Frequency response analysis

ABSTRACT:This is one of a series of white papers on systems modeling, analysis and control,

prepared by control system. In control system there are a number of generic systems

and methods which are encountered in all areas of industry and technology.There white

papers aim to explain these important systems and methods in straight forward

terms.the white paper describes what makes a particular types of system ,how it works

and then demonstrate how to control or use it. The frequency response of a system can

be viewed two different ways: via the Bode plot or via the Nyquist diagram. Both

methods display the same information; the difference lies in the way the information is

presented. We will study both methods in this tutorial.

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Frequency response analysis

CONTENTS:1) Abstract …………………………………………………..12) Introduction ………………………………………………33) Types of responses ……………………………………..44) Impulse response

a) Impulse fuctionb) Frequency responsec) Estimating and plotting ………………………………7d) Advantages

e) Disadvantages 5) Frequency response theory…………………………… 10 a) Using laplace transfer function b) without using laplace transfer function 6) frequency response system characteristics …………13 7) polar graphs ……………………………………………. 14 8) Nyquist plots ……………………………………………..15 a) Basic rules for constructing N. P. b) Relative stability assessment using the nyquist plots

Nyquist stability criteriaSolved examples

9) Applications …………………………………………….18 10) References …………………………………………….19

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Frequency response analysis

INTRODUCTIONFrequency response is a measure of magnitude and phase of the output as a function

of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into

a system at a given frequency, a linear system will respond at that same frequency with

a certain magnitude and a certain phase angle relative to the input. Also for a linear

system, doubling the amplitude of the input will double the amplitude of the output. In

addition, if the system is time-invariant, then the frequency response also will not vary

with time.

Two applications of frequency response analysis are related but have different

objectives. For an audio system, the objective may be to reproduce the input signal with

no distortion. That would require a uniform (flat) magnitude of response up to

the bandwidth limitation of the system, with the signal delayed by precisely the same

amount of time at all frequencies. That amount of time could be seconds, or weeks or

months in the case of recorded media. In contrast, for a feedback apparatus used to

control a dynamical system, the objective is to give the closed-loop system improved

response as compared to the uncompensated system. The feedback generally needs to

respond to system dynamics within a very small number of cycles of oscillation (usually

less than one full cycle), and with a definite phase angle relative to the commanded

control input. For feedback of sufficient amplification, getting the phase angle wrong can

lead to instability for an open-loop stable system, or failure to stabilize a system that is

open-loop unstable. Digital filters may be used for both audio systems and feedback

control systems, but since the objectives are different, generally the phase

characteristics of the filters will be significantly different for the two applications.

TRANSFER FUNCTION

A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions and equilibrium point to be zero. If we have an input function of X(s), and an output function Y(s), we define the transfer function H(s) to be:

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Frequency response analysis

IMPULSE RESPONSE

For comparison, we will consider the time-domain equivalent to the above input/output relationship. In the time domain, we generally denote the input to a system as x(t), and the output of the system as y(t). The relationship between the input and the output is denoted as the impulse response, h(t).

We define the impulse response as being the relationship between the system output to its input. We can use the following equation to define the impulse response:

Impulse Function

It would be handy at this point to define precisely what an "impulse" is. The Impulse Function, denoted with δ(t) is a special function defined piece-wise as follows:

The impulse function is also known as the delta function because it's denoted with the

Greek lower-case letter δ. The delta function is typically graphed as an arrow towards

infinity, as shown below:

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Frequency response analysis

Impulse signal

It is drawn as an arrow because it is difficult to show a single point at infinity in any other

graphing method. Notice how the arrow only exists at location 0, and does not exist for

any other time t. The delta function works with regular time shifts just like any other

function. For instance, we can graph the function δ(t - N) by shifting the function δ(t) to

the right, as such:

Delayed impulse signal

An examination of the impulse function will show that it is related to the unit-step

function as follows:

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Frequency response analysis

And

The impulse function is not defined at point t = 0, but the impulse response must always

satisfy the following condition, or else it is not a true impulse function:

The response of a system to an impulse input is called the impulse response. Now, to

get the Laplace Transform of the impulse function, we take the derivative of the unit

step function, which means we multiply the transform of the unit step function by s:

Frequency Response

The Frequency Response is similar to the Transfer function, except that it is the

relationship between the system output and input in the complex Fourier Domain, not

the Laplace domain. We can obtain the frequency response from the transfer function,

by using the following change of variables:

Because the frequency response and the transfer function are so closely related,

typically only one is ever calculated, and the other is gained by simple variable

substitution. However, despite the close relationship between the two representations,

they are both useful individually, and are each used for different purposes.

