PDEs in a high degree of
accuracy. In this work, the thermal response of the pyroelectric device SPH-43 has been
modelled using the Heat Transfer Module contained within the COMSOL Multiphysics
package.
3.1.1.1 Geometric modelling
Finite element analysis (FEA) is often used to study physical phenomenal on a part or
assembly and compute its response to a given set of environmental conditions. The
results obtained, are then used to verify its performance, and/or improve and optimise
the design. However, successful analysis relies on accurate geometric representation of
the model and properly defined boundaries. In this case, the exact dimensions and
material properties of SPH-43 were provided by the manufacturer (Spectrum Detectors),
to reproduce an identical geometric model of the device. A complete 3D model is
illustrated in Fig. 3.1a, whereas Fig. 3.1b depicts a closer view around the sensitive
element. The constituent parts of the 3D model are numbered in Fig. 3.1. The
pyroelectric crystal (8) has thickness of 60 μm and is mounted on a ceramic substrate,
usually alumina (6). Each corner of the crystal sits on four conical shaped pillars (7)
creating an air gap between the crystal and the ceramic.
54
Figure 3.1 Geometric models created within COMSOL Multiphysics. (a) whole structure of SPH-
43, (b) geometric model used in the FEA, (c) snapshot of SPH-43
The top and bottom surface of the crystal is covered with a thin (~20[nm]) and partially
absorbing metal electrode (typically Chromium) (9). A transimpedance amplifier (TIA)
is soldered on a printed circuit board (PCB) (2) located below the ceramic substrate.
The header pins 4 and 11 provide conductivity between the PCB and substrate, to
enable the current flow in a close loop circuit. In turn, the pins are connected to the top
and bottom electrodes by a small diameter gold wire (10) and a conductive epoxy that
runs to the header pin (5) respectively. A metallic can (1) surrounds the assembly to
protect the integrated electronics as well as the pyroelectric crystal. The rest of the pins
(3) are used to connect the power supplies and provide access to the feedback elements
of the TIA. Initially, the whole structure was modelled, and results have shown that heat
flow was concentrated only around the pyroelectric crystal. Furthermore, the analysis of
such complex geometries requires the use of computers with high processing power and
large memory cards. Therefore, the geometric model is simplified to the one depicted in
Fig. 3.1b resulting a less demanding and time consuming FEA.
3.1.1.2 Subdomains and material properties
The domain of the FEA is confined within the volume of the geometric model whereas
subdomains represent the assembled parts. The material properties assigned for each
subdomain, are loaded manually or from libraries provided by the FEA software. The
symbols and units of three important thermal properties of a heat transfer problem are
shown in Table 3.1-I.
55
TABLE 3.1-I MATERIAL PROPERTIES USED IN THE ANALYSIS
Property Symbol Units
Thermal conductivity k Km
W
Specific heat Cp Kkg
J
Density ρ 3m
kg
The property of specific heat is often referred to as the mass heat capacity defining the
amount of heat required to change the temperature of the material per unit mass.
Similarly, the volumetric heat capacity represents the amount of heat required per unit
volume. This is shown in (3.1) where ρ defines the density of the material.
As shown in Fig. 3.2, the model under study comprises six subdomains; the pyroelectric
crystal (S1), four silver pillars (S2, S3, S4, and S5) and the ceramic (S6). The
dimensions of the geometry are given in appendix C. To ensure room temperature
conditions, the initial temperature of each subdomain is assigned to 25 degree Celsius
Co.
Figure 3.2 Subdomains of the geometric model used in the FEA.
pmp CρC , (3.1)
56
3.1.1.3 Boundary conditions
Physical phenomena are often described by ordinary or partial differential equations.
Their solution must satisfy a set of additional constraints. From a mathematical point of
view, a differential equation has a unique solution when the derivatives or integrals of
the governing equation, in this case the heat balance equation (HBE) shown in (3.2), are
eliminated.
This is achieved by forcing the solution to satisfy additional mathematical equations
that are used to characterize the temperature conditions of each bounding surface of the
3D model. COMSOL provides all the necessary boundary constraints to solve the HBE
directly or iteratively, without needing to elaborate in complicated mathematics. These
conditions are described in the following subsections.
3.1.1.3.1 Specified temperature
In microelectromechanical systems, Alumina or Aluminium Oxide (Al2O3) is often used
as a thermal insulator in which the temperature gradient is zero. Therefore, all of its
surfaces can be assumed that are subjected to a constant temperature, which in this case
is equal to the initial temperature of the subdomain.
3.1.1.3.2 Specified heat flux
The top surface of S1(crystal) along with the top surfaces of S2-S5 (silver pillars) are
assigned to a prescribed heat flux measured in W/m2. Thus, the total incident power
must be divided by the area of the corresponding irradiated surface. The area of the top
surface of each pillar and the area of the crystal were given by the manufacturer to be
7.07 and 0.09 mm2. In COMSOL this boundary is describe by (3.4).
where n is the normal vector of the boundary and q0 is the inward heat flux in W/m2,
normal to the boundary.
Qk
tCρ p
ΤΤ
(3.2)
zyx
ΤΤΤΤ
(3.3)
0Τn- qk (3.4)
57
3.1.1.3.3 Highly conductive layer
The significant benefit of heat transfer in highly conductive layers is that the two
chromium electrodes attached on the two sides of the crystal, can be represented as
boundaries instead of domains keeping the geometry simple and with reduced number
of mesh elements. This constraint is described by (3.5) where the subscript s denotes the
projection of the general heat equation shown in (3.2), onto the plane of the highly
conductive layer, multiplied by its thickness ds.
3.1.1.3.4 Convective cooling
This boundary regulates the heat transfer between a solid surface and an adjacent, non-
static fluid or gas, by combining the effects of conduction and fluid motion. The natural
or forced convection is described in (3.6) where Tinf is the temperature of the external
fluid or gas far away from the surface and hc represents all the physics occurring
between that surface and the space coordinate where Tinf is obtained.
The heat transfer module of COMSOL provides build-in functions to determine these
coefficients, taking into account the type of the convection condition (forced or natural)
and the type of geometry. For the PED model, we only consider natural convection.
This implies that the gas motion (air) around the detector is caused by natural buoyancy
forces, induced by density differences due to temperature variations. With respect to the
geometry of the bounded surface, we considered three types of convection; the vertical,
horizontal and incline convection boundary. These are shown in Fig. 3.3 where the
black arrows denote the orientation of each surface with respect to the xyz-plane.
Figure 3.3 Illustration of convection on vertical (a), upside/downside horizontal (b) and incline
surface (c)
ΤΤ
Τn- , ssssspss kdt
Cρdk
(3.5)
ΤΤΤn- inf chk (3.6)
58
The vertical boundary condition is illustrated in Fig. 3.3a and includes surfaces with a
normal vector, either pointing in ±x or ±y-axis. The surfaces with their normal vector
pointing towards the positive and negative z-axis are assigned to upside and downside
horizontal convection respectively (Fig. 3.3b). Lastly, inclined convection is assigned to
the surfaces that correspond to the sides of the four conical shaped pillars. These are
shown in Fig. 3.3c, where the angle a defines the slope of the tilted surface. Finally,
COMSOL estimates hc according to geometric and material properties of the surface
and convective fluid, which in this case is air.
3.1.1.3.5 Surface to surface or surface to ambient radiation
During the heat transfer process, some energy is lost due to the emission of radiation
from one surface to another or to the ambient. This constrain is particularly useful when
modelling an array of pyroelectric crystals allowing to investigate the thermal influence
(thermal crosstalk) between the adjacent elements. The outward heat flux of surface-to
surface radiation (SSR) and surface-to-ambient radiation (SAR) is given by (3.7) and
(3.8) respectively
where G is the incoming irradiation in W/m2, Tamb is the ambient temperature, ε is the
surface emissivity (0 ≤ ε ≤ 1) and σ is the Stefan-Boltzmann constant. With respect to
the temporal temperature distribution of the 3D model, COMSOL estimates the values
of G and qss (qsa for SAR conditions) when the emissivity and ambient temperature are
specified.
Figure 3.4 Representation of the finite-element meshed model of SPH-43 PED in 3D (a), top (b)
and side (c) view
4ΤσGεq ssss (3.7)
44 ΤΤ ambsa εσq (3.8)
59
3.1.1.4 Meshing
The FEM is based on the solution of a physical problem that is defined by a numerical
model, combined with additional set of parameters called “simulation attributes”. The
refinement (finite-element mesh) of each subdomain consists of group of cells and
nodes, with triangular, quadrilateral or tetrahedral topologies. The simulation attributes
are then associated to the finite-element mesh that corresponds to a sequence of the
nodes that belong to each element. The model comprises 15655 tetrahedral elements,
5676 are triangular elements, 697 edge elements and 68 vertex elements.
3.1.1.5 Simulation parameters and setup
The FEM simulation yields the temporal temperature distribution within the
pyroelectric crystal when exposed to incident radiation. The simulation time (tsim) and
time step (Δtsim) are specified in COMSOLs’ graphical user interface (GUI). The
simulation solver, direct or iterative solver, can either be manually tuned or
automatically selected by COMSOL, based on the chosen space dimension (3D,2D, and
1D), physics (boundaries) and study type (stationary, time-dependent etc). Even though
COMSOL is capable to choose optimal settings, it is sometimes necessary to modify
and manually tune the solver especially in a multi-physics problem. In our case, the
PED model is based only on the heat transfer process and COMSOL selects a direct
method to obtain the temporal thermal response.
3.1.1.6 Modeling a pyroelectric detector array
One of the main objectives of the TempTeT project, was to utilise a linear array of eight
pyroelectric elements to detect a flat beam of THz pulsed radiation. Single pyroelectric
devices of the SPH-40 series6 have been studied by the TempTeT team, concluding that
the SPH-43 was the optimal choice for THz detection. The 3D model of SPH-43 was
modified to create an array of four identical elements evenly distributed in a row on a
rectangular ceramic substrate of which the geometric structure being identical to the
bridge configuration used in the single 3D model. Conveniently, the same boundary
conditions are assigned to all the corresponding subdomains surfaces, as illustrated in
Fig. 3.5.
6 Spectrum detectors by Gentec-EO have developed a series of PEDs of different types and crystal dimensions. SPH-40 series are specifically design for THz detection due to their high sensitivity
60
Figure 3.5 Boundary conditions used 3D PED array
However, surfaces that were previously assigned to SAR are now switched to SSR, to
investigate the thermal crosstalk between the crystals. The incident radiation is applied
to the top surfaces of the two outer crystals whereas the inner ones are prescribed to
convection boundaries only. Thus, depending on the emissivity and distance between
the elements, it is hypothesized that heat will propagate through the inner crystals from
the outer side surfaces due to surface-to-surface emitted radiation. Thermal crosstalk
was studied under three different geometries where the separating distance between the
crystals was set to 1, 1.5 and 2 mm.
61
3.1.2 Modelling in Laplace domain
The contents of this section describes in more detail the Laplace transfer functions (TF)
of each conversion stage, followed by an introduction to the noise analysis of a current-
mode PED. As discussed previously, the transient response of a PED can be simulated
in the Laplace domain, by cascading (multiplying) the TF that correspond to each
individual stage.
3.1.2.1 The transfer function of PED thermal model
The TF of the “lumped” capacitance model of a PED was given in the previous chapter
by (2.9). The expressions (2.4) and (2.5) are often used to estimate the total thermal
resistance Rth and total thermal capacitance Cth. The product of Rth and Cth yields the
thermal time constant of the detector τth.
Visualizing the geometry of SPH-43 in 2D it is obvious that, under steady-state
conditions the heat capacitive elements are equivalent to open circuits permitting the
parallel combination of the thermal resistance of the air gap and silver posts. The
resulting resistance is then added in series with the rest of the materials, as shown in
(3.9).
Multilayer structures with one heat source and one heat sink are usually described by
grounded or non-grounded capacitor RC-network models. In literature [53], these
networks are known as the Cauer and Foster ladders and they depicted in Fig. 3.6a and
Fig. 3.6b respectively
Figure 3.6 Cauer ladder (a) and Foster ladder (b) equivalent thermal RC networks where n
represents the number of thermal layers
subpostsaircrystTth RRRRR (3.9)
62
The main difference between the Cauer and Foster ladder is that the former resembles
the actual physics of the thermal system, whereas the latter has no physical meaning
regarding any thermal/electrical analogies [53]. According to [54], when a Foster ladder
network is employed to simulate the thermal transient response of a thermal system, the
magnitude and time constant of the RC sub-circuits are required. Similarly, if the Cauer
ladder is used, the exact values of the R’s and C’s must be given. However, it is often
the case that a physical model of a complicated geometry cannot be adequately
represented by the Cauer ladder network. Thus, even with the knowledge of the RC
components, the output transient response of the equivalent circuit will be subjected to
significant errors. The most importance feature between the Cauer and Foster network is
that the resulting equivalent thermal time constant and magnitude response is the same.
The idea behind the Foster principle is used to relate the thermal response of SPH-43 to
an equivalent 1st order RC network, where the values of Rth and Cth are to be
determined. This is accomplished by assuming that the thermal step response of SPH-43
resembles the response of a 1st order RC network. An exemplified normalised response
of a generic thermal model of a PED is shown in Fig. 3.7, where the pulse duration was
set long enough to allow the output to saturate and hence obtain the thermal time
constant τth. The thermal time constant represents the time required for the output
temperature to reach the 63% of its saturated value. A necessary assumption in
estimating the value of Rth, is that the thermal model of the PED is linearly time-
invariant, causal and asymptotically stable. Hence, if H1(s) is the TF of this system,
given by (3.11), according to [41] its Fourier transform is obtained by substituting the
Laplace operator s with jω. The result is shown in (3.12) in polar form where ΔTd(jω)
is the output temperature response and Φ(jω) is the input radiation.
Figure 3.7 Thermal response of a PED to a pulse input of 50% duty cycle
FosterCauer RCRC (3.10)
th
th
thth
thd
τs
R
CsR
R
s
ssH
11Φ
ΔΤ1 (3.11)
63
If the phase angles are ignored and the frequency responses (FR) of ΔTd and Φ are
obtained, then the value of Rth is estimated by (3.13), with the condition that f0 is equal
to the frequency of the fundamental harmonic of the input square wave.
Further, the value of Cth can be obtained by dividing τth and Rth. However, this
procedure suffers from major difficulties. Firstly, the values of Rth and Cth can only be
estimated if the thermal transient response of the detector is given. The nearest
alternative is to obtain the thermal time constant and voltage response from datasheets,
if available. Obviously, purchasing the detector before modelling it, does not offer any
advantage, unless experimental results are necessary to verify an on-going simulation
model. The second difficulty is that the actual output of the PED is a voltage signal and
therefore, the temperature transient can only be approximated by reverse signal
processing. The degree of approximation depends on how accurately the last two
conversion stages of the PED are modelled. The impracticality of FEM software
(COMSOL) to model electrical circuits and DSP algorithms was the driving motive to
use the Foster network in this work. Following this approach, the thermal transient
response of the geometric model of SPH-43 is first accurately obtained by FEMs and
subsequently reproduced by the Foster network in LabVIEW where further signal
processing algorithms are introduced investigate possible measurement schemes.
3.1.2.2 Voltage responsitivity of a current-mode pyroelectric detector
From theory, the overall TF of a PED is equal to the product of the TF of each
conversion stage. Since the PED of choice operates in current mode, this section
describes the non-ideal TF of a transimpedance amplifier (TIA) and the significance of
a compensating capacitor. Details on TIA amplifiers can be found in literature [31, 55-
61], and the therefore fundamental theory of the op-amp is out of the scope of this
thesis.
thth
dωτRj
th
thωjj
ωjjd e
τω
R
eωj
eωj 11 tantan
22Φ
ΔΤ
1Φ
ΔΤ
(3.12)
0
22 2for 1Φ
ΔΤfπωτω
ωj
ωjR th
dth
(3.13)
64
Figure 3.8 Equivalent circuit of a PED (in dashed frame) in series with a non-ideal representation
of a transimpedance amplifier
3.1.2.3 Transfer function of a TIA
The non-ideal circuit of a TIA in series with a PED is shown in Fig. 3.8. Excluding the
noise sources, the non-ideal TF of a TIA is given by (3.14) with all the existing
components and sources listed in Table 3.1-II. The PED is considered as a high
impedance current source with an electric capacitance Cp, usually specified in
datasheets.
