Extraction of Noise Parameters for Single-Ended Components

University of Calgary

PRISM: University of Calgary's Digital Repository

Graduate Studies The Vault: Electronic Theses and Dissertations

2018-07-09

Extraction Of Noise Parameters For Single-Ended

Components Inside A Differential Circuit Using

Single-Ended Equipment

Huang, Yuxiang

Huang, Y. (2018). Extraction Of Noise Parameters For Single-Ended Components Inside A

Differential Circuit Using Single-Ended Equipment (Unpublished master's thesis). University of

Calgary, Calgary, AB. doi:10.11575/PRISM/32349

http://hdl.handle.net/1880/107127

master thesis

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THE UNIVERSITY OF CALGARY

Extraction Of Noise Parameters For Single-Ended Components Inside A Differential

Circuit Using Single-Ended Equipment

by

Yuxiang Huang

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING

CALGARY, ALBERTA

JULY, 2018

c© Yuxiang Huang 2018

Abstract

This thesis proposes an approach of investigating electrical and noise parameters of sub-

components inside a fully differential system. To find the single-ended noise parameters,

two sets of single-ended noise-parameter measurements andone set of S-parameter mea-

surement are performed. A proof-of-concept PCBs is designed, fabricated, and tested after

the algorithms are verified using Matlab. The circuit is designed to have a bandwidth from

500MHz to 1.5GHzand be unconditionally stable at all frequencies. The designed circuit

has the gain of 9.4dBat 500MHzand 9.5dBat 1.5GHzin schematic simulations, while in

measurements, it has the gain of 8.1dB at 500MHz and 4.3dB at 1.5GHz. The minimum

noise figure is 2.4dB at 500MHz and 2.4dB at 1.5GHz in simulations, while in measure-

ments, it is 3.1dB at 500MHz and 3.2dB at 1.5GHz. This thesis presents schematic and

measurement results for the electrical and noise parameters. The measurement results are

analyzed in Matlab and compared with the relevant single-ended measurement results to

verify the operation of the method.

ii

Acknowledgements

Foremost, I would really like to thank my supervisor Dr. Leonid Belostotski for his guid-

ance, patience, and interest throughout my master studies.I appreciate his broad knowledge

and experience in the RF circuit and layout design. He is always helpful and spent as much

time as I needed to discuss the problems of my research work. Iwould also like to thank

him for his patience and spending days and days helping me write and edit my thesis.

Second, I would like to thank all the MiNT lab students, post-doctoral fellows, my

friends throughout the ECE department and department staffs for their help and support.

Many thanks to Donuwan Navaratne, Zhixing Zhao, Nan Zhang, Hao Xie and Vahid Asgari

for their helpful discussion and support regarding this thesis work.

Last but not least, I would like to thank my parents for their support and encouragement.

iii

For my parents and my supervisor.

Thank you for your encouragement and support.

iv

Table of Contents

Abstract ii

Acknowledgments iii

Dedication iv

Table of Contents vi

List of Tables vii

List of Figures x

Glossary xi

1 Introduction 1

1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . .. 1

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

2 Extraction of Electrical Parameters for Single-Ended Amplifiers Inside a Dif-

ferential Amplifier 4

2.1 Introduction and Objectives . . . . . . . . . . . . . . . . . . . . . . .. . . 4

2.2 Proposed Method of the Circuit Electrical-Parameter Extraction . . . . . . 4

2.3 Verification of the Proposed Method . . . . . . . . . . . . . . . . . .. . . 11

2.4 Schematic Design Process and Simulation Results . . . . . .. . . . . . . . 12

2.5 Differential Amplifier Design and Simulation Results . .. . . . . . . . . . 16

2.5.1 Schematic Simulation and Extraction Results . . . . . . .. . . . . 17

2.5.2 Layout Design Process and Comparison of Momentum Extraction

Results with Momentum Single-Ended Simulation Results . . .. . 18

2.6 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

3 Extraction of Noise Parameters for Single-Ended Amplifiers Inside a Differen-

tial Amplifier 26

3.1 Introduction and Objectives . . . . . . . . . . . . . . . . . . . . . . .. . . 26

3.2 Single-Ended Noise-Parameter Extraction Algorithm . .. . . . . . . . . . 27

3.3 Schematic Extraction and Simulation Results . . . . . . . . .. . . . . . . 32

v

3.3.1 Algorithm Verification Using Ideal Components . . . . . .. . . . 32

3.3.2 Comparison Between Schematic Extraction Results andSchematic

Single-Ended Simulation Results . . . . . . . . . . . . . . . . . .34

3.4 Measurement Noise Extraction Results . . . . . . . . . . . . . . .. . . . . 35

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

4 Conclusions and Future Work 41

4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

A Forming Matrix to Solve S-Parameters for Single Amplifier Inside a Differen-

tial Amplifier 55

B Z-Parameters Conversion from 2-Port to 1-Port 58

C Amplifier B with Input and Output Noise Source for Noise Correlation Matrix

in Z Form Calculation 60

D Z-Representation for Combined Common Network and “One Side Terminated”

Differential Amplifier 62

vi

List of Tables

2.1 Circuit Component Values. . . . . . . . . . . . . . . . . . . . . . . . . .. 13

2.2 DC Biasing Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

2.3 Schematic Design Specifications. . . . . . . . . . . . . . . . . . . .. . . . 16

4.1 Comparison of S-parameters from Schematic SimulationsMeasurement

Results to the Relevant Single-Ended Simulation and Measurement Results

for Single-Ended Amplifiers and the Common Network Impedance . . . . . 42

4.2 Comparison of S-Parameters from EM-Schematic Momentumto the Rele-

vant Single-Ended Simulation for Single-Ended Amplifiers and the Com-

mon Network Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . .43

4.3 Comparison of Ideal Differential-Circuit Extraction Results with Single-

Ended Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . .45

4.4 Comparison ofNFmin, Rn, andΓopt Obtained from Schematic Simulations

to the Relevant Single-Ended Simulation Results . . . . . . . . .. . . . . . 46

4.5 Comparison of S-Parameters from Measurement Results tothe Relevant

Single-Ended Measurement Results for Single-Ended Amplifiers and the

Common Network Impedance . . . . . . . . . . . . . . . . . . . . . . . .47

4.6 Comparison ofNFmin, Rn, andΓopt Obtained from Measurements to the

Relevant Single-Ended Measurement Results. . . . . . . . . . . . .. . . . 48

vii

List of Figures

2.1 4-Port Differential Amplifier. . . . . . . . . . . . . . . . . . . . . .. . . 5

2.2 4-Port Differential Amplifier with Ports 2 and 4 Terminated with Loads. . . 5

2.3 Overall 2-port Network for Terminated Differential Amplifier. . . . . . . . 7

2.4 Terminated TransistorB Cascaded with Common Network. .. . . . . . . . 9

2.5 System Level Differential Circuit. . . . . . . . . . . . . . . . . .. . . . . 11

2.6 Comparison of Databox Extracted Results and Single-Ended Results.Top

Graphs: Pre-Stored Data. Middle Graphs: Extracted Pre-stored Data for

DataboxA. Bottom Graphs: Extracted Pre-stored Data for DataboxB. (300MHz

to 3GHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

2.7 Differential Amplifier ADS Schematic. . . . . . . . . . . . . . . .. . . . . 13

2.8 Differential Gain. (300MHz to 3GHz) . . . . . . . . . . . . . . . . .. . . 13

2.9 Differential Circuit Gain, Source and Load Stability Circles, and Stability

Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

2.10 Schematic of the Single-Ended Amplifier. . . . . . . . . . . . .. . . . . . 15

2.11 Schematic of a Single-Ended Amplifier S Parameters in dB. (300MHz to

3GHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

2.12 Stability Factor, Schematic Single-Ended Source and Load Stability Circles. 16

2.13 Schematic S-parameters Extraction Results vs Schematic Single-Ended Sim-

ulation Results. Red Line: Schematic Simulated Single-Ended S-parameters.

Blue Line: Schematic Extracted Results. (300MHz to 3GHz) . .. . . . . . 17

2.14 Substrate Parameters and Substrate Transverse Plane Picture. . . . . . . . . 18

2.15 3D View of Two-Layer PCB Layout in ADS with Input and Output Trans-

mission Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.16 Top View for the Two-Layer PCB. Top Layer: Brown. BottomLayer: Yellow. 19

2.17 ATF 35143 Footprint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

2.18 Comparison between Simulated Single-Ended and Momentum Extraction

Results (300MHz-1.5GHz). Sky Blue and Red Line: Momentum Extrac-

tion Results. Dark Blue Line: Momentum Single-Ended Simulation Re-

sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2.19 Top View and Bottom View of Surface-Mounted Two-Layer PCB. . . . . . 22

2.20 ADS De-Embedding Schematic (Left Databox: Measurement Raw Data.

Right Databox: SMA S Parameters from HFSS.). . . . . . . . . . . . . .. 22

viii

2.21 Comparison of Measured S-Parameters Before De-Embedding and After

De-Embedding. Blue line: S-Parameter for the Measured 4-port before

De-Embedding. Red line: S-Parameter for the Measured 4-port after De-

Embedding. (300MHz to 3GHz) . . . . . . . . . . . . . . . . . . . . . . .23

2.22 Comparison of De-Embedded Measured Single-Ended Amplifier S-Parameters,

and Extracted Single-Ended Amplifier S-Parameters from theDifferential

Amplifier. (Smith chart) Red Line: Measured Single-Ended S-Parameters.

