ECE 402 Controls Lab JSG - November 30, 2001
ECE 402 - Lab #5 Gain Compensation - Bode Plots & Nichols Charts Purpose: This lab looks at using using frequency-domain techniques techniques to find a feedback gain. Specifically, gain compensation using Bode Plots and Nichols Charts will be presented.
Introduction If you are trying to control a dynamic system, it can be represented by a transfer function in 's'. One way to interpret a gain, G(s), is to let s → j . This results in the system behaving as a filter with the gain of G( j ω). If a dynamic system has feedback, as shown below, two questions arise: "How large can you make the gain and have a stable system?" "How large should the gain be for some desired damping ratio?" If interpreted using root-locus techniques, stability stability results from keeping the poles in the left-half plane. If interpreted using frequency domain techniques, stability results from the closed-loop gain being finite. R
G(jw) -
Y
ECE 402 Controls Lab JSG - November 30, 2001
Nichols Charts The relationship which converts the gain of the open-loop system to the closed-loop gain is G( j ω) →
G( jω) 1+G( j ω)
For example, if the gain of G is 0.9∠ − 1500 at some frequency, the closed-loop gain at that frequency will be 1.7958∠ − 86.11 0
= 0.9 ∠−150 . 1+0.9∠−150 0
0
Since this is a nonlinear mapping, a Nichols Chart is useful to show
G
how G maps into 1 +G .
To read a Nichols chart, The open-loop gain, G, is read on the outer coordinates. For example, the gain 0.9∠ − 1500 is marked as point 'A' on the following plot. The closed-loop gain is read from the ovals (termed M-circles). The point marked is slightly outside of the 6dB M-circle. Hence, the closed-loop gain is about 5dB (1.79). Note: The M-circles are for convenience. All points on the 6dB M-circle have a closed-loop gain of 6dB.
Gain (dB) 20
G(jw) 15 10 5
1d B
3d B 0d B
6d B
ECE 402 Controls Lab JSG - November 30, 2001
The resonance can be read by finding the point closest to -1 as read by the M-circles. (0dB, -1800.) For this system, the largest closed-loop gain is close to point, 'A', with a resonance of about 5dB closed-loop.
To design a gain compensator using a Nichols chart, Translate the design requirement to the desired resonance (Mm) using 2nd-order approximations. Plot G(jw) on the Nichols Chart. Adjust the gain until the open-loop gain is less then 1.000 when the phase is 180 degrees. Adjust the gain (slide G(jw) up and down on the Nichols chart) until G(jw) is tangent to the desired M-circle. k is then how much you slid G(jw) up or down. For example, if a resonance of 12dB is desired for the closed-loop system, you need to slide G(jw) up +7dB as shown below. K is then +7dB = 2.2387.
Gain (dB) 20
G(jw) 15 10 5
1d B
3d B 0d B
6d B 9d B 12dB
0
15dB
A
ECE 402 Controls Lab JSG - November 30, 2001
Bode Plots: A Bode Plot is a plot of gain and phase vs. frequency. Ideally, you would like the gain to be 1.000 at al frequencies, meaning that the output tracks the input. Unfortunately, this often is not the case. To make the gain 1.000, feedback can be used with high gains. With feedback, the gain becomes G( j ω) →
G( jω) 1+G( j ω)
This is approximately:
G( jω) ≈ 1+G( j ω)
1 G
G G
>> <<
Hence, if you make the gain large, the closed-loop gain will be approximately 1.000 as desired. From previous results, there is a limit on how much gain you can tolerated, however. As a result, the trick to designing a feedback system using frequency domain techniques is to Crank up the gain as much as possible. A Nichols chart is a useful tool to determine how large the gain can be. This results in the closed-loop system having a gain of about 1.000 when the open-loop gain is greater than 1. This will likewise be the 'passband' of your closed-loop system.
For example, consider G(s) = (s+1)(s+10 )(s+50 ) . The open-loop and closed-loop gain 5000s
G( j ω) are plotted 1+G( j ω)
below. Note that When |G|>1,
G ≈ 1 1+G
At the corner (when G=1), a resonance occurs. The amplitude of this can be seen on the Nichols chart if G(jw) were plotted on this chart.
ECE 402 Controls Lab JSG - November 30, 2001
30
Open-Loop
20
10
dB 0 Closed-Loop
-1 0
-2 0
-3 0 0.01
0.1
1
rad/sec
10
100
1000
ECE 402 Controls Lab JSG - November 30, 2001
Lab #5: Prelab Step 1) Build the following circuit
R
1 M
100k
R
1 0 k 2
0.01 741
100k
100k
100k
Y
741
6
3 100k
0.01
0.01
0.01
R k =
1M
1 0 k
1 0 k
Open in Step 2 Close in Step 4+
1 0 k 2 6
741 3
Lab #5: Procedure: Step 2) With the feedback gain removed, find the gain and phase from point R to Y. Be careful to measure the gain with the phase is close to -180 degrees. Step 3) Plot the gain on a Bode plot. Plot the gain and phase of G(s) on a Nichols chart. From the
Gain (dB) 20 15
1dB
3dB
10
0d B
5
6dB 9dB 12dB
0
15dB
-5 -10 -15 -20 -220
-210
-200
-190
- 180
- 170
-160
-150
-140
-130
- 120
-110
-100