The difference between an ideal and a non-ideal TIA circuit, is that under realistic
conditions, the open loop gain (OLG) of the op-amp is not spectrally flat but instead,
drops 20 dB per decade similarly to a 1st order LPF with a cut-off frequency defined by
ωn.
TFDF
FoDn
TFDF
FDToFDFn
TF
no
p
p
CCRR
RARω
CCRR
RRCACRRωss
CC
ωA
sI
sV
122
(3.14)
11
n
o
ωs
AsA (3.15)
65
TABLE 3.1-II COMPONENTS AND SOURCES INCLUDED IN A NON-IDEAL TRANSIMPEDANCE AMPLIFIER CIRCUIT
3.1.2.3.1 Stability and compensation
In literature, the feedback factor βf(s), which is simply the non-inverting closed-loop
gain, is often related to the transient response of TIAs. However, in this work we
consider the feedback factor only in noise analysis of TIAs. Stability considerations are
assessed directly by the dynamics of the 2nd order TF of which the general form is
shown in (3.16) where ωc is the natural frequency, ζ is the damping ratio, Lo denotes the
gain of the system and s is the Laplace operator. For comparison purposes, we assume
that (3.17) is the TF of the TIA with the variables a, b representing the polynomial
coefficients and G the output gain.
Second order systems exhibit oscillatory behaviour that manifests itself when the roots
of the quadratic polynomial in (3.16) are complex. In control theory, the value of ζ
defines the stability of a system as overdamped, critically damped, underdamped and
undamped. If ζ is greater than one, the system is overdamped and stable.
Component Symbol Value Units
Common-mode capacitance of the amplifier CCM 200 fF Feedback capacitance CF User F Feedback stray capacitance CS 200 fF Pyroelectric crystal capacitance CP 54 pF Capacitance at the differential inputs CDIFF 2.5 pF Total input capacitance CT = CDIFF + CCM + CP 56.7 pF Feedback resistance RF User Ω Pyroelectric crystal resistance RP 1 TΩ
Amplifier voltage noise eA 16 nV/√Hz Amplifier current noise iA 0.5 fA/√Hz Johnson noise eRF - V/√Hz Amplifer’s open loop gain (OLG) A(s) 250000 OLG cut-off frequency ωn 20 Hz Output pyroelectric voltage Vp V Input pyroelectric current Ip A
22 2 cc
o
ωsζωs
LsL
(3.16)
bass
GsHTIA
2 (3.17)
66
For values less than one, the system begins to oscillate (underdamped) and when
reaches zero, it behaves like an undamped harmonic oscillator. A critical condition is
defined at the border between the overdamped and underdamped state defines a critical
condition, which occurs when ζ is equal to one. The value of ζ, must however by related
to the sensitivity and bandwidth of the charge amplifier (TIA). Empirically, when RF is
very high (>500 MΩ) the damping ratio is above one and the response is overdamped.
For lower values of RF, ζ becomes less than one and hence the oscillatory behaviour
becomes apparent. Using equations (3.16) and (3.17), ζ can be estimated with respect to
the actual components of the circuit. This is shown in (3.18), with a and b denoting the
coefficients of (3.14).
In real time applications, it is desirable to establish the value of RF that will drive the
circuit to the critically damped state. This essentially determines the maximum
bandwidth threshold (MBT), where the TIA will operate without overshooting. In the
need of higher bandwidth than the one specified by the MBT, the RF must be reduced
even further which results in pushing the TIA into the underdamped state. Under this
condition, an external feedback capacitor CF is employed to increase the damping ratio
and bring the poles of the TF closer or exactly on the real axis of the s-plane. The
dynamics and stability performance of a TIA are controlled and optimized directly by
(3.18), with the assumption that the intrinsic stray capacitances and resistances of the
detector and the amplifier are measured or given by the manufacturer.
3.1.2.3.2 Overall TF and voltage responsitivity of a PED
The voltage responsitivity of the PED is obtained by cascading the TFs of each
conversion stage as shown in Fig. 3.9.
Figure 3.9 Cascade representation of the PED conversion stages in Laplace domain. Each block
represents the TF of the corresponding conversion stage.
b
αζ
2 (3.18)
TFDF
FDToFDFnc CCRR
RRCACRRωζωα
2
2 (3.19)
TFDF
FoDnc CCRR
RARωω
12
(3.20)
67
Figure 3.10 Voltage responsitivity of SPH-43 (black line) illustrating the overshoot when RF is
reduced to achieve higher bandwidth (coloured lines)
The black sold line in Fig. 3.10 represents the FR of SPH-43 whereas the coloured lines
show the response as reproduced for different values of RF. The red line plot shows the
critically damped case where equation (3.18) was used to estimate the value of RF that
results to a damping ratio of one. Further reduction of RF causes the electrical cut-off
frequency fel of the PED to increase and consequently ζ to decrease. The peaking effect
becomes apparent when ζ<0.707 as illustrated by the green plot, where ζ is equal to 0.1.
In this particular case, even though the compensating capacitor CF is not included, the
bandwidth of the PED is still limited by the stray capacitance CS; however, due to its
small value the peaking effect cannot be prevented. Adding a parallel capacitor CF with
RF provides the prerequisite compensation to increase the damping ratio and thus reduce
the overshoot. This is illustrated by the purple dashed-line plot, where a compensating
capacitor of 2.03 pF was added to increase the damping ratio and drive the circuit to a
critically damped state.
3.1.2.4 Noise considerations
This section, gives an insight to the noise performance of a PED with a primary
objective to investigate the noise contribution of the integrated electronics of SPH-43.
Derivations of fundamental noise concepts are exhaustively described in literature [31,
55, 56] and are out of the scope of this thesis. A TIA circuits suffers from three main
noise sources: Johnson noise, shot noise, and the op-amp voltage noise.
3.1.2.4.1 Johnson noise
Johnson noise is frequently detected in electronic systems and is always a serious
factor to consider when designing low-level (below μVolt) detection schemes.
68
This type of noise is observed in all resistive components in which the output voltage
across the terminals suffers from random fluctuations due to thermal agitation of
electrons. The Johnson noise density is quantified as
where kb is the Boltzmann constant, T is the absolute temperature of the resistive
components, R is the resistance in question (in this case RF). Noise density is measured
in V/√Hz, reflecting the noise contained in a 1 Hz bandwidth at a given frequency. The
frequency spectrum of Johnson noise is uniform, implying that the energy within a
certain bandwidth is the same at any position in the frequency spectrum. Under this
condition, the noise is said to be white. Other types of noise may possess different
frequency spectra and they are often named after colours, such as pink noise, brown
noise, blue noise etc. To combine all the noise contributions, each noise source must be
converted in root mean square (RMS) units by integrating its spectral noise effect over
frequency. According to [31], the RMS output Johnson noise is given by (3.22)
where An represents the corresponding noise gain applied to the input-referred noise in
the bandwidth limit specified by f1 and f2. In particular, constant unity-gain implies that,
the effect of eniR is transferred directly to the circuit output. However, the bandwidth
limit of the TIA rolls off this ideal response to a higher frequency fBW, specified by the
RFCTF feedback network where CTF denotes the parallel addition of the compensating
capacitor CF and stray capacitance CS. Thus if:
then, the resulting RMS output is given by (3.24).
According to (3.24), the amplitude of the Johnson noise is increasing by a square-root
relationship with RF while the output signal increases linearly. Thus, the resulting SNR
improves by the square-root of RF as shown in (3.25).
Rke bnoR Τ4 (3.21)
2
1
22f
f
niRnnoR dfeAV
(3.22)
BW
n
f
fj
sA
1
1
(3.23)
BWFnoR fkTRπV 2 (3.24)
Fp
noR
FpR
kT
I
V
RISNR
4 (3.25)
69
Even though maximising RF might improve the output SNR of the PED, other noise
effects may dominate the noise performance.
3.1.2.4.2 Op-amp input current noise (shot noise)
The value of RF also affects the TIA’s noise through interaction with the op-amp’s noise
sources. The noise current ini, represents the amplifier’s shot noise of the input bias
current IB and its noise density is given by (3.26)
where q is the electron charge. Consequently, this noise current flows directly through
RF, producing a voltage noise signal that is eventually transferred to the circuit output
with unity gain and is defined by (3.27)
Similar to the Johnson noise, this type of noise is considered being independent to
frequency (white) and band limited by the cut-off frequency of the feedback network.
However, it is possible that, shot noise and the bias current of the op-amp has 1/f
characteristic, which is often specified within the datasheet of the component. The
resulting RMS shot noise voltage is given by
The noise component can be minimised by ensuring that the selected op-amp has input
bias current IB in the pico-ampere range or lower. However, this is only valid for
practical values of RF and therefore careful design is necessary when high resistive
feedback is used to achieve extremely high sensitivity.
3.1.2.4.3 Op-amp input voltage noise
To estimate the contribution of the input noise voltage (eni), to the total output noise of
the op-amp (Vno), one must acknowledge that eni has variable density along the
frequency spectrum and experiences a frequency-dependent amplification. At low
frequencies, the noise voltage eni displays magnitude decay proportional to 1/f followed
by white noise above the certain frequency.
Bni qIi 2 (3.26)
BFFninoI qIRRie 2 (3.27)
BWBFnoI fqIπRV (3.28)
70
Figure 3.11 (a) Illustration of the 1/f noise curve (b) Magnitude response of the noise gain as a
function of frequency
This frequency is often referred to as the corner frequency ff while the average
magnitude of the white noise represents the noise floor voltage (enif). Graph (a) of
Fig. 3.11, illustrates a typical 1/f noise denoting the corner frequency ff and noise floor
enif. The values of these parameters are usually available in datasheets and their use is
crucial when computing the total RMS noise of the circuit. As reported in literature [31,
55, 56], the eni is amplified by a feedback factor βf (ω) (often called noise gain).
According to [31], βf (ω) is essentially the closed loop gain of the op-amp circuit and is
given by (3.29).
Examples of various noise gains are illustrated in Fig. 3.11b where each coloured plot
represents a noise gain for a specific value of RF. The plots were generated by
evaluating (3.29) as a function of frequency (2πf) while RF was given values of 100
(red), 10 (green), 1 (purple), and 0.1 (blue) GΩ. As RF increases the zero and pole
frequencies of the TF are shifting towards the OLG of the op-amp resulting to a
decrease of the noise gain plateau.
FF
TFDF
DF
D
FDf CRωj
CCRR
RRωj
R
RRωβ
1
1(3.29)
71
Figure 3.12 Segregated output noise density of a TIA produced by the product of the feedback
factor βf (ω) and eni (ω)
The latter defines the flat region above the natural frequency due to the zero of the TF
and below the frequency where the NG interconnects with the OLG. Analytical
estimation of the RMS output noise (VnoV) due to eni requires evaluating the integral in
(3.30).
Undoubtedly, the complexity of (3.30) creates the need to break the whole integration
task into segments allowing a more manageable and insightful evaluation. To obtain the
net RMS noise produced by the op-amp input noise voltage, the RMS voltage in each
region is computed and subsequently combined in an RMS manner as shown in (3.31).
Naturally, the number of regions is dependent on the position of the poles and zeros of
the noise gain as well as the corner frequency of the input noise voltage of the op-amp.
3.1.2.4.4 Total RMS output noise of the TIA
The RMS combination of VnoR, VnoI, and VnoV is once again necessary to estimate the
overall output noise of the TIA given by (3.32).
In this work, the noise analysis of the TIA is implemented within the LV-based SPS,
where Laplace transient analysis is used to simulate the time domain noise components
and subsequently add them to the resulting pyroelectric voltage.
0
2
0
2ωdωeωdωβωeV noVfninoV
(3.30)
2
42
32
22
1 noVnoVnoVnoVnoV eeeeV (3.31)
222
noInoVnoRno VVVV (3.32)
72
3.1.3 Modelling of a PED in NI LabVIEW
In this work, Matlab and Simulink have been initially utilised to model PEDs with
satisfactory results. In the recent years of development, LV has evolved significantly,
allowing the execution of algorithms with performance being identical to
Matlab/Simulink. However, implementing complex mathematics involving large
matrices, a text-based language like Matlab is usually preferred. Nonetheless, the
capability of LV to perform demanding mathematical operations is not limited by the
functionality of the software but it might be limited from the inability of the
programmer to use the software adequately. It must be stated here, even though LV was
the final software of choice, it does not mean that satisfactory results could have not
been obtained by Matlab. Time management and convenience have been the two most
important factors in making such a choice.
In LV, a virtual instrument (VI) refers to any algorithm or subroutine that is coded
within the LV environment. Each VI consists of a block diagram and a front panel. The
former is a graphical programing panel where LV functions (sub-VIs) are represented
by blocks with predefined number of inputs and outputs. Data flow occurs sequentially
or in a pseudo-parallel manner between sub-VIs by interconnecting wires and nodes.
The front panel is essentially the graphical user interface (GUI) in which controls and
indicators allow the user to input data into or extract data from a running VI. The
deployment of a GUI while coding was one of the most appealing characteristics of LV
in terms of time consumption and convenience. Thus, a fully interactive and operational
PED simulator is implemented and subsequently cascaded with SP sub-VIs to identify
optimal DSP methods to measure the pyroelectric voltage response.
3.1.3.1 Transient and frequency modelling of a PED
Using basic Laplace theory, the temporal voltage response Vp(s) relates to the radiation
input signal Φ(s) by the following expression:
where in this case H(s) represents the TF of the PED. The combination of the second
order TF of the non-ideal TIA circuit with the first order equivalent Laplace thermal
model, results to an overall third order TF.
sΦsHsVp (3.33)
73
Figure 3.13 Canonical realization of a 3rd order transfer function.
A general representation of a third order TF is given by (3.34). According to theory, the
TF of the PED has only one “zero” and therefore the coefficients b3, b2, and b0 are zero.
The TF of the PED was initially realised in canonical form as shown in Fig. 3.13.
Having access to the initial conditions of each integrator (s-1 box) allows pausing and
restarting the simulation manually. Additionally, the simulator permits continuous
operation by breaking down the simulation in smaller time windows (acquisition
frames). At the end of each frame, the final value of the last integrator is temporarily
stored and fed back to the first integrator as the initial conditions for the following
acquisition frame. A simplified block diagram of the simulation process is shown in Fig.
3.14. The overall algorithm runs within a main while-loop in which the simulation
process is initiated and configured by the user.
Figure 3.14 Simplified representation of the block diagram of the PED simulator in LabVIEW
012
23
012
23
3
asasas
bsbsbsbsY
(3.34)
74
Figure 3.15 Snapshot of the LV front-panel of the “PED Settings” sub-VI
In the current version of the simulator, the “Transfer Function” block incorporates
dedicated LV functions, optimised to simulate the time response of a TF to any given
input function. The coefficients in (3.34) are computed in the “PED Setting” sub-VI,
which must always be called before the simulation initiates. Upon execution of the
“PED Settings” sub-VI, a separate LV-front-panel (GUI) appears on the screen allowing
the user to configure the PED properties. In this sub-VI, the Laplace TFs of the PED’s
conversion stages are composed and finally combined to produce the final PED TF. The
LV front-panel of this sub-VI consists of four sections. Section A, displays the FRs of
an ideal PED TF (black line), the OLG A(s) (blue line), and the feedback factor βf (s)
(green line). The red line plot is controlled by a string enumerator in section C, which
allows the user to select between the FRs of the PED thermal model, the TIA and lastly
the non-ideal PED TF. Section B contains all the controls associated with the TIA of the
PED.
75
In section C, the thermal properties and temperature responsitivity of the PED are
determined in a separate sub-VI, invoked by pressing the control button “Calculate”.
This sub-VI is called only when the string enumerator above the “Calculate” control is
set to “Estimate the thermal time constant according to the FEA PED model”.
Alternatively, manually control of the thermal time constant and response of the
detector is possible by specifying the values of Rth and Cth. The simulator can process
the ideal TF of the PED by neglecting the OLG of the TIA. This option is available in
section D where the damping ratio indicator becomes important when the OLG is
included in the simulation.
3.1.3.2 Noise simulation of a PED in LV
The amplitudes of Johnson noise (enoR) and shot noise (enoI) are independent of
frequency and are given by (3.21) and (3.26) respectively. However, the amplitude of
the op-amp input noise (enoV) has a variable voltage density and therefore a single scalar
value does not define its magnitude. Thus, dedicated sub-VIs invoked from LV’s
libraries, are used to simulate the required noise signals, corresponding the noise
components of the PED. Johnson and shot noise are passed through a unity gain 1st
order RC filter, since the spectrum of both is limited by the transimpedance bandwidth,
fBW. Similarly, the input noise voltage of the op-amp is shaped by the magnitude
response of the feedback factor βf (s). The resulting noise signals are then added to the
voltage response of the PED. This procedure is graphically illustrated in the block
diagram shown in Fig. 3.16.