Blue Line: Measured Amplifier A S-Parameters. Purple Line: Measured

Amplifier B S-Parameters. (300MHz to 3GHz) . . . . . . . . . . . . . . .24

2.23 Comparison of De-Embedded Measured Single-Ended Amplifier S-Parameters,

and Extracted Single-Ended Amplifier S-Parameters from theDifferential

Amplifier. (Magnitude) Red Line: Measured Single-Ended S-Parameters.

Blue Line: Measured Amplifier A S-Parameters. Purple Line: Measured

Amplifier B S-Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . .25

3.1 Divisions of 2-Port Networks. . . . . . . . . . . . . . . . . . . . . . .. . 27

3.2 2-Port Upside Down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

3.3 Cascade AmplifierB with Common Network . . . . . . . . . . . . . . .. . 29

3.4 Serial Connection of AmplifierA and Common Network . . . . .. . . . . 29

3.5 Ideal Differential Amplifier. . . . . . . . . . . . . . . . . . . . . . .. . . 32

3.6 Comparison ofRn between Extracted Results and Prestored Data.. . . . . .32

3.7 Comparison ofNFminbetween Extracted Results and Prestored Data. . . .33

3.8 Comparison ofΓopt between Extracted Results and Prestored Data. . . . . .33

3.9 Schematic ExtractedNFmin vs Schematic Single-Ended Simulation Re-

sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

3.10 Schematic ExtractedRn vs Schematic Single-Ended Simulation Results. . .35

3.11 Schematic ExtractedΓopt vs Schematic Single-Ended Simulation Results. .35

3.12 One-Side TerminatedNFmin Raw Data from Measurement in Figure3.4.

(300MHz to 3GHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

3.13 One-Side TerminatedRn Raw Data from Measurement in Figure3.4. (300MHz

to 3GHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

3.14 One-Side TerminatedΓopt Raw Data from Measurement in Figure3.4.

(300MHz to 3GHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

3.15 Comparison ofNFmin between Measurement Extracted Results and Mea-

surement Single-Ended Results (300MHz to 3GHz). . . . . . . . . .. . . 38

ix

3.16 Comparison ofRn between Measurement Extracted Results and Measure-

ment Single-Ended Results (300MHz to 3GHz). . . . . . . . . . . . . .. 38

3.17 Comparison ofΓopt between Measurement Extracted Results and Measure-

ment Single-Ended Results (300MHz to 3GHz). . . . . . . . . . . . . .. 39

B.1 Diagram for conversion of a 2 Port Z Parameters into 1 Port. . . . . . . . . 58

C.1 AmplifierB with Input and Output Noise Source for Noise Correlation Ma-

trix in Z form Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . .60

D.1 AmplifierB with Input and Output Noise Source Cascaded with the Com-

mon Network for Noise Correlation Matrix in Z Form Calculation. . . . . . 62

x

Glossary

Acronom Definition

ADS Advanced Design Systems

AC Alternating Current

BIT Built-In Test

DC Direct Current

DLNA Differential Low Noise Amplifier

FR4 A NEMA grade designation for glass-reinforced epoxy laminate

material

Γ1 Reflection Coefficient at Port1 when Port2 and Port4 are terminated




Γopt Reflection coefficient which optimal noise figure can be found

accordingly

HFSS A commercial finite element method solver for electromagnetic

structures from Ansys

LNA Low Noise Amplifier

NFmin Minimum Noise Figure

PCB Printed Circuit Board

PNA-X Microwave Network Analyzer

Rn Equivalent Noise Resistance

SDUT31 , SDUT42 S parameters for Device Under Test when measuring one side ofthe

differential amplifier while terminating the other side

SMA Subminiature version A connector

VIA Vertical Interconnect Access

ZDUT31, ZDUT42 Z parameters for Device Under Test when measuring one side ofthe

differential amplifier ad terminatingthe other side

Zin Termination resistor placed at input port of one amplifier inthe

differential amplifier

Zout Termination resistor placed at output port of one amplifier in the

differential amplifier

xi

Chapter 1

Introduction

1.1 Motivation and Objectives

Differential circuits provide designers with significant benefits of supply, ground, and

common-mode coupling noise rejection and linearity improvement in integrated analog

and RF circuit designs. However, it is hard to examine their operation because they usu-

ally have differential inputs and outputs but external measurement equipment is typically

single-ended. It is also not possible to measure the behaviour of the subcircuit components

inside the differential circuit to verify whether they are operating as expected as they have

hidden internal nodes. To determine the faults in a differential amplifier, it is needed to

tune the circuit simulation until the measured behaviour isthe same as in simulations.

Differential amplifier noise behaviour analysis requires differential noise parameters

measurement. And then, a simulation-assisted fault analysis can be carried out. In order to

help the fault analysis process, this paper provides a method that can provide an insight into

the electrical and noise parameters of the subcomponents inside the differential system,

which is designed with following specifications: (a) Passive common network; (b) Only

one stage; (c) No internal grounding. This method only needssingle-ended measurement

equipment, which means there are no transformers, hybrids or baluns included.

Some previous works have been performed to analyze the performance of the amplifiers

relying on the knowledge of the internal configurations of the differential amplifiers [1–

9]. Because the differential measurement equipment is usually unavailable in the market,

various approaches have been developed to extract the differential parameters using single-

ended equipment [2,8–12].

The work in [10] presents a mathematical theory for mixed-mode S-parameters that is

developed for characterization of microwave differentialcircuits. The work in [11] demon-

strated a method to characterize the noise parameters of multiport devices with considera-

1

tion of the correlated input and output noise sources. Procedures of measuring differential

LNAs with correlated output noise sources are discussed in [8] and [9]. [1–4] talk about the

ways of measuring the differential noise figures without adding baluns. The work in [12]

demonstrated a method of determining the differential noise parameters of differential am-

plifiers using hybrids. [13] shows that using hybrids, transformers and baluns is the most

general approach currently available as long as the baluns are de-embeded properly. The

work in [5] demonstrates a built-in test (BIT) circuit for radio frequency differential low

noise amplifiers (DLNAs). However BIT circuits do not apply when the circuit is already

fabricated.

The work in [14] shows a theory for combined differential and common mode normal-

ized power waves developed in terms of even and odd mode impedances and propagation

constants for a microwave coupled line system.

The above works, which have been done previously, allow to examine the operation of

the differential amplifier using single-ended measurementequipment. However, they did

not provide a way of estimating the operation of components inside the circuit. This topic of

“dissecting” a differential circuit into its subcomponents with the purpose of investigating

the behaviour of the subcomponents is discussed in this thesis.

1.2 Thesis Outline

The thesis starts with an introduction of the algorithms that are needed for extracting single-

ended noise parameters inside a differential circuit. A differential amplifier is used as the

differential circuit to extract the single-ended amplifiers’ electrical parameters as shown in

Chapter2 and noise parameters as shown in Chapter3.

Chapter2 presents the design and the extraction results of a differential amplifier con-

stituted by two single amplifiers and a common network. Section2.2presents the algorithm

that has been applied to extract the S parameters for single-ended amplifiers and the com-

2

mon network, which in this example, is an inductor. Section2.3 demonstrates the system

level extraction results, which are used to verify the algorithms. Section2.4 shows the

process of designing the schematic for the differential circuit, including meeting the design

specifications and the selection of the circuit components when taking consideration of the

circuit gain and stability. The comparison of the extraction results for the single amplifiers

from ADS and single-ended simulation results are also discussed in this section. Section

2.5 includes the layout design process and EM momentum simulation results. Section2.6

shows the measurement simulation and extractions results.A summary is provided in Sec-

tion 2.7 to conclude the chapter and briefly summarize the performance of the designed

circuit.

Chapter3 demonstrates the idea of extracting noise parameters for the single amplifiers

inside the differential amplifier by using the electrical parameters extracted in Chapter2and

the overall noise parameters, which are taken from terminating one side of the differential

amplifier with 50Ω loads. Section3.2 shows the algorithm that has been used to extract

the single-ended noise parameters from the differential circuit. Section3.3.1provides the

extraction results when using ideal databoxs, which have noise parameters data and elec-

trical parameters data prestored. Section3.3.2gives the comparison between schematic

extraction and single-ended simulation results. Section3.4shows the measurement extrac-

tion and single-ended results. Summaries are also providedto conclude the chapter and to

discuss the overall performance of the algorithms and the design.

Chapter4 provides a summary of the whole thesis by making tables to compare the

extraction results between schematic and measurement. It also talks about the future work

that can be done in addition to this project.m

3

Chapter 2

Extraction of Electrical Parameters for Single-Ended Amplifiers Inside a Differential

Amplifier

2.1 Introduction and Objectives

The target of this chapter is to extract electrical parameters for single-ended circuits inside

a differential amplifier. The chapter starts with the derivation of the governing algorithm

and the associated equations in Section2.2. To verify the algorithms for the extraction, a

system level differential circuit, which contains pre-stored S-parameter data that is selected

randomly for each subcircuit components, is built and discussed in Section2.4. After the

extraction, a comparison is made between the extracted results and pre-stored S-parameters.

Then, the schematic extraction results as shown in Section2.4, schematic momentum ex-

traction results as shown in Section2.5 and measurement extraction results as shown in

Section2.6are made and compared to their relevant single-ended simulation results.