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Frequency response analysis

Estimation and plotting

Frequency response of a low pass filter with 6 dB per octave or 20 dB per decade

Estimating the frequency response for a physical system generally involves exciting the

system with an input signal, measuring both input and output time histories, and

comparing the two through a process such as the Fast Fourier Transform(FFT). One

thing to keep in mind for the analysis is that the frequency content of the input signal

must cover the frequency range of interest or the results will not be valid for the portion

of the frequency range not covered.

The frequency response of a system can be measured by applying a test signal, for

example:

applying an impulse to the system and measuring its response (see impulse response)

sweeping a constant-amplitude pure tone through the bandwidth of interest and

measuring the output level and phase shift relative to the input

applying a signal with a wide frequency spectrum (for example digitally-

generatedmaximum length sequence noise, or analog filtered white noise equivalent,

like pink noise), and calculating the impulse response by deconvolution of this input

signal and the output signal of the system.

The frequency response is characterized by the magnitude of the system's response,

typically measured in decibels (dB) or as a decimal, and the phase, measured

in radians or degrees, versus frequency in radians/sec or Hertz (Hz).

These response measurements can be plotted in three ways: by plotting the magnitude

and phase measurements on two rectangular plots as functions of frequency to obtain

a Bode plot; by plotting the magnitude and phase angle on a single polar plot with

frequency as a parameter to obtain a Nyquist plot; or by plotting magnitude and phase

on a single rectangular plot with frequency as a parameter to obtain a Nichols plot.

For audio systems with nearly uniform time delay at all frequencies, the magnitude

versus frequency portion of the Bode plot may be all that is of interest. For design of

control systems, any of the three types of plots [Bode, Nyquist, Nichols] can be used to

infer closed-loop stability and stability margins (gain and phase margins) from the open-

loop frequency response, provided that for the Bode analysis the phase-versus-

frequency plot is included.

The frequency response is a representation of the system's open loop response to

sinusoidal inputs at varying frequencies.  The output of a linear system to a sinusoidal

input is a sinusoid of the same frequency but with a different amplitude and phase.  The

frequency response is defined as the amplitude and phase differences between the

input and output sinusoids.   The open-loop frequency response of a system can be used to predict the behaviour of the closed-loop system . 

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Frequency response analysis

The frequency response method may be less intuitive than other methods.    However, it

has certain advantages, especially in real-life situations such as modeling transfer

functions from physical data.    The frequency response of a system can be viewed two

different ways: via the Bode plot or via the Nyquist diagram.   Both methods display the

same information; the difference lies in the way the information is presented.    

To plot the frequency response, it is necessary to create a vector of frequencies

(varying between zero (DC) and infinity) and compute the value of the system transfer

function at those frequencies.  If G(s) is the open loop transfer function of a system

and ω is the frequency vector, we then plot G(j.ω) vs. ω.  Since G(j.ω) is a complex

number, we can plot both its magnitude and phase (the Bode plot) or its position in the

complex plane (the Nyquist plot).   Harmonic response methods can be completed

using algebraic manipulation and can also be completed by testing actual systems.

Advantages

The tests involve measurements under steady state conditions which

are more simpler to analyse compared to measurements of transient

responses

The tests are made on open loop systems which are not subject

instability problems

The results give convenient access to control system order, gain,

error constants, resonant frequencies etc

Disadvantages

It is not always easy to deduce transient response characteristics from

a knowledge of the frequency response

In completing tests it can be difficult to generate low-frequency signals

and obtain the necessary measurements. Normal frequencies of 0,1

to 10 Hz are used. However for process control frequencies of one

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Frequency response analysis

cycle over several hours may be required while for fluid servos

frequencies of > 100 Hz may be experienced

Frequency Response Theory Using Laplace Transfer Functions

Consider a basic open loop transfer function..

The system response to a sinusoidal input is considered by providing an input signal =

a. sin(ω t)

G( jω) and G(-jω) are complex values of the Open loop transfer function with s replaced

by jω.