Figure 3.16 Block diagram of the simulation of the noise performance of the PED
76
However, it must be clarified that the above procedure does not compute the output
RMS noise of the PED but instead it simulates the output noise waveform by taking into
account the three most dominant noise sources of the TIA. Further, this waveform is
added selectively at the simulated transient response of the PED. The purpose behind
simulating the noise performance of the PED is to identify the relative significance of
the feedback resistor and capacitor to the output pyroelectric signal. Without this
information, traditional intuition alone may lead to suboptimal design choices.
3.1.4 Pulsed performance of PEDs
PEDs are AC coupled devices and therefore, their use is effective only in modulation
schemes. In existing literature [8, 17, 20], the response of PEDs has been studied almost
exclusively with slow varying sinusoidal or square wave signals. Evidently, high SNR
output is achieved by either maximizing the amplitude of the output pyroelectric signal
or by reducing the output noise. In the case of a current-mode PED, high values of RF
ensure high transimpedance gain but also a narrow bandwidth around the baseband
region of the frequency spectrum. Under these circumstances, pyroelectric detection of
modulated pulsed radiation below the cut-off frequency of the TIA, results in a
spectrally weak output signal and hence poor SNR. The term “spectrally” is used here to
emphasize that pulsed signals are not single-tone but instead, they inherit a rich
spectrum in which multiple harmonics spread along the frequency as the pulse width
becomes narrower. Thus, any alteration of the magnitude and phase of these harmonics
due to the frequency response of the PED, results in a distorted time domain signal at
the output of the device.
In this work, we are mainly interested in the waveform integrity of the pulsed
pyroelectric output, and how it influences the selection between SP methods with
respect to their SNR performance. In turn, this reflects to the feedback components of
the TIA, which are the key parameters to achieve the desired shape of the pulse train.
Ideally, the resulting pulse response of a PED would be a perfect rectangular train under
the assumption that the CF is zero. The resulting pulsed signal is amplified with respect
to RF, while the shape of the pulse remains intact. In the frequency domain, this
corresponds to an FR where the fel extends to infinity and hence, allowing all the
harmonics of the input signal to be transferred at the output of the op-amp. Now,
assume the case where the stray capacitance of the PED is not ignored but instead it is
as small as 200 fF.
77
Figure 3.17 Transient response of a PED to an input pulse train (a) for RF = 100 GΩ (b) and RF =
188 MΩ whereas (c) and (e) illustrate a closup view of the two responses, respectively.
Intuitively, if the pulse rate is higher than fel, the harmonics of the pulsed signal will be
subjected to a magnitude and phase change.
As illustrated in Fig. 3.17b, the output of the PED suffers from sever damping and high
transient overshoot, due to accumulated excess of charge within the CF. Increasing the
78
bandwidth fel (smaller value of RF) implies smaller electrical time constant and thus
preserving the shape of the pulse train. As shown in 3.17d and Fig. 3.17e, the damping
effect in each transmitted pulse is reduced as well as the transient overshoot. However,
since the voltage responsitivity is directly proportional to the RF, a trade-off between the
PED sensitivity and bandwidth is unavoidable. Applications in which high SNR
response is desired, the relationship between the bandwidth and sensitivity becomes an
important detail in the selection of optimal SP methods.
3.2 Pulsed detection and SP with PEDs
High quality measurements are often described by high SNR values, obtained at the
output of a system. SP algorithms are employed to extract a parameter of the detected
signal, which relates to the physical meaning of the experiment. In this work, we
investigate suitable SP methods able of measuring the strength of pulse-modulated
pyroelectric signals. The two methods under consideration are the static-mode gated
integrator (SMGI) and quadrature synchronous demodulation QSD. Their advantages
and disadvantages had led to a novel SP method, which combines the principles of QSD
and SMGI in achieving better SNR in pulsed signal measurements. We therefore
introduce this method as Gated Quadrature Synchronous Demodulation (GQSD),
emphasizing the synergy between GI and QSD in this case.
3.2.1 Measuring the pulsed response of a PED with QSD and SMGI
Methods that are based on the SD principle are capable of measuring the magnitude and
phase of a selected frequency component of the input signal. SD has maximum
sensitivity when the input signal is purely sinusoidal or has a duty cycle of 0.5.
Alternatively, a static-mode gated integrator (SMGI) with narrow gating and adjustable
triggering delay, can measure the peak voltage of the pulsed pyroelectric response.
However, the signal-to-noise improvement ratio (SNIR) between these two methods is
debatable when taking into account the observed time constant (OTC) of the SMGI in
comparison to the effective bandwidth of the QSD. Additionally, the pulsed
performance and noise conditions of the PED also play a fundamental role to the
achievable SNIR.
79
Figure 3.18 Pulse performance of PED. In case study A, RF is equal to 100 [GΩ] and in case study
B, RF is reduced to 1 GΩ. The graphs (a), (b) and (c) corresponds to case study A and (d), (e) and (f) in B. For both studies the transient response of the PED along with the excitation signal are
depicted in (a) an (d) with their frequency spectra shown in (b) and (e) respectively. The graphs in (c) and (f) depict the frequency response of the PED in case A and B.
To investigate the advantages and disadvantages of QSD and SMGI in pulsed
measurements with PEDs, two case studies have been considered. In both cases, the LV
based PED simulator was used to obtain the response of a PED model to a pulsed
modulated rectangular train. The pulse modulation frequency, duty cycle and amplitude
of the input signal were set to 200 Hz, 0.06 and 0.5 mW respectively. In Fig. 3.18,
graph (a) illustrates the pyroelectric transient response (solid line) and input pulse train
(dotted line), (b) depicts their corresponding frequency spectrum and (c) shows the FR
of the PED for case A. The plots on the right hand side (RHS) of Fig. 3.18 represent the
same analysis for case B. A noticeable similarity between the peak voltages of the PED
response in A and B is observed, even though the gain was decreased by approximately
a factor of four.
80
The underlying reason behind this behaviour can be determined by observing the
frequency spectra for each case. In case A, all the dominant harmonics of the input
signal are attenuated by a factor of 1/f since they are all located above the electrical cut-
off frequency of the PED. In case B, the first three harmonics of the input signal have
almost constant gain, as they are located within the pass band region of the PED. It is
evident that severe and unequal attenuation of the dominant frequencies of the PED
response in case A, yields a spectrum in which the fundamental harmonic carries most
of the energy. Contrary, the shape of the input pulse train in Case B is preserved while
the energy is more homogeneously spread along the frequency spectrum. In addition to
this, the peak-to-peak voltage (Vpp) of the PED response in case A is approximately
twice the Vpp of the response in case B.
Assuming that the output noise is constant in both case A and B, one may observe that
SMGI improves the SNR, approximately by a factor of two compared to QSD. In the
example shown, the peak voltage is 0.14V whereas the magnitude of the first
fundamental is approximately 70mV. In case B, even if the gain is reduced, the SNR
performance of SMGI is speculated to be about 6 times higher than the one obtained by
QSD (Vpp ~ 0.11V and Vf ~ 0.18mV). However, the ability of SMGI to gate the detected
pulse at the highest observed voltage point allows signal maximisation with the
condition that the gate length is extremely narrow and perfectly located at the peak
voltage of the signal. The implication of very narrow gating to the OTC of the
measurement is severe, therefore the outperformance of SMGI to QSD with longer
integration time constants can be arguable.
Until now, the two methods have been compared with respect to their effective time
constant (ETC) and not to their OTC. According to (2.52) the OTC of the SMGI is
equal to the effective time constant (ETC) of the integrator, multiplied by the number of
samples (gate triggers) needed to integrate over the whole period of the signal. Thus,
narrow gating results in maximisation of the output signal while sacrificing the speed of
the transient response. Moreover, as mentioned in background theory, the SMGI is
susceptible to 1/f noise, since its operational bandwidth covers the base-band region of
the frequency spectrum regardless the gate size. Considering the noise performance of
the PED and the example shown in Fig. 3.12, the output of the PED will most likely be
subjected to Flicker noise (most probably pink), and hence deteriorating the
performance of SMGI.
81
By increasing the time constant of the QSD towards the OTC of SMGI, the resulting
bandwidth becomes narrower and consequently reducing the output measured noise.
Therefore, if the output noise of the QSD is reduced by equal amount to the signal
maximisation of SMGI then equal SNR performance is obtained. For example, in case
A, there is a higher probability that the resulting SNR of the QSD will be greater than
the SMGI due to the smaller difference between the output signal levels, when
compared to case B. Moreover, apart from the bandwidth narrowing, the ability of QSD
to avoid 1/f is an additional contribution to noise reduction and hence, to the
improvement of SNR. The study of the advantages and disadvantages mentioned in this
chapter as well as in Chapter 3, has led to the development of a new approach to
measure pulsed signal and ultimately use it to quantify the pulsed response of a PED.
3.3 Gated quadrature synchronous demodulation
As already mentioned, the sensitivity and efficiency of QSD is greatly reduced when
low duty cycle pulse trains are considered. In this section, we introduce an algorithm for
implementing GQSD suitable of measuring pulsed signals of low duty cycle. Based on
the principles of GI and SD, this algorithm suppresses Flicker noise while enhancing the
signal, and thus improving the output SNR. The utilization of QSD ensures minimum
contributions from flicker noise while the gating function conditions the signal for the
application of synchronous detection.
3.3.1 Theory of operation
The structure of GQSD is broken down in three sections as illustrated in Fig. 3.19. The
signal processing throughout each section of GQSD is illustrated in Fig. 3.20, where for
each section the resulting discrete time domain signals are presented on the left hand
side of the figure. The right hand side of the figure describes the process in the Fourier
domain showing the effect of gating.
Figure 3.19 Block diagram of a digital GQSD separated in three sections: gating signal generation,
gating function and QSD
82
Figure 3.20 Illustration of the resulting discrete time domain signals of each section of GQSD along
with the Fourier spectrum. The notation used in the figure is also used throughout the text and equations of this paper.
3.3.1.1 Generation of the gating signal
As shown in Fig. 3.19, the first block is responsible for the acquisition (from
measurements or simulation) of the pulsed input signal s[k]. The remaining two blocks
are used to generate the gating signal g[k]. Therefore, to introduce this block properly, it
is necessary to involve also the effect of g[k] on the output of the next section (Gating
Function), which is the gated signal sG[k]sB k . The opening of the gate (low to high
transition of g[k]) and its duration (g[k] high) are given in units of k by D and M
respectively (see Fig. 3.20). Since the output from the gating stage sG[k] is formed by a
continuous concatenation of the gated segments taken from the input signal, the purpose
of block 2 in Fig. 3.19, is to estimate the value of M which maximizes the magnitude of
the fundamental frequency in sG[k]. This is accomplished by computing the discrete
time Fourier series (DTFS) of the pulse input train. Following [41] the general
expression of discrete Fourier series representation of a function f[k] is:
where k and h enumerate respectively the samples and harmonics of f[k]; h and N0
denote respectively the magnitude of h-th constituent harmonic and the period (in units
of k) of f[k].
1
0
20
0][
N
h
kN
πjh
hekf
(3.35)
1
0
2
0
0
0][1
N
k
kN
πjh
h ekfN
(3.36)
83
According to [41] the DTFS of the pulsed input train s[k] is given by (3.37):
where A is the pulse amplitude and X is the pulse width (in units of k). The second term
gives the magnitude sum of the harmonics that are contained within the input signal.
The h=1 term in (12) represents the magnitude 1 of the fundamental harmonic of s[k]
given by:
The function 1(N0) in (3.38) has a maximum at N0max = M = 2X, which corresponds to a
duty cycle δ = 0.5.
3.3.1.2 Gating function
A single period of the gating signal g[k] consists of L samples. By setting L = N0, each
sample of the input signal s[k] can be selected or discarded according to the value of the
corresponding element of g[k]. The algorithm for the digital gating (block 4) in Fig.
3.19 is described by the flowchart in Fig. 3.21.
Figure 3.21 Flowchart of the gating function algorithm
kN
πjh
N
h
e
hN
π
hN
πX
N
A
N
AXks 0
0 21
00
0
00 sin
sin
(3.37)
)/sin(
)/sin(
0
0
001 Nπ
NπX
N
AN
(3.38)
84
The open gate is represented by unity values of g[k] while the closed gate is represented
by zeros. Since L = N0, the signal index k and gating index i are identical and the loop
runs only on a single index i. Under an open gate condition (g[i]=1), the ith element of
the input signal s[i] is stored in a register REG at a position specified by a separate
index p. Thus, the loop index i enumerates the input signal samples, while p enumerates
the samples in the resulting gated output signal sG[k]. When the loop index i runs
through the complete period ofg k , the content of the register is passed to the QSD and
the process is repeated. Consequently, REG contains the new discrete signal sG[k] for
the current period only and concatenates adjacent periods by discarding the incoming
samples when the gate is closed. Consider now that the gate length is optimized in
section 1 (see Fig. 3.19) and N0 is replaced by M. Then the DTFS of sG[k] is expressed
by (3.39).
The magnitude G1 of the fundamental harmonic of sG[k] is represented again by the h =
1 term and the duty cycle of the modified signal sG[k] is defined by δG as shown in
(3.40).
3.3.1.3 Quadrature synchronous detection
QSD takes place within blocks 6 and 7 of section 3 in Fig. 3.19. The principle of
operation of a QSD is discussed in chapter 2, and here we focus on aspects relevant only
to the gated character of sG[k] and GQSD in general. Block 6 generates the reference
signals i[k] and q[k]. The product of each of these references reference with sG[k] yields
the real (I[k]) and imaginary (Q[k]) components of the referenced harmonic. The period
length of the reference signals (in units of k) is set equal to M and the optimal gate
length is determined in block 2 (see Fig. 3.19). Referencing towards M ensures selective
demodulation from a discrete carrier wave specified by the optimized gating period, as
opposed to the pulse repetition rate. Infinite impulse response (IIR) LPFs are used to
extract the DC term of I[k] and imaginary Q[k] and subsequently compute the
magnitude of the demodulated complex vector according to [1].
kM
πjr
M
h
G eh
M
π
hM
πX
M
A
M
AXks
21
0 sin
sin
(3.39)
5.0;/sin
sinmax1
M
Xδ
Mπ
πδ
M
AG
GG
(3.40)
85
To validate and compare the GQSD with an SMGI, IIR filters are used, to mimic
classical analogue LPF structures. The algorithm employs two 1st order Butterworth
filters with a cut-off frequency defined by (2.44). In addition to the LPFs of the PSD, a
1st order Bessel high pass filter (HPF) is utilized in block 5 of Fig. 3.19, to attenuate the
DC component of sG[k], diminishing its contribution to the output signal. Similarl to a
SMGI, the time response of GQSD is affected by the gate length resulting to an
observed time constant (OTC) as described by (3.41)
where τsd, T0, and Ts are the time constant of the LPF, the pulse period, and the sampling
time, respectively (in sec), whereas M specifies the gate width (in number of samples).
3.3.1.4 Determining M for an arbitrary pulse shape
The fundamental component G1 reaches maximum when δG becomes 0.5, therefore M
= 2X can be calculated from X. However, X is easy to quantify only in the trivial case of
a train composed by ideal rectangular pulses. In practice, (e.g. radiation measurements)
the pulse shape will depend on the characteristics of the emitter and receiver. When the
pulse shape is complex but reproducible, G1 can be estimated by comparing its value
with M as parameter. The flowchart show in Fig. 3.22, describes the algorithm used to
determine the value of M and D.
Figure 3.22 Flowchart describing the estimation of the optimal value of M and D
ssds
sd MTτ for T
T
M
τOTC 0
(3.41)
86
Figure 3.23 Estimation of the optimal value of M. A rectangular pulse pattern (solid line) and a
steady-state response of a PED (dashed line) are shown in (a). For each input, the maximum value of G1 signifies the optimal value of M (b) for D = 0.
