Unit 4.7: Transfer Functions for Circuit Analysis

Unit 4.7: Transfer Functions for Circuit Analysis#

The preparatory reading for this section is Chapter 4.4 [Karris, 2012] which discusses transfer function models of electrical circuits. We have also adapted content from 3.6 The System Function from [Hsu, 2020].

Follow along at cpjobling.github.io/eg-150-textbook/laplace_transform/7/tf_for_circuits

QR Code for this lecture

Agenda#

In this unit, we will explore how transfer functions introduced in Unit 4.6: Transfer Functions can be applied to the analysis of circuits.

Transfer Functions for Circuits
Exercises 14

% Initialize MATLAB
clearvars
cd ../matlab 
format compact

Transfer Functions for Circuits#

When doing circuit analysis with components defined in the complex frequency domain, the ratio of the output voltage $V_{out} (s)$ to the input voltage $V_{in} (s)$ under zero initial conditions is of great interest.

This ratio is known as the voltage transfer function denoted $G_{v} (s)$ :

G_{v} (s) = \frac{V_{out} (s)}{V_{in} (s)}

Similarly, the ratio of the output current $I_{out} (s)$ to the input current $I_{in} (s)$ under zero initial conditions, is called the current transfer function denoted $G_{i} (s)$ :

G_{i} (s) = \frac{I_{out} (s)}{I_{in} (s)}

In practice, the current transfer function is rarely used, so we will use the voltage transfer function denoted:

G (s) = \frac{V_{out} (s)}{V_{in} (s)}

Exercises 14#

We will work through these and demonstrate the MATLAB solutions in class.

Exercise 14.1#

Derive an expression for the transfer function $G (s)$ for the circuit shown in Fig. 68.

In this circuit $R_{g}$ represents the internal resistance of the applied (voltage) source $v_{s}$ , and $R_{L}$ represents the resistance of the load that consists of $R_{L}$ , $L$ and $C$ .

Sketch of Solution for Example 14.1#

Replace $v_{s} (t)$ , $R_{g}$ , $R_{L}$ , $L$ and $C$ by their transformed (complex frequency) equivalents: $V_{s} (s)$ , $R_{g}$ , $R_{L}$ , $s L$ and $1 / (s C)$

Use the Voltage Divider Rule to determine $V_{out} (s)$ as a function of $V_{s} (s)$

Form $G (s)$ by writing down the ratio $V_{out} (s) / V_{s} (s)$

Have a go for the next five minutes.

Worked solution for Example 14.1#

Pencast: ex6.pdf - open in Adobe Acrobat Reader.

Answer for Example 14.1#

G (s) = \frac{V_{out} (s)}{V_{s} (s)} = \frac{R_{L} + s L + 1 / s C}{R_{g} + R_{L} + s L + 1 / s C} .

Exercise 14.2#

MATLAB Example

This is based on Example 4.7 from [Karris, 2012].

This is the basis for the mini project in MATLAB Lab 5.

Compute the transfer function for the op-amp circuit shown in Fig. 69 in terms of the circuit constants $R_{1}$ , $R_{2}$ , $R_{3}$ , $C_{1}$ and $C_{2}$ .

Then replace the complex variable $s$ with $j ω$ , and the circuit constants with their numerical values and plot the magnitude

| G (j ω) | = \frac{| V_{out} (j ω) |}{| V_{in} (j ω) |}

versus radian frequency $ω$ rad/s.

Sketch of Solution for Example 14.2#

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 $R_{3}$

Sketch of Solution for Example 14.2 (continued)#

Note that the voltage at the input to the op-amp is a virtual ground

Solve for $V_{out} (s)$ as a function of $V_{in} (s)$

Form the reciprocal $G (s) = V_{out} (s) / V_{in} (s)$

Have a go for the next five minutes

Answer for Example 14.2#

(41)#

G (s) = \frac{V_{out} (s)}{V_{in} (s)} = \frac{- 1}{R_{1} ((1 / R_{1} + 1 / R_{2} + 1 / R_{3} + s C_{1}) (s C_{2} R_{3}) + 1 / R_{2})} .

Worked solution for Example 14.2#

Pencast: ex7.pdf - open in Adobe Acrobat Reader.

Sketch of Solution for Example 14.2 (continued)#

Use MATLAB to calculate the component values, then replace $s$ by $j ω$ .

Compute $| G (j ω) |$ and plot on log-linear “paper”.