2.2 Proposed Method of the Circuit Electrical-Parameter Extraction

This section demonstrates the derivation of equations thatare used in this thesis to extract

the electrical-parameters of single-ended circuits forming a differential circuit. A concep-

tual diagram of the differential amplifier is shown in Figure2.1. The assumptions made

in this section are that a) the 4-port S-parameters of the differential amplifier are available

from either simulations or measurements and b) the single-ended circuits do not have a

hidden connection to ground, or in other words, all single-ended circuit ground connec-

tions are tied to the common network of the differential circuit. The goal is using the

4-port S-parameters to find 2-port electrical parameters for single-ended amplifiers and the

common-network inductor numerically. The key idea explored in this thesis is that by

terminating the 4-port network with some carefully selected loads, it is possible to create

4

Figure 2.1: 4-Port Differential Amplifier.

Figure 2.2: 4-Port Differential Amplifier with Ports 2 and 4 Terminated with Loads.

enough linearly independent equations that would allow theextraction of internal param-

eters of the subcomponents in the differential circuit. Following this idea, two of the four

ports of the differential circuit are terminated by loads turning the 4-port network into a

2-port network. The differential amplifier can be considered as a 2-port network as shown

in Figure2.2with two ports (Port 2 and Port 4) terminated for illustration purposes.

The reflection coefficient at the terminated ports can be described by

Γ2 =V+2 /V−

2 (2.1)

and

Γ4 =V+4 /V−

4 (2.2)

whereV+i andV−

j represent wave amplitudes entering port i and exiting from port j. Γ2

5

represents reflection coefficient at Port 2 when looking fromPort 2 into some loads,Γ4

represents reflection coefficient at Port 4 when looking fromPort 4 into some loads.

If Port 2 and Port 4 are terminated in the differential amplifier in Figure2.7, the newly

created 2-port network can be described by its 2-port S-parameters as

S′

a =

S′

a11 S′

a12

S′

a21 S′

a22

, (2.3)

whereS′

a represents the 2-port S-parameters for the 4-port differential amplifier with

Port 2 and Port 4 terminated.

By repeating this process, several sets of 2-port S-parameter matrices can be formed to

find S′

ai j = V−i

V+j

∣

∣

∣

∣

V+k6= j

.

For example, (2.4) is used to solve forS′

a11.

−S11

−S21

−S41

=

−1 S12Γ2 S12Γ4

0 S22Γ2−1 S24Γ4

0 S42Γ2 S44Γ4−1

S′

a11

V−2

V+1

V−4

V+1

, (2.4)

whereSi j represents measured S-parameters of the 4-port network. Ifthe loads for Port

2 and Port 4 are known, this matrix can be solved using standard linear algebra approach

since there are three equations and three unknowns in this system.S′

a12 , S′

a21, andS′

a22 can

be found in similar ways. Detailed derivation procedures can be found in Appendix A.

When one set of input and output ports of the differential amplifier are terminated with

known loads, the resultant 2-port S-parameters,SDUT31 (or SDUT42 when Port 1 and 3

are terminated), of the unterminated single-ended amplifier in series with a cascade com-

bination of the terminated amplifier and the common network as shown in Figure2.3 are

found.

Once the common-network value is calculated later in (2.19), it is possible to form the

6

Figure 2.3: Overall 2-port Network for Terminated Differential Amplifier.

de-composition equations of the system so that the electrical parameters (ZA andZB) for

single-ended amplifiers can be found. The following sectionintroduces the way to calculate

ZC, and thenZA andZB.

Because Amplifier A is in series connection with the rest of the system,SDUT31 or

SDUT42 are needed to be converted to Z representation. The unknown Z-parameters for

Amplifier A and Amplifier B are defined as

Za =

Z11,a Z12,a

Z21,a Z22,a

(2.5)

and

Zb =

Z11,b Z12,b

Z21,b Z22,b

. (2.6)

Assuming that the Amplifier B is terminated, then 1-port representation of this amplifier is

needed for describing the cascade of it with the common network. The 1-port representation

Zb,T is expressed as

Zb,T =

Zb,T Zb,T

Zb,T Zb,T

(2.7)

7

where

Zb,T =

(

Zin+Z11,b)(

Zout+Z22,b)

−Z12,bZ21,b

Zin +Z11,b+Zout+Z22,b−Z12,b−Z21,b. (2.8)

Detailed calculation procedures can be found in AppendixB. If the input and output of

the amplifier are terminated withZin andZout, the overallZ11becomesZin+Z11, the overall

Z22 becomesZout+Z22.

When cascaded with common network as shown in Figure2.4, the matrix of the cascade

can be represented as

Zb,Toverall =

(Z−1b,T +Z−1

c )−1 (Z−1b,T +Z−1

c )−1

(Z−1b,T +Z−1

c )−1 (Z−1b,T +Z−1

c )−1

, (2.9)

whereZC represents the impedance for the common network.

And similarly, if Amplifier A is terminated, the 1-port representationZa,T is expressed

as

Za,T =

Za,T Za,T

Za,T Za,T

, (2.10)

where

Za,T =(Zin+Z11,a)(Zout+Z22,a)−Z12,aZ21,a

Zin +Z11,a+Zout+Z22,a−Z12,a−Z21,a. (2.11)

When cascaded with the common network, the matrix of the cascade can be represented

as

Za,Toverall =

(Z−1a,T +Z−1

c )−1 (Z−1a,T +Z−1

c )−1

(Z−1a,T +Z−1

c )−1 (Z−1a,T +Z−1

c )−1

. (2.12)

If Port 2 and Port 4 are terminated byZin andZout as shown in Figure2.3, ZDUT31 can

8

Figure 2.4: Terminated TransistorB Cascaded with Common Network.

be represented as

ZDUT31 =

Z11,a Z12,a

Z21,a Z22,a

+

Zb,Toverall,11 Zb,Toverall,12

Zb,Toverall,21 Zb,Toverall,22

, (2.13)

whereZb,Toverall,11, Zb,Toverall,12, Zb,Toverall,21 and Zb,Toverall,22 are Z parameters for

Zb,Toverall.

If, on the other hand, Port 1 and Port 3 were terminated,ZDUT42 is found from

ZDUT42 =

Z11,b Z12,b

Z21,b Z22,b

+

Za,Toverall,11 Za,Toverall,12

Za,Toverall,21 Za,Toverall,22

, (2.14)

whereZa,Toverall,11, Za,Toverall,12, Za,Toverall,21 and Za,Toverall,22 are Z parameters for

Za,Toverall.

There are 9 complex unknowns, which are Z parameters for Amplifier A and Amplifier

B, as well asZC. By substituting differentZin andZout into Equation2.8, unknowns can be

solved using

ZDUT31(Zin,Zout) = Za+Zb,Toverall(Zin,Zout)

ZDUT42(Zin,Zout) = Zb+Za,Toverall(Zin,Zout) .

(2.15)

When open, i.e. infinite impedance, and short, i.e. zero ohm impedance, terminations are

9

substituted into the (2.15), the following two systems of equations are obtained

ZDUT31(∞,0) = Za+Z22,bZC

Z22,b+ZC

1 1

1 1

ZDUT42(∞,0) = Zb+Z22,aZC

Z22,a+ZC

1 1

1 1

(2.16)

and

ZDUT31(∞,∞) = Za+ZC

1 1

1 1

ZDUT42(∞,∞) = Zb+ZC

1 1

1 1

.

(2.17)

From the above equations, an observation can be made that

ZDUT31(∞,∞) (1,1) = Z11,a+ZC

ZDUT42(∞,∞) (2,2) = Z22,b+ZC

ZDUT31(∞,0) (1,1) = Z11,a+Z22,bZC

Z22,b+ZC.

(2.18)

From this system of equations,

Z2C =

[

ZDUT31(∞,∞) (1,1)−ZDUT31(∞,0) (1,1)]

ZDUT42(∞,∞) (2,2) (2.19)

and ZC can be determined. In real calculation, there are two roots for ZC. The imagi-

nary part of the correct root should be positive and close to the expected impedance of

the common-mode network. OnceZC is determined, Z parameters for Amplifier A and

Amplifier B can be found from solving (2.20) and (2.17) by substitutingZC into them:

10

Z11,a = ZDUT31(∞,∞) (1,1)−ZC

Z12,a = ZDUT31(∞,∞) (1,2)−ZC

Z21,a = ZDUT31(∞,∞) (2,1)−ZC

Z22,a = ZDUT31(∞,∞) (2,2)−ZC

(2.20)

Z11,b = ZDUT42(∞,∞) (1,1)−ZC

Z12,b = ZDUT42(∞,∞) (1,2)−ZC

Z21,b = ZDUT42(∞,∞) (2,1)−ZC

Z22,b = ZDUT42(∞,∞) (2,2)−ZC

(2.21)

2.3 Verification of the Proposed Method

In order to verify the correctness of the equations derived in Section2.2, an ideal differential

amplifier was implemented in Agilent’s Advanced Design System (ADS). Figure2.5shows

the simulated schematic. Figure2.6shows that the extraction results of the 4-port network

are exactly the same as the original values used to build the differential amplifier. Since the

extraction results are perfect, the extraction procedure can be considered correct given the

assumptions made in its derivations.

