ece201 sp2010 solutions
TRANSCRIPT
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ECE 201 Spring 2011
Homework Sets (All problems are from the 3rd edition of DeCarlo and Lin)
HW Set #1 Due W 1/12
1. Chapter 1 – prob. 1 (For part e, please use Figure P1.1b.) 2. Chapter 1 – prob. 7(a) 3. Chapter 1 – prob. 8
HW Set #2 Due F 1/14
1. Chapter 1 – prob. 11 2. Chapter 1 – prob. 13 3. Chapter 1 – prob. 17
HW Set #3 Due W 1/19
1. Chapter 1 – prob. 19 (Be careful. Table 1.2 lists relative, not absolute, resistivities!)
2. Chapter 1 – prob. 22(a) 3. Chapter 1 – prob. 27 4. Chapter 1 – prob. 37
HW Set #4 Due F 1/21
1. Chapter 2 – prob. 3 2. Chapter 2 – prob. 6 3. Chapter 2 – prob. 14 There are two typos:
1) In 14.(a), “Figure P2.13” should be “Figure P2.14” 2) In 14.(b), “I1 = 4A” should be “Iin = 4A”
4. Chapter 2 – prob. 17
HW Set #5 Due M 1/24
1. Chapter 2 – prob. 26 (It is the current out of the positive terminal of the source)
2. Chapter 2 – prob. 32 (a) 3. Chapter 2 – prob. 40
HW Set #6 Due W 1/26
1. Chapter 2 – prob. 39 2. Chapter 2 – prob. 46 3. Chapter 2 – prob. 63
HW Set #7 Due F 1/28
1. Chapter 3 – prob. 2 2. Chapter 3 – prob. 6(a)
HW Set #8 Due M 1/31
1. Chapter 3 – prob. 13 2. Chapter 3 – prob. 18 3. Chapter 3 – prob. 26
HW Set #9 Due F 2/4
1. Chapter 3 – prob. 37 2. Chapter 3 – prob. 42 3. Chapter 3 – prob. 52
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HW Set #10 Due M 2/7
1. Chapter 5 – prob. 1 2. Chapter 5 – prob. 21(a) 3. Chapter 5 – prob. 26
HW Set #11 Due W 2/9
1. Chapter 5 – prob. 41 2. Chapter 5 – prob. 43
HW Set #12 Due M 2/14
1. Chapter 6 – prob. 2 2. Chapter 6 – prob. 5
HW Set #13 Due W 2/16
1. Chapter 6 – prob. 9(a) and (b) 2. Chapter 6 – prob. 17 3. Chapter 6 – prob. 21 4. Chapter 6 – prob. 37
HW Set #14 Due F 2/18
1. Chapter 6 – prob. 48 2. Chapter 6 – prob. 53
In Figure P6.53, change the symbol inside the circle on the left of the circuit to an upward arrow for is1.
HW Set #15 Due M 2/21
1. Chapter 7 – prob. 2 (Change part (b) to read “Find and plot the instantaneous absorbed power.”)
2. Chapter 7 – prob. 4
HW Set #16 Due W 2/23
1. Chapter 7 – prob. 12 2. Chapter 7 – prob. 16 3. Chapter 7 – prob. 17
HW Set #17 Due F 2/25
1. Chapter 7 – prob. 27 2. Chapter 7 – prob. 38 3. Chapter 7 – prob. 41
HW Set #18 Due M 2/28
1. Chapter 8 – prob. 4 2. Chapter 8 – prob. 5 3. Chapter 8 – prob. 9(b)
HW Set #19 Due W 3/2
1. Chapter 8 – prob. 18 (a) and (b) 2. Chapter 8 – prob. 19 (a) and (b) 3. Chapter 8 – prob. 20 (a) and (b)
HW Set #20 Due F 3/4
1. Chapter 8 – prob. 18 (c) - (e) 2. Chapter 8 – prob. 19 (c) - (e) 3. Chapter 8 – prob. 20 (c) and (d)
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HW Set #21 Due M 3/7
1. Chapter 8 – prob. 31 2. Chapter 8 – prob. 33
Note that −V0 and V1 are marked on the ordinate of Figure P8.33(a).
