Reference no: EM131377901

Assignment: Signals and Systems

Class Room Activity: Transfer Function Analysis of Continuous Systems in s Domain Using MATLAB

Objective of the lab experiment:

The objective of this experiment is to create continuous (s domain) transfer functions in MATLAB and explore how they can be manipulated to extract relevant data.

We shall first present an example of how MATLAB is used for s (Laplace) domain analysis, and then the student shall be required to perform specified analysis on a given circuit.

Equipment list:

MATLAB version 7.0 or higher

Software needed: Sdomainanalysis.m : This file is available in Doc Sharing.

Theory

Brief Explanation of Creating Transfer Functions of Circuits in Laplace Domain

We shall illustrate suchanalysis by calculating theTFofa filter.

Aside from resistors, there are two other common analog components that are found in circuits: capacitors and inductors. These are reactive components; that is, they are capable of storing (and giving back) energy, in contrast to resistors, which can only convert electrical energy into heat but cannot store it. The volt ampere relationship of the two reactive components are given by

v(t)=L(di(t)/dt)
i(t)=C((dv(t))/dt)

where L is the inductance and C is the capacitance. This is in contrast with the resistive case, where v(t) = i(t) R, (Ohm's law), a linear and simple algebraic relationship without any derivative terms.

If one of these reactive components is located in a circuit, then a differential equation would need to be solved in order to see their output over time. The presence of two reactive components would mean a second-order differential equation; three would mean a third order, and so on. As the number of reactive components increases, generating a solution by hand is very tedious, if not impossible. In this case, computer software such as MATLAB is utilized to generate the desired answer. This lab will explore how to convert differential equations into a MATLAB-compatible form, using Laplace transforms and what MATLAB commands can do to find information of interest about circuits and systems.

In order to model the previous derivative relationships in MATLAB, Laplace transform operations will have to be performed on the volt ampere relationship of capacitor and inductor.

V(s)=L*I(s)*s
I(s)=C*V(s)*s

This is true where V(s) and I(S) are the voltages across and the current through the reactive components respectively and s is the Laplace operator , a complex number that we write as σ+j ω. The impedance of the reactive components can now be defined in Laplace domain as algebraic ratios (rather than as derivative relations).

V(s)/I(s) =Ls=Z_L
V(s)/I(s) =1/Cs=Z_c

Laplace Domain Transfer Function Analysis Using MATLAB

We illustrate such analysis using the example of an analog circuit. Given the circuit below, find the transfer function. Note that the transfer function is defined as Vout / Vin. In order to find the TF, we apply Kirchhoff's laws or use the voltage divider rule, with the circuit impedances represented in Laplace domain. In this example we will apply the voltage divider rule.

Figure 1. An analog filter that uses no active components (such as op amps, etc.)

This is not a simple circuit. So we must break it down into something easier to calculate. Aim to use the voltage divider rule and break the circuit into the following components:

Figure 2. A symbolic representation of two complex impedances connected as shown

Therefore, the transfer function is defined by
transfer function=V_out/V_in =Z_2/(Z_2+Z_1 ).

In order to solve this equation, the circuit must be broken down even further. If we apply parallel and series impedance rules, we can get the following. Note that in Z1, R and L are in series, while C is parallel, and the impedance of a resistor is just the resistance because it is not a reactive component.

Z_1=((Z_L+Z_R )*Z_C)/((Z_L+Z_R )+Z_C )=((Ls+R)*1/Cs)/(Ls+R+1/Cs)

Substituting numerical values of L and C, we get the equation below.

Z_1=(s/s+R/s)/(s+R+1/s)*s/s=(s+R)/(s^2+Rs+1)

Substituting numerical value of R, we get the equation below.

Z_1=(s+0.1)/(s^2+0.1s+1)

All three components of Z2 are in series, so we add them together.

Z_2=Z_L+Z_R+Z_C=Ls+R+1/Cs=sL+sCR+1/Cs

Substirtuting numerical values of R, L, and C, we get the following.