When the pulse shape is complex but reproducible, G1 can be estimated by comparing
its value with M as parameter. The solid line in Fig. 3.23a represents a rectangular pulse
pattern, repeating every 250 samples (i.e. N0=250). In this example, the width of the
emitter pulse contains 25 discrete samples resulting in a duty cycle of 0.1. The dashed
line plot illustrated in Fig. 3.23a shows the receiver’s steady-state response. Figure
3.23b shows how M is determined numerically by finding the maximum from multiple
FFT calculations of G1 for a gate size that varies in unit increments from 1 to N0.
Noticeable in Fig. 3.23b, is the difference between the two plots corresponding to the
rectangular excitation pulse (bold) and the PED response (dashed). In the former case,
concatenation with new samples of the same value (up to k=25) does not contribute to a
harmonic function and G1 is zero. The latter plot manifests a double peak – the first
maximum is caused by the PED response peaking half way through the excitation pulse
while the second corresponds to a harmonic contribution (k=45) resulting from the
positive to negative swing. The position of the second maximum is determined by the
value of k for which the area under the signal becomes zero, resembling δG of 0.5. This
procedure repeats for all possible values of delay D. This is achieved by shifting and
rotating the input signal s[k] one sample at a time. Each value of G1 is estimated and
saved in a 2D matrix as shown in (3.42). The row and column indices of the element
containing the maximum value of G1, corresponds to the optimal value of D and M,
respectively.
M
M
M
k
DG
DG
DG
GGG
GGG
111
11
11
11
01
01
01
21
21
21
(3.42)
87
Figure 3.24 The four rainbow coloured plots show the values of [k] for four trigger delays. The black connecting plot indicates the maximum value of [k] for all possible triggering delay plots.
Take for example the case of a rectangular pulse waveform with period of 624 samples
and pulse width of 50 samples. The resulting matrix shown in (3.42) is graphically
illustrated on the 3D graph shown in Fig. 3.24. For clarity purposes, [k] is plotted only
for 0, 200, 400, and 624 trigger delays (rainbow coloured plots) whereas the black-
sphere plots indicates the maximum value of G1 for all the triggering delays. From the
3D plot, it can be observed that as the trigger delay departs from the zero the value of
[k] is be kept constant as long as the value of M is adjusted to withhold δG at 0.5.
Consider the rectangular pulse input shown in Fig. 3.25. A pulse width of 50 samples
implies that maximum value if [k] is achieved for M = 50 and D = 100. Reducing D to
50, the value of δG still remains 0.5 and hence [k] at maximum. Similarly, a smaller
GW of 20 or 50 samples yields the same results by adjusting D to the appropriate value.
For arbitrary, and most importantly, non-symmetric shaped pulses, the possibility of
multiple combinations between M and D is minimized
Figure 3.25 Illustration of various combinations of M and D that result to a maximum value of [k]
88
3.4 Signal processing software
All the SP methods discussed in this thesis are developed in a fully interactive LV-
based SP software (SPS). This section contains snapshots of the SPS’s interface along
with explanatory details regarding the operation of the SP algorithms. The last
subsection describes how the response of a PED is simulated within the SPS and
subsequently measured by one of the implemented methods.
3.4.1 Overview of SPS
The block diagram shown in Fig. 3.26 illustrates the structure of the SPS. The blocks
above the dotted line correspond to the source code (block diagram) of the software and
the ones below represent all the controls and indicator of the front panel (interface) of
the SPS. The source code comprises two main blocks; the signal generation/acquisition
and SP environment (SPE). Due to the high complexity of the SPS’s coded algorithm,
this section only describes the important parts of the algorithm with the aid of
flowcharts and block diagrams. LV controllers are used within the main front panel of
the SPS to modify settings and various other features of the program. Scalar or graph
indicators, also located in the main front panel, are used to illustrate results either in
scalar form (grey filled arrow) or in a waveform (white filled arrow). The solid line
arrows represent the input variables (array or scalar) that control functions (sub-VIs)
within the code of the program. Block 1 of Fig. 3.26 is responsible for the acquisition
or simulation of a desired input signal. SP algorithms are implemented in block 2 to
measure the generated signals with the results presented in either graphs or scalar form.
Figure 3.26 Block diagram of the SPS
89
Figure 3.27 Snapshot of the main front panel of the simulator
A snapshot of the main front panel (MFP) is shown in Fig. 3.27. The simulated
input/output signals are plotted on a graph indicator in section A of Fig. 3.27. In this
particular example, the black line denotes the simulated transient response of a PED,
with the blue and orange plots signifying the gating signal g[k] and the gated
pyroelectric response sG[k]. Execution and termination of the simulation or acquisition
is controlled in section B. Section C displays a folder which consists of 3 tabs. The
graph shown in the first tab plots the FFT of the generated signal on a graph indicator.
The rest of the tab-pages demonstrate the output processed signals and statistical
analysis of the SP methods that are implemented within SPS. The parameters of the
simulated pulse trains are controlled in section D. The software either permits the
selection between simulating ideal rectangular pulses or it utilises these pulses to
generate the transient response of a CM-PED. All the parameters, including noise, of
the PED model can be fully adjusted by a separate sub-VI (PED Settings) which is
invoked under user command within section D. The gating properties of GQSD and
SMGI are linked to controllers in section D while the rest are located in E. Lastly, this
statistics of the SNR performance the simulated or real time measurement are displayed
in section E.
90
3.4.2 Signal generation/acquisition
Discrete-time domain signals are generated or acquired within block 1 of Fig. 3.26.
Upon initial execution of the main program, a nested sub-VI is evoked with its front
panel emerging on top of the main interface of the SPS allowing the user to select
between acquisition and simulation mode. Discrete-time domain signals are simulated
by LV sub-VIs, or else refer to as signal generators. The signal generation algorithm is
described by the flowchart shown in Fig. 3.28. The sampling frequency (fs) defines the
time step between generated samples and is accessible from the MFP of SPS. The
algorithm estimates automatically the number of samples per period (L), with the
assumption that the input frequency f0 is provided in the MFP.
Figure 3.28 Flowchart of the signal generation algorithm within the SPS
91
This is achieved by evaluating the expression given in (3.43)
where T0 and Δt are the input signal period and simulation time step respectively. The
overall generation process is divided in smaller simulation frames, with duration
defined by the product of Δt, L and the number of periods per frame (NP). Consequently,
the number of frames (r) required for desired number of simulated periods (NT) is
estimated. As long as the number of simulated periods (rNP) is less than NT, signal
generation is repeated until the condition is met. If NT is set to infinity, the outer loop of
the algorithm runs indefinitely until manual termination. For the purposes of this
research, the SPS incorporates only pulse generators with variable pulse width (X),
frequency (f0), amplitude (S0), delay (φ) and offset (DC). Nonetheless, other types of
signal generators are available within national instruments (NI) libraries allowing the
simulation of various waveforms, such as triangular, sinewave, sawtooth etc. In addition
to periodic waveforms, noise generation is also possible within the SPS as already
mentioned in earlier section.
3.4.3 Implementation of signal processing methods
Comparing the theory of operation between GQSD and SMGI, one may notice that
these two methods have the gating function as their common feature. The main
principle of SMGI is based on a periodic averaging of the input signal only at the time
instant where the gate window is open (g[k] = 1). If the gated signal sG[k] is applied to a
QSD instead of a integrator, then the whole system is rearranged to GQSD. This
arrangement is illustrated in Fig. 3.29, where SW1 and SW2 are controlled from the
MFP, allowing to switch between the two methods.
Figure 3.29 Block diagram illustrating how the SP methods are implemented within the SPS
t
T
f
fL s
Δ0
0
(3.43)
92
The GQSD operates as a conventional QSD (CQSD) by equalizing the gate duration to
input signal’s period. The resulting signal from either method is statistically processed
to obtain the output SNR of the measurement.
3.4.3.1 Estimation of M and D for GQSD and SMGI
The algorithm that is used to estimate the gate size (M) and trigger delay (D) of GQSD
is explained in detail in section 4.2.2.1. The same logic is followed to obtain the optimal
gating properties suitable for SMGI. Instead of locating the magnitude of the 1st
fundamental harmonic, the algorithm extracts the magnitude of the DC component of
the spectrum for every possible combination of the M and D. Thus, the matrix given in
(3.42) yields
where represents the magnitude of the DC component for a gate length of M
and trigger delay of D. Take for example the case where the optimal value of M and D
needs to be found for a rectangular train of pulses, each one containing 6 discrete
samples.
Figure 3.30 Estimation of M and D using the Gate Estimation algorithm. The blue dotted line
corresponds to the SMGI and red dotted line to the GQSD.
M
M
M
k
DG
DG
DG
GGG
GGG
000
10
10
10
00
00
00
21
21
21
(3.44)
93
For this example, it is assumed that the pulse train has no delay. Therefore, the Gate
Estimation algorithm is used to estimate the magnitude of the DC and fundamental
components of the input signal, for all values of M and D = 0. According to the selected
method (GQSD or SMGI), the resulting optimal values are displayed in section B of
Fig. 3.30. Section C outputs the value pointed by the graph cursor and section D
controls the axis of the plot. As expected, for the GQSD, the maximum value of G1
occurs when M = 2X (12 samples – see the value pointed by the cursor in section C)
whereas for the SMGI, the maximum value of G0 occurs at any value of M below 6
samples.
3.4.3.2 Gating function
The flowchart describing the gating function is explained in Section 4.2.2.1b. Even
though the gating principle is the same for both, GQSD and SMGI, the resulting gated
signal differs fundamentally. Their difference lies on the following principle: the
optimal gated signal for SMGI must be DC signal, whereas for the GQSD must be an
AC signal with 50% duty cycle.
Figure 3.31 Illustration of the input (black), gated (orange) and gating (blue) signals for SMGI (a)
and GQSD (b)
94
As shown in Fig. 3.31 a, when the gating signal (blue plot) is perfectly aligned with
each input rectangular pulse, the resulting gated signal is a DC. While continuously
feeding the LPF, the output of the SMGI yields an output, reflecting the peak voltage of
the pulse train. On the contrary, the GQSD requires a longer window in order to capture
the pulse transition and hence obtain a 50% duty cycle AC signal with a frequency fG
defined by the reciprocal of the gate duration. Subsequently, the QSD is used to
demodulate the gated signal and obtain the magnitude of its fundamental component.
3.4.3.3 Statistical analysis of the SP outputs
As mentioned earlier, the overall duration of a measurement is broken down to smaller
simulation frames. Regardless of the method used, the output signal from each period is
statistically processed, to obtain the output SNR of the measurement. This is achieved
by continuously measuring the joint standard deviation and mean of each output frame.
Additionally, the software collects all the values of the measurement and plots a
histogram as illustrated by the example in Fig. 3.32. More details regarding the SNR
estimation are included in the chapter 5.
Figure 3.32 (a) Histogram of the output signal for a measurement duration of approximately 3
minutes (b) display of the estimated and measured SNR.
95
3.4.4 Modeling of PEDs and measurements in SPS
Valuable achievement in these work, was to embed the existing PED simulator
explained in section 4.1.3 within the SPS, allowing the evaluation of SP algorithms
under simulated but a realistic response of a PED model. However, a slight modification
of the flowchart shown in Fig. 3.28 was necessary. Moreover, the use of dedicated LV
sub-VIs to model the transient response of a TF instead of using the canonical
realization, had increased the efficiency and decreased the complexity of the code.
Figure 3.33 Modified version of the flowchart described in Fig. 3.28. Modifications are indicated by
the red markings. This version allows the generation of the simulated response of the PED to be used within the SPS.
96
Figure 3.34 Snapshot of the MFP, illustrating the controllers responsible for configuring the noise
and TF of the PED model
The front panel of the “PED Settings.vi” emerges upon its execution. After configuring
the properties of the PED model, the generated TF is propagated to the sub-VI that
simulates the transient response of the detector to a generated pulsed train signals. The
noise performance of the PED model can also be modified within the SPS, as well as
selectively turned on and off by the Boolean controller shown in Fig. 3.34
97
CHAPTER
4 Evaluation of methods
This chapter is organised as follows. The first topic discussed in section 4.1 describes
the procedure followed to consolidate the finite element model of the pyroelectric
device SPH-43. Subsequently, the thermal modeling of a linear pyroelectric array
comprising four elements identical to SPH-43, are simulated in COMSOL Multiphysics
to investigate the influence of heat transfer between adjacent elements. Further, we
discuss the importance of the lumped-mass approach (LMA) when simulating the
voltage response of PEDs. Section 4.2 outlines the evaluation performance of GQSD on
data trains that are simulated or acquired experimentally from a radiation measurement.
All the evaluations in this work were performed within the SPS environment. Finally,
section 4.3 describes the simulations curried out to investigate the performance of the
implemented SP methods when used to measure the pulsed response of SPH-43 PED.
4.1 Evaluation of PED models
Accurate modelling of sensors minimises the probability of suboptimal design of signal
conditioning circuits. Additionally, precise modelling and reproduction of sensor signals
aids the selection of suitable devices that would satisfy most of the requirements of a
particular application. The SPH-43 was modelled in two environments. First, the
thermal performance of the device was studied in COMSOL Multiphysics package
within the Heat Transfer Module to obtain the transient response of SPH-43. Secondly,
the voltage response of the detector was simulated within LV where SP algorithms are
conveniently employed to investigate optimal solutions in signal recovery. Results
obtained from FEA are used in the LV-based PED simulator, enabling accurate
reproduction of the thermal transient response of SPH-43, without the need of
continuously running FEM.
98
Figure 4.1Block diagrams of experimental setups, used to (a) to measure the incident radiation by
utilising an energy meter with an optical sensor (LM-2-VIS) and (b) record the PED transient voltage response by replacing the LM-2-VIS sensor with SPH-43
4.1.1 Validating the FEM model
4.1.1.1 Extracting the experimental thermal response of SPH-43
The transient response of SPH-43 was recorded to compare its performance with
simulated data. The block diagrams of the experimental procedures are shown in Fig.
4.1. The PED was irradiated within an aluminium case containing a laser pointer and a
plastic sensor support. Horizontal adjustments of the sensor’s base were necessary,
permitting accurate alignment between the pyroelectric crystal and the transmitted laser
beam. Initially, the incident radiation was measured by a standard energy meter
(FieldMaster), with its associated sensor (LM-2-VIS). The latter was mounted on the
same base support, allowing a distance of 5cm between the sensitive element and the
emitting laser source. LM-2-VIS was then replaced by SPH-43 while ensuring that the
distance between the crystal and the laser source remained the same. The transient
response of the device is obtained to a step illumination of known power. A 3D
representation of the apparatus is shown in Fig. 4.2 where (a) and (b) depicts the
measurements with the LM-2-VIS and SPH-43 respectively.
Figure 4.2 Illustration of the hardware assembly designed using a demo version of Solidworks
2009. (a) measurement with LM-2-VIS and (b) measurement with SPH-43.
99
A standard laser pointer operating at 660nm, was used to illuminated the detector. When
placed 5cm from the target, the pointer projects a red dot with a radius of approximately
2mm, and therefore an exposure area of 12.56mm2. The total power delivered by the
laser pointer was measured by the Field Master while ensuring that the sensitive area of
LM-2-VIS was larger than the exposure area (sensitive area of LM-2-VIS is 19.63 mm2.
The SPH-43 was therefore exposed to red illumination of approximately 142μW/m2.
According to datasheets, the sensitive area of SPH-43 is 7.07 mm2 and thus, the total
dissipated power on the top surface of the crystal is translated into a coupled power
density of 79.93 μW/m2. The voltage response of the detector to an ON/OFF
illumination is recorded by a digital oscilloscope and saved for further analysis.
According to [41], the response of LTI systems to a sinusoidal input x(t) = cos(ωt + θ) is
given by (4.1), with the assumption that the system is causal and asymptotically stable.
Rearranging (4.1), the input signal x(t) can be determined with the assumption that the
output response y(t) and frequency response (FR) of the system H(jω) are known. In
this case, the experimental voltage response of SPH-43 (Vp) corresponds to the output
signal y(t) while x(t) represents the unknown variable, which is the rate of temperature
change ΔTd(t). The product between the FR of the system describing the temperature to
pyroelectric current conversion stage of the PED and the FR of the transimpedance
stage results to a FR of a system, denoted by H2,3(jω). Performing the Fourier transform
in both sides of (4.1) and substituting X(jω), Y(jω) and H(jω) with ΔTd(jω), Vp(jω) and
H2,3(jω) respectively, yields
The Fourier spectrum of ΔTd(t) is then obtained by (4.3)
Finally, the inverse Fourier transform of (4.3) gives the time domain representation of
ΔTd, which is then compared to the temporal temperature distribution of the FEA.