Figure 2.5: System Level Differential Circuit.

11

Figure 2.6: Comparison of Databox Extracted Results and Single-Ended Results.TopGraphs: Pre-Stored Data. Middle Graphs: Extracted Pre-stored Data for DataboxA. Bot-tom Graphs: Extracted Pre-stored Data for DataboxB. (300MHz to 3GHz)

2.4 Schematic Design Process and Simulation Results

The next step is to verify the extraction procedure with simulations of a real amplifier and

then experimentally. To verify the extraction algorithm with a real circuit, a differential

system is built using two amplifiers and a common network. In order to make a virtual

ground at the drain terminals, a big choke inductor is neededto be placed after load resis-

tors at the drain terminals. Coil Craft inductors are selected, which have a self-resonant

frequency of 1.15GHz. It is also needed to have a big inductor at the source terminal to

create differential ground at the source terminals. In order to reduce the potential instability

and the inductor’s self-resonant frequency effect on the extraction results, a 4.7nH inductor

is chosen with a self-resonant frequency of 12.7GHz. The circuit is constructed in ADS as

shown in Figure2.7.

The amplifiers are constructed with Avago ATF-35143 transistors because this type

of amplifier has low noise figures and large available gain in the frequency range from

300MHz to 3GHz. The biasing network is also formed to make circuit stable, to have wide

bandwidth, and to have gain. Table2.1 lists all circuit component values. The differential

gain for the differential amplifier is shown in Figure2.8. The circuit has around 9.5dB

12

Table 2.1: Circuit Component Values.

Circuit Component SizeGate resistor (R8 and R9) 50Ω

Feedback resistor(R1 and R4) 332ΩGate biasing resistor (R10 and R11)10000Ω

Choke inductor (L1, L2 and L3) 220nHDrain resistor (R12 and R13) 15Ω

Capacitors 56pF

Figure 2.7: Differential Amplifier ADS Schematic.

Figure 2.8: Differential Gain. (300MHz to 3GHz)

differential gain from 300MHz to 3GHz. It can also been observed from Figure2.9 that

the differential circuit is unconditionally stable at all frequencies.

Because the threshold voltage for ATF 35143 is−0.95V and the source biasing voltage

is 0V, a 10kΩ resistor is connected between the gate and ground to bias thegate voltage

also at 0V. Therefore, only one power supply is needed to be connected to the drain

13

(a) Differential Circuit Stability Factor.

(b) Differential Circuit Source and Load Stability Circles.

Figure 2.9: Differential Circuit Gain, Source and Load Stability Circles, and StabilityFactor.

Table 2.2: DC Biasing Conditions.

DC biasing conditions VoltagesGate voltage 0VDrain voltage 1.6VSource voltage 0V

terminals of the amplifiers. The DC biasing conditions can befound in Table2.2. Since the

maximum DC current is 80mA for ATF 35143 transistor, the biasing conditions are chosen

so that the transistor DC current is 62.6mA, which is less than the maximum.

A single-ended schematic is constructed in Figure2.10for comparison purpose. Be-

cause ideal 220nH choke inductors are applied to the schematic, the DC biasing voltage at

drain terminal is 1.6V. When the choke inductors are replace with real inductors(0402AF),

there will be a 0.08V voltage drop across the inductors. The S-parameters forthe single-

ended circuit are reported in Figure2.11.

14

Figure 2.10: Schematic of the Single-Ended Amplifier.

Figure 2.11: Schematic of a Single-Ended Amplifier S Parameters in dB. (300MHz to3GHz)

The selection of the gate resistors and feedback resistors is based on the consideration

of the stability for the circuit at all frequencies. The stability circles showed in Figure2.12b

are all outside the unity circle, which means the circuit is unconditionally stable.

The single-ended simulation results can be concluded in Table 2.3. Under current bias-

ing condition, the single-ended amplifier has gain of 11.2dB. It also has a large bandwidth.

In the next section, an electrical parameters extraction typology is introduced.

15

(a) Stability Factor for Single-Ended Amplifier.

(b) Input and Output Stability Circles for Single-Ended Amplifier.

Figure 2.12: Stability Factor, Schematic Single-Ended Source and Load Stability Circles.

Table 2.3: Schematic Design Specifications.

Design Parameters SpecificationsGain at 700 MHz 11.2dB

Stability unconditionaly stableBandwidth 8GHz

2.5 Differential Amplifier Design and Simulation Results

In this section, the layout design process for the differential amplifier is implemented. The

extraction results for the schematic and momentum simulation, as well as their relavant

single-ended simulation results are discussed and compared.

16

2.5.1 Schematic Simulation and Extraction Results

Matlab is used as the tool to simulate the extraction procedures numerically following the

equations in Section2.2.

In Figure2.13, extracted S-parameters for single-ended amplifiers inside the differential

amplifier as shown in Figure2.7are compared with single-ended amplifier S-parameters as

shown in Figure2.10.

Figure 2.13: Schematic S-parameters Extraction Results vs Schematic Single-Ended Sim-ulation Results. Red Line: Schematic Simulated Single-Ended S-parameters. Blue Line:Schematic Extracted Results. (300MHz to 3GHz)

17

From Figure2.13, it can be observed that extractedS11, S12 andS21 are very close to

the single-ended simulation results both in magnitude plots and on the Smith chart.S22 has

the largest variation of 1dB between extracted results and single-ended results. The trend

of S22 on the Smith chart is also visible. It is not clear at this timewhy S22 does not agree

exactly with expectations.

2.5.2 Layout Design Process and Comparison of Momentum Extraction Results with

Momentum Single-Ended Simulation Results

A two-layer printed circuit board (PCB) was selected to construct the differential amplifier

and to verify the extraction algorithms. The PCB substrate is FR4 material, whose substrate

parameters and structure are shown in Figure2.14. By using the ADS line calculation tool,

it was found that on a 1mm (i.e. 40mil) thick substrate, the 50Ω transmission-line width

should be 1.94mm (i.e. 76.4mils).

(a) PCB Substrate Parameters.

(b) Substrate Transverse Plane Picture.

Figure 2.14: Substrate Parameters and Substrate Transverse Plane Picture.

18

Figure2.15shows a 3D view of the circuit layout with input and output transmission

lines. The transmission lines are designed as coplanar waveguides due to low dispersion

and the broadband performance. Figure2.16shows the center area for both top and bot-

tom layer of the designed PCB. Vias are placed through out thePCB for good grounding

condition.

Figure 2.15: 3D View of Two-Layer PCB Layout in ADS with Input and Output Trans-mission Lines.

Figure 2.16:Top View for the Two-Layer PCB. Top Layer: Brown. Bottom Layer: Yellow.

The assumption made during the derivation of the extractionalgorithm is that there are

no references to the PCB ground from the single-ended subcomponents inside the differen-

tial amplifier. In particular, this includes various transmission lines that interconnect circuit

19

Figure 2.17: ATF 35143 Footprint.

components as the transmission lines may couple to ground planes. To reduce the effects

from the internal transmission lines on the extraction results, all circuit components must

be placed as close to each other as possible. The feedback loops, which contain feedback

resistors and DC block capacitors, are placed on the bottom layer due to the diagonal po-

sitions for the drain terminals and gate terminals as shown in Figure2.17. The soldering

pads are connected with top layers through vias. These linesare unavoidable and have to be

de-embedded during measurements. To make de-embedding process less complicated, all

transmission lines are designed to have the same lengths. Since each transistor package has

4 legs as shown in Figure2.17. To make the input line and output lines fully symmetrical,

the legs of the transistor on the right-hand side of the PCB are bent over.

Figure2.18 shows the comparison of extracted results obtained from schematic mo-

mentum with a single-ended amplifier also from schematic momentum within a selected

bandwidth (500MHz-1.5GHz). It can be observed from Figure2.18, S11, S21 andS12 are in

good agreement between expectations and extraction. They all go higher as the frequency

increases. Although the momentum single-ended simulationresults for those parameters

have some variation in magnitude, the shapes for them can still be considered as good

since they are similar to each other.S22 is larger in value and shows high variation between

what is obtained with direct simulation and what is obtainedwith extraction. The variation

of about 2dB between the single-ended momentum results and momentum extraction re-

20

sults is observed. It can be concluded that although the extraction results are not as good

as schematic extraction results, the values for S parameters within the bandwidth that is

selected are still reasonable. The next section demonstrated experimental results used to

verify the extraction process.

Figure 2.18: Comparison between Simulated Single-Ended and Momentum ExtractionResults (300MHz-1.5GHz). Sky Blue and Red Line: Momentum Extraction Results. DarkBlue Line: Momentum Single-Ended Simulation Results.

21

2.6 Measurement Results

Figure2.19shows the fabricated PCB with circuit components and SMA connectors. To

obtain accurate S-parameters for the differential network, de-embedding process needs to

be performed. HFSS (High Frequency Structure Simulator) isused as the tool to construct

the S-parameters for the SMA connector. Since the input and output transmission lines

have the same length, they are de-embedded in the same way.

De-embedding process followed in this work is:

1. Measure S-parameters of the transmission line with SMA connectors on each side.

2. In ADS, use the SMA S-parameter data to de-embed one SMA from the transmission

line measured in step 1 as shown in Figure2.20.

3. Use the results from step 2 to de-embed transmission linesfrom each port of the

differential circuit.