Z_2=(s^2+0.1s+1)/s

Now combine the impedances to obtain the transfer function using the voltage divider rule.

V_out/V_in =Z_2/(Z_2+Z_1 )=((s^2+0.1s+1)/s)/((s^2+0.1s+1)/s+(s+0.1)/(s^2+0.1s+1))*s/s=(s^2+0.1s+1)/(s^2+0.1s+1+s(s+0.1)/(s^2+0.1s+1))*(s^2+0.1s+1)/(s^2+0.1s+1)=(s^2+0.1s+1)^2/((s^2+0.1s+1)^2+s^2+0.1s)

Expand and simplify to get the following.

V_out/V_in =(s^4+0.2s^3+2.01s^2+0.2s+1)/(s^4+0.2s^3+3.01s^2+0.3s+1)

Now that we have our transfer function, we must enter it into MATLAB. Because a transfer function is a special type of function, we must use the proper syntax to let MATLAB know that we want a transfer function. A special command called tf is used to generate our transfer function. The syntax for the tf command is as follows.

tf(num,den)

The num and den items are arrays that represent the coefficients of the polynomials in the numerator and denominator respectively, in descending order (i.e., starting with the coefficient of the highest power of s). Note also that if a particular power of s is missing in the polynomial, its coefficient must be entered as 0. The following MATLAB code will generate a desired transfer function for our circuit.

num = [1,0.2,2.01,0.2,1];
den = [1,0.2,3.01,0.3,1];
circuitTF = tf(num,den)

This will generate the following transfer function.

Transfer function:

s^4 + 0.2 s^3 + 2.01 s^2 + 0.2 s + 1
------------------------------------
s^4 + 0.2 s^3 + 3.01 s^2 + 0.3 s + 1

Having obtained the transfer function, we can now use the power of MATLAB to study its properties-among them step and impulse response-as well as a Bode plot and pole zero constellation.

You can study the supplied MATLAB file sdomainanalysis.m to see how the whole process can be automated.

Example Commands (Impulse, Step, Bode, Pzmap)

For obtaining key properties of the transfer function, we use certain special commands of MATLAB. Some commands to note are impulse, step, bode, andpzmap. These operations will tell us a number of properties of the circuit in both time and frequency domain.

The key word impulse is used to obtain the impulse response of a transfer function. Note that because the Laplace transform of a unit impulse is unity, the impulse response is simply the inversion of the transfer function back to timedomain.

The syntax for this command is impulse(name of the variable representing the transfer function). In our case, it becomes impulse(circuitTF).

The plot below shows the graph presented by MATLAB in response to this command.

Figure 1. Impulse response of circuit as computed by using the impulse command in MATLAB

The key word step is used to obtain the unit step response of a transfer function. Note that because the Laplace transform of a unit step is (1/s), the step response is simply the inversion of the transfer function back to timedomain after multiplying it with 1/s.

The syntax for this command is step(name of the variable representing the transfer function). In our case, it becomes step(circuitTF).

The graph presented by MATLAB in response to this command is shown below.

Figure 2. Step response of circuit as computed by using the step command in MATLAB

The key word pzmap is used to obtain the pole zero constellation associated with the transfer function. The syntax for this command is pzmap(name of the variable representing the transfer function). In our case, it becomes pzmap(circuitTF).

The graph presented by MATLAB in response to this command is shown below.

Figure 3. Pole zero constellation of the circuit transfer function as computed by using the pzmap command in MATLAB.Note that both pairs of complex zeroes are located at the same place in the s plane.

The key word bode is used to obtain the Bode plot of a transfer function. Note that the frequency is plotted in radians/sec and amplitude is in decibels.

The syntax for this command is bode (name of the variable representing the transfer function). In our case, it becomes bode(circuitTF).

The graph returned by MATLAB in response to this command is shown below.

Figure 4. Bode plot or frequency response(FR) of the circuit as computed by using the bode command in MATLAB. Note that the frequency scale is in radians.