ωjHθtωωjHtxωjHty cos
(4.1)
ωjωjHωjV
ωjXωjHωjYtxωjHty
dp ΔΤ3,2
3,23,2
(4.2)
ωjH
ωjVωj
pd
3,2
ΔΤ
(4.3)
100
Figure 4.3 3D grid used in COMSOL to extract the temperature distribution
4.1.1.2 FEM results of the thermal response of SPH-43
The thermal response of the SPH-43 model was simulated in COMSOL using the 3D
geometric model and specifications described in Chapter 4. To compare the results with
the experimental thermal response, the heat flux boundary condition was set equal to the
power obtained from the experiments (79.93μW/m2). Details on boundary condition
assigments are already mentioned in Chapter 3. As shown in Fig. 4.3, a 3D grid is used
to extract the simulated temperature distribution for various points within the volume of
the pyroelectric crystal. The data are then exported within a text file (.csv format) which
is then used by the LV-based simulator to obtain the average temperature distribution of
the pyroelectric crystal.
4.1.1.3 Comparing results within the LV-based Simulator
The block diagram shown in Fig. 4.4 illustrates the steps followed to compare the
measurement-derived thermal transients with the one modelled by FEM.
Figure 4.4 Block diagram showing the procedure followed to compare the thermal transient
response obtained from COMSOL to the experimental data derived from the voltage response of the detector.
101
The blocks enclosed by a dotted line correspond to the COMSOL Multiphysics
simulations while the blocks enclosed by the dashed-line rectangle represent the LV-
based simulator. The thermal transient response simulated in COMSOL is exported to
an .csv file which is then loaded within the LV simulator to extract the data and estimate
the average temporal response of the simulated temperature profiles. The experimental
thermal transient of SPH-43 is obtained using the FR of H2,3(jω). The parameters of the
variables comprising the H2,3(jω) are taken according to the components of the TIA
(Table 3.1-II - see Chapter 3), including the area (7.07μm2) and the pyroelectric
coefficient (-2.3 (sA)/(Km2)) of the LiTaO3 crystal. However, these parameters can be
modified in the front-panel of the “PED Settings” sub-VI to satisfy the needs of the
simulation.
The LV process shown in Fig. 4.4, is essentially a subroutine within the “PED Settings”
sub-VI called after the name “Experimental vs Simulated Response”. When this
subroutine is evoked, it is continuously running until the control button “Return” is
pressed to return to the “PED Settings” sub-VI. A snapshot of the subroutine’s front
panel (GUI) is illustrated in Fig. 4.5. The graph indicator is used to plot the results and
allowing visual comparison of the measurement-derived thermal transient (green) with
the one simulated in COMSOL (blue). The transient response obtained from FEA is
used in (4.2) to simulate the voltage response of the detector (red-line plot) and compare
it with the experimental voltage response (black-line plot) of the device.
Figure 4.5 Snapshot of the front panel LV subroutine used to compare the experimental thermal
transient to that obtained by FEM
102
Finally, the algorithm calculates the equivalent thermal components (Rth and Cth),
including the thermal time constant (RthCth), from either the experimentally derived or
simulated thermal response, as described in Section 3.1.2.1. These parameters are
transferred to the main sub-VI of the simulator where the Foster network is
implemented to reproduce the thermal response of the detector.
4.1.1.4 Summary of the objectives
The aim of this analysis is to verify the FEA results, where a 3D thermal model of SPH-
43 was used within COMSOL to simulate its thermal transient response. The
experimental response of the detector was recorded for 7.3sec to a step illumination of
3.65 sec. The data are then imported in the LV sub-VI to derive the actual thermal
transient response of the detector. Ultimately, results are compared with simulated data
and the Foster network developed is used to regenerate the thermal transient response of
the detector. Based on the verified FEM model of SPH-43, three different geometric
models of detector arrays are designed within the COMSOL package to investigate their
thermal behaviour. The distance between the crystals was set 1, 1.5 and 2mm to
examine the effect of crosstalk between the side surfaces of adjacent crystals. The
constraints used in the array geometries are described in Section 3.1.1.3 while the rest of
the boundaries and initial conditions are identical to the FEM model of SPH-43.
103
4.2 Evaluation of SP methods
This section describes the evaluation of the performance of GQSD on pulse trains,
which are either simulated or acquired experimentally from a radiation experiment. All
evaluations in this work were performed within the SPS environment: the digitally
simulated or physically acquired signals are presented in an identical manner to the
input of GQSD and/or alternative methods implemented in SPS, regardless of the mode
of operation (simulation or acquisition). As discussed in chapter 3, the output signal
level of a conventional QSD (CQSD) drops significantly as the duty cycle of the input
pulse train decreases, consequently deteriorating the output SNR. Contrary, the
contained gating function of GQSD maximises the output signal while noise levels
remain the same. Consequently, the output SNR is improved when compared to CQSD
and an SMGI. The obvious SNR improvement in GSQD against CQSD has been
confirmed by processing of pulsed trains. For duty cycle of 10%, GQSD outperforms
CQSD with around 5dB and with much more at lower values of the duty cycle.
Nevertheless, GQSD was evaluated against a CQSD and SMGI, using SNR as the main
criterion.
4.2.1 Simulated data
The simulation setup refers to a pre-specified set of input signal parameters, noise
levels, and settings with respect to the type and objectives of the simulation.
4.2.1.1 Cases for evaluation
A number of cases are considered, depending on the type of the input pulsed signal. The
first one (Case A) investigates performance when detecting rectangular pulse trains
while the second (Case B) deploys the two methods in a realistic environment, with
realistically shaped pulsed signals generated by SPS.
TABLE 4.2-I INPUT SIGNAL PARAMETERS VALUES
Description Symbol Value Units
Pulse Peak Voltage A 50 mVPulse modulation frequency f0 2 kHzPulse modulation period T0 0.5 msDuty cycle δ ≈10 %Samples per period N0* 624 samplesSamples per pulse X* 63 samples
* The discrete period of the signal is estimated from the product T0·fs, where “fs” denotes the sampling rate. The number of samples per pulse is estimated by N0·δ.
104
Figure 4.6 Simulated pulsed signals of ideal (top) and realistic (bottom) shape. (a) and (d) – noise
free; (b) and (e) – with Gaussian noise; (c) and (f) – with flicker noise.
In both cases, the input signals are accompanied with Gaussian white and/or coloured
noise, synthesized by fully adjustable noise generators. The effect of various types of
coloured noise on the output SNR of GQSD and SMGI is examined in Case C. The time
domain representation and parameters of the signals used in case A, B and C, are shown
in Fig. 4.6 and Table 4.2-I. To allow comparison between performance between
acquired and simulated data, a forth case (Case D) simulates the voltage response of a
PIN photodiode exposed to a pulsed modulated radiation. The input signal in Case D,
matches the experimentally acquired one, described in a later section, along with the
experimental setup.
4.2.1.2 Input signal
Rectangular pulsed trains (Case A) are produced by standard function generators
available within the LV libraries. The generation of arbitrary pulsed signals (Case B), is
achieved by feeding the rectangular pulse train to the input of a TF within the SPS.
Therefore, the shape of the resulting time domain signal depends on the characteristic
polynomial of the TF. This allows the simulation of sensors, provided that their TF is
realizable [34, 35]. Under ideal input conditions (noiseless input), the output of the
CQSD, SMGI and GQSD methods represents the “true” output which is subsequently
used to estimate the relative error when noise is taken into account. The synthesized
ideal and non-ideal signals are shown in Fig. 4.6a and Fig. 4.6d. Further, Fig. 4.6b and
Fig. 4.6e illustrate the two signals with additional Gaussian noise. Lastly, Fig. 4.6c and
Fig. 4.6f depict that same signal immersed in 1/f noise.
105
Figure 4.7 Time (top) and frequency (bottom) domain representation of simulated coloured noise.
The PSD in (a) is proportional to 1/f β for β = 1.58, (b) β = 2.04 and (c) β = 2.23
Noise is synthesized by corresponding LV subroutine algorithms and then is added to
the input to construct noisy signals. Gaussian white noise generation is achieved by
converting uniformly distributed random numbers into a Gaussian distributed sequence.
Unlike to the flat spectrum of white noise, the power spectral density P(f) of coloured
noise is inversely proportional to the frequency [62] as shown in (4.4):
The colour of the noise is defined by the exponent β (0 < β < 3), including white (β = 0),
pink (β = 1) and brown (β = 2). The noise generating algorithm permits the synthesis of
any noise colour specified within the range of β by feeding white noise though a
dynamic system, usually a shaping filter. While Case A and Case B involve white and
pink noise respectively, Case C introduces the variety of noise signals, as shown in Fig.
4.7 a,b,c, where β takes values between 1.5 <β <2.5. Within this range we aim to
demonstrate the SNR improvement obtained by GQSD as the noise colour shifts from
pink to brown (β = 1.5), brown (β = 2) and beyond (β = 2.5). The log-log power spectra
of Fig. 4.7a, Fig. 4.7b and Fig. 4.7c exhibit the three values of slope β, as estimated by
(4.5), with Δy denoting the magnitude difference in dB and Δx corresponding to a
frequency range of 1 octave.
ffS
1 (4.4)
x
y
ff
yβ
Δlog10
Δ
log10log10
Δ
12
(4.5)
106
4.2.1.3 Sampling and processing settings
The generation or acquisition of the discrete input signals is achieved under a pre-
specified sampling rate (fs) and time frame (tsim) defined by the amount of processed
pulsed periods. Assuming a hypothetical sampling rate of 250kHz, the overall discrete
size of a 60 sec input signal becomes very large (15MSamples). To avoid degradation of
computational performance, the simulation/acquisition of the input signal is broken into
smaller segments utilizing single feedback nodes to pass the final values as initial
conditions for the next simulation/acquisition segment. This enables continuous data
processing without excessive strain on computer memory (RAM).
The main goal of the evaluation procedure is to determine and compare the
SNIR of each method under various types of pulsed signals and noise. According to
(2.52) and (3.41), the OTC of SMGI and GQSD is proportional to the effective
bandwidth (feff) of their internal LPFs and the size of the gate M. Useful comparison
between the two methods is possible only if their OTCs are equal. With respect to the
gate size M, the feff must be adjusted so that the OTC reaches the desired value. Table
4.2-II shows the calculated correspondence between gate size and bandwidth settings
for three values of the sampling rate measured in units of samples per gate duration. In
the example on the bottom row, assume that the GQSD of 50Hz effective bandwidth
utilizes a gating signal of 136 samples gate width (M) to detect a rectangular pulsed
train: this setting yields an OTC of 14.628ms. In an example comparison with the
performance of SMGI for M = 4 samples, the feff must increase to 1.7kHz to meet the
required OTC of 14.628ms. Alternatively, decreasing the feff to 1.47of GQSD while the
feff of SMGI remains at 50Hz the two methods can be compared at a higher OTC (≈
497ms). Following this approach we compute the output SNR of GQSD and SMGI for
various values of M at a specified feff. For each gate length the SNR is computed for all
possible values of OTC by adjusting the feff accordingly.
TABLE 4.2-II GATE SIZE AND BANDWIDTH SETTINGS
M [samples per gate]
feff
[Hz]
ETC [ms]
OTC [ms]
4 50.000 3.18000 497.36 4 775.01 0.20536 32.087 4 1700.0 0.09362 14.628 62 3.2260 49.3380 497.36 62 50.000 3.18000 32.087 62 109.679 1.45110 14.628 136 1.4700 108.230 497.36 136 22.790 6.98210 32.087 136 50.000 3.18000 14.628
107
Figure 4.8 Temporal response of GQSD to a simulated input pulsed signal immersed in flicker
noise. The probability density distribution of its steady state response is shown on the RHS histogram.
4.2.1.4 Estimation of the output SNR and relative error
The computation of the output SNR requires the analysis of signals obtained at the
output of the method under test (MUT). Under noisy conditions, statistical analysis is
imperative to ensure the fidelity of the output average to the quantity being measured.
Consider the case of a continuous measurement on a time-domain output signal VSP
shown in Fig. 4.8. The estimated output SNR (SNRest), in dB, is defined by the ratio of
the mean (μest) to the standard deviation (σ) of the output measured signal (4.6).
To find the true (expected) SNR value it is necessary first to run the simulation without
noise and record the mean (μexp) of the response. Subsequently the ratio between μexp
and the estimated standard deviation of the noise (σ) yields the expected SNR (SNRexp)
as shown in (4.7) while the relative error (Er) is estimated by (4.8).
where the statistical terms μexp/est and σ are estimated by (4.9) and (4.10) respectively.
σ
μSNR est
est log20 (4.6)
σ
μSNR
expexp log20
(4.7)
exp
expestr μ
μμE
(4.8)
2
112
1t
t
SPexp/est dttVtt
μ
(4.9)
2
1
2
12
1t
t
SP dtμtVtt
σ (4.10)
108
Figure 4.9 Block diagram of the experimental setup
4.2.2 Experiments
For this task, SPS operates in acquisition mode and uses NI acquisition hardware (NI
USB-6251), permitting up to 1.25MHz sampling rate. The block diagram of the
hardware setup is shown in Fig. 4.9. The digital function generator (DFG) TTi 1010 is
used to drive a basic red laser diode with a rectangular pulsed signal of 2 kHz frequency
and duty cycle of 0.0048. Detection of the emitted radiation was achieved by a PIN
photodiode transducer (PDA10CS) operating in trans-impedance mode. The signal is
amplified by a generic non-inverting amplifier and subsequently digitized by the NI
USB-6251 at the specified sampling rate. To ensure synchronous detection and avoid
custom made triggering circuits, a reference signal (TTL level) originating from the
DFG was connected to the digital ports of the NI USB-6251 to trigger the acquisition.
The detected signal is processed by the two methods to measure the output signal and
noise levels. Subsequently, these values are used to compute their SNR performance.
The results obtained in Case D of the simulations are compared with the experimental
findings. Pictures from the hardware setup are shown in Fig. 4.10.
Figure 4.10 Snapshots of the experimental setup.
109
4.3 Simulated measurements with SPH-43
As discussed in Section 3.1.4, the trade-off between sensitivity and bandwidth depends
on RF and CF which they also affect the noise performance of the PED. Even though the
thermal properties of the detector also affect the voltage responsitivity of the detector, in
this work they are kept constant, since in a real-time application they cannot be changed
or modified to alter the performance of the device. However, the design of
commercially available PEDs often allows gain and bandwidth manipulation by adding
external electronic components in parallel to the feedback network of the device.
Obviously, any reduction of the TIA gain will significantly decrease the PED voltage
response and possibly deteriorate the SNR performance of the following signal recovery
method. As this may apply for the CQSD, the involvement of the gating function of
SMGI and GQSD creates the need to examine their performance from a
multidimensional point of view. Considering the RF and CF as the only variable
components, this evaluation includes various simulations investigating the SNR
performance of CQSD, SMGI and GQSD when used to measure the simulated pulsed
response of SPH-43.
4.3.1 Simulation settings
4.3.1.1 Excitation signal settings
The response of SPH-43 model to a rectangular pulse train was simulated within the
SPS which was continually feeding the input of each SP algorithm for a pre-specified
number of periods. The parameters of the excitation signal are shown in Table 4.3-I.
Subsequently, the SNR output of each method is statistically obtained as discussed in
Section 4.2.1.4. The sampling frequency (fs) and duty cycle (δ) are set to 124.8kHz and
1% respectively, resulting a pulsed period of 624 samples and pulse width of 6 samples.
The pulse modulation frequency was selected below the corner frequency of 1/f noise in
order to see the performance of the DSP algorithms under the influence of pink noise.
