Final Exam
ME 0031
Student Name:
April 24, 2008
Score:
Problem 1. (60%)
1) If R=20 , C= 2mC, L= 40mH, calculate the impedance of each component if angular frequency
=500 rad/s.
2)
(a) If i(t)=20 cos(t+45o) (A), the current phasor can be written as
I=______________(A)
(b) If v(t)=20 sin(t-40o) (V), the voltage phasor can be written as
V=______________(V)
(c) If voltage phasor V=20-j20 (V), =500rad/s, express the steady-state voltage v(t) (i.e., in
time domain) as v(t)=_____________________(V)
(d) If current phasor I=835o (A), =300rad/s, express the steady-state current i(t) (i.e., in
time domain) as i(t)=_____________________(A)
3)
For an inductor L=20 mH has a steady-state voltage across it of v(t)=20 cos(t+50o) (V),
Find the steady-state current in the inductor.
4)
If the impedance of an element is Z=-j10 , and =200 rad/s, what is the type and value for this
circuit element?
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Final Exam
5)

ME 0031
April 24, 2008
Find R and L of the shown circuit, when
v(t)=12 cos(t +40o) (V)
i(t)=2 cos(t -20o) (A), and
=1000 rad/s
6) For a circuit element, the voltage and current adhere to the passive convention, and are given as
v(t)=12 cos(100t +30o) (V), and
i(t)= 0.3 sin(100t +30o) (A).
Indicate whether the element is capacitive, inductive, or resistive and find its value.
7) Represent the shown circuit in frequency domain using impedance and phasors.
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Final Exam
ME 0031
April 24, 2008
8) For the circuit in 7), calculate the voltage v(t) (expressed in both phasor form and time domain).
9) Represent the shown circuit in frequency domain using impedance and phasors, and calculate the
steady-state current i(t) for the inductor (in both phasor form and time domain).
10) For the circuit in 9), calculate the steady-state voltage v(t) for the capacitor (in both phasor form
and time domain).

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Final Exam
ME 0031
April 24, 2008
11). For the following circuit, find the equivalent impedance Zeq seen by the voltage source.

12). For the shown circuit, find the Thévenin equivalent circuit seen from the two terminal, assume
Vs=10035o (V)
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Final Exam
ME 0031
April 24, 2008
Problem 2. (10%) Determine the steady-state current, i(t), for the circuit shown below.
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Final Exam
ME 0031
April 24, 2008
Problem 3 (10%) For the shown RC circuit, find the response v(t) for t > 0. The voltage source
vs(t)=10cos20t (V), and the initial voltage v(0)=0.
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Final Exam
ME 0031
April 24, 2008
Problem 4 (20%)For the parallel RLC circuit, R=6, L=7H, and C=1/42 F. When i s  4e 2t (A), initial
conditions v(0) =0 (V), i(0)=0 (A),
a) Derive the differential equation to describe the circuit using inductor current i(t) as variable for t>0.
b) Find the complete response i(t) for t>0.
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Final Exam
ME 0031
April 24, 2008
Equation Sheet:

For inductor: v L t   L
d
i L t 
dt
d
 For capacitor: iC t   C
vC t 
dt

For first-order circuit with a constant source
Differential equation is:
d
x( t )
x( t ) 
K

dt
The general solution is:
x( t )  x   x0  x  e
or: x( t )  B  A e


t

t

or: x( t )  x n t   x f t  ,
where x n ( t )  A e


t

, xf (t )  B
For first-order circuit with nonconstant source:
d
x( t )
x( t ) 
 f t 
dt

x( t )  x n t   x f t 
Differential equation:
The general solution is:

t
where:
xn ( t )  A e
and ,
x f ( t )  Be bt ,
if f ( t )  Ce
or,
x f ( t )  B1 cos t  B2 sin t ,
if f ( t )  C cos t or f ( t )  C sin t

bt
1 t  t 0
u( t  t 0 )  
0 t  t 0

Unit step forcing function

For second-order circuit:
The natural response will be the solution of the equation:
a
d 2 x( t )
dx( t )
 a1
 a0  0
2
dt
dt
The general form of the natural response depends on the two roots of the characteristic equation.
If the two root are real and distinct,
x n ( t )  A1 e s1t  A2 e s2t
If the two root are real and equal,
x n ( t )  A1 e st  A2 te st
If the two roots are complex numbers,
x n ( t )  et B1 cos  d t  B2 sin  d t 
The two unknown in the natural response solution are determined by the initial conditions.

For sinusoidal steady-state response, the circuit needs to be converted to phasor circuit (impedance & and
phasor current and phasor voltage) [from time domain to frequency domain), and then the general circuit
analysis methods can be applied. After the parameter (in phasor form) being solved, the parameter can be
converted back to time domain.
 Sin=cos(90o-)=cos(-90o)=-cos(+90o)
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