Circuits
ECSE 2010
Spring 2014
Section _________
Circuits
Quiz 2
Spring 2014
/45
1.
(do 9 of 10)
Not graded: Q_____
2.
/15
3.
/20
4.
/20
Total
/100
Name __________________
Notes:
1) If you are stuck on one part of the problem, choose ‘reasonable’ values on the following
parts to receive partial credit
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
1
Circuits
ECSE 2010
Spring 2014
Section _________
1) Short Analysis
Question 1 (5 points)
For both series and parallel RLC with step function sources
T / F a) The differential equation associated with voltage across or current through an
arbitrary component is always a second order differential equation
T / F b) The Laplace expression for voltage across or current through an arbitrary
component always has a pole at s = 0.
T / F b) The current through the resistor at t = 0+ is zero.
T / F c) The voltage across the capacitor at t = 0+ is zero.
Question 2 (5 points)
In the above pole-zero plot, one pole of a complex conjugate pair, a double pole and a zero are
shown. Determine a complete expression for F(s). To maintain simplicity, express your solution
s  z1  . (Remember, one pole is not shown on the plot and you need to
in the form F s  
s  p1 
include it in your expression.)
F(s) = ___________________________________________________________ (3 pts)
Does this set of poles/zeros represent a stable circuit? Explain your answer. (2 pts)
_____________________________________________________________________________
_____________________________________________________________________________
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
2
Circuits
ECSE 2010
Spring 2014
Section _________
Question 3 (5 points)
The voltage across the resistor in an RLC series circuit with a 10 V step function source is given
as
VR t   100t exp  4t 
Determine the differential equation for the capacitor voltage, using coefficients instead of
component values (ie. do not use R, L, and C in your expression).
Question 4 (5 points)
R1
R2
2
C1
1E-6
L1
1E-3
1
Vs
4/s
V2
1E-7
For a 10V source that turns off at t = 0, a Laplace equivalent circuit is shown above. The circuit
was at DC steady state at t = 0-. Determine values for R1 and R2.
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
3
Circuits
ECSE 2010
Spring 2014
Section _________
Question 5 (5 points)
2.0V
1.0V
0V
-1.0V
0.1s
0s
0.2s
0.3s
0.4s
0.5s
0.6s
0.7s
0.9s
0.8s
1.0s
V(R:1,R:2)
Time
The above plot is the voltage across some unknown component in a second order circuit.
Determine the steady state voltage as t → ∞, the attenuation constant (α), the oscillation
frequency (β).
I(t→∞)
___________________________________________________ (1 pt)
α
___________________________________________________(2 pts)
β
___________________________________________________(2 pts)
Question 6 (5 points)
The Laplace expression for the voltage across some component is given as
1
V R t   3
2
s  As  Bs  C
Assuming A, B and C are nonzero and real,
How many poles are in the above expression? _________________ (2 pts)
T / F b) There is a pole at zero.
T / F b) It is possible there is a complex conjugate pair of poles.
T / F b) It is possible there are three real and negative poles.
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
4
Circuits
ECSE 2010
Spring 2014
Section _________
Question 7 (5 points)
R1
2
Vs
Is
C
L
R2
1
RC circuit
RL circuit
The above plot is either the current through or voltage across a component in the above RC and
RL circuits. Both Vs and Is are step function sources. Indicate which of the following
measurements may be represented by the plot. (Circle the possible answers for each component.)
Resistor R1:
VR1
Capacitor C: VC
IR1
Resistor R2:
VR2
IR2
IC
Inductor L:
VL
IL
Question 8 (5 points)
U1
+
0
Vin
OUT
-
Vout
OPAMP
Z1
Z2
The above circuit is a differentiator
T / F a) Z1 is an inductor, Z2 is a resistor
T / F b) Z1 is an resistor, Z2 is a inductor
T / F a) Z1 is an capacitor, Z2 is a resistor
T / F b) Z1 is an resistor, Z2 is a capacitor
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
5
Circuits
ECSE 2010
Spring 2014
Section _________
Question 9 (5 points)
10
4u(t)
2
0.1
10
1
For the above circuit, determine the differential equation for the current through the inductor.
Question 10 (5 points)
10
4u(t)
10
2
0.1
1
For the above circuit, determine the Laplace expression for the current through the inductor,
IL(s).