TABLE 4.3-I INPUT SIGNAL PARAMETERS VALUES
Description Symbol Value Units
Pulse Peak Voltage A 500 μVPulse modulation frequency f0 200 HzPulse modulation period T0 5 msDuty cycle δ ≈1 %Samples per period N0 624 samplesSamples per pulse X 6 samples
110
Figure 4.11 Frequency response simulation of SPH-43 for RF = 100GΩ (a) and RF = 1GΩ (c). The corresponding noise gains are plotted as a function of frequency in (b) and (d) respectively. This
process was repeated for various values of feedback capacitance CF (0.1 – 10 pF)
4.3.1.2 PED settings
The voltage responsitivity of the PED model was configured within the LV simulator by
assigning set of values for the feedback resistor (RF) and capacitor (CF) while the rest of
the components were set according to the specifications of the SPH-43 device. The
thermal/physical and electrical properties of the pyroelectric crystal were provided by
the manufacturer who also disclosed all information regarding the electronic
components within the device. FEM modelling and experimental results of the SPH-43
confirmed a thermal time constant of 684ms. Further information of the TIA were found
from the op-amp’s datasheet (AD8627), also listed in Table 3.1-II. The output SNR of
each method was recorded for two cases; in Case I, RF takes a value of 100GΩ and in
Case II, is decreased to 1GΩ. In each case, the simulations were repeated for various
values of feedback capacitance CF, ranging from 0.1pF (common value of parasitic
capacitance) to 10pF. Figure 4.11 shows the simulated magnitude response and noise
gain (NG) as a function of frequency for Case I (top graphs) and Case II (bottom
graphs). A noticeable difference between Fig. 4.11a and Fig. 4.11c, is the magnitude
drop due to RF reduction. The trade-off between gain and bandwidth is illustrated by
observing the electrical cut-off frequency (fel) of the detector shifting from 15.91Hz to
2.22kHz as RF decreases from 100 to 1GΩ.
111
Assuming a modulation frequency of 200Hz, the magnitude of the detected fundamental
component will reduces approximately by a factor of 8. In parallel, as the feedback
resistor decreases, the natural frequencies of the feedback factor given in (3.29), tend to
shift towards the open loop gain of the op-amp yielding equal attenuation of the noise
and signal gain at 200Hz. Zero SNR improvement (SNIR) is therefore expected under
the assumption that noise and signal magnitudes are extracted exactly at the frequency
of reference. Therefore, it can be speculated that SP methods used to extract the
magnitude of frequency components that are located within a region where the noise is
amplified linearly with increasing frequency, will be highly prone to SNIR below 1dB.
As mentioned earlier, the magnitude and noise gain response of the detector can also be
modified by varying CF. Comparing Fig. 4.11a and Fig 4.11b, it is observed that in Case
A, varying CF from 0.1 to 10pF reduces the signal gain by approximately a factor 100
(2220/22.2) whereas the noise gain only drops by a factor ~85. Similarly, for Case B the
reduction of signal gain is slghtly higher than the noise gain reduction for the same set
of capacitance values. Therefore, the probability of SNR improvement is higher when
the CF is optimised. The explanation given here, neglects the Johnson and shot noise
since the predominant noise source at 200Hz is mainly the op-amp input noise which is
amplified by the noise gain. Nonetheless, the LV simulator includes all the noise
sources allowing a more realistic simulation of the PED response.
4.3.2 Digital signal processing
As explained in the previous section, altering RF or CF will either degrade or neutralise
the SNR performance. However, if further reduction of noise is not possible, the
remaining option to improve the output SNR is to amplify the output signal. One way of
accomplishing this, is by feeding the output signal of the PED through an ultra-low
noise amplifier. However, the inherent noise of the input signal will also be amplified
yielding an insignificant SNIR. In this work, we use the GQSD and SMGI to maximise
the output by gating the pyroelectric output signal, while the output noise remains
unchanged. However, since the two methods take into account the temporal shape of the
input signal, the two methods, along with the CQSD are discussed individually,
signifying the importance of the feedback components RF and CF to their performance.
112
Figure 4.12 Simulated steady-state response of the SPH-43 model to a pulse train input. (a) Case A:
RF is 100GΩ and (b) Case B: RF is 1GΩ.
4.3.2.1 GQSD settings
As explained in section 3.1.4, the temporal shape of the PED pulsed response is mainly
influenced by its electrical time constant τel. For high values of τel (>20ms) the PED
response resembles a triangular waveform whereas for smaller values the shape of the
pulse pattern is preserved. Figure 4.12a and Fig. 4.12b, illustrate the simulation of the
steady-state pulsed response of SPH-43 for Case I and II respectively. The effect of CF,
is observed by comparing the red and black-lined plots, which correspond to a CF of
0.1pF and 2pF respectively. As expected, the effect is more obvious in Case II. Keeping
RF to 1GΩ and gradually increasing CF, the temporal response of the PED will
eventually approximate the shape of the signal depicted by the black-lined plot in
Fig. 4.12a. Since the shape of the resulting time domain signal is altered for every
combination of RF and CF, the optimal estimation of the gate size (M) and triggering
delay (D) is required for each value of CF.
TABLE 4.3-II GQSD: ESTIMATION OF M AND D FOR CASE I (RF = 100GΩ)
Case I CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
I1GQSD
M (optimal) 57 47 42 42 41 41 41 41 41 D (optimal) 601 605 607 607 607 607 607 607 607
OTC [ms] 348.47 422.61 472.92 484.45 484.45 484.45 484.45 484.45 484.45
I2GQSD
M 57 57 57 57 57 57 57 57 57
D 601 600 600 599 599 599 599 599 599
OTC [ms] 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 feff 5 5 5 5 5 5 5 5 5
113
TABLE 4.3-III GQSD: ESTIMATION OF M AND D FOR CASE II (RF = 1GΩ)
For both Case I and Case II, the optimal values of M and D were obtained for each
value of CF using the gate estimation algorithm described in section 3.3.1.4. According
to (3.41), the OTC increases as M becomes smaller. To maintain the OTC constant,
simulation I2GQSD considers a gate size of 57 samples for all values of CF. Results
showed negligible difference between the I1GQSD and I2GQSD (~0.4dB), therefore only
I2GQSD was considered. In Case II, the well-preserved shape of the pulsed response of
SPH-43 gives a good perspective for SNR improvement for GQSD or SMGI.
Interestingly, the optimal value of M increases with CF, implying possible SNR
improvement with a decreasing OTC. Thus, in simulation II1GQSD the OTC is
maintained to 993.13ms by adjusting the effective bandwidth (feff), whereas in
simulation II2GQSD, the OTC varies with M by fixing feff to 5Hz. Lastly, II3GQSD
considers the case where feff and M are kept constant to obtain a constant OTC of
993.13[ms].
4.3.2.2 SMGI settings
The gate size and trigger delay of SMGI was optimised in a similar fashion to the
GQSD setup. Naturally, the optimal value of M is 1 for all values of CF, with the
condition that the trigger delay is adjusted so that the SMGI will continuously integrate
the peak voltage of the PED response. A gate size of 1 sample corresponds to an OTC
of 19.86[s]. Slow response systems are often not desirable, unless severe noise
conditions demand such a long integration time constant. Alternatively, we keep M
equal to 1 and increase the feff to meet the OTC obtained by GQSD. Under these
conditions, feff is estimated 284.74[Hz] for Case I and 100[Hz] for Case II.
Case II CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
II1GQSD
M (optimal) 20 22 25 29 31 37 39 41 41
D (optimal) 618 617 615 613 612 609 608 607 607
OTC [ms] 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13
feff [Hz] 5 4.54 4 3.44 3.22 2.7 2.56 2.43 2.43
II2GQSD
M (optimal) 20 22 25 29 31 37 39 41 41
D (optimal) 618 617 615 613 612 609 608 607 607
OTC [ms] 993.13 902.84 794.50 684.91 640.72 536.82 509.29 484.45 484.45
feff [Hz] 5 5 5 5 5 5 5 5 5
II3GQSD
M 20 20 20 20 20 20 20 20 20
D 618 618 618 618 618 618 618 618 618
OTC [ms] 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 feff 5 5 5 5 5 5 5 5 5
114
TABLE 4.3-IV SMGI: ESTIMATION OF M AND D FOR CASE I
TABLE 4.3-V SMGI: ESTIMATION OF M AND D FOR CASE II
A processing bandwidth of 284[Hz] is wide enough to allow a significant amount of
noise passing through the filter and hence this scenario is not considered. Table 4.3-IV
lists the gating properties for Case I, where equal OTCs between the GQSD and SMGI
are maintained by applying identical gating of 57 samples. In Case II (SII1SMGI) the
gate width was set to 20 samples to meet the OTC of GQSD whereas in SII2SMGI the
SMGI is evaluated at its optimal gate width (1 sample) with an effective bandwidth of
100[Hz]. These settings are summarised in Table 4.3-V.
4.3.2.3 CQSD settings
The GQSD operates as a CQSD if the gate duration is set equal to the period of the
input signal. The configuration settings of CQSD for Case I and II are summarized in
Table 4.3-VI and 4.3-VII. Since gating is not involved in CQSD, the OTC is equal to
the ETC of the QSD. Comparison between the three methods is valid, only under equal
OTCs.
TABLE 4.3-VI CQSD: ESTIMATION OF M AND D FOR CASE I
TABLE 4.3-VII CQSD: ESTIMATION OF M AND D FOR CASE II
Case I CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
I1SMGI
M 57 57 57 57 57 57 57 57 57 D 9 8 6 6 6 6 6 6 6
OTC [ms] 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 feff 5 5 5 5 5 5 5 5 5
Case II CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
II1SMGI
M 20 20 20 20 20 20 20 20 20
D 4 5 6 6 6 6 6 6 6
OTC [ms] 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13
feff [Hz] 5 5 5 5 5 5 5 5 5
II2SMGI
M (optimal) 1 1 1 1 1 1 1 1 1
D (optimal) 7 7 7 7 7 7 7 7 7
OTC [ms] 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 feff [Hz] 100 100 100 100 100 100 100 100 100
Case I CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
I1CQSD
OTC [ms] 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 feff 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456
Case I CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
II1CQSD
OTC [ms] 993.12 993.12 993.12 993.12 993.12 993.12 993.12 993.12 993.12 feff 0.160 0.160 0.160 0.160 0.160 0.160 0.160 0.160 0.160
115
Therefore, feff was adjusted to 0.456 and 0.16[Hz] for Case I and II respectively. It is
interesting to note here that the bandwidth of the CQSD is reduced approximately by a
factor of 10 for Case I and 30 for Case II. This suggests a considerable amount of noise
reduction at the output and hence an improved SNR response.
4.3.2.4 Simulations summary
To avoid confusion between the evaluations, the following scenarios are summarized in
Table 4.3-VIII and 4.3-IX for Case I and Case II respectively.
TABLE 4.3-VIII SUMMARY OF SIMULATIONS FOR CASE I
TABLE 4.3-IX SUMMARY OF SIMULATIONS FOR CASE II
The simulations have been categorised with respect to the OTC regardless the values of
M and feff. However, the scenario SII3GQSD shown at the bottom of Table 4.3-IX is
examined and presented with results from Case II, to investigate the possibility of SNR
improvement of GQSD for a decreasing OTC (993.13 to 484.45 μs).
Case I CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
SI1GQSD
M 57 57 57 57 57 57 57 57 57
D 601 600 600 599 599 599 599 599 599
feff 5 5 5 5 5 5 5 5 5
SI1CQSD
M 57 57 57 57 57 57 57 57 57
D 9 8 6 6 6 6 6 6 6
feff 5 5 5 5 5 5 5 5 5
SI1CQSD
M 57 57 57 57 57 57 57 57 57
D 9 8 6 6 6 6 6 6 6
feff 5 5 5 5 5 5 5 5 5
OTC [ms] 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47 348.47
Case II CF [pF] 0.1 0.2 0.4 0.8 1 2 4 8 10
SII1GQSD
M (optimal) 20 22 25 29 31 37 39 41 41
D (optimal) 618 617 615 613 612 609 608 607 607
feff [Hz] 5 4.54 4 3.44 3.22 2.7 2.56 2.43 2.43
SII2GQSD
M 20 20 20 20 20 20 20 20 20
D 618 618 618 618 618 618 618 618 618
feff 5 5 5 5 5 5 5 5 5
SII1CQSD feff 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16
SII1SMGI
M 20 20 20 20 20 20 20 20 20
D 4 5 6 6 6 6 6 6 6
feff 5 5 5 5 5 5 5 5 5
SII2SMGI
M (optimal) 1 1 1 1 1 1 1 1 1
D (optimal) 7 7 7 7 7 7 7 7 7
feff 100 100 100 100 100 100 100 100 100
OTC [ms] 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13 993.13
SII3GQSD
M (optimal) 20 22 25 29 31 37 39 41 41
D (optimal) 618 617 615 613 612 609 608 607 607
OTC [ms] 993.13 902.84 794.50 684.91 640.72 536.82 509.29 484.45 484.45
feff [Hz] 5 5 5 5 5 5 5 5 5
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CHAPTER
5 Results and discussion
This chapter is organised as follows. Results obtained by the PED FEA, are
demonstrated in the first section while subsequently compared to experimental data for
verification. The next section compares the SNR performance of the three SP
algorithms, the CQSD, SMGI and GQSD. The suitability of these methods in measuring
the pulsed response of a commercially available pyroelectric device (SPH-43) is
discussed in the last section, where their evaluation under noise conditions is taken into
account.
5.1 PED modelling
The following subsections include the verification of the FEM model with experimental
data recorded using the actual device (SPH-43). Results obtained from the FEA of the
array models are discussed, focusing on the crosstalk between adjacent elements.
5.1.1 Verification of the FEM model
The resulting temperature distribution of the SPH-43 3D FEM model is shown in Fig.
5.1.
Figure 5.1 3D temperature distribution of the SPH-43 PED model
117
Figure 5.2 (a) Comparison between the measurement-derived thermal response with the one
modelled by FEM (b). The temperature profile obtained by the FEA is then used to simulate the voltage response of the SPH-43 by taking into account only the second and third conversion stage TFs of the SPH-43. The resulting transient is then compared to the experimentally derived voltage
response of the device.
As discussed in section 4.1.1.2 the average temperature distribution of the 3D model
along with the experimentally derived temperature response of the device are extracted
using the LV-based simulator. The results are shown in Fig. 5.2a, indicating excellent
match between the thermal response obtained from FEM and the actual thermal
response derived from recorded data. In fact, this verifies the boundary conditions and
constraints that where included in COMSOL. Subsequently, the obtained thermal
response from COMSOL was used within the SPS to simulate the voltage step response
of the detector. As shown in Fig. 5.2b, a noticeable difference between the peak values
of the two transients is observed whereas the error between their exponential growth
and decay is negligible. This difference could be due to the unrealisable sharp transition
of the step input signal that is provided by the laser pointer while on the contrary the
excitation signal used in COMSOL has a perfect sharp edge. Since the PED is
responsive to temperature change, faster transitions could possibly increase the detail of
the output voltage transient.
118
Figure 5.3 Meshing of the 3D PED array models. The distance between their elements was set to 1
(a), 1.5 (b) and 2mm (c)
5.1.2 Simulation results of PED arrays
Figure 5.3 illustrates the meshing of three geometric models of three linear
arrangements of pyroelectric crystal of SPH-43. In each case, the distance between the
the elements was varied, taking values of 1, 1.5 and 2mm in order to investigate thermal
transients of radiation emission of adjacent surfaces
Figure 5.4 Results from FEM analysis of three PED array geometries. The middle elements of the arrays were only assigned natural convection conditions to examine the thermal influence between
their adjacent surfaces. The simulation was repeated for the each geometries where in (a) the crystals were spaced by 1, in (b) by 1.5 and in (c) by 2mm.
119
Figure 5.5 Simulated average temperature transients of the middle left crystal of each array
geometries (red plots – left axis). The response was identical for each outer element of the arrays making the plots indistinguishable (black plots). The right hand side axis corresponds to the
temperature transient of the outer crystals of the arrays.
The effect of thermal crosstalk is observed by assigning the side elements to irradiation
conditions while the middle ones were set to natural heat convection. The duration of
the step response was set to 4 seconds for an input heat flux of 80μW/m2. Temperature
disturbance of the non-irradiated crystals was observed on the left and right sides of the
inner crystals due to surface-to-surface radiation emission. To illustrate this
phenomenon, the spatial temperature response of each array geometry was recorded and
plotted in 3D graphs as shown in Fig. 5.4. The effect was predominantly apparent in (a)
due to the close proximity of the crystals (1mm). However, a significant reduction of
the effect is noticeable for the other two cases where the distance between the crystals
was increased to 1.5 and 2mm respectively. The spatial temperature distribution of the
two leftmost crystals was extracted within the SPS, to obtain the average temperature
distribution as a function of time. Results are plotted in Fig. 5.5 where the red and black
plots correspond to the thermal transient of the inner and outer crystals of each array
respectively. Furthermore, the resulting voltage responses due to surface-to-surface
radiation and the input heat flux emission, where derived within the SPS.