HW Set #22 Due W 3/9
1. Chapter 9 – prob. 1 2. Chapter 9 – prob. 10
HW Set #23 Due M 3/21
1. Chapter 9 – prob. 16(b) 2. Chapter 9 – prob. 18 (use C = 10mF instead of C = 8mF) 3. Chapter 9 – prob. 20
HW Set #24 Due W 3/23
1. Chapter 9 – prob. 28 2. Chapter 9 – prob. 32(a) (compute the response only; no need to plot)
HW Set #25 Due F 3/25
1. Chapter 9 – prob. 43 2. Chapter 9 – prob. 49
Please use the following figure in place of Figure P9.49 in the textbook.
HW Set #26 Due M 3/28
1. Chapter 4 – prob. 3 2. Chapter 4 – prob. 5
HW Set #27 Due W 3/30
1. Chapter 4 – prob. 7 2. Chapter 4 – prob. 13
HW Set #28 Due F 4/1
1. Chapter 6 – prob. 29 2. Chapter 6 – prob. 31
Change answer to part b to “(b) 3R”
HW Set #29 Due M 4/4
1. Chapter 8 – prob. 40(a) 2. The op amps in the following circuit are ideal
For vIN(t) as shown in the plot, find and plot vout(t). Label the axis.
(continue on next page…)
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HW Set #30 Due W 4/6
1. Add the complex numbers j / 41z 10e π= and j5 / 8
2z 5e π= . Express your answer (a) in real and imaginary parts, and (b) as a magnitude and phase. Draw z1, z2 and z1 + z2 on the complex plane. Label the real part, the imaginary part, and magnitude and the phase of z1 + z2.
2. Multiply the complex numbers z1 = 12-16 j and z2 = 1+0.75j. Express your answer (a) in real and imaginary parts, and (b) as a magnitude and phase. (c) Compare 1z 2z to 1 2z z . (d) What is the relationship between the phase of z1, the phase of z2, and the phase of z1z2?
3. For z1 and z2 given in problem 2, determine z1/z2. Express your answer (a) in real and imaginary parts, and (b) as a magnitude and phase. (c) Compare 1z and 2z to 1 2z / z . (d) What is the relationship between the phase of z1, the phase of z2, and the phase of z1/z2?
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HW Set #31 Due F 4/8
1. Chapter 10 – prob. 7 2. Chapter 10 – prob. 10 3. Chapter 10 – prob. 14
HW Set #32 Due M 4/11
1. Chapter 10 – prob. 21 2. Chapter 10 – prob. 23 3. Chapter 10 – prob. 25
HW Set #33 Due W 4/13
1. Chapter 10 – prob. 30 2. Chapter 10 – prob. 40
HW Set #34 Due M 4/18
1. Chapter 10 – prob. 46 2. Chapter 10 – prob. 54 3. Chapter 10 – prob. 60
HW Set #35 Due W 4/20
1. Chapter 10 – prob. 66 2. Chapter 10 – prob. 68
HW Set #36 Due F 4/22
1. Chapter 11 – prob. 1 2. Chapter 11 – prob. 2
HW Set #37 Due M 4/25
1. Chapter 11 – prob. 5 2. Chapter 11 – prob. 7 3. Chapter 11 – prob. 9 4. Chapter 11 – prob. 10
HW Set #38 Due W 4/27
1. Chapter 11 – prob. 17 2. Chapter 11 – prob. 20
HW Set #39 Due F 4/29
1. Chapter 11 – prob. 22 2. Chapter 11 – prob. 23 3. Chapter 11 – prob. 26
The last sentence before (a) should be: “This constitutes a pf of 0.866 lagging.”
HW Set #40 Not collected. Solutions will be posted.
