1. No category

 Allison
Thongvanh
Section
B
Thursday
12:10 - 3PM
Spring EE324 Lab 13
13 Lab 13
Thongvanh 1 Purpose
The purpose of this lab is to observe how the use of feedback influences the stability and
bandwidth of an amplifier. We analyze the root locus of the system in Figure 1 and verify the
Nyquist stability criterion.
Fig. 1 Feedback system with compensator network
Lab Assignment
(3) We are given the Laplace transfer function of Figure 1 as A(s), Equation 1. From A(s) and the
compensator network we know that the open-loop transfer function, L(s), is the function as shown
in Equation 2 and closed-loop transfer function, T(s), is Equation 3.
A( s) =
Eq. 1
L(s) =
R2
10 4
A(s) =
R1 + R2
901(1+ 10−4 s)(1+ 4 ⋅10−4 s)(1+ 10−6 s)
Eq. 2
T(s) =
A(s)
901×10 4
=
1+ L(s) 901(1+ 10−4 s)(1+ 4 ⋅10−4 s)(1+ 10−6 s) + 10 4
Eq. 3
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10 4
(1+ 10−4 s)(1+ 4 ⋅10−4 s)(1+ 10−6 s)
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(4) Now we verify the results with the MatLab function rlocus.
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Fig. 2 Root locus of A(s)
Fig. 3 Root locus of L(s)
Lab 13
Thongvanh 2 (5) (6) Next we use the rlocfind function to find the value of R2 when the system is on the
verge of instability and what happens when the value is increased further.
>> rlocfind(aS)
Select a point in the graphics window
selected_point =
6.4516e+03 - 1.0067e+04i
ans =
0.0010
*For K=0.0010 and R1=9kΩ:
k=
R2 *
R2 *
⇒ 0.0010 =
⇒ R2* = 9.009
R1+ R2 *
9000 + R2 *
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Fig. 4 R2 = R2*
Fig. 5 R2 = 1000 Ω
When the value of R2 is greatly increased the location of the poles is real only.
(7) (8) The next step is to represent the bode plot and analyze L(s) and T(s).
Fig. 6 Bode plot of L(s)
Fig. 6 Bode plot of L(s)
Fig. 7 Bode plot of T(s)
Lab 13
Thongvanh 3 The phase margin of the open-loop system L(s) is 42 degrees, and that of the closed-loop T(s) is -24.
T(s) is stable, and its bandwidth (from using the bandwith() function) is 2.4604e+04.
(9) Now we represent the Nyquist diagram of L(s) (using the nyquist() function) to verify the
location of the poles and then comment on the stability of T(s).
Fig. 9 Bode plot of T(s)
Fig. 8 Bode plot of L(s)
*Both plots are STABLE.
(10) The next step is to map the poles of T(s) on the s-plane and to comment on its stability.
Fig. 10 Pole-zero map of T(s)
*Poles and zeros are in the right-half plane. This indicates INSTABILITY.
Lab 13
Thongvanh 4 (11) Finally we represent the step response of T(s) and comment on stability.
*This system is STABLE.
Conclusion
The purpose of this lab was to observe how the use of feedback influences the stability and
bandwidth of an amplifier. We analyzed the root locus of the system in Figure 1 and verified the
Nyquist stability criterion.
Lab 13
Thongvanh 5 Appendix
>> aS=tf([40e-15 40.5e-9 501e-6 1],[1e4])
Transfer function:
4e-14 s^3 + 4.05e-08 s^2 + 0.000501 s + 1
----------------------------------------10000
>> aS=tf([1e4], [40e-15 40.5e-9 501e-6 1])
Transfer function:
10000
----------------------------------------4e-14 s^3 + 4.05e-08 s^2 + 0.000501 s + 1
>>
>>
>>
>>
rlocus(aS)
r1=9000;
r2=10;
lS=(r2/(r1+r2))*aS
Transfer function:
11.1
----------------------------------------4e-14 s^3 + 4.05e-08 s^2 + 0.000501 s + 1
>> rlocus(lS)
>> rlocfind(aS)
Select a point in the graphics window
selected_point =
-9.8710e+05 - 1.0067e+04i
ans =
0.0630
>> rlocfind(aS)
Select a point in the graphics window
selected_point =
6.4516e+03 - 1.0067e+04i
ans =
0.0010
Lab 13
Thongvanh 6 >> r2=9.009;
>> lS2=(r2/(r1+r2))*aS
Transfer function:
10
----------------------------------------4e-14 s^3 + 4.05e-08 s^2 + 0.000501 s + 1
>> rlocus(lS2)
>> r2=1000;
>> ls3=(r2/(r1+r2))*aS
Transfer function:
1000
----------------------------------------4e-14 s^3 + 4.05e-08 s^2 + 0.000501 s + 1
>> rlocus(ls3)
>> margin(lS)
>> tS=aS/(1+lS)
Transfer function:
4e-10 s^3 + 0.000405 s^2 + 5.01 s + 10000
----------------------------------------------------------------1.6e-27 s^6 + 3.24e-21 s^5 + 1.68e-15 s^4 + 4.11e-11 s^3 +
7.815e-07 s^2 + 0.006562 s + 12.1
>> margin(tS)
>> bandwidth(tS)
ans =
2.4604e+04
>>
>>
>>
>>
>>
nyquist(lS)
nyquist(lS2)
nyquist(tS)
pzmap(tS)
step(tS)