120
Figure 5.6 Voltage transients of the middle left crystal of each array geometries (red plots – left
axis). Again, the response was identical for the leftmost crystals, making the plots indistinguishable (black plots). The axis on the right hand side corresponds to the voltage transient of the outer
crystal of the arrays.
Results indicate that thermal crosstalk between adjacent elements does exists due to
surface-surface radiation, but when translated to temperature the effect is insignificant
compared to the resulting voltage response of the crystals that were assigned to heat
flux conditions (black line plot). According to Fig. 5.6, it is confirmed that linear arrays
of evenly spaced pyroelectric elements, of a structure based on the commercial PED
device SPH-43, can be constructed without any substantial influence of thermal
crosstalk. However, in the need of high-resolution measurements, the space (d) between
crystals must be reduced to accommodate higher number of elements within the array.
Under these circumstances (d<500μm), thermal crosstalk might be intensified
especially for the elements located in the middle of the array.
121
5.2 DSP Evaluation: GQSD, CQSD & SMGI
5.2.1 Simulation results
The SNR performance of GQSD, CQSD and SMGI was recorded for Case A, B and C
under various input conditions. The output SNR values for each method are plotted in
Fig 5.7, Fig 5.8, and Fig. 5.9 as bar charts where the blue, black and grey bar series
correspond to the performance of CQSD, GQSD and SMGI respectively. The results of
Case A are shown in Fig. 5.7a for white noise and Fig. 5.7b for 1/f noise. Similarly,
Fig. 5.8a and Fig. 5.8b depict the performance of the two methods for Case B. Results
for Case C are presented in Fig. 5.9. The results are consistent for varying gate size,
varying OTC, as well as type of pulse. The variations depending on the type of noise are
presented and discussed below.
5.2.1.1 White noise
The results in Fig. 5.7a and Fig. 5.8a imply that for white noise, the SNR performance
of SMGI is superior to GQSD for the lower two gate sizes. In the example case shown
in Fig. 5.7a.2, even though for a gate size of 136 samples the SNR output of GQSD
(49.5dB) is higher than SMGI (46.2dB), this is not sustained for a smaller number of
samples.
Figure 5.7 Simulated SNR performance of CQSD (blue bars), GQSD (black bars) and SMGI (grey bars) for Case A. The methods where evaluated under white (a) and 1/f (b) noise conditions of three
OTCs.
122
Figure 5.8 Simulated SNR performance of CQSD (blue bars), GQSD (black bars) and SMGI (grey bars) for Case B. Similar to Case A the methods were evaluate under white and 1/f noise conditions. The output SNR of the GQSD for M = 1 is not estimated since the reconstruction of the gated signal
is non-periodic. On the other hand, a gate of 1 sample is used by the SMGI to detect the peak voltage of the input pulse.
The SNR performance of SMGI increases to 52.5dB when the gate size is decreased to
62 samples. The reduction of the gate size caused the output signal of SMGI to increase
from 18.054mV to 44.961mV and the output noise from 88.45μV to 106.74μV. In
comparison, the GQSD output increases from 15.85mV to 15.94mV whereas the noise
from 53.86μV to 82.48μV. In both methods, the output noise was observed to be higher
since the feff was increased to maintain the same OTC for SMGI. Even with lower
output noise, the SNR of GQSD is overall less than the SMGI under Gaussian white
noise conditions. These are expected results, since the SMGI measures the peak
amplitude of the pulse which is substantially higher than the magnitude of the
fundamental harmonic of the gated signal sG(k) used in GQSD.
5.2.1.2 1/f Noise
The SNR performance of GQSD improves in the case of 1/f noise as shown in Fig. 5.7b
and Fig. 5.8b. This is exemplified in charts (2) and (3) where the performance of the
two methods becomes comparable already for the intermediate gate sizes. A significant
improvement in GQSD against SMGI is notable for higher bandwidths where the OTCs
values are reduced to achieve higher detection rates.
At slower rates, the SNR of SMGI improves due to a substantial decrease of the output
noise while the output signal level is maintained at a voltage much higher than the
123
GQSD. Interestingly, the performance of the SMGI and GQSD is challenged by the
CQSD for the intermediate gate sizes of Case A. Under 1/f noise conditions, the SNR
performance of the CQSD is approximately 5dB higher than the SMGI and 4dB higher
than the GQSD. However, for all OTCs the GQSD can achieve higher SNR than CQSD
by manipulating the gate width.
The results obtained in Case C are shown in Fig. 5.9 where the GQSD and SMGI were
tested further, under various types of coloured noise proportional to 1/f β by varying the
exponent factor β. For consistency with previous simulations, the evaluation settings
and pulse type are identical to Case B. The bar charts in Fig. 5.9a, Fig. 5.9b, and Fig.
5.9c represent the SNR performance of the two methods for β = 1.5, β = 2, and β = 2.5
respectively.
Figure 5.9 SNR performance of GQSD and SMGI to an arbitrary shaped pulse train signal with
accompanied noise proportional to 1/f (β). The value of the exponent β was set to 1.58, 2.04, and 2.51 for (a), (b), and (c) respectively. Similarly to Case B, the output SNR of the GQSD for M = 1 is not
estimated, since the reconstruction of the gated signal is non-periodic.
124
As speculated, the performance of SMGI exacerbates as β increases while the SNR
response of GQSD remains unaffected. Naturally, the low frequency components of
1/f (β) noise are dominating critically the baseband region causing the SMGI output
noise to increase. On the other hand, the GQSD operates at 2kHz where the 1/f (β) noise
is much less, hence the improved SNR.
5.2.2 Performance comparison on acquired and simulated data
To partially validate our evaluation procedure for comparing SP methods, and
particularly the deployment of our noise models, GQSD and SMGI were applied
separately and in real time on two types of data trains: acquired from an experiment, as
well as simulated as described in section 4.2.2. The time domain signals shown in
Fig. 5.10a and Fig. 5.10b represent the acquisition and simulation, respectively, of the
detected pulsed modulated signal. The simulated signal used for comparison was
obtained by feeding an ideal pulse train to the LV trans-impedance amplifier (TIA)
model mimicking the conversion of the pulsed photodiode current to a voltage signal.
Figure 5.10 Acquisition (a) and simulation (b) of temporal voltage response of PIN Photodiode to a pulsed laser diode emission. The FFT of both signals is shown in (c) where linear RMS averaging is
performed to estimate the exponent β as the slope of the noise curve.
125
The timing properties of the ideal pulse train are set equal to the real time signal that
drives the laser source. Subsequently, the gain and time constants of the LV TIA model
were appropriately configured to match the output of the acquired real-time photodiode
signal.
To recreate the noise contribution in the acquired signal, the voltage response of the PIN
photodiode was recorded under no irradiation (dark) conditions. The fast Fourier
transform (FFT) of the detected signal was then passed through a root-mean-square
(RMS) averaging algorithm to reduce the fluctuations of the constituent noise
harmonics without reducing the actual noise floor. Figure 5.10c shows the
approximation of the noise floor achieved after taking 1000 averages. The spectrum of
the input noise is proportional to 1/f (β) where Δx and Δy are extracted from the graph
and substituted in (4.5) yielding β ≈ 1.7, which was then passed to the noise generation
subroutine to simulate accurately the experimental noise conditions. In Fig. 5.10c. the
error between the simulated (dotted line) and real time (solid line) noise spectra is
negligible, making the two plots indistinguishable.
The overall comparison between the SNR performance of GQSD and SMGI, on
acquired and simulated data, is shown in Fig. 5.11a and Fig. 5.11b respectively. The
excellent match of the results justifies the use of the SPS to investigate the performance
of GQSD and SMGI under various types of pulsed signals. Secondly, the obtained
results are in line with those obtained for Case C on simulated data only, where we
concluded that for β ≥ 1 the performance of SMGI becomes poor as opposed to GQSD.
Indeed, for β ≈ 1.7 the SNR obtained by GQSD is improved by approximately 15 dB.
Figure 5.11 SNR performance of GQSD and SMGI in a real-time experimental setup. Experimental
SNR results are shown in (a.1) and (a.2) corresponding to an OTC of 1.989s and 49.7359ms respectively. Similarly the SNR performance of the simulated input signal is shown in (b.1) and
(b.2)
126
5.3 Simulated measurements with SPH-43
The LV based PED model was utilised within the SPS to simulate the pulsed behaviour
of SPH-43. The simulated pulsed response of the model was measured within the SPS,
by existing (CQSD, SMGI) and a novel (GQSD) SP methods. Their performance, was
recorded according to the evaluation procedure discussed in section 4.3.2. The gating
feature of GQSD and SMGI complicates the “actual” (observed) rate of detection
(OTC) since the effecting time constant (ETC) of the measurement is scaled according
to the gate width. Multidimensional scenarios, are therefore considered to investigate
the performance of GQSD and SMGI, taking into account not only the ETC but also the
OTC of the overall measurement. The aim of these simulations is to investigate the
effect of the feedback capacitor CF, for various values of observed and effective time
constant (i.e. effective bandwidth; feff) on the output SNR. Finally, the SNR
performance of each method is recorded under the scenarios discussed in chapter 5,
section 4.3
5.3.1 Case I: High voltage responsitivity (RF = 100GΩ)
During the initial stages of the TempTeT project, research was focused on high
sensitivity sensors allowing detection of weak radiation signals of micro to nano-watt
average pulse power. PEDs were the detectors of choice and specifically the SPH-43
model. The device was delivered with a feedback resistor (RF) of 100GΩ allowing
maximum sensitivity of approximately 25kV/W at 5Hz. In Case I, the SPS was
calibrated to simulate maximum voltage responsitivity, by setting RF to 100GΩ. For
various values of CF, the SNR performance of GQSD and SMGI was recorded for a
gate window M of 57 samples which results to an OTC of 0.348s. In order to compare
results, the effective bandwidth (feff) of the CQSD was reduced from 5Hz to 0.456Hz to
equalise its ETC (for CQSD the ETC = OTC) with the OTC obtained by GQSD and
SMGI. The configuration parameters and labels of each simulation are listed in Table
4.3-VIII. The results shown in Fig. 5.12a, demonstrate the SNR performance of GQSD
(black plot), CQSD (red plot) and SMGI (blue plot) for a range of feedback capacitance
(CF) as described in the previous chapter.
127
Figure 5.12 SNR performance of the SP methods when used to measure a simulated pulsed
response of a PED (SPH-43). The black, red and blue plots in each graph correspond to GQSD, CQSD and SMGI respectively. The resulting SNR performance of each method is shown in (a), the
output measured signal in (b) and noise in (c).
In addition to the SNR results, the averaged values of the output signal and noise were
plotted separately with respect to CF, on log-log scaled graphs, as shown in Fig. 5.12b
and Fig. 5.12c respectively. From Fig. 5.12c, it is observed that CF has less effect on the
output noise of SMGI compared to the CQSD and GQSD. As already discussed in
section 5.2, the susceptibility of SMGI to Flicker noise is much higher than GQSD and
CQSD and.since the output of the PED is dominated by 1/f noise, the performance of
SMGI was expected to be worse than GQSD or even the CQSD.
128
Figure 5.13 Simulated noise spectrum of SPH-43 under dark conditions for Case I. Each plot is
recorded for each of the capacitance values depicted on the right hand side of the graph. The red and blue markings indicate the bandwidths of the GQSD/CQSD and SMGI respectively.
As discussed in section 3.1.32, the output noise of a PED comprises several
components; the shot noise, Johnson noise and the noise generated at the input of the
operational amplifier. It is important to recall here that the power spectral density of the
later noise component is proportional to 1/f, which is also proportional to the noise gain
of the TIA. According to (3.29), variation of CF affects not only the bandwidth but also
the noise gain (NG) of the PED. The magnitude level of the plateau of the NG (see
section 3.1.2.4.3).begins to drop as CF is increasing. Additionally the pole of the NG is
shifted at a lower frequency specified by the electric cut-off frequency of the TIA. Thus,
to examine the noise performance and influence of NG for each method, the output
noise spectrum of SPH-43 was simulated for all given values of CF, under dark
conditions (the amplitude of the simulated pulses was set to zero watts). The results are
plotted on log-log graphs as shown in Fig. 5.13. The operational bandwidth of GQSD
and CQSD is signified by the red markings while the blue markings indicate the
bandwidth of SMGI. The advantage of SD methods is obvious by observing the
comparatively lower noise around the demodulation frequency (in our case 200Hz). On
the contrary, the passband region of SMGI is located at higher noise levels, which
verifies the increased output noise compared to the other two methods. In addition, the
tendency of the output noise spectra of SPH-43 to merge at lower frequencies verifies
the less negative slope of the output noise of the SMGI. In other words, while the CF is
increasing, the output signal of the SMGI drops faster than the noise yielding a poor
SNR performance.
129
5.3.2 Case II: Low voltage responsitivity (RF = 1GΩ)
Case II is slightly more complicated than Case I, but also more interesting. It is evident
that as RF decreases, the bandwidth of SPH-43 widens allowing the detector to capture
high frequency transitions of the pulsed signal. The preservation of the pulsed
pyroelectric train allows various usability scenarios of the two gating methods (GQSD
and SMGI) when used to quantify the, in this case, simulated energy transmission. In
this work, we considered the two scenarios discussed below.
5.3.2.1 Scenario 1: SNR performance for constant OTC and varying feff
Various cases have been considered, where the effective bandwidth feff was adjusted to
achieve the desired OTC, with respect to the selected values of M and D. Table 4.3-IX
lists a summary of the simulations, each one labelled by a distinct name to avoid
confusion. SII1GQSD corresponds to the GQSD method where the optimal value of M
was estimated for each value of CF. The feff of GQSD was modified for each value of M
to maintain the OTC to 0.993s. Similarly, SII2SMGI corresponds to the SMGI method,
where the optimal gate length is 1 sample regardless of the value of CF. Again, the same
OTC of 0.993s is obtained by setting the feff of SMGI to a fixed frequency of 100Hz for
all values of CF, since M remains unchanged. In SII1CQSD, the feff of CQSD, was adjusted
to 0.16Hz. The results are shown in Fig. 5.14 (next page), demonstrate the SNR
performance of each method. As opposed to SMGI, while CF increases the output SNR
of GQSD improves slightly (approximately by 2dB). Interestingly, the SNR obtained by
the CQSD also remains flat but lower than GQSD (~5dB).
As before, the averaged output signal and noise are plotted on a log-log graph as shown
in Fig. 5.14b and Fig. 5.14c. Compared to the results in case I, the noise performance of
SMGI inherits the same behaviour. As CF is increased, the output noise of SMGI is
reduced at a slower rate than in GQSD and/or CQSD. Similarly to Case I, the output
noise spectra of SPH-43 were simulated under dark conditions, for each given value of
CF (Fig. 5.15). The output noise spectrum of the PED reduces approximately by 40dB
for the frequency range of 1-10kHz. This was also observed in Case I, and it is mainly
caused by the noise gain reduction, as CF increases. One may speculate that
optimisation of feedback capacitor could significantly improvement the SNR of GQSD
or CQSD if the modulation frequency is selected to be higher than a 1kHz. On the other
hand the substantial SNR improvement cannot be achieved by the SMGI since the
baseband noise is only partially influenced by the noise gain variations, and hence any
change of the feedback capacitor.
130
Figure 5.14 Results for Case II – Scenario 1. The SNR results of each method (black-GQSD, red-
CQSD and blue-SMGI) are plotted in (a). The averaged magnitude of the output signal and noise of the corresponding methods are depicted in (b) and (c) respectively.
Figure 5.15 Simulated noise spectrum of SPH-43 under dark conditions for Case II.
131
Figure 5.16 Results for Case II – Scenario 2. The plot legend is identical to Fig. 5.12 and Fig. 5.14.
while the additional dashed plot in (a), (b) and (c) corresponds to the simulation SII3GQSD where the GQSD was configured to measure the pulsed response of SPH-43 for a varying OTC and M.
5.3.2.2 Scenario 2: SNR performance for a constant feff and varying OTC
In this scenario I examined/compared the performance of the three methods for a
constant feff and also a varying OTC. The latter was achieved by optimally varying the
value of M. The results are shown in Fig. 5.16. Interestingly, the output SNR of SMGI
improves in SII1SMGI where M is set to 20 samples and feff was decreased from 100 to
5Hz. This improvement relies to its ability to obtain a higher output signal than GQSD
or CQSD.
