EE324 Lab 12 - Allison Thongvanh

 Allison
Thongvanh
Section
B
Thursday
12:10 - 3PM
Spring EE324 Lab 12
13 Lab 12
Thongvanh 1 Purpose
In this lab we observe how the use of feedback influences the stability of an amplifier and what
measures can be taken to improve this effect. To make said observations, we experiment with and
manipulate the compensator network in Figure 1.
Fig. 1 Feedback system with compensator network
Lab Assignment
We are given the Laplace transfer function of Figure 1 as A(s), Equation 1, and that L(s) is the
function as shown in Equation 2.
10 5
A(s) =
Eq. 1
(1+ 10−4 s)(1+ 10−6 s)
L(s) =
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R2
A(s) = R ⋅ A(s)
R1 + R2
Eq. 2
First we use MatLab to verify the Bode plot of A(s) and L(s). Then we verify the phase margin and
step-response of T(s), Equation 3. (See Appendix A2 for derivation of T(s)).
€
T(s) =
A(s)
A(s)
=
10000
1+ A(s) ⋅ R 1+ A(s)
10000 + 90000
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Fig. 2 Bode plot of A(s)
Fig. 3 Bode plot of L(s)
Eq. 3
Lab 12
Thongvanh 2 Fig. 4 Gain and Phase Margins of T(s) (zoomed on margins on right)
Fig. 5 Step Response of T(s)
Now we improve the response using a lead compensator system. The improved system is shown in
Figure 6.
Fig. 6 Improved compensator network
Lab 12
Thongvanh 3 Lastly we find a value for the capacitor in the improved compensator network. To do so we find
the transfer function for the network itself (Equation 4), play with numbers for the capacitor value,
and observe the values' effects on L(s).
R2
R2
R2
Eq. 4
R( s) =
=
=
1
R1sC + 1
Z
+
R
2
R1 ||
+ R2
+ R2
sC
R1
•
R1 = 9k Ω , R2 = 1k Ω , C = 1pF
Transfer functions:
€
9e11
L(s)=-----------------------------------90 s^3 + 9e16 s^2 + 9.09e12 s + 9e06
9e06 s^3 + 9e21 s^2 + 9.09e17 s + 9e11
T(s)=-------------------------------------------------------------------9e11 s^5 + 9e26 s^4 + 1.818e23 s^3 + 9.009e21 s^2 + 9.09e17 s + 9e11
Fig. 6 Bode Plot of L(s)
Fig. 7 Step Response of T(s)
Fig. 8 Bode Plot of T(s)
Lab 12
•
Thongvanh 4 R1 = 9k Ω , R2 = 1k Ω , C = 5pF
Transfer functions:
9e11
L(s)=------------------------------------450 s^3 + 9e16 s^2 + 9.09e12 s + 9e06
4.5e07 s^3 + 9e21 s^2 + 9.09e17 s + 9e11
T(s)=---------------------------------------------------------------------4.5e12 s^5 + 9e26 s^4 + 1.818e23 s^3 + 9.009e21 s^2 + 9.09e17 s + 9e11
Fig. 9 Bode Plot of L(s)
Fig. 10 Bode Plot of T(s)
Fig. 11 Step Response of T(s)
Lab 12
•
Thongvanh 5 R1 = 9k Ω , R2 = 1k Ω , C = 10pF
Transfer functions:
9e11
L(s)=------------------------------------900 s^3 + 9e16 s^2 + 9.09e12 s + 9e06
9e07 s^3 + 9e21 s^2 + 9.09e17 s + 9e11
T(s)=-------------------------------------------------------------------9e12 s^5 + 9e26 s^4 + 1.818e23 s^3 + 9.009e21 s^2 + 9.09e17 s + 9e11
Fig. 12 Bode Plot of L(s)
Fig. 13 Bode Plot of T(s)
Fig. 14 Step Response of T(s)
Lab 12
•
Thongvanh 6 R1 = 9k Ω , R2 = 1k Ω , C = 15pF
Transfer functions:
9e11
L(s)=-------------------------------------1350 s^3 + 9e16 s^2 + 9.09e12 s + 9e06
1.35e08 s^3 + 9e21 s^2 + 9.09e17 s + 9e11
T(s)=----------------------------------------------------------------------1.35e13 s^5 + 9e26 s^4 + 1.818e23 s^3 + 9.009e21 s^2 + 9.09e17 s + 9e11
Fig. 15 Bode Plot of L(s)
Fig. 16 Bode Plot of T(s)
Fig. 17 Step Response of T(s)
Conclusion
The purpose of this lab was to observe how the use of feedback influences the stability of an
amplifier and what measures can be taken to improve this effect. We observe changes in phase of
L(s) when changing the capacitor in the compensator network. I would choose the 10 pF capacitor.
Lab 12
Thongvanh 7 Appendix
A1: Various A(s), L(s), and T(s) plots
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r1=9000;
r2=1000;
c=1e-12;
z=tf([r1*c 1],[r1]);
RS=r2/(z+r2);
AS=tf([0 0 0 0 0 1e5],[10000000000 1010000 1]);
LS1=AS*RS
TS1=AS/(1+LS1);
bode(LS1)
margin(TS1)
step(TS1)
%Repeat the same process for varied capacitor (c) values. A2: Gain = 10 calculation
Given R=R2/(R1+R2)
Vo(s) = A(s)(Vp(s)−Vn(s))=A(s)(Vp(s)−RVo(s))
Therefore,
Vo(s)
A(s)
----- = ---------Vp(s)
1+A(s)*R
From here, manipulate values in R so that gain is equal to 10.