Note that the Bode plot is represented in radians/second. Most people are more comfortable with hertz instead, so here is a trick to get the Bode plot to display in hertz.

How to Plot Frequency Response in Hertz Rather than Radians

First, generate an array of frequencies in radians to use as a scale for both magnitude and phase plots. This can be done using the colon operator of MATLAB. It can be named anything, but this example will use the letter W as the variable.

W = [0.1:0.01:10];

This command creates a row vector (a 1 x N matrix or a matrix of 1 row and N columns) of frequencies, W, which starts at 0.1 radians/sec and increments with astep size of 0.01 until it reaches the ending upper value of 10 radians/sec.

The reason we chose this range is that the most interesting part of the Bode plot of our circuit occurs between 0.1 rad/sec and 10 rad/sec, and we chose a reasonably small step size to give the graph a good resolution, 0.01 rad/sec. Now, we want to extract the data from the Bode plot and manipulate it. This can be accomplished by putting an array before the bode command. We use MAG and PHASE as our variables, but any variable name that is not a key word in MATLAB can be used instead.

[MAG,PHASE] = bode(circuitTF,W);

This will extract the data for the magnitude and phase parts of the graph and store them in the variable MAG and PHASE. Note that the outputs are three-dimensional matrices. In order to change the format to a more manageable one, type in the code below.

MAG1D = MAG(:);
PHASE1D = PHASE(:);

When the data are extracted using this method, magnitude is in a linear form. Standard Bode plots have the magnitude displayed using the decibel scale. Use the normal format to change magnitude to decibels.

MAG1DB = 20*log10(MAG1D);

The final thing of note is that the reference array W is interpreted to be in radians per second. In order to change that to frequency in hertz, we use the frequency and radian relationship.

Wfreq = W/(2*pi);

Now use the subplot command to generate a Bode plot with frequency on the X axis.

hold on
subplot(2,1,1), plot(Wfreq,MAG1DB)
title('Bode plot in frequency')
ylabel('Magnitude (dB)')
subplot(2,1,2), plot(Wfreq,PHASE1D)
xlabel('Frequency (Hz)')
ylabel('Phase (deg)')
hold off

This will generate the figure below.

Figure 4. Bode plot or frequency response of the circuit as computed by using the bode command in MATLAB with further modifications to plot the frequency response in hertz rather than radians.

Procedure for completing the lab requirements:

An analog circuit is given below . Manually derive the transfer function of this circuit shown below using the voltage divider rule. Express the transfer function with the coefficients of the highest power of s unity in both the numerator and denominator. Show your various steps by entering them on this page using the equation editor of Word. Then, using MATLAB, determine its step and impulse response, pole zero map,and Bode plot with frequency scale in hertz.Paste the answers below. Be sure to give the legend of the figure describing what the figure is showing and label the axis, including the units as was done in the figures above.

Now answer the questions below.

Why does the unit step response of the circuit given in the theory part of this lab settle out at 1 and not at 0? Justify and briefly explain your answer based on the transfer function derived in the theory part. (Hint: it has to do with the DC gain of the circuit and in circuit terms with the impedance of capacitor and inductor at DC.) What kind of a filter does this circuit represent (bandpass, bandstop, low pass, high pass)?
When s goes to infinite it goes to one it's the transfer functions

Why does the unit step response of the circuit given in the procedure part of the lab go to zero rather than unity? Briefly explain and justify your answer.

What kind of filter (bandpass, bandstop, low pass, high pass) does the circuit shown in the procedure part represents? Briefly justify your answer.

Why does swapping the arms of the filter change the type of filter that is realized?

What is the numerical value of impedance of an ideal inductor at DC and as frequency approaches infinity?

What is the numerical value of impedance of an ideal capacitor at DC and as frequency approaches infinity?

Why does the pole zero map shown above have only two zeroes, while the transfer function has four zeroes? (Hint: use the roots command of MATLAB to find out.)

Attachment:- Lab.rar