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ECE 2100 Homework 4
Solution
Due April 1, 2010
Professor Alyosha Molnar
Subjects: Sinusoidal steadystate analysis, phasors, complex impedance, complex circuit
analysis, AC power transfer, transformers, mutual inductance.
1)
Prelab: Looking at the overlaid input/output sinewaves below, extract:
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
1.2
time, microseconds
Voltage
Input
Output
a.
The frequency in Hz.
F=1/0.5
μ
s =
2MHz
b.
The frequency in radians per second.
ω
= f·2
π
=
12.56Mrad/s
c.
The change in amplitude between input and output (that is the ratio of
output to input amplitudes)
Vout/Vin = 1.4/2 =
0.7
d.
The change in phase between input and output (Remember that time delay
corresponds to negative phase)
~ 
π
/4 (45 degrees)
e.
Write the transformation from input to output as a phasor, and as a
complex number.
That is, Vout = (A+jB)Vin.
What are A and B?
Vout =
0.5(1j)
Vin = Vin(
0.7exp(j
π
/4)
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Prelab: You will be analyzing the impedance and voltagedivision characteristics
of a several “black box” circuits in lab.
Several example circuits are given below.
In each case analyze the impedance across each pair of terminals (a complex
number, as a function of
ω
), and in particular analyze Z for f=0Hz.
Also use
voltage divider analysis to find the phase and amplitude of V
2
for
V
1
=1V·cos(2
π
ft), as a function of f, and find it explicitly for f=1kHz and 100kHz.
Also, find the frequency (in the range from 1kHz to 100kHz) at which the
amplitude of V
2
is maximized.
Perform the same analyses for V
1
when
V
2
=1V·cos(2
π
ft).
a.
Pure resistors: analyze the circuit in Fig. 2a
i.
What are Zac, Zbc and Zab?
What are they at f=0Hz?
Zac = 2k
Ω
, Zbc = 3k
Ω
and Zab= 3k
Ω
?
at all frequencies
ii.
What is V
2
if V
1
=1V·cos(2
π
ft)?
V
2
= R3/(R1+R3)V
1
= V
1
/2 =
0.5V·cos(2
π
ft)
iii.
What is V
1
if V
2
=1V·cos(2
π
ft)?
V
2
= R3/(R2+R3)V
1
= V
1
2/3 =
0.66V·cos(2
π
ft)
b.
RC: analyze the circuit in Fig. 2b
i.
What are Zac, Zbc and Zab?
What are they at f=0Hz?
Zac = 1k
Ω
+1/(j·f·628nF)
b
∞
when f=0
Zbc = 2k
Ω
+1/(j·f·628nF)
∞
when f=0
Zab = 3k
Ω
at all frequencies
ii.
What is V
2
if V
1
=1V·cos(2
π
ft)?
In phasors: V
2
= 1/(1+j
2
π
f·100
μ
s
)V
1
In time:
V
2
=1V/(1+f
2
·3.9·10
7
)
1/2
cos(2
π
ft+atan(f·628
μ
s))
iii.
What is V
1
if V
2
=1V·cos(2
π
ft)?
In phasors: V
1
= 1/(1+j
2
π
f·200
μ
s
)V
2
In time:
V
1
=1V/(1+f
2
·15.6·10
7
)
1/2
cos(2
π
ft+atan(f·1256
μ
s))
c.
Based on the zerofrequency impedances and voltage divider results, how would
you extract the values of R1 and R2 for figure 2b?
Their series resistance tells you that R1+R2=3k
Ω
, for frequencies f>>1/(2
π
RC),
the amplitude of V1 = 1/(2
π
R1C), V2 = 1/(2
π
R2C), so the ratio of the results of ii
and iii above gives the ratio of the resistors:
V2/V1=R2/R1=2
R1=1k
Ω
, R2=2
k
Ω
.
d.
RL: analyze the circuit in figure 2d
i.
What are Zac, Zbc and Zab?
What are they at f=0Hz?
( 29
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This note was uploaded on 11/21/2011 for the course ECE 2100 taught by Professor Kelley/seyler during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 KELLEY/SEYLER
 Impedance

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