Figure 2.19: Top View and Bottom View of Surface-Mounted Two-Layer PCB.

Figure 2.20: ADS De-Embedding Schematic (Left Databox: Measurement RawData.Right Databox: SMA S Parameters from HFSS.).

22

Figure2.21shows the 4-port de-embedding structure for S-parameters obtained from

the measurement and the comparison of the 4-port S-parameters before and after the de-

embedding process.

Figure 2.21: Comparison of Measured S-Parameters Before De-Embedding and After De-Embedding. Blue line: S-Parameter for the Measured 4-port before De-Embedding. Redline: S-Parameter for the Measured 4-port after De-Embedding. (300MHz to 3GHz)

From Figure2.22and Figure2.23, it can be found thatS11 andS12 are good both in

magnitude and phase.S21 andS22 demonstrate the correct shape but the magnitude for

23

them are both higher than the single-ended simulation results. S21 has 1.017dB difference

at 502.5MHzand 0.751dBdifference at 1.515GHzfor Amplifier A, while it has 0.633dB

difference at 502.5MHz and 0.303dB difference at 1.515GHz for Amplifier B . S22 has

2.41dB difference at 502.5MHz and 5.125dB difference at 1.515GHz for Amplifier A,

while it has−2.045dBdifference at 502.5MHzand−5.242dBdifference at 1.515GHzfor

Amplifier B. It can still be concluded that the S-parameters extracted from measurement

results are good enough to be used as the inputs to the noise calculation in next chapter.

Figure 2.22: Comparison of De-Embedded Measured Single-Ended AmplifierS-Parameters, and Extracted Single-Ended Amplifier S-Parameters from the DifferentialAmplifier. (Smith chart) Red Line: Measured Single-Ended S-Parameters. Blue Line:Measured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters.(300MHz to 3GHz)

24

Figure 2.23: Comparison of De-Embedded Measured Single-Ended AmplifierS-Parameters, and Extracted Single-Ended Amplifier S-Parameters from the Differential Am-plifier. (Magnitude) Red Line: Measured Single-Ended S-Parameters. Blue Line: Mea-sured Amplifier A S-Parameters. Purple Line: Measured Amplifier B S-Parameters.

2.7 Summary

To summarize this chapter, the extraction algorithm is presented and experimentally veri-

fied. Within the bandwidth (500MHz to 1.5GHz), the schematic extraction results are very

close to the schematic single-ended simulation results. Momentum extraction results are

good in low frequencies. But as frequency goes higher, the agreement gets worse. The

measurement results are good in magnitude and phase forS11 andS12, while the biggest

variation is the magnitude forS21 andS22. Since the target of this project is to extract noise

parameters of single amplifiers inside the differential amplifier, it can be examined later to

see that whether the variations ofS22 have significant influences to the noise parameters

extractions. The next chapter will talk about the extraction algorithms and experimental

measurement results for the noise parameters of the single-ended amplifiers inside the dif-

ferential amplifier.

25

Chapter 3

Extraction of Noise Parameters for Single-Ended AmplifiersInside a Differential

Amplifier

3.1 Introduction and Objectives

After the electrical-parameter extraction method, which is introduced in Chapter2, the next

stage is to determine the noise parameters, which are the minimum noise figureNFmin,

the equivalent noise resistanceRn and the optimum source reflection coefficient, which

corresponds to minimum noise figure achievement,Γopt, for the subcomponents inside the

differential circuit. There are a few approaches to measurethe noise parameters for single-

ended circuits.

Some of the approaches model noise in terms of power waves [15–21]. There are also other

approaches, which perform single noise figure measurementsand try to put the results into a

DUT noise model determined by using other techniques [22–24]. The most commonly used

techniques are performed by using source impedance tuners to generate different signal-

source admittances at the DUT input port. And then use the receivers to measure the noise

powers at the output port [25–36]. The noise parameters are found by using data fitting

techniques as described in [37–41].

The purpose of this chapter is to demonstrate a method of extracting noise parameters of

single-ended amplifiers inside a differential amplifier. The same assumptions as in Chapter

2 are considered to be applied to the same type of amplifiers here. This chapter starts with

the discussion of the noise-parameter extraction algorithm in Section3.2and proceeds with

simulation and experimental verification of the algorithm in Section3.3and Section3.4.

26

3.2 Single-Ended Noise-Parameter Extraction Algorithm

In this noise-parameter extraction algorithm, 2-port single-ended noise-parameter measure-

ment equipment is used. Because a differential amplifier hasfour ports that can interface

single-ended equipment, when measuring noise parameters,there are always two unused

ports. These unused ports are terminated with 50Ω terminations. In this way, the 4-port

network becomes 2-port network as shown in Figure3.1. The following extractions assume

that the electrical parameters as derived in Chapter2 are available.

Based on Figure3.1, the Z representation of the overall noise correlation matrix for the

measured 2-port circuit can be written as

C13,overall = CZ,a+C′

Z,b

C24,overall = CZ,b+C′

Z,a

(3.1)

Figure 3.1: Divisions of 2-Port Networks.

whereCZ,a represents the Z-represenntation of the noise correlationmatrix of Amplifier

A if Amplifier B is terminated as shown in Figure3.1, CZ,b represents the Z-reprensentation

noise correlation matrix of Amplifier B if Amplifier A is terminated,C′

Z,b represents the

Z-reprentation noise correlation matrix of the terminatedAmplifier B cascaded with the

common network,C′

Z,a represents the Z-representation noise correlation matrixof the ter-

27

minated Amplifier A cascaded with the common network.

CZ,a andCZ,b can be represented as

CZ,a =

CZ,a,11 CZ,a,12

CZ,a,21 CZ,a,22

(3.2)

CZ,b =

CZ,b,11 CZ,b,12

CZ,b,21 CZ,b,22

(3.3)

and then

CZa,T = CZ,a+2kT

RZin 0

0 RZout

(3.4)

CZb,T = CZ,b+2kT

RZin 0

0 RZout

(3.5)

whereCZa,T represents the Z-representation noise correlation matrixfor terminated

Amplifier A, CZb,T represents the Z-representation noise correlation matrixfor terminated

Amplifier B, T is the absolute temperature, k is Boltzmann’s constant.

Since the ground terminals of the single-ended amplifiers are connected to the common

network of the differential circuit, the 2-port network forthe terminated amplifiers needs

to be “turned upside down” as shown in Figure3.2 for further derivations. As can be seen

from Figure3.1, the overall structure of the 2-port network whose noise parameters are

measured is an input and output terminated amplifier cascaded with the common network

as shown in Figure3.3, then in series with the amplifier connected to the measurement

equipment as shown in Figure3.4. Seeing from the source terminal of this single-ended

amplifier, the terminated single-ended amplifier becomes an1-port network with identical

4 entries in its noise correlation matrix.

28

Figure 3.2: 2-Port Upside Down.

Figure 3.3: Cascade AmplifierB with Common Network

Figure 3.4: Serial Connection of AmplifierA and Common Network

If assuming that Amplifier A is “turned upside down” and is terminated, its resultant

29

input and output referred noise voltages are found as

vn

vn

=

vna,T

vnb,T

+Za,T

1

−1

ia (3.6)

wherevna,T refers to the input-referred noise voltage at the terminated input port of

Amplifier B, vnb,T refers to the input-referred noise voltage at the terminated output port

of Amplifier B, vn refers to the open circuit noise voltage at Port 1,ia refers to the current

flowing into the input port of terminated Amplifier A,Za,T refers to the 2-port Z-parameters

of the terminated Amplifier A when turned upside down, derivation can be found in Chapter

2, Section2.2.

In the following analysis,vn,T =

vna,T

vnb,T

is used. From (3.6), it can be shown that

vn =1

∆Za,TZ

′

a,Tvn,T I2by1, (3.7)

whereZ′

a,T=

[

Za,T,22−Za,T,21 Za,T,11−Za,T,12

]

, ∆Za,T=

[

1 −1

]

Za,T

[

1 −1

]T

.

Detailed calculation procedures can also be found in Appendix C.

The Z-representation of the noise correlation matrix can befound from

CZa,T = vnvHn I2by2

=Z′a,Tvn,T vH

n,TZ′Ha,T

|∆Za,T |2I2by2

=Z′a,T(CZa,T)Z

′Ha,T

|∆Za,T |2I2by2

=Z′a,T(CZ,a+CT)Z

′Ha,T

|∆Za,T |2I2by2

(3.8)

whereCZa,T represents the noise correlation matrix in Z-representation for the termi-

nated 2-port network,CT =2kT

RZin 0

0 RZout

, I2by2 is a 2×2 all-ones matrix.

The noise correlation matrix of the common network in Z-representation is found in

[14]

30

CZC = 2kT

RZC RZC

RZC RZC

, (3.9)

whereCZC represents the Z-representation of the noise correlation matrix of the com-

mon network as shown in Figure3.3andZC is the common-network impedance.

So the overall Z-representation form of noise correlation matrix can be derived as

C′

a =CZa,T |ZC|

2

|Za,T +ZC|2+

CZC|Za,T |2

|Za,T +ZC|2, (3.10)

whereC′

a is the overall noise correlation matrix in Z-representation for Amplifier A

cascaded with the common network. Detailed derivation procedures can be found in Ap-

pendixD.