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
6
Circuits
ECSE 2010
Spring 2014
Section _________
3) First Order Circuits (15 points)
R1
6k
Vs
1
2
U1
C1
1.25E-6
R2
12k
C2
1.25E-6
In the above circuit, at t = 0 the source turns on with voltage 9u(t) and the switch is closed.
Determine the t = 0+ initial conditions and indicate them in the following table. (4 pts)
Component
R1
R2
C1
C2
Voltage [V]
Current [mA]
Assuming the switch stays closed permanently, determine the voltage across C1 as t → ∞. ( 1 pt)
Using any method, determine the voltage across C1 (the ‘left’ capacitor) as a function of time. (4
pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
7
Circuits
ECSE 2010
Spring 2014
Section _________
At t = 0.00405 [s] (4.05 [ms]), switch U1 opens. Using any method, determine the voltage across
C2 (the ‘right’ capacitor) as a function of time. (4 pts)
For t > 0.00405 [s], does the expression for the voltage across C1 as a function of time change?
If so, describe what changes in the expression. (Calculations/math are not required.) (1 pt)
Is opening the switch an ‘unsafe’ operation? Why? (1 pt)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
8
Circuits
ECSE 2010
Spring 2014
Section _________
4) Second Order Laplace Equations (20 points)
R
1
3.5
L
0.5
2
C
0.2
Vs
At t = 0+, the following voltages are known
Vs = 20V (and is constant for t > 0)
VR(0+) = 7V
VL(0+) = 1V
VC(0+) = 8V
Draw the s-domain equivalent circuit. Include the Laplace transform of the source and all nonzero initial condition terms. Use numerical value instead of symbols (ie. ‘7’ instead of ‘R’)(4 pts)
As t → ∞, what is the voltage across the resistor? ____________________ (1 pt)
Determine the Laplace expression for the voltage across the resistor, VR(s). Your expression
should be in the form VR(S) = N(s)/D(s), where N(s) and D(s) are polynomials. (4 pts)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
9
Circuits
ECSE 2010
Spring 2014
Section _________
Determine the Laplace expression for the current through the resistor, VR(s). Your expression
should be in the form IR(S) = N2(s)/D2(s), where N2(s) and D2(s) are polynomials. (1 pt)
Determine the poles and zeroes of your VR(s) expression (voltage across the resistor). If any
roots are repeated or complex conjugates, indicate that. (2 pts)
Zeroes:
Poles:
Apply partial fraction expansion to your VR(s) expression. (4 pts)
Determine the resistor voltage as a function of time. (3 pts)
What two quick checks with regard to expected voltages can you use to test the accuracy of your
above result? (1 pt)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
10
Circuits
ECSE 2010
Spring 2014
Section _________
5) Second Order Differential Solutions (20 points)
2
Is
L1
3.715
R1
5
C1
0.01
1
In the above circuit, the current source is defined as
4 A t  0
I s t   
2 A 0  t
Determine the initial conditions, IL(0+) and VL(0+). (2 pts)
Is the circuit underdamped or overdamped? Justify your answer. (3 pts)
What resistance value would make the circuit critically damped? (2 pt)
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
11
Circuits
ECSE 2010
Spring 2014
Section _________
What is the form of the solution for the current through the inductor, IL(t), with arbitrary
coefficients A1, etc. (2 pts)
Determine the coefficients (A1, etc.) in the above expression. (4 pts)
Determine the current through the inductor as a function of time, IL(t). (2 pts)
If the source turns off at t = 50s, sketch the current through the inductor as a function of time for
0<t<100s. Your plot does not have to be perfect, though should generally reflect the type of
damping you determined. On your plot, if the circuit is overdamped, indicate the time constant of
the dominant pole. If the circuit is underdamped, indicated the attenuation constant, α, and the
period of oscillation, T. (5 pts)
IL
50s
J. Braunstein
Rensselaer Polytechnic Institute
100s
t
Revised: 2/8/2016
Troy, New York, USA
12
Circuits
ECSE 2010
Spring 2014
Section _________
5) Extra Credit (5 points)
1
L
2
0.5
5u(t)
C
0.5
Determine the voltage across the capacitor as a function of time.
J. Braunstein
Rensselaer Polytechnic Institute
Revised: 2/8/2016
Troy, New York, USA
13