UNIVERSITY OF CALIFORNIA AT BERKELEY
EECS Department
EECS 40, Fall 2006
Prof. Chang-Hasnain
Homework #7



Posted: 10/25/06
Due at 5 pm in 240 Cory on Wednesday, 11/1/06
Total Points: 100
Put (1) your name and (2) discussion section number on your homework.
You need to put down all the derivation steps to obtain full credits of the problems.
Numerical answers alone will at best receive low percentage partial credits.
No late submission will be accepted expect those with prior approval from Prof.
Chang-Hasnain.
Second-order circuits
1. 2nd order bode plot [15 points]
Consider the following circuit:
C
+
Vin
R1
L
Vout
-
R2
R1 = 5k, R2 = 10k, C = 20uF, L = 50mH
a. Write the transfer function H(ω).
b. What is the value of the resonant frequency ω0?
c. Draw the Bode plot. Make sure to label the value at ω=ω0.
2. 2nd order bode plot [15 points]
Repeat exercise 1 for the following circuit:
L
+
Vin
C
R
Vout
-
R = 30k, C = 0.04uF, L = 40nH
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UNIVERSITY OF CALIFORNIA AT BERKELEY
EECS Department
3. Bode plots with large phase shifts [10 points]
Draw the bode plot of the following transfer function:
1
H ( ) 
1 
2000 
1 j 


2  2000
 
4. Transfer functions [10 points]
Hambley P6.13. When it says to plot magnitude and phase, draw a bode plot (not
linear scale).
5. Extreme frequencies [10 points]
a. What is the impedance of a capacitor for very low frequencies (ω0)? For
very high frequencies (ω∞)?
b. What about inductors?
Now consider the following circuit:
C
C
+
R
L
R
Vout
-
Vin
L C
R
R
L
c. Re-draw the circuit for very low frequencies (ω0). What is H(0)?
d. Re-draw the circuit for very high frequencies (ω∞). What is H(∞)?
6. 2nd order bode plots versus first-order cascades [20 points]
Consider the following transfer function:
10
H 1 ( ) 
j 1000
11 

100
j
a. Find the value of ω0.
b. Find the value of Q.
c. Draw the bode plot
Now consider the following transfer function:
H 2 ( ) 
j

1
100 
j  
j 

1 
 1 

 100   1000 
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UNIVERSITY OF CALIFORNIA AT BERKELEY
d.
e.
f.
g.
EECS Department
Draw the bode plot for the first half of this transfer function.
Draw the bode plot for the second half.
Use superposition to combine the two bode plots.
Show that H1(ω) = H2(ω).
7. Review of complex arithmetic[12 points]
Perform the following calculations (without using a calculator):
j

j

a. 2  e  2  e 6
b. (1  j 2)(3  j 5)  (6  j 7)
c.
2
2 e
j

4
 (1  j 2)
d. (5  j 5)  e
j
e.
f.
j
3
2
 (5  j 2)
3
8
e
 2  j2
3  j3
1 j 3
8. Symbolic manipulation of complex numbers [8 points]
j
Reduce the following expressions to polar form ( A  e ). Your final expression
should only have one arctan() in it (Only convert from rectangular to polar once).
1
R
jC
a.
1
R
 jL
jC
1
b. jL // R //
jC
 1 
  jL
c. R // 
 jC 
 1

1
d. 
 R  //
 jL // R
 jC
 jC
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