Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
Circuits
Quiz 2
Spring 2012
1.
/20
2.
/20
3.
/20
4.
/20
5.
/20
Total
/100
Name __________________
Notes:
1) If you are stuck on one part of the problem, choose ‘reasonable’ values and continue with
the following parts to receive partial credit
2) You don’t need to simplify all your numerical calculations. For example, you can leave
square root terms in radical form.
3) If you feel there is not enough room to do your work, you are likely overcomplicating the
problem
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
1
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
1) Short Answers (20 points)
R
C
1
L
2
V1
10V
5V
0V
0s
2s
V(R1:1,R1:2)
4s
6s
8s
10s
12s
14s
16s
Time
Question 1 (4 points)
The above plot is a measurement (voltage or current) across one of the components in the RLC
series circuit. The initial conditions in the circuit are zero and the source is a step function that
turns on at t = 0. Determine which of the follow is a possible measurement. (Circle all correct
answers.)
a) The voltage across the capacitor
b) The voltage across the inductor
c) The voltage across the resistor
d) None of the above
Question 2 (4 points)
For the measurement shown above, which of the following is true?
a) The circuit is overdamped
b) The circuit is critically damped
c) The circuit is underdamped
d) Not enough information
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
2
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
Question 3 (4 points)
Which of the following transfer functions corresponds to a possible voltage measurement across
an unknown component in a second order RLC circuit with a step function source? There are no
‘typos’ in the expressions. (Circle all correct answers)
1
a) V s  
ss  4s  8
1
b) V s  
s  4s  8
s
c) V s  
s  4s  8
s
d) V s  
s  4s  8
Question 4 (4 points)
Which of the following transfer functions corresponds to a possible voltage measurement across
an arbitrary component for an underdamped second order RLC circuit with a step function
source? There are no ‘typos’ in the expressions. (Circle all correct answers)
1
a) V s  
s  42
1
b) V s  
s  4  2  9
s
c) V s   2
s  2s  4
s
d) V s  
s  2  j 4s  2  j 4
Question 5 (4 points)
Determine the differential equation associated with the following transfer function. The source is
an arbitrary source, VS, and the initial conditions are zero.
b sV s 
VC t   3 1 S
a4
s  a1 s  a0
Equation:
________________________________________________________
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
3
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
2) Short Problems (20 points)
Question 1 (5 pts)
A
+
IC
C
V1
2
R
L
1
B
-
In the above figure, V1 is the voltage between node A and node B. IC is the current through the
capacitor. Determine the expression for V1(s) in terms of IC(s). All initial conditions are zero.
Answer:
VC s   I C s 

J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
4
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
Question 2 (6 pts)
2
I1
R
L
8
1
0
4.0A
3.0A
2.0A
1.0A
0A
0s
2s
4s
6s
8s
10s
I(L1)
Time
The above plot represents the current through inductor for the time period 0 < t < 10s. The source
is a 5A pulse that exists for 0 < t < 5s.
a) Determine the value of the resistor. (Check the plot axis carefully.) (3 pts)
b) Determine the expression for the current through the inductor as a function of time for t > 5s.
Your answer should be exact (no symbolic terms). (3 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
5
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
Question 3 (9 points)
2.0V
1.0V
0V
-1.0V
-2.0V
0s
1s
V(R1:1,R1:2)
2s
3s
4s
5s
6s
7s
8s
9s
10s
Time
a) Based on the above plot, is the response underdamped, overdamped or criticitally
damped? (1 pt)
_______________________________
b) Determine the attenuation constant, α, and the harmonic oscillation constant, β. (4 pts)
c) Based on your results for part b), determine the coefficients (a, b and c) for the Laplace spolynomial shown below. (4pts)
as 2  bs  c  0
a=
b=
J. Braunstein
Rensselaer Polytechnic Institute
c=
Revised: 2/8/2016
Troy, New York, USA
6
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
3) RC circuits (20 points)
R3
10
10
2
R1
1
U1
15
Vs
C1
2
R2
10
TCLOSE = 20
C2
6
0
In the above circuit, the source turns on at t = 0 with a voltage of 15V, Vs=15u(t)V.
Additionally, at t = 20s the switch in series with C2 is closed. You can (should) ignore C2 for
part a) of this problem.
a) For 0 < t < 20s, determine the voltage across C1 as a function of time, VC(t). If you do
any circuit reduction/transformation, include a drawing of your circuit (This is required
for full credit). (7 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
7
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
b) At t = 20s, the switch is closed and C2 is safely connected to the circuit. “Safely” can be
interpreted as saying there are no sparks or sudden charges/discharges when C2 is
connected. What is the voltage across C2 at t = 20-, just before it is connected to the
circuit. (6 pts)
c) For 20s < t, determine the voltage across C1 as a function of time. Use the resistor and
capacitor values in your solution. If you do any circuit reduction/transformation, include
a drawing of your circuit (This is required for full credit). (7 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
8
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
4) Series Circuits/Differential Equations (20 points)
+
R1
-
+
3.5
10
1
L1
1/2
-
2
+
C1
-
1/6
V1
In the above RLC circuit, the DC source was turned on some time before t = 0, but the circuit has
not reached steady state (do not overanalyze this statement). At t = 0, the following
measurements are taken across the resistor and the inductor (based the polarity indicated in the
schematic):
Resistor:
VR(0) = -3.5 V
Inductor:
VL(0) = 8.5 V
a) Determine the DC steady voltage across the capacitor as t →∞. (2 pts)
b) Determine the initial voltage and current across the capacitor at t = 0+, VC(0+) and IC(0+) (4
pts)
c) Based on the circuit, determine the differential equation for the voltage across the
capacitor.(4 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
9
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
d) Based on the homogeneous component of the differential equation, determine the associated
s-polynomial and find the roots to the polynomial. (5 pts)
e) Determine the voltage across the capacitor as a function of time. (5 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
10
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
5) Laplace Solutions (20 points)
TCLOSE = 0
1
2
C
U1
1
R
1
1/2
5
L
2
1
V1
0
In the above circuit, a 5V DC source is shown. At t = 0, the switch in series with the capacitor is
closed.
a) Draw the s-domain equivalent circuit. Include all initial condition source components. (5
pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
11
Circuits
Name _______________________ ECSE 2010
Spring 2012
Section _________
b) Determine the transfer function for the voltage across the inductor, VL s  
N s 
, where
Ds 
N(s) and D(s) are polynomials. (5 pts)
c) Apply partial fraction expansion to your above expression. (5 pts)
d) Based on your result in part c), determine the voltage across the inductor as a function of
time. (5 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
12