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Frequency response analysis

M the Magnification factor (Magnitude ratio) =y(t) / r(t).  This is a function of the frequency ω.The frequency of y(t) is the same as (r(t) but the phase is advanced by α. 

A frequency response analysis (without using Laplace Transforms )

The response includes a transient response (complimentary function) and a steady

state (forced) response(particular integral).   In frequency response analysis only the

steady state response needs to be considered.   The system is linear and the output will

thus be sinusoidal at a frequency ω.

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Frequency response analysis

If G (j ω ) = -1 ( unity gain at a phase shift of 180 ) it can be proved that the closed loop

system with unity feedback will be unstable.

It can be proved that the value of ω for a value of |G(jω)| = 1 =

Phase difference between input and output =

The Nyquist plot is simply a plot of G(jω) on an argand diagram for a range of

frequencies from

Frequency Response Systems Characteristics

In closed loop system , if at some frequency the signal undergoes no change in

amplitude but is shifted in phase by 180 deg. ( π ) the system will be unstable.   The

feedback signal arriving at the summing point will reinforce the input signal and the

progressive cumulative input will result in a theoretically infinite output.  The feedback

signal will be effectively a 100%positivefeedback..It can be therefore concluded that in the open loop version of the above system if the signal is modified resulting in unity amplitude change and a change in a phase shift

of π then the system will be unstable..

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Frequency response analysis

In an open loop positioning control system a D.C. test signal (ω = 0) will result in the

drive running continuously resulting in the output position increasing without limit.  A

high frequency input signal would produce zero output because the inertia of system

would prevent oscillatory movement.

In velocity controlled systems the output speed is (ideally) proportional to the applied

signal.- the drive acts as and integrator and the phase lag at ω =0is p. /2. 

In torque controlled systems the output acceleration is (ideally) proportional to the

applied signal.- the drive acts as and double integrator and the phase lag at ω = 0 is π .

 

Polar graphsThere are a number of polar graph options for studying control systems including the

nyquist, inverse polar plot and the nichols plot.  The nyquist open loop polar plot

indicates the degree of stability, and the adjustments required and provides stability

information for systems containing time delays.  Polar plots are not used exclusively

because,without powerful computing facilities, they can be difficult to generate at a

detailed level and they do not directly yield frequency values.

The Nyquist plot is obtained by simply plotting a locus of imaginary(G(j ω)) versus

Real(G(j ω)) at the full range of frequencies from ( - ¥ to + ¥ ) It is very easy to produce

nyquist plots by hand or by using proprietary software packages such as Matlab.   Links

below show how bode and nyquist plots can be produced using Excel and using

Mathcad.  The plots below have been produced in minutes using Mathcad..

Nyquist plot

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Frequency response analysis

Basic Rules for constructing Nyquist plotsIn control systems a transfer function to be assessed is often of the form

This transfer function is modified for frequency response analysis by replacing the s

with jω

Assuming the function is proper and n > m..he Nyquist plot will have the following

characteristics. Crude plots to be may be produced relatively easily using these

characteristics.

Asymptotic behavior.. For n - m > 1, the Nyquist plot approaches the

origin at an asymptotic angle of -(n-m) p/2...

Assuming G(s) = K(s)/s k.  For k poles at zero, the Nyquist plot comes

in from infinity at an angle of -(n-m) p/2

In a system with no OL zeros, the plot of G(jω) will decrease

monotonically as ω rises above the level of the largest imaginary part

of the poles; This will also be true for large enough ω even in the

presence of zeros.(Ref plot 1 below).

The plot will cross the imaginary axis when Real G(jω) =0 and will

cross the real axis when Imaginary G(jω) = 0,   ( for crossing of the

negative real axis use Arg- G(jω)= p )

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Frequency response analysis

Relative stability assessments Using the Nyquist Plot:As identified in the page on frequency response Frequency response The nyquist plots

are based on using open loop performance to test for closed loop stability.  The system

will be unstable if the locus has unity value at a phase crossover of 180 o ( p ). 