1. Chapter 11 – prob. 30 2. Chapter 11 – prob. 37
Homework #3
Prob. 1-19
(a) 2.575 Ω 800 2.575 0.8 Ω 2.06Ω
(b) From table 1.2, 200 feet of 14-gauge nickel wire has resistance:
5.1 2.575 Ω1000 200 5.1 2.575 0.001 200 2.6265Ω
(c) !""#$ % &'!(#)
!""#$ % &'!(#) 2.06 % 2.6265 4.6865Ω
Prob. 1-22
(a) + ,- 400 sin20πt 1034- % 2 % 5 0.16 sin20πt- 8
1.28 5,6-20πtW
>> t=[0:0.0001:0.5];
>> p=1.28*(sin(20*pi*t)).^2;
>> plot(t,p);
>> grid;
>> xlabel('t (time)');
>> ylabel('p(t) (instantaneous power)');
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
t (time)
p(t)
(in
stan
tane
ous
pow
er)
Prob 1-27
(a) + 89:;
< -
+ 120 12060 240Ω
(b) + 89:; 2
< 2 -
+ 2 120 120150 192Ω
Prob 1-37
'& '&
> ?'&-
?'&-
+;- >- - ?-'&- -
-- - ?-'&- -
-
ECE 201 Spring 2010
Homework 5 Solutions
Problem 26
Since V2 = 60 V , the current through 60 Ω branch is 1 A. The resistors 90 Ω
and 180 Ω are in parallel. Their equivalent resistance is
Req =90 × 180
90 + 180= 60
Now 60 Ω and 60 Ω are in series. Thus the voltage drop across the parallel
combination of 40 Ω and 120 Ω is 120 V. Thus current through 40 Ω resistor
is 120/40=3 A. Hence,
Is = 3 + 1
= 4 A
Applying KVL around the loop containing the source and 40 Ω resistor,
Vs − 180 × 4 − 120 = 0
⇒ Vs = 840 V
Power delivered by the source is given by,
Ps = Vs × Is
= 840 × 4
= 3360 W
Problem 32(a)
Using voltage division, the following equations can be written,
R1 + R2 + Rs = 2400 (1)
R1 + R2
R1 + R2 + Rs
= 0.75 (2)
R2
R1 + R2 + Rs
= 0.25 (3)
1
Using (1) and (3), R2 = 600 Ω. Using (2), R1 = 1200 Ω. Using (1), Rs =
600 Ω.
Problem 40
The resistors 9 kΩ and 18 kΩ are in parallel. Thus their equivalent resistance
of 6 kΩ is in series with 6 kΩ. Now the combination 4 kΩ and 12 kΩ are in
parallel. The current Iin divides into I1 and I2 through the parallel resistors
in inverse ratio of their resistances. Similarly, the current I2 divides in inverse
ratio of resistances 9 kΩ and 18 kΩ in parallel. Thus we get,
I1 = 120 ×12
12 + 4= 90 mA
I2 = 120 − 90 = 30 mA
I3 = 30 ×9
9 + 18= 10 mA
Vin = 0.09 × 4 × 103
= 360 V
Psource = VinIin
= 43.2 W
2
ECE 201 Spring 2010
Homework 9 Solutions
Problem 37
(a)
The following loop equation can be written,
−(I1 − 0.75)200 − 300I1 − (I1 + 0.1)500 = 0
⇒ I1 = 0.1 A
P = V I
Ps1 = 200(0.75 − 0.1)0.75
= 97.5 W
Ps2 = 500(0.1 + 0.1)0.1
= 10 W
(b)
Again, writing the loop equation and comparing the equation obtained with
that given in the problem,
−(I1 − 0.4)R1 − 600I1 − (I1 + 0.1)R2 = 0
(R1 + R2 + 600)I1 = 0.4R1 − 0.1R2
⇒ R1 + R2 = 1400
4R1 − R2 = 600
⇒ R1 = 400 Ω
R2 = 1000 Ω
1
Problem 42
(a)
The following loop equations can be written,
21 − 20I1 − 80(I1 − I2) = 0
24 + 80I2 − 80(I1 − I2) = 0
⇒ I1 = 0.15 A
I2 = −0.075 A
VR3= (I1 − I2)R3
= 0.225 × 80 = 18 V
PR3= 18(0.225) = 4.05 W
(b)
Again, writing the loop equations,
21 − (I1 + I2)20 − 80I2 − 24 = 0
21 − (I1 + I2)20 − 80I1 = 0
⇒ I1 = 0.225 A
I2 = −0.075 A
VR3= I1R3
= 0.225 × 80 = 18 V
PR3= 18(0.225) = 4.05 W
Problem 52
(a)
Assume clockwise loop currents I1 and I2 in the two rightmost loops respec-
tively. The following equations can then be written for the two loops,
−(I1 − Is1)R1 − rmIx − R2(I1 − I2) = 0
Ix = Is1 − I1
−R2(I2 − I1) − R3I2 − Vs2 = 0
2
Putting in matrix form,
rm − R1 − R2 R2
R2 −(R2 + R3)
I1
I2
=
(rm − R1)Is1
Vs2
(b)
Putting the respective literal values,
−80 40
40 −120
I1
I2
=
−40
40
⇒ I1 = 0.4 A
I2 = −0.2 A
(c)
VA = (1 − 0.4)100 = 60 V
VB = 0.6(40) = 24 V
(d)
Ps1 = 60 × 1 = 60 W
Ps2 = 40 × 0.2 = 8 W
(e)
Pdep = 60 × 0.6 ×−0.4 = −14.4 W
3
ECE 201 Spring 2010
Homework 13 Solutions
Problem 9
(a)
2R and 6R in series gives 8R. 8R and 8R in parallel gives 4R. Thus 12R is in
series with Vs in the simplified circuit. Thus the Thevenin voltage is given
by
Voc =Vs
12R×
1
2× 6R
=Vs
4= 30 V
To find the Thevenin resistance, we short circuit Vs. Thus 8R and 8R are
in parallel, which gives 4R, which in turn is in series with 2R. This gives 6R
and 6R in parallel. Thus Rth = 3R = 900 Ω.