C′

b, which represents the unknown noise correlation matrix in Z-representation for Am-

plifier B cascaded with the common network can also be found insimilar way

C′

b =CZb,T |ZC|

2

|Zb,T +ZC|2+

CZC|Zb,T |2

|Zb,T +ZC|2. (3.11)

Then the system of matrixes can be formed as below

C13−|ZC|

2

|Zb,T+ZC|2Z′

b,TCTZ′Hb,T

|∆Zb,T |2I2by2−

|Zb,T |2CC

|Zb,T+ZC|2= CZ,a+

|ZC|2

|Zb,T+ZC|2Z′

b,TCZ,bZ′Hb,T

|∆Zb,T |2I2by2

C24−|ZC|

2

|Za,T+ZC|2Z′a,TCTZ

′Ha,T

|∆Za,T |2I2by2−

|Za,T |2CC

|Za,T+ZC|2= CZ,b+

|ZC|2

|Za,T+ZC|2Z′a,TCZ,aZ

′Ha,T

|∆Za,T |2I2by2

(3.12)

whereC13 refers to the measured 2-port overall noise correlation matrix in Z-representation

terminating Port 2 and Port 4,C24 refers to the measured 2-port overall noise correla-

tion matrix in Z-representation with Port 1 and Port 3 terminated and similar to (3.7),

Z′

b,T=

[

Zb,T,22−Zb,T,21 Zb,T,11−Zb,T,12

]

, ∆Zb,T=

[

1 −1

]

Zb,T

[

1 −1

]T

.

In 8 equations described by the system in (3.12), there are 8 unknows, which are the

terms in Amplifiers A and B noise correlation matrices. Once the matrix system is solved,

these noise correlation matrices should be converted to their ABCD-representation in order

to calculate their noise parameters [14].

31

3.3 Schematic Extraction and Simulation Results

3.3.1 Algorithm Verification Using Ideal Components

When performing extraction for an ideal circuit, which is shown in Figure3.5, the noise

parameters data are stored in the “databoxes” with different values in order to make the

comparison more convincing. The extracted and original values forNFmin, Rn andΓopt are

shown in Figure3.6, Figure3.8and Figure3.7.

Figure 3.5: Ideal Differential Amplifier.

Figure 3.6: Comparison ofRn between Extracted Results and Prestored Data..

32

Figure 3.7: Comparison ofNFminbetween Extracted Results and Prestored Data.

Figure 3.8: Comparison ofΓopt between Extracted Results and Prestored Data.

From the figures above, it can be seen thatRn , NFmin andΓopt in the ideal circuit extrac-

tion have exactly same value for the extraction and originaldata, which means the algorithm

perfectly recovers the noise parameters for subcircuit inside the differential circuit.

33

3.3.2 Comparison Between Schematic Extraction Results andSchematic Single-Ended

Simulation Results

The simulation described in Section3.2were used to verify the noise parameters extraction

algorithm.

From Figure3.9, it can be observed that the extraction results for AmplifierA and Am-

plifierB are exactly the same as each other.NFmin from the extraction is 0.67dB lower than

the single-ended simulation results at 500MHz. As the frequency goes higher, theNFmin

from the extraction is 0.62dB lower than the single-ended simulation results at 1.5GHz.

From Figure3.10, it can be observed thatRn from the extraction has a variation of 0.99Ω

when compared with the single-ended simulation results at 500MHz. However, as the fre-

quency increases, the difference between the extraction results and single-ended results gets

smaller. They have a variation of 0.62Ω at 1.5GHz. ForΓopt in Figure3.11, the extracted

results and the single-ended simulation results are both reasonable and very close to each

other.

Figure 3.9: Schematic ExtractedNFmin vs Schematic Single-Ended Simulation Results.

34

Figure 3.10: Schematic ExtractedRn vs Schematic Single-Ended Simulation Results.

Figure 3.11: Schematic ExtractedΓopt vs Schematic Single-Ended Simulation Results.

3.4 Measurement Noise Extraction Results

The measurement process is performed using PNA-X network analyzer located in a shielded

room to reduce interference. The PNA-X is capable of measuring both electrical and noise

parameters.

Because of PNA-X limitations, when in low frequency noise-parameter measurements,

measured noise parameters, which are shown in Figures3.12-3.14, exhibits several points

35

that have extremely high values and are not continuous with the other points around. To

make extraction results reasonable, smoothing function inMatlab forNFmin, Γopt andRn is

applied to remove all odd data points and to take average for the rest data points with their

nearby data points in future extraction procedures.

Figure 3.12: One-Side TerminatedNFmin Raw Data from Measurement in Figure3.4.(300MHz to 3GHz)

Figure 3.13: One-Side TerminatedRn Raw Data from Measurement in Figure3.4.(300MHz to 3GHz)

36

Figure 3.14: One-Side TerminatedΓopt Raw Data from Measurement in Figure3.4.(300MHz to 3GHz)

As can be seen, averaging reduced the trace noise onNFmin andRn as expected. Al-

thoughΓopt for both amplifiers are not perfectly continuous at all frequencies, the trend for

the curve on the Smith chart can still be found. Comparison between the extraction results

and the single-ended results are made in Figure3.15and Figure3.16. It can be observed

that at 500MHz, the single-endedNFmin has a value of 2.05dB, while the extracted Am-

plifier A has a value of 1.84dB and the extracted Amplifier B has a value of 2.04dB. At

1.5GHz, the single-endedNFmin is 1.88dB, while the extracted Amplifier A has a value

of 2.01dB and the extracted Amplifier B has a value of 2.08dB. NFmin for the extracted

Amplifier A and the extracted Amplifier B are close to each other. They are also both close

to the single-ended results. ForRn, at 500MHz, the single-ended simulation result has a

value of 38.73Ω, while the extracted Amplifier A has a value of 23.55Ω and the extracted

Amplifier B has a value of 45.71Ω. At 1.5GHz, the single-ended simulation result has

a value of 30.76Ω, while the extracted Amplifier A has a value of 24.51Ω and the ex-

tracted Amplifier B has a value of 28.58Ω. Overall, the extraction results and single-ended

measurements ofNFmin andRn are reasonably similar.

From Fig.3.17, it can be seen that both extractedΓopt are located close to what is ex-

pected from measurements of the single-ended amplifier.

37

Figure 3.15: Comparison ofNFmin between Measurement Extracted Results and Measure-ment Single-Ended Results (300MHz to 3GHz).

Figure 3.16:Comparison ofRn between Measurement Extracted Results and MeasurementSingle-Ended Results (300MHz to 3GHz).

38

Figure 3.17: Comparison ofΓopt between Measurement Extracted Results and Measure-ment Single-Ended Results (300MHz to 3GHz).

3.5 Summary

When comparing the schematic extraction performance with the measurement extraction

performance forNFmin, it can be found that at 500MHz, the variation between the ex-

traction results and single-ended simulation results fromschematic, which is 0.67dB, is

larger than the variation from measurement results, which is 0.21dB for Amplifier A and

0.01dB for AmplifierB. At 1.5GHz, the variation between the schematic single-ended sim-

ulation result and extraction result, which is 0.62dB, is also larger than variation from the

measurement of the extracted Amplifier A, which is 0.13dB, and Amplifier B which is

0.04dB. It can be concluded that the performance of measurement extraction ofNFmin

for both AmplifierA and AmplifierB is better than the schematic extraction result. So the

overall performance forNFmin extraction works reasonably well.

ForRn, the schematic extraction results are perfect at 500MHz, while the measurement

extraction result for Amplifier A is smaller than the single-ended simulation result with

a value of 9.994Ω. The extraction result for Amplifier B is also smaller than the single-

ended simulation result with a value of 8.429Ω. At 1.5GHz, the schematic extraction

result is larger than the single-ended simulation result with a value of 3Ω, while for mea-

39

surement, the extraction result for Amplifier A is larger than the single-ended result with

a value of 5.549Ω. For Amplifier B, the extraction value is smaller than the single-ended

measurement result with a value of 1.321Ω. The performance for extraction ofRn is also

very good.

For Γopt, the extraction results from schematic are close to the single-ended results not

only in magnitude, but also in shape on Smith Chart. For the measurement results, the

extraction results are even closer to the single-ended simulation results in magnitude and

shape.

Overall it is concluded that the measurement extraction results verify the noise-parameter

extraction algorithm.

40

Chapter 4

Conclusions and Future Work

4.1 Conclusion

This thesis presents a novel algorithm of extracting electrical and noise parameters of

differential-amplifier single-ended subcomponents. The thesis provides experimental ev-

idence to verify the algorithm. In this thesis, the differential amplifier is formed by two

single-ended amplifiers and a common network as shown in Figure 2.1. The bandwidth of

the design is set be from 500MHzto 1.5GHzin order to avoid the self-resonant frequencies

for choke inductors and reduce the the influence of the resonance in the common-network

inductor as much as possible.