However, the output SNR begins to drop as CF departs from 1pF while the performance
of GQSD and CQSD remains relatively unchanged, as in previous cases. According to
132
Table 4.3-IX (last row) the optimal value of M increases with an increasing value of CF
which in turn reduces significantly the OTC. Results indicate that, despite the OTC
reduction, the SNR performance of GQSD was maintained flat by optimally gating the
pyroelectric signal. This performance was simulated in SII3GQSD, which is depicted by
the dashed line plots in Fig. 5.16.
133
CHAPTER
6 Conclusions and future work
The proposed modelling methodologies and studies had led to the development of
various set of tools, allowing accurate emulation of the performance of PEDs. In this
work, 3D geometric models of a commercial PED (SPH-43) have been created
according to detail specifications provided by the manufacturer. Accurate reproduction
of the thermal response of the device has been verified experimentally, with the aid of a
novel and fully interactive LabVIEW based PED simulator [35]. The need to detect and
measure pulsed incident radiation with PEDs was certainly a motivational impact to this
research that led to the involvement of various DSP algorithms such as a CQSD and
SMGI. Furthermore, a novel DSP algorithm, named as Gated Quadrature Synchronous
Demodulation (GQSD), was implemented and evaluated within a Signal Processing
Software (SPS) also developed in LV. Finally, the PED simulator was additionally
embedded within SPS permitting the simulation, detection and measurement of realistic
pyroelectric signals as well as real time processing and acquisition. This chapter
discusses conclusions and future work that might provide improvements regarding
either the simulation models or the generated DSP algorithm GQSD.
6.1 Conclusions
The overall content of this thesis is circulating over three main areas. The first involves
the simulation of a pyroelectric device, and more specifically the SPH-43. The second
part focuses on studies of DSP algorithms and their performance in pulse regime
applications. Lastly, the two were combined to form a unique environment to satisfy the
requirements of both problems. Modeling the behaviour of sensors is often be extremely
useful when designing a front-end detection block.. Moreover, accurate and reliable
modelling prior implementation becomes a cost effective solution that can also provide
substantial headroom for improvement.
134
In this work, we have initially modelled PEDs by solving the traditional heat balance
equation. Comparison between simulated and experimentally derived data signified that,
for the particular sensor of choice, the LMA approach yields inaccurate results with
respect to the thermal sensitivity and temporal thermal response. Thus, the next best
alternative was to utilise more dedicate software packages and methods such FEM or
FDM. The former modelling approach was considered the optimal choice to model the
thermal behaviour of the complicated structure of PED in used (SPH-43). The Heat
Transfer Module within the COMSOL Multiphysics package was used to accurately
reproduce the model in order to examine the thermal behaviour of the actual device.
Results showed excellent match between the experimentally derived thermal response to
the one obtain in FEM.
The accuracy of FEA has encouraged this research to proceed with the design and
simulation of linear pyroelectric arrays with evenly spaced pyroelectric elements.
Surface-to-surface radiation boundary constraints were set to investigate the effect of
crosstalk between adjacent and opposite surfaces of the crystals. Even though the effect
was noticeable, the translated voltage contribution due to thermal leakage ranged
between 0.2 to 6 μV while the distance between the crystal was varied from 2[mm] to
1mm. For the purposes of the TempTeT project, the team was in close collaboration
with Spectrum Detectors, who developed an eight-element PED array as shown in Fig.
6.1. According to the simulation, the crystals could have be spaced even closer than
2mm without significant effect from thermal crosstalk.
Figure 6.1 Snapshot of a PED array based on the requirements specified by the TempTeT team.
The distance between the adjacent crystal is 2mm as tested in the simulations.
135
In this work, we use a novel approach that uses the results from the FEA to associate the
3D thermal model with a single RC network. This enables us to emulate the transient
response of the detector with same accuracy in a more convenient environment where
DSP algorithms can be applied to investigate possible solutions to measure its pulsed
response. The novelty is the complete performance model of a commercially available
PED, capable to predict the transient response to a variety of input signals with
adjustable parameters (i.e. duty cycle, frequency).
Further, an accurate PED simulator is developed within LV allowing the reproduction
of a continuous transient analysis while in parallel DSP algorithms are applied to
measure the response. A final version of a Signal Processing Software (SPS) includes
the existing PED simulator along with some modifications allowing the import of 3D
temperature profile solutions from COMSOL. In addition, DSP algorithms are easy to
integrate within the software, as a first and necessary step to analyse or optimize
established and new techniques for processing the data.
Regarding the SP part of these studies, two methods have been taken into account.
Synchronous Demodulation (SD) is the signal recovery method of choice when the
input envelope signal is modulated by either a pure sine wave or a square wave. It is
however, less efficient for pulsed periodic signals with a low duty factor. For the latter
signals, we introduce a data processing algorithm, which applies gating on a part of the
signal period to achieve optimum conditions for recovering the pulse amplitude by
quadrature SD. The proposed method is evaluated for signal-to-noise performance
against Boxcar-type gated integrators in cases of simulated data, as well as data
acquired from physical measurements, in the presence of 1/f and Gaussian noise. It is
shown that by combining the gating and SD principles, our suggested Gated
Synchronous Demodulation outperforms other routine signal processing methods under
typical 1/f noise conditions. It is interesting to comment here on the possible real-time
performance of systems implementing GQSD. In the simplest and least demanding case
of low repetition rate, stationary pulse trains, an initial calibration and setup phase is
required to establish the optimal GW for a certain shape of the detector response; that is
adequate to manage the real time acquisition and processing. In the other end of
complexity is the case of high repetition rate, dynamic pulse trains.
136
The calibration and setup provides only the starting optimal GW value. The latter may
change in real time if the shape of the response changes across the dynamic range of the
detector. Therefore, analyses and corrections need to be applied in real time to avoid
unacceptable errors in recovering the gated signal (G1). The suitability of any
algorithm for this will depend heavily on the character of every particular case, and in
general will consume more resources, e.g field programmable logic and/or other
hardware acceleration.
6.2 Future work
The perspective for future work is considerably high, not so much for the PED
modelling but for the novel method GQSD. This method combines two fundamental
techniques gaining the benefits from the advantages of both. The last section of chapter
5 was not mentioned in the conclusions but instead we discuss is in this section due to
the high potential of improvement.
With aid of the SPS, the response of the PED was measured with three SP methods,
CQSD, SMGI and finally the GQSD. Results showed a significant SNR improvement
of GQSD against the other two methods for specific scenarios. It is crucial to note here
that the methods were evaluated only for two cases; for RF = 100GΩ and RF = 1GΩ.
Even though results were satisfactory and as expected, other combinations of RF’s and
CF’s may lead to better or worst performance. Additionally, the results are based on
simulation and not real data. The noise performance of the PED was simulated
according to equivalent TFs that were employed within the simulator to emulate the
effect of the TIA noise. Future experiments can be performance to investigate the
performance GQSD for difference parameter values and different modulation
frequencies.
Undoubtedly, there is a lot of room of improvement for GQSD. Although the method
was tested experimentally, it must be mentioned here that the SNR of the input signal
was relatively high allowing accurate estimation of the gate length and trigger delay by
using the FFT Gate Estimation algorithm. Unfortunately, there are limitations to this
process.
137
The first limitation arises when the input pulsed signal has ultra-narrow pulse. This
implies an extremely high sampling rate in order to capture the fidelity of each
repetitive pulse pattern. Under these circumstances, the Gate Estimation algorithm will
suffer from an enormous amount of computations with a consequence to withhold the
GQSD on stand-by until the optimal values of M and D are obtained. A solution for this
problem has already been implemented in the simulator but is not optimised and
therefore is not considered in the contents of this thesis. Future work on this FFT Gate
Estimation module is therefore suggested.
Another limitation is again related with the estimation of M and D. In the case where the
input signal is buried in noise, the Gate Estimation algorithm will need to obtain an FFT
average for every possible combination of M and D and consequently degrading the
speed of optimisation. A possible solution for this problem is to use a scanning mode
gated integrator to first recover the shape of the signal and then proceed with Gate
Estimation. With advances of technology and digital design, these problems can be
significantly reduced by taking into advantage the parallel computation of FPGAs. In
the current version the algorithm is 100% sequential implying that while the algorithm
is estimating the optimal values of M and D all the rest of the processes are on-hold.
Finally, the implementation of this method in a real embedded system and can be an
additional asset to existing lock in amplifiers, especially those that can accommodate
high sampling rates. At this current state, we have already prepared a manuscript
intended to be published in the IEEE transactions on Signal Processing. Due to the on-
going patenting procedure of the novel algorithm, the manuscript will be published as
soon as the IP issues are resolved.
138
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143
A APPENDIX
A.1 Solution of the heat transfer ODE for a time invariant input Φ(t)
The analytical solution of (A.1) yields to the temporal thermal response of the PED
when excited with a time variant input function Φ(t). The solution of a first order ODE
is described in the following steps [63].
1. Rearranging the equation
To solve a non-homogeneous ODE it is first necessary to rearrange the consisting
variable in the form shown in (A.2).
2. Estimating the integrating factor F
The integrating factor F is used to ensure an exact solution of (A.3). The integrating
factor is obtained by integrating both parts of (A.5) and solving for F. The result is
shown in (A.6).
Multiply the F with the original ODE
tΦη
G
t
dt
tdC
Tth
ddTth
ΔΤΔΤ(A.1)
0ΔΤ PdttQd d (A.2)
0ΔΤ
ΔΤ
dt
C
tΦη
τ
ttd
Tthth
dd
(A.3)
where
1Q
and
Tthth
d
C
tΦη
τ
tP
ΔΤ
(A.4)
dt
dQ
td
dP
dt
dF
F d
ΔΤ
1
(A.5)
thτt
eF (A.6)
where
G
Cτ
Tth
Tth
th
(A.7)
144
3. Use the product rule to convert (A.8) to a separable ODE
According to the product rule the left hand side (LHS) of (A.8) is equivalent to (A.9).
4. Obtain the solution of the heat transfer ODE by integrating
The solution of the heat transfer problem is obtained by integrating (A.9) for any input
Φ(t) at a particular instant of time t. The result is shown in (A.10).
A.2 Solution of the heat transfer ODE to a sinusoidal input Φ(t)
According to Euler’s formula [41] sinusoidal signals are more conveniently expressed
as the real part of exponential functions with imaginary exponents. Therefore, the polar
representation of the sinusoidal input variable Φ(t) is shown in (A.12).
Inserting (A.12) in (A.10), the result of the integration yields to the solution of the heat
transfer to a sinusoidal input with zero initial conditions.
ththth τt
Tth
dτ
t
th
τt
d eC
tΦηte
τe
dt
td ΔΤ
1ΔΤ
(A.8)
th
th
τt
Tth
τt
d
eC
tΦη
dt
etd
ΔΤ
(A.9)
thth
thτ
tτ
t
Tth
τt
d cedtetΦC
eηt
ΔΤ
(A.10)
where c is the initial condition (A.11)
tωj0eΦtΦ (A.12)
dteΦC
eηdteeΦ
C
eηtωj
tωjC
G
0Tth
C
tG
C
tG
tωj0T
th
C
tG
d
Tth
TthT
th
Tth
Tth
TthT
th
Tth
ΔΤ
(A.13)
145
The solution of the heat transfer ODE to a sinusoidal input is shown in (A.15) and the
magnitude of the resulting thermal harmonic is shown (A.16).
A.3 Solution of the heat transfer ODE in Laplace domain
The ODE in (A.1) is often described by an equivalent resistor-capacitor (RC) network
where the input voltage V(t), electrical resistance R and capacitance C correspond to
the input radiant flux Φ(t), thermal resistance (Gth)-1 and thermal capacitance Cth. The
Laplace transform of (A.1) yields to:
Assuming zero initial conditions and the ratio of the output ΔΤd(s) to the input Φ(s)
denotes the transfer function H(s) of the network.
Tth
TthT
th
Tth
C
tG
tωj
TthT
th
Tth
C
tG
d ee
CωjC
G
eηtωj
ΔΤ
(A.14)
tωj
Tth
Tth
0d e
CωjG
Φηtωj
ΔΤ
(A.15)
22ΔΤ
Tth
Tth
0Tth
Tth
0d
CωG
Φη
CωjG
Φηtωj
(A.16)
sΦηsGCssC dTthd
Tthd
Tth ΔΤ0ΔΤΔΤ (A.17)
sΦηsCGs Tth
Tthd ΔΤ (A.18)
Tth
Tth
d
sCG
η
sΦ
ssH
ΔΤ1
(A.19)
146
B APPENDIX
B.1 Fourier series representation of a rectangular pulse train f(t)
Figure 6.2 A rectangular pulse train
The Fourier series states that any periodic signal can be decomposed to a sum of
sinusoids of different frequencies and magnitudes. The general form of the
trigonometric Fourier series is shown in (B.1) and the compact form in (B.2).
where 102 Tπω and a0, ah , and bh are the Fourier series coefficients defined by the
expressions (B.3), (B.4), and (B.5) respectively.
1
000 sincos
h
hh tωhbtωhaatf
(B.1)
1
00 cos
h
hh θtωhatf
(B.2)
h
hhhhh a
b θba 122 tan and (D.1)
00
01
T
dttfT
a
(B.3)
3,2,1cos
2
0
00
hdttωhtfT
aT
h
(B.4)
147
where0T
means that the integration is performed over any interval of T0 seconds.
δAT
χAt
T
AAdt
Ta
Tχ
00
0000
01
πδhπh
A
T
πh
T
πh
T
A
T
πh
χT
πh
T
A
ωh
tωh
T
Adttωh
T
Aa
χχ
h
2sin2
02
sin2
2
2sin
2
sin2cos
2
0
0
0
0
0
0
00
0
00
00
πδhπh
A
πδhπh
A
πh
A
T
πh
T
πh
T
A
T
πh
χT
πh
T
A
ωh
tωh
T
Adttωh
T
Ab
χχ
h
2cos1
2cos2
02
sin2
2
2cos
2
cos2sin
2
0
0
0
0
0
0
00
0
00
00
πδhπh
Ah 2cos12
πδh
πδhθ h
2sin
2cos1tan 1
Therefore, substituting the coefficients in the compact trigonometric Fourier series
yields:
1
10 2sin
2cos1tancos2cos12
hπδh
πδhtωhπδh
πh
AδAtf
3,2,1sin
2
0
00
hdttωhtfT
bT
h
(B.5)
148
B.2 Fourier transform of a rectangular pulse
Figure 6.3 A rectangular pulse
The Fourier transform (FT) and inverse FT (IFT) of a function f(t) are given by tade and
tade respectively.
Therefore:
ωχχ
ωχ
ωχχ
ω
ωχ
eeωj
dteVT
ωF ωχjωχjχ
χ
tωjpp
5.0sinc
5.0
5.0sin5.0sin2
11 5.05.05.0
5.00
dtetfT
ωF tωj
0
1(B.6)
ωdeωFπ
tf tωj
2
1
(B.7)
149
C APPENDIX
C.1 Layout of a geometric model of pyroelectric detector SPH-43
Figure 6.4 Layout drawing of the geometric model of a PED SPH-43 used in finite
element analysis
150
C.2 Geometric model of an array of multiple pyroelectric elements based on
SPH-43
Figure 6.5 Layout drawing of a geometric model of an array of four pyroelectric
elements, of which their structure is based in the single element SPH-43
151
D APPENDIX
D.1 Derivation of a non-ideal transfer function of transimpedance amplifier
Where FF
FF CsR
RZ
1 and
TD
DIN CsR
RZ
1. Substituting ZF and ZIN in (D.1) yields
where
n
o
ω
sA
sA
1
and DCMDIFFT CCCC . Substituting A(s) in (D.2) yields
Dividing (D.3) by Fno
TF RωA
CCa
the final transfer function of TIA is given by.
TFDF
FoDn
TFDF
FDToFDFn
TF
no
p
p
CCRR
RARω
CCRR
RRCACRRωss
CC
ωA
sI
sV
122
(D.7)
0111
AZ
Zs
Z
sA
sA
sI
sV
IN
F
F
p
p
(D.1)
FFD
TDF
FF
F
p
p
CsRR
CsRRsA
CsR
sAR
sI
sV
1
11
1
(D.2)
cbsas
R
sI
sV F
p
p
2
(D.3)
Fno
TF RωA
CCa
(D.4)
o
FFT
D
F
nooFF A
RCC
R
R
ωAACRb
1
111 (D.5)
Do
F
o RA
R
Ac
11 (D.6)
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