Question 12.6: For this example, use the transfer function introduced in Ex...

For this example, use the transfer function introduced in Example 12.5:

T(s) =\frac{10}{2s + 1} .

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Table 12.3. Summary of MATLAB sinusoidal transfer-function-related commands

Syntax Description Returns Comments
R=Evalfr (sys, w) Evaluates frequency response of a TF object for a single frequency w Single complex number that is the value of the TF at frequency w Appropriate for quickly checking the response at one or a few frequencies
H= Freqresp (sys, w) Evaluates the frequency response of a TF object over a grid of frequencies (contained in the vector w) A vector of complex numbers (H) that are the TF evaluated at each frequency contained in w Generates the response for a range of user-specified frequencies
Bode (sys) Plots the Bode plot of a TF object. Can return the Bode data (see help) Generates the Bode plot
Nyquist (sys) Plots a polar (Nyquist) frequency-response plot of the TF object Can return response data (see help) Generates the polar frequency-response plot

First, enter the transfer function as a TF object:

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>>extf = tf([10], [2 1])
Transfer function:
10
2s+1

Use the evalfr command to verify entries in Table 12.2 for a frequency of 0.25 rad/s.

>>evalfr(extf, 0.25*i)
ans =
8.0000 — 4.0000i

Note that the frequency passed to the function is specified as an imaginary number.
This is consistent with the interpretation that the sinusoidal transfer function is found by substituting s = jω. Note also that the result is the same that can be found in Table 12.2 for a frequency of 0.25 rad/s.
Now, use the freqresp function to generate the entire table with a single computation.
The first step is to build a vector of frequencies:

>>wex = i*[0 .1 .25 .5 1 5 10 1000]

Use the function to generate the response:

>>h=freqresp(extf,wex)

It is left to the reader to verify that the results correspond to the entries previously presented in Table 12.2.

\omega \left(rad/s\right) 0.0 0.1 0.25 0.5 1.0 5.0 10.0 \infty
Re\left[T(j\omega )\right] 10 9.62 8.0 5.0 2.0 0.10 0.025 0
Im\left[T(j\omega )\right] 0 −1.92 -4.0 −5.0 −4.0 -1.0 -0.5 0
T(\omega ) 10 9.8 8.94 7.07 4.47 1.0 0.5 0
\phi _{T}(\omega ),\deg 0 -11.3 -26.6 -45.0 -63.4 -84.3 -87.1 -90

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