ECE 2100 Homework 4 Solution
Professor Alyosha Molnar
Due March 31, 2011
Subjects:
Sinusoidal steadystate analysis, Phasors, frequencydomain analysis.
1)
Prelab: Looking at the overlaid input/output sinewaves below, estimate:
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)
Prelab: For problems 25
, You will be analyzing the impedance and voltage
division 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).
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View Full DocumentFigure 2:
Figure 3:
Figure 4:
Figure 5:
2)
Pure resistors: analyze the circuit in Fig. 2
a.
What are Zac, Zbc and Zab?
At f=0Hz?
Zac = 3k
Ω
, Zbc = 4k
Ω
and Zab= 3k
Ω
?
at all frequencies
b.
What is V
2
if V
1
=1V·cos(2
π
ft)?
V
2
= R3/(R1+R3)V
1
= 2/3V
1
=
0.66V·cos(2
π
ft)
At all frequencies
c.
What is V
1
if V
2
=1V·cos(2
π
ft)?
V
2
= R3/(R2+R3)V
1
= V
1
/2 =
0.5V·cos(2
π
ft)
At all frequencies
3)
RC: analyze the circuit in Fig. 2b
a.
What are Zac, Zbc and Zab?
At f=0Hz?
Zac = 2k
Ω
+1/(j·f·628nF)
b
∞
when f=0
Zbc = 500
Ω
+1/(j·f·628nF)
∞
when f=0
Zab = 2.5k
Ω
at all frequencies
b.
What is V
2
if V
1
=1V·cos(2
π
ft)?
In phasors:
V
2
= 1/(1+j
2
π
f·200
μ
s
)V
1
In time:
V
2
=1V/(1+f
2
·1.6·10
6
)
1/2
cos(2
π
ft+atan(f·1.26ms))
At 1kHz:
V
2
=0.63Vcos(6.28krad/s·t0.9)
At 100kHz:
V
2
=0.008Vcos(6.28krad/s·t1.56)
Max at f = 1kHz
c.
What is V
1
if V
2
=1V·cos(2
π
ft)?
In phasors:
V
2
= 1/(1+j
2
π
f·50
μ
s
)V
1
In time:
V
2
=1V/(1+f
2
·10
7
)
1/2
cos(2
π
ft+atan(f·314
μ
s))
At 1kHz:
V
2
=0.95Vcos(6.28krad/s·t0.3)
At 100kHz:
V
2
=0.032Vcos(6.28krad/s·t1.53)
Max at f = 1kHz
d.
Based on the zerofrequency impedances and voltage divider results, how
would you extract the values of R1 and R2 for figure 2b?
Based on resistance, you know component 3 is a capacitor, and R1+R2 =
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 Spring '05
 KELLEY/SEYLER
 Frequency, Volt, Voltage divider, ZAC, ωRC, jf 7.5mH

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