In Chapter2, the mathematical algorithms for extracting the electrical parameters of

the single-ended amplifiers inside a differential amplifierare presented and proven with

schematic simulations, EM and schematic simulations, and measurements. Section2.3

shows the extraction results for the ideal case. The differential circuit is formed with

“databoxes” having random selected S parameters prestoredas shown in Figure2.5. The

algorithm perfectly recovers the S-parameters for the single amplifiers, which proves that

the mathematical algorithm works very well. Table4.1 shows comparison results from

schematic which are also good since the variation for all S-parameters between the ex-

pected values and extracted values are small. The biggest variation happens inS22, which

is about 1.28dBat 502.5MHzand 1.355dBat 1.515GHz. However, the shape ofS22 is still

good as it is very close to the single-ended simulation result. It also provides very reason-

able extraction results for the common network impedance. Table4.2 shows comparison

of momentum results, the extraction results are good around502.5MHz, which has little

variation between expected and extracted results for all S parameters and the common net-

work. As the frequency approaches 1.515GHz, the variations are larger. Table4.5 shows

the comparison of measurement results, the extraction results for S-parameters are also

41

good within the bandwidth not only in magnitude, but also in shape. The variation of the

extraction results for the common-network impedance are also good at 502.5MHz, which

is about−1.986− j5.28Ω. The real part for the common network is larger than its actual

value in data sheet, which is about 0.989Ω. But since the real part of the common net-

work also includes losses of traces on the PCB, the extra lossis expected. Therefore, it can

be concluded that the algorithm designed for extracting S-parameters for the single-ended

amplifiers inside a differential amplifier works very well.

Schematic (Ideal Common Network L)S-Parameters Frequency Single-Ended Extracted Variation

S11502.5MHz −8.419dB −8.491dB 0.072dB1.515GHz −8.520dB −8.646dB 0.126dB

S12502.5MHz −20.875dB −21.474dB 0.599dB1.515GHz −20.75dB −21.646dB 0.896dB

S21502.5MHz 9.351dB 9.558dB −0.207dB1.515GHz 9.503dB 9.719dB −0.216dB

S22502.5MHz −10.774dB −9.494dB −1.280dB1.515GHz −10.628dB −9.273dB −1.355dB

Directly Measured

ZC502.5MHz 0+ j14.839 0.717+ j15.115 −0.717− j0.2761.515GHz 0+ j44.739 1.146+ j44.102 −1.146+ j0.637

Table 4.1: Comparison of S-parameters from Schematic Simulations Measurement Re-sults to the Relevant Single-Ended Simulation and Measurement Results for Single-EndedAmplifiers and the Common Network Impedance

42

Momentum (Real Common Network L)S-Parameters Frequency Single-Ended Extracted Variation


S12502.5MHz −21.56dB −22.429dB 0.869dB1.515GHz −21.234dB −12.249dB −8.985dB

S21502.5MHz 8.559dB 7.608dB 0.951dB1.515GHz 8.658dB 10.073dB −1.415dB


Directly Measured

ZC502.5MHz 0.555+ j14.862 0.875+ j15.982 −0.32− j1.121.515GHz 0.989+ j45.357 9.909+ j73.525 −8.92− j28.168

Table 4.2: Comparison of S-Parameters from EM-Schematic Momentum to the Rele-vant Single-Ended Simulation for Single-Ended Amplifiers and the Common NetworkImpedance

In Chapter3, the mathematical algorithms for extracting the noise parameters of the

single-ended amplifiers inside a differential amplifier using electrical parameters obtained

from Chapter2 and the overall noise parameters and S parameters as shown inFigure3.4

are presented and proven by using the simulation and measured data from schematic and

measurements. Section3.3 shows the extraction results for the ideal circuit, which isob-

tained by forming the differential circuit by using “databoxes” having randomly selected

reasonable noise parameters stored. The extraction results for the noise parameters of the

single “databox” inside the differential system can be found in Table4.3, which shows

that the mathematical algorithms work perfectly since there are no variations between the

extracted results and the pre-stored data forNFmin , Rn andΓopt at all frequencies. In Ta-

ble 4.4, for the schematic simulation results at around 502.5MHz, the variation between

the single-ended simulation results and extraction results for NFmin is around 0.67dB. As

the frequency increases, the variation becomes about 0.62dB at 1.515GHz. For Rn , at

502.5MHz, the variation between the single-ended simulation results and the extraction

results is around 0.99Ω. At 1.515GHz, the variation becomes 0.62Ω. Γopt also has little

variation at both 502.5MHz and 1.515GHz. Since the variation for the singl-ended sim-

43

ulation results and the extracted results is small, it can beconcluded that the schematic

extraction results are reasonable. Table4.6shows the measurement results. The extraction

results forNFmin are good at 502.5MHz and 1.515GHz, which have variation of 0.21dB

and 0.13dB for AmplifierA, and variation of 0.01dB and 0.04dB for AmpliferB. Their

variation is smaller than the variation between schematic simulation and extraction results

at 502.5MHz and 1.515GHz. The variation forRn at 502.5MHz and 1.515GHz is larger

than the variation between the schematic single-ended simulation and extraction results for

both AmplifierA and AmplifierB. From Figure3.17, it can be seen that both the magnitude

and the shape ofΓopt for both amplifiers are similar to the expectations. Therefore, it can be

concluded that the extraction of the noise parameters for the amplifiers works reasonably

well.

4.2 Future Work

TheS22 from the schematic extraction is reasonable. But in measurement extraction results,

from Figure2.22, the variation forS22 between the extracted results and the single-ended

results gets bigger as frequency goes higher. The reason forthis inaccuracy may be caused

by having coupling to ground problems for the designed PCB orthe choke inductors are

not large enough to be considered as open circuit when in parallel with the ouput port

with open termination in the simulations. Because the internal transmission lines has big

influence to the extraction results, the layout has to be designed in the way that least internal

transmission lines are used as shown in Figure2.16. The future work for this project would

be to find a way of improving the algorithms to accommodate grounding and finite choke

inductor problems, which are found in the circuit.

44

Ideal Differential CircuitNoise ParametersFrequency Single-Ended(A/B) Extracted(A/B) Variation

NFmin500MHz 1.023dB/1.052dB 1.023dB/1.052dB 0dB/0dB1.5GHz 1.047dB/1.102dB 1.047dB/1.102dB 0dB/0dB

Rn500MHz 11Ω/14.5Ω 11Ω/14.5Ω 0Ω/0Ω1.5GHz 11Ω/13Ω 11Ω/13Ω 0Ω/0Ω

Γopt500MHz 0.905+ j0.095/0.838+ j0.059 0.905+ j0.095/0.838+ j0.059 0/01.5GHz 0.715+ j0.380/0.594+ j0.371 0.715+ j0.380/0.594+ j0.371 0/0

Table 4.3: Comparison of Ideal Differential-Circuit Extraction Results with Single-Ended Simulation Results.

45

SchematicNoise Parameters Frequency Single-Ended Extracted Variation

NFmin502.5MHz 2.38dB 1.71dB 0.67dB1.515GHz 2.36dB 1.74dB 0.62dB

Rn502.5MHz 21.28Ω 20.29Ω 0.99Ω1.515GHz 20.39Ω 21.01Ω −0.62Ω

Γopt502.5MHz 0.220+ j0.056 0.209+ j0.055 -1.515GHz 0.202+ j0.072 0.207+ j0.076 -

Table 4.4: Comparison ofNFmin, Rn, andΓopt Obtained from Schematic Simulations tothe Relevant Single-Ended Simulation Results .

46

Measurement Amplifier A Measurement Amplifier BS-

ParametersFrequency Single-

EndedExtracted Variation Single-

EndedExtracted Variation

S11502.5MHz −8.645dB −7.692dB −0.953dB −8.645dB −8.105dB −0.54dB1.515GHz −9.596dB −9.532dB −0.064dB −9.596dB −8.396dB −1.2dB

S12502.5MHz −19.176dB −23.372dB 4.196dB −19.176dB −22.488dB 3.312dB1.515GHz −13.299dB −17.289dB 3.99dB −13.299dB −16.529dB 3.23dB

S21502.5MHz 8.067dB 8.451dB −1.017dB 8.067dB 9.084dB −0.633dB1.515GHz 4.272dB 5.023dB −0.751dB 4.272dB 4.575dB −0.303dB

S22502.5MHz −6.639dB −4.229dB −2.41dB −6.639dB −4.594dB −2.045dB1.515GHz −6.185dB −1.060dB −5.125dB −6.185dB −0.943dB −5.242dB

DirectlyMeasured

DirectlyMeasured

ZC502.5MHz 0.555+

j14.8622.541+j20.142

−1.986−j5.28

same asAmplifier

A

same asAmplifier

A

same asAmplifier

A1.515GHz 0.989+

j45.35718.579+j48.983

−17.59−j3.626

same asAmplifier

A

same asAmplifier

A

same asAmplifier

A

Table 4.5: Comparison of S-Parameters from Measurement Results to theRelevant Single-Ended Measurement Results for Single-Ended Amplifiers and the Common Network Impedance

47

Measurement Amplifier A (after smooth function)Measurement Amplifier B (after smooth function)Noise Parameters Frequency Single-Ended Extracted Variation Single-Ended Extracted Variation

NFmin502.5MHz 2.05dB 1.84dB 0.21dB 2.05dB 2.04dB 0.01dB1.515GHz 2.10dB 1.97dB 0.13dB 2.10dB 2.14dB −0.04dB

Rn502.5MHz 38.73Ω 23.55Ω 15.18Ω 38.73Ω 45.71Ω −6.98Ω1.515GHz 25.89Ω 21.44Ω 4.45Ω 25.89Ω 27.56Ω −1.67Ω

Γopt502.5MHz 0.350+ j0.016 0.226+ j0.086 - 0.350+ j0.016 0.444+ j0.136 -1.515GHz 0.125− j0.033 0.066+ j0.164 - 0.125− j0.033 0.108+ j0.185 -

Table 4.6: Comparison ofNFmin, Rn, andΓopt Obtained from Measurements to the Relevant Single-Ended Measurement Results.