Two relative stability indicators "Gain Margin" and "Phase Margin" may be determined

from the suitable Nyquist Plots.   The degree of gain margin is indicated as the amount

the gain is less than unity when the plot crosses the 180 o axis (Phase crossover). The

phase margin is the angle the phase is less than 180 o when the gain is unity.  The

values are generally identified by use of Bode plots

The phase margin is clearly illustrated below

Nyquist Stability Criterion:

In the nyquist plots below the area covered to the right of the locus(shaded) is the Right

Hand Plane (RHP)

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Frequency response analysis

A closed loop control system is absolutely stable if the roots of the characteristic

equation have negative real parts.  This means the poles of the closed loop transfer

function, or the zeros of the denominatior ( 1 + GH(s)) of the closed loop tranfer

function, must lie in the (LHP).   The nyquist stability criterion establishes the number of

zeros of (1 + GH(s) in the RHP directly from the Nyquist stability plot of GH(s) as

indicated below.

The closed loop control system whose open loop transfer function is GH(s) is stable

only if..

N = -Po ≤ 0

where 

1) P o = the number of G(s) poles in the RHP ³ 0 

2) N = total number of CW encirclements of the (-1,0) in the G(s) plane. 

If N > 0 the number of zeros (Z o) in the RHP is determined by Z o = N + P o

If N ≤ 0 the (-1,0) point is not enclosed by the nyquist plot. 

If N ≤ and P 0 then the system is absolutely stable only if N = 0.   That is if and only if the

(-1,0) point does not lie in the shaded region..

Considering the LH plot above of 1/s(s+1). The (-1,0) point is not in the RHP therefore

N<= 0. The poles are at s =0, and s=-1, both outside of the RHP and therefore P o = 0. 

Thus N = -P o = 0 and the system is therefore stable. 

Considering the RH plot above of 1/s(s-1).   The (-1,0) point is enclosed in the RHP and

therefore N > 0 (N= 1).  The poles of GH are at s= 0 and s = +1 .   S= +1 is in the RHP

and therefore P o = 1.

N ¹ - P o Indicating that this system is unstable.. 

There are Z o = N + P o zeros of 1+GH in the RHP.

Nyquist Plots exampales:

Nyquist Plots A number of typical nyquist plots are shown below to illustrate the various

shapes. 

Plot 1..... 1 /(s + 2)

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Frequency response analysis

 

1 /(s + 2)

Note that G(i0) = 0.5 and as ω increases to ¥ the plot approaches zero along the

negative locus.

G(jω) moves from 0 to 0.5 as ω - ¥ to 0

G(j ¥) = 0

The asymptotic angle approaching 0 is = -90 o and

Plot 2.....1 /(s 2 + 2s + 2)

 

1 /(s 2 + 2s + 2)

The zero-frequency behaviour is:G(j0) =0.5

G(j ¥) = 0

The asymptotic angle is = -180 o

Plot 3.....s(s+1) /(s 3 + 5.s 2 +3.s + 4 )

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Frequency response analysis

 

s(s+1) /(s 3 + 5.s 2 +3.s +4 )

Plot 4.....(s+1) /[(s+2)(s+3)]

 

((s+1) /[(s+2)(s+3)]

Plot 5.....1 /s(s-1).. an unstable regime

(1 /s (s-1)

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Frequency response analysis

Applications

In electronics this stimulus would be an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cable, twisted-pair cable, video switching equipment, wireless communications devices, and antenna systems. Infrasonic frequency response measurements include earthquakes andelectroencephalography (brain waves).

Frequency response requirements differ depending on the application.[2] In high fidelityaudio, an amplifier requires a frequency response of at least 20–20,000 Hz, with a tolerance as tight as ±0.1 dB in the mid-range frequencies around 1000 Hz, however, in telephony, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.

Frequency response curves are often used to indicate the accuracy of electronic components or systems. When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.

Once a frequency response has been measured (e.g., as an impulse response), providing the system is linear and time-invariant, its characteristic can be approximated with arbitrary accuracy by a digital filter. Similarly, if a system is demonstrated to have a poor frequency response, a digital or analog filter can be applied to the signals prior to their reproduction to compensate for these deficiencies.

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Frequency response analysis

References

1. ↑ 1.0 1.1 1.2 1.3 1.4 Riggs, J. B., & Karim, M. N. (2006). Chemical and Bio-Process Control. Lubbock, TX, USA: Ferret Publishing.

2. ↑ Perry, Robert H., Don W. Green, and James O. Maloney. Perry's Chemical Engineers' Handbook. New York, NY: McGraw-Hill, 1973.

3. ↑ Stephanopoulos, G. (1984). Chemical Process Control: An Introduction to Theory and Practice. Englewood Cliffs, NJ, USA: Prentice-Hall, Inc.

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