Frequency Response

Understanding how amplitude and phase change at each frequency connects Bode plots to actual signals.

Starting conditions

$G(s)=\frac1{s^2+0.5s+1}$, $u=\sin(\omega t)$, $\omega\in\{.2,1,3\}$ rad/s. $0\le t\le 200$ s, $\Delta t=.01$ s.

Procedure

1. Calculate the output using ctl.forced_response, andcalculate the output using lsim, then plot the input and output for only the final two periods.
2. Fit the output to $a\sin(\omega t)+b\cos(\omega t)$ by least squares to obtain gain $\sqrt{a^2+b^2}$ and phase $\operatorname{atan2}(b,a)$.
3. Compare with the magnitude and phase from ctl.frequency_response. Phase is returned in radians.Compare with the gain and phase calculated using abs and angle from the complex response of squeeze(freqresp(G,omega)). angle returns radians. Convert both phase calculations to degrees in the table.

Results to retain

• A frequency table with 3 rows and 3 plots comparing the final two periods.

Starting conditions

Set the denominator to $s^2+as+1$ and compare $a\in\{.2,.5,1\}$. Use ω=logspace(-2,2,400).

Procedure

1. Plot the gains and phases of the three models on the same frequency axis.
2. Tabulate each model's maximum gain, its frequency, and DC gain.
3. Choose one sinusoidal condition near the maximum gain and check it using the method from the first activity.

Results to retain

• 1 set of Bode plots and a resonance comparison table.

Starting conditions

$\omega=1$ rad/s differs from $f=1$ Hz. The second input is $\sin(2\pi t)$.

Procedure

1. First record the periods of both inputs, then calculate their responses for the same G.
2. Record the phase difference both as a time delay and as an angle.
3. Explain what you would misread on a Bode plot if you confused rad/s with Hz.

Results to retain

• A unit comparison table with 2 rows and output plots.