48

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Appendix A

Forming Matrix to Solve S-Parameters for Single Amplifier Inside a Differential

Amplifier

To form the matrix mentioned in (2.4) with the purpose of finding the 2-port S-parameters

for the terminated differential Ampplifier B (which means Port 2 and Port 4 are terminated)

as mentioned in Section2.2. Firstly, the 4-port S parameters can be represented in the

following way

V−1

V−2

V−3

V−4

=

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

V+1

V+2

V+3

V+4

, (A.1)

whereV+i are the wave amplitudes entering Port i,V−

j are the wave amplitudes exiting Port

j,

V−1 = S11V

+1 +S12V

+2 +S13V

+3 +S14V

+4 (A.2)

V−2 = S21V

+1 +S22V

+2 +S23V

+3 +S24V

+4 (A.3)

V−3 = S31V

+1 +S32V

+2 +S33V

+3 +S34V

+4 (A.4)

V−4 = S41V

+1 +S42V

+2 +S43V

+3 +S44V

+4 (A.5)

If both sides of (A.2) are divided byV+1 , then

V−1

V+1

= S11+S12V+

2

V+1

+S13V+

3

V+1

+S14V+

4

V+1

(A.6)

SinceV+

2

V+1

=V+

2

V−2

×V−

2

V+1. (A.7)

Similar transforms can be applied to other similar terms, so(A.6) can be represented in the

way as shown in (A.8)

S′

11 = S11+S12V+

2

V−2

V−2

V+1

+S13V+

3

V−3

V−3

V−2

+S14V+

4

V−4

V−4

V+1

(A.8)


V−2

V+1

= S21+S22V+

2

V−2

V−2

V+1

+S23V+

3

V−3

V−3

V−2

+S24V+

4

V−4

V−4

V+1

, (A.9)


V−4

V+1

= S41+S42V+

2

V−2

V−2

V+1

+S43V+

3

V−3

V−3

V−2

+S44V+

4

V−4

V−4

V+1

(A.10)

With (A.8), (A.9), and (A.10), the following matrix that was used in (2.4) can be formed

(when calculatingS11 of a two port,V+2 which represents the wave going into Port 2 will

set to be 0):

−S11

−S21

−S41

=

−1 S12Γ2 S14Γ4

0 S22Γ2−1 S24Γ4

0 S42Γ2 S44Γ4−1

S′

11

V−2

V+1

V−4

V+1

, (A.11)

whereΓ2 =V+

2V−

2represents the input reflection coefficient when looking from the termi-

nated input port of Amplifier B into the load,Γ4 =V+

4V−

4represents the output reflection

coefficient when looking from the terminated output port of Amplifier B into the load,

S′

i j =V+

iV−

j|V+

k =0 f or k6= j representing the 2-port S parameters for the differential amplifier with

one side terminated. This matrix can be solved to findS′

11.

Similarly, to findS′

22, it needs to divide both sides of the (A.3), (A.4) and (A.5) by V+3 .

Similarly, this timeV+1 = 0. The following matrixes are found

−S33

−S23

−S43

=

−1 S32Γ2 S34Γ4

0 S22Γ2−1 S24Γ4

0 S42Γ2 S44Γ4−1

S′

22

V−2

V+3

V−4

V+3

(A.12)

Dividing (A.2), (A.3), and (A.5) by V+3 gives

56

−S13

−S23

−S43

=

−1 S12Γ2 S14Γ4

0 S22Γ2−1 S24Γ4

0 S42Γ2 S44Γ4−1

S′

12

V−2

V+1

V−4

V+1

. (A.13)

Dividing (A.3), (A.4), and (A.5) by V+1 gives

−S31

−S21

−S41

=

−1 S32Γ2 S34Γ4

0 S22Γ2−1 S24Γ4

0 S42Γ2 S44Γ4−1

S′

21

V−2

V+3

V−4

V+3

. (A.14)

And finally,

S′

a =

S′

11 S′

12

S′

21 S′

22

(A.15)

The above procedures are designed to find the overall 2-port Sparameters (S′

a) measured

from the input and output ports of Amplifier A with Amplifier B terminated as shown in

Figure (2.3). If measuring Amplifier B with Amplifier A terminated, by following the above

steps,S′

b can be found.Γ2 andΓ4 are the two variables, which would change according to

different load situations.

57

Appendix B

Z-Parameters Conversion from 2-Port to 1-Port

Solve for 1-port Z-parameters for Amplifier B when looking from the source,Z1port:

Figure B.1: Diagram for conversion of a 2 Port Z Parameters into 1 Port.

From FigureB.1, a system of equations can be formed as

V1 =Va

i1 = ia+ ib

Va =−Z11ia−Z12ib

V2 =−Z21ia−Z22ib

V1 =V2

, (B.1)

and the matrix can be formed as below

0

i1

0

0

0

=

1 0 −1 0 0

0 0 0 1 1

0 0 1 Z11 Z12

0 1 0 Z21 Z22

−1 1 0 0 0

V1

V2

Va

ia

ib

(B.2)

Det1 = Z11+Z22−Z12−Z21 (B.3)

whereDet1 represents the determinant of matrix above,

the above matrix can be transformed into the form like below by switching the position of

the column on the left hand side of the quation with the first column on the right hand side:

0 0 −1 0 0

i1 0 0 1 1

0 0 1 Z11 Z12

0 1 0 Z21 Z22

0 1 0 0 0

(B.4)

which can be used to solve forV1 andV2, with

Det2 = i1(Z12Z21−Z22Z11) (B.5)

whereDet2 represents the determinant of matrix above.

V1 andV2can be determined by dividing the two determinants,

V1 =V2 =Det2Det1

=−i1(Z12Z21−Z11Z22)

Z22+Z11−Z12−Z21(B.6)

Z1port =V1

i1(B.7)

whereZ1port represents the 1-port representation of the 2 port is Port 1 and Port 2 are

connected together.

Z1port =−(Z12Z21−Z22Z11)

(Z22+Z11−Z12−Z21)(B.8)

59

Appendix C

Amplifier B with Input and Output Noise Source for Noise Correlation Matrix in Z

Form Calculation

For calculating noise voltagevn of Amplifier A is “turned upside down”, the following

system of equations is written as:

Figure C.1: AmplifierB with Input and Output Noise Source for Noise Correlation Matrixin Z form Calculation.

As can be seen from FigureC.1

V1 =Vn1+Va

Va = Z11ia+Z12ib

Vb = Z21ia+Z22ib

V1 =V2

V2 =Vn2+Vb

ia =−ib

(C.1)

whereZ11, Z12, Z21, andZ22 represent the Z-parameters for the 2-port network of Amplifier

A. Sinceia =−ib, then it can be shown that (C.1) becomes

Va = Z11ia−Z12ia

Vb = Z21ia−Z22ia,

(C.2)

V ≡

Va

Vb

= Z2−port

ia

−ia

, (C.3)

V1

V2

=

Vn1

Vn2

+Z2−port

ia

−ia

. (C.4)

Rearranging the equations as

ia =V1−Vn2

Z21−Z22, (C.5)

V1 =Vn1+Z11V1−Vn2

Z21−Z22−Z12

V1−Vn2

Z21−Z22, (C.6)

V1(Z21−Z22) =Vn1(Z21−Z22)−Vn2(Z11−Z12)+V1(Z11−Z12), (C.7)

V1(Z21−Z22−Z11+Z12) =Vn1(Z21−Z22)−Vn2(Z11−Z12), (C.8)

V2 =V1 =Vn1(Z22−Z21)+Vn2(Z11−Z12)

Z11+Z22−Z12−Z21. (C.9)

(C.9) can be transformed into

V2 =V1 =1

∆Z2−portZ

′

2−portVn,T , (C.10)

whereZ′

2−port=

[

Z22−Z21 Z11−Z12

]

, ∆Z2−port=

[

1 −1

]

Z2−port

[

1 −1

]T

, and

Vn,T=

[

Vn1 Vn2

]

−1 as was shown in (3.7).

61

Appendix D

Z-Representation for Combined Common Network and “One SideTerminated”

Differential Amplifier

Solve for noise correlation matrix in Z form for Amplifier B with input and output noise

source cascaded with the common network.

Figure D.1: AmplifierB with Input and Output Noise Source Cascaded with the CommonNetwork for Noise Correlation Matrix in Z Form Calculation.

From FigureD.1, following equations can be obtained

V1+Z1i =VC−ZCi (D.1)

i =VC−V1

Z1+ZC(D.2)

Vn =V1+Z1VC−V1

Z1+ZC(D.3)

Vn =ZCV1+Z1VC

Z1+ZC(D.4)

Vn =Vn[ 1 1 ]T (D.5)

And the Z-representation of the noise correlation is

Cn = 2kT×VnV†n (D.6)

where † denotes the Hermitian conjugate.

Finally from (D.4) and (D.6), the elements for the Z-representation of the noise correlation

matrix can be shown as

V2n =

|ZC|2

|Z1+ZC|2V2

1 +|Z1|

2

|Z1+ZC|2V2

C (D.7)

63

Documents

Extraction of Noise Parameters for Single-Ended Components