11/02/2023, 10:54 Unit 3.
4: Transfer Functions — EG-247 Signals and Systems
Unit 3.4: Transfer Functions
Contents
Colophon
Agenda
Transfer Functions for Circuits
Examples
Using Transfer Functions in MATLAB for System Analysis
Using Transfer Functions in Simulink for System Simulation
Reference
The preparatory reading for this section is Chapter 4.4 [Karris, 2012] which discusses transfer
function models of electrical circuits.
Colophon
An annotatable worksheet for this presentation is available as Worksheet 7.
The source code for this page is laplace_transform/4/transfer_functions.ipynb.
You can view the notes for this presentation as a webpage (HTML).
This page is downloadable as a PDF file.
Agenda
Transfer Functions
A Couple of Examples
Circuit Analysis Using MATLAB LTI Transfer Function Block
Circuit Simulation Using Simulink Transfer Function Block
% Matlab setup
clear all
cd ../matlab
pwd
format compact
ans =
'/Users/eechris/code/src/github.com/cpjobling/eg-247-
textbook/laplace_transform/matlab'
Transfer Functions for Circuits
When doing circuit analysis with components defined in the complex frequency domain, the
ratio of the output voltage to the input voltage
Vout(s) under zero initial conditions is of
Vin(s)
great interest.
This ratio is known as the voltage transfer function denoted : Gv(s)
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
Vout(s)
Gv(s) =
Vin(s)
Similarly, the ratio of the output current to the input current
Iout(s) Iin(s) under zero initial
conditions, is called the cuurent transfer function denoted : Gi(s)
Iout(s)
Gi(s) =
Iin(s)
In practice, the current transfer function is rarely used, so we will use the voltage transfer
function denoted:
Vout(s)
G(s) =
Vin(s)
Examples
See Worksheet 7 for the worked solutions to the examples. We will work through these in
class. Here I’ll demonstrate the MATLAB solutions.
Example 6
Derive an expression for the transfer function for the circuit below. In this circuit
G(s) Rg
represents the internal resistance of the applied (voltage) source , and represents the vs RL
resistance of the load that consists of , and . RL L C
Sketch of Solution
Replace , , , and by their transformed (complex frequency) equivalents:
vs(t) Rg RL L C
, , , and
Vs(s) Rg RL sL 1/(sC)
Use the Voltage Divider Rule to determine as a function of Vout(s) Vs(s)
Form by writing down the ratio
G(s) Vout(s)/Vs(s)
Worked solution.
Pencast: ex6.pdf - open in Adobe Acrobat Reader.
Answer
Vout(s) RL + sL + 1/sC
G(s) = = .
Vs(s) Rg + RL + sL + 1/sC
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
Example 7
Compute the transfer function for the op-amp circuit shown below in terms of the circuit
constants , , , and .
R1 R2 R3 C1 C2
Then replace the complex variable with , and the circuit constants with their numerical
s jω
values and plot the magnitude
|Vout(jω)|
|G(jω)| =
|Vin(jω)|
versus radian frequency rad/s. ω
Sketch of Solution
Replace the components and voltages in the circuit diagram with their complex
frequency equivalents
Use nodal analysis to determine the voltages at the nodes either side of the 50K resistor
R3
Note that the voltage at the input to the op-amp is a virtual ground
Solve for as a function of
Vout(s) Vin(s)
Form the reciprocal G(s) = Vout(s)/Vin(s)
Use MATLAB to calculate the component values, then replace by . s jω
Plot on log-linear “paper”:
|G(jω)|
Worked solution.
Pencast: ex7.pdf - open in Adobe Acrobat Reader.
Answer
Vout(s) −1
G(s) = = .
Vin(s) R1 ((1/R1 + 1/R2 + 1/R3 + sC1) (sC2R3) + 1/R2)
The Matlab Bit
See attached script: solution7.m.
Week 3: Solution 7
syms s;
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
R1 = 200*10^3;
R2 = 40*10^3;
R3 = 50*10^3;
C1 = 25*10^(-9);
C2 = 10*10^(-9);
den = R1*((1/R1+ 1/R2 + 1/R3 + s*C1)*(s*R3*C2) + 1/R2);
simplify(den)
ans =
100*s*((7555786372591433*s)/302231454903657293676544 + 1/20000) + 5
Simplify coefficients of s in denominator
format long
denG = sym2poly(ans)
denG =
0.000002500000000 0.005000000000000 5.000000000000000
numG = -1;
Plot
For convenience, define coefficients and : a b
a = denG(1);
b = denG(2);
−1
G(jω) =
2
aω − jbω + 5
w = 1:10:10000;
Gs = -1./(a*w.^2 - j.*b.*w + denG(3));
Plot
semilogx(w, abs(Gs))
xlabel('Radian frequency w (rad/s')
ylabel('|Vout/Vin|')
title('Magnitude Vout/Vin vs. Radian Frequency')
grid
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
Using Transfer Functions in MATLAB for System
Analysis
Please use the file tf_matlab.m to explore the Transfer Function features provide by MATLAB.
Open the file as a Live Script to see a nicely formatted document.
Using Transfer Functions in Simulink for System
Simulation
The Simulink transfer function (Transfer Fcn) block implements a transfer function
The transfer function block represents a general input output function
N (s)
G(s) =
D(s)
and is not specific nor restricted to circuit analysis.
It can, however be used in modelling and simulation studies.
Example
Recast Example 7 as a MATLAB problem using the LTI Transfer Function block.
For simplicity use parameters , and
R1 = R2 = R3 = 1 Ω F. C1 = C2 = 1
Calculate the step response using the LTI functions.
Verify the result with Simulink.
The Matlab solution: example8.m
MATLAB Solution
From a previous analysis the transfer function is:
Vout −1
G(s) = =
Vin R1 [(1/R1 + 1/R2 + 1/R3 + sC1)(sR3C2) + 1/R2]
so substituting the component values we get:
Vout −1
G(s) = =
2
Vin s + 3s + 1
We can find the step response by letting vin(t) = u0(t) so that Vin(s) = 1/s then
−1 1
Vout(s) = .
2
s + 3s + 1 s
We can solve this by partial fraction expansion and inverse Laplace transform as is done in the
text book with the help of MATLAB’s residue function.
Here, however we’ll use the LTI block.
Define the circuit as a transfer function
G = tf([-1],[1 3 1])
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
G =
-1
-------------
s^2 + 3 s + 1
Continuous-time transfer function.
step response is then:
step(G)
Simples!
Simulink model
See example_8.slx
open example_8
Result
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11/02/2023, 10:54 Unit 3.4: Transfer Functions — EG-247 Signals and Systems
Let’s go a bit further by finding the frequency response:
bode(G), grid
Reference
See Bibliography.
By Dr Chris P. Jobling
© Copyright Swansea University (2019-2022).
This page was created by Dr Chris P. Jobling for Swansea University .
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