The MATLAB Bit#

Set up the symbols we will be using. In this case just the Laplace complex frequency $s$ .

syms s

Now define the values of the components

R1 = 200*10^3; % 200 kOhm
R2 = 40*10^3;  %  40 kOhm
R3 = 50*10^3;  %  50 kOhm

C1 = 25*10^(-9); % 25 nF
C2 = 10*10^(-9); % 10 nF

Define the transfer function derived from analysis (Eq. (41))

den = R1*((1/R1+ 1/R2 + 1/R3 + s*C1)*(s*R3*C2) + 1/R2)

d e n = 100 s (\frac{7555786372591433 s}{302231454903657293676544} + \frac{1}{20000}) + 5

Simplify coefficients of $s$ in the denominator. Note sym2poly converts a symbolic polynomial with numerical coeficients into a MATLAB polynomial.

format long
denH = sym2poly(den)

denH = 1×3 double
   0.000002500000000   0.005000000000000   5.000000000000000

Now define the numerator

numH = -1;

Plot the frequency response

For convenience, define coefficients $a$ and $b$ :

a = denH(1)
b = denH(2)

a =      2.500000000000000e-06

b =    0.005000000000000

G (j ω) = \frac{- 1}{a ω^{2} - j b ω + 5}

w = 1:10:10000;
Hw = -1./(a*w.^2 - j.*b.*w + denH(3));

Plot $| H (j ω) |$ against $ω$ on log-lin “graph paper”.

semilogx(w, abs(Hw))
xlabel('Radian frequency w (rad/s')
ylabel('|Vout/Vin|')
title('Magnitude Vout/Vin vs. Radian Frequency')
grid

../../_images/0e28b7317f9e929757bbb0345a7aeb8c66a86108177dee39641cfc0a25608acb.png

Note

Note that this is a low-pass filter. Sinusoids at low frequencies are passed with a gain of 0.2. For frequencies above around 100 rad/s, the filter starts to increase the attenuation of the passed signal. At 10,000 rad/s, the attenuation is 1/10 of the attenuation at 1,000 rad/s.

MATLAB Solutions#

For convenience, single script MATLAB solutions to the examples are provided and can be downloaded from the accompanying MATLAB folder.

Exercise 14.2 [example_14.2.mlx]

open example_14_2

Summary#

In this unit, we will explored how transfer functions introduced in Unit 4.6: Transfer Functions can be applied to the analysis of circuits.

Transfer Functions for Circuits
Exercises 14

Unit 4.7: Take Away#

The ratio of the output voltage $V_{out} (s)$ to the input voltage $V_{in} (s)$ under zero initial conditions is of great interest. We call this ratio the voltage transfer function

G_{v} (s) = \frac{V_{out} (s)}{V_{in} (s)}

We can consider other ratios such as the current transfer function

G_{i} (s) = \frac{I_{out} (s)}{I_{in} (s)}

but in practice this is rarely used.

Next time#

We explore the facilties provided by other toolboxes in MATLAB, most notably the Control Systems Toolbox and the simulation tool Simulink in Unit 4.8: Computer-Aided Systems Analysis and Simulation. We will also look at some of the problems you have studied in EG-152 Analogue Design hopefully confirming some of the results you have obbserved in the lab.

Unit 4.8: Computer-Aided Systems Analysis and Simulation

References#

[Hsu20]

Hwei P. Hsu. Schaums outlines signals and systems. McGraw-Hill, New York, NY, 2020. ISBN 9780071634724. Available as an eBook. URL: https://www.accessengineeringlibrary.com/content/book/9781260454246.

[Kar12] (1,2)

Steven T. Karris. Signals and systems with MATLAB computing and Simulink modeling. Orchard Publishing, Fremont, CA., 2012. ISBN 9781934404232. Library call number: TK5102.9 K37 2012. URL: https://ebookcentral.proquest.com/lib/swansea-ebooks/reader.action?docID=3384197.

Unit 4.7: Transfer Functions for Circuit Analysis

Contents

Unit 4.7: Transfer Functions for Circuit Analysis#

Agenda#

Transfer Functions for Circuits#

Exercises 14#

Exercise 14.1#

Sketch of Solution for Example 14.1#

Worked solution for Example 14.1#

Answer for Example 14.1#

Exercise 14.2#

Sketch of Solution for Example 14.2#

Sketch of Solution for Example 14.2 (continued)#

Answer for Example 14.2#

Worked solution for Example 14.2#

Sketch of Solution for Example 14.2 (continued)#

The MATLAB Bit#

MATLAB Solutions#

Summary#

Unit 4.7: Take Away#

Next time#

References#