1. Frequency Response Techniques
Islamic University of Gaza
Faculty of Engineering
Electrical Engineering Department
CH 10:
Frequency Response Techniques
Control Systems Design, Dr. Moayed Almobaied

2. Nyquist diagram showing gain and phase margins
Gain margin, GM, The gain margin is the change in open-loop gain, expressed in decibels (dB), required at 180o of phase shift to make the closed-loop system unstable.
Phase margin, , The phase margin is the change in open-loop phase shift required at unity gain to make the closed-loop system unstable.
GM = 20log a.

3. Finding gain and phase margins
Problem: find the gain and phase margins of system of previous example. If K = 6.
Solution: to find the gain margin, first find the frequency where the Nyquist diagram crosses the negative real axis.
The Nyquist diagram crosses the real axis at frequency of The real part is calculated to be Thus, the gain can be increased by (1/0.3) =3.33 before the real part becomes -1. Hence the gain margin is GM = 20log 3.33 = dB.
To find the phase margin, find the frequency for which the magnitude is unity. We need to solve the equation to find frequency = rad/s. at this freq. the phase angle is o. The difference between this angle and -180o is 67.7o, which is the phase margin.

4. Range of gain for stability via Bode plots
Problem: use Bode plots to determine range of K within which the unity feed-back system with G(s) = K/[(s+2)(s+4)(s+5)] is stable.
Solution: The closed loop system will be stable if the frequency response has a gain less than unity when the phase is +/-180o
We sketch the Bode plots as shown. The low-frequency gain of G(s)H(s) is found by setting s=0. So the Bode magnitude plot starts at K/40. For convenience let K = 40 so that the log-magnitude plot starts at 0 dB.
At frequency 7rad/s, when phase is 180o, the magnitude is -20dB. So an increase in gain of +20dB is possible before system becomes unstable.
Since the gain plot was scaled for a gain of 40, +20dB(a gain of 10) represents the required increase in gain above 40. Hence, the gain for instability is 40X10 = <K<400 for stability. Compare to actual results 378 at freq rad/s.

5. Evaluating Gain and Phase Margins

6. Gain and Phase Margins from Bode plots
Problem: If K =200 in the system of previous example, find the gain and phase margins
Solution: The Bode plot in Figure is scaled to a gain of 40. If K=200, the magnitude plot would be 20log5 = dB higher.
At phase = 180, on the magnitude plot the gain is -20. So the gain margin = = 6.02 dB
To find phase margin, we look on the magnitude plot for freq where the gain is 0 dB ( dB normalized) occurs at 5.5 rad/s freq. At this freq the angle is 165 so the phase margin is -165-(-180) = 15 degrees

7. Using the open loop transfer function
Relation between Closed-Loop Transient and Closed-Loop Frequency Responses
Using the open loop transfer function
And closed-loop transfer function
We evaluate the magnitude of the closed-loop freq response as
A representative sketch of the log plot is shown in the Figure

8. To relate the peak magnitude to the damping ratio we find
Relation between Closed-Loop Transient and Closed-Loop Frequency Responses
To relate the peak magnitude to the damping ratio we find
At a frequency, of
Since damping ratio is related to percent overshoot we can plot MP vs. percent overshoot

9. Response Speed and Closed-Loop Frequency Response
We can relate the speed of the time response ( as measured in settling time, rising time, and peak time) and the bandwidth , of the Frequency response.
The Bandwidth is defined as the frequency at which the magnitude response curve is 3 dB down from its value at zero frequency.
The Bandwidth for a two-pole system can be found by finding that freq for M=1/√2 (that is -3dB)

10. We relate to settling time and peak time by substituting And We get,
Response Speed and Closed-Loop Frequency Response
We relate to settling time and peak time by substituting
And We get,
Figure Normalized bandwidth vs. damping ratio for: a. settling time; b. peak time; c. rise time

11. Relation Between Closed-Loop Transient and Open-Loop Frequency Responses

12. Steady-State Error Characteristics from Frequency Response
For type 0 system (the initial slope is 0)
For un-normalized un-scaled Bode plot, the low frequency magnitude is 20 log KP
For type 1 system (the initial slope is -20dB/decade)
For un-normalized un-scaled Bode plot, the intersection of the initial -20dB/decade slope with the frequency axis is KV
For type 2 system (the initial slope is -40dB/decade)
For un-normalized un-scaled Bode plot, the intersection of the initial -40dB/decade slope with the frequency axis is √Ka

13. Steady-State Error Characteristics from Frequency Response
Example: For Bode Log plots shown, find system type, and static error constants
Solution:
In (a) system is type 0, since the initial slope is 0. And 20 log KP =25, or KP = 17.78
In (b) system is type 1, since the initial slope is -20dB/decade. And the value of Kv is the value of frequency that the initial slope intersects at the zero dB crossing of the frequency axis. Hence Kv = 0.55
In (c) system is type 2, since the initial slope is -40dB/decade. And the value of √Ka is the value of frequency that the initial slope intersects at the zero dB crossing of the frequency axis. Hence Ka = 32 = 9