(b)
Using the Thevenin equivalent circuit to simplify our calculations,
PRL=
(
30
900 + RL
)2
RL
= 0.1875 W, 0.24 W, 0.244898 W (RL = 300 Ω, 600 Ω, 1200 Ω)
The Thevenin equivalent circuit analysis allows us to analyze the circuit
without the load once and then plug in the various load resistor values to
compute the relevant quantities. However, using earlier techniques would
require 3 separate analyses to compute these values. Hence the use of a
Thevenin equivalent reduces the effort needed to obtain the answers.
1
(c)
Using Maximum Power Transfer Theorem, for maximum power trans-
fer, RL = 900 Ω and resultant power delivered to the load is given by
Pmax = (30)2/(4 × 900) = 0.25 W .
(d)
If Vs is doubled, power becomes four times, as Pmax ∝ V 2
s . Thus power
delivered to the load becomes Pload = 0.25 × 4 = 1 W .
Problem 17
We proceed to find the Thevenin voltage Voc and Thevenin resistance Rth.
Norton current is then given by Isc = Voc/Rth and Norton resistance is same
as Thevenin resistance. Since this problem involves a dependent source, we
cannot apply the conventional method. We write KVL equations to find the
required values.
vAB − 300iA − vx = 0
vx − 200(iA − vx/18000) −4
3vx = 0
⇒ vx = −18000
29iA
vAB = −320.689655iA
vAB = RthiA + Voc
⇒ Voc = 0
Rth = −320.69 Ω
⇒ Isc = 0
Rnorton = −320.69 Ω
Problem 21
(a)
Again, we write KVL equations to find the Thevenin and Norton equivalent
circuits.
vAB − 40iA − 200(iA + kvx + vx/50) = 0
Vs − vx + 40iA − vAB = 0
2
⇒ vAB [1 + 200(k + 0.02)] = iA [240 + 8000(k + 0.02)] + 200Vs(k + 0.02)
vAB = 60iA + 18
vAB = iARth + Voc
⇒ Voc = 18 V
Rth = 60 Ω
Isc = 0.3 A
(b)
Clearly, using the previous set of equations, Voc is zero for k = −0.02 and for
this value of k Rth = 240 Ω.
Problem 37
(a)
vAB = iARth + Voc
54 = 0.01Rth + Voc
66 = 0.04Rth + Voc
⇒ Rth = 400 Ω
Voc = 50 V
Isc = 0.125 A
(b)
RL = 400 Ω, Pmax = V 2
oc/4RL = 2500/(4 × 400) = 1.5625 W .
3
P 4 (a)
P 4 (c)
0 0.005 0.01 0.015-10
-8
-6
-4
-2
0
2
4
6
8
10
t(s)
i out(t
) (m
A)
0 0.005 0.01 0.0150
10
20
30
40
50
60
70
80
90
100
t(s)
WL(t
) (n
J)
ECE 201 Spring 2010
Homework 17 Solutions
Problem 27
(a)
Let V be the common voltage across C1 and C2. Thus
(C1 + C2)dV
dt= is(t)
C2
dV
dt= iC2(t)
⇒ iC2(t) =C2
C1 + C2
is(t)
(b)
Using KVL across the second loop and relation from part (a), we get
rmiC2(t) − (L1 + L2)di
dt= 0
vout(t) = L2
di
dt
= rm
L2C2
(L1 + L2)(C1 + C2)is(t)
Problem 38
(a)
Starting from the right end and combining the inductors in series and parallel
consecutively, we get Leq as
Leq = 4 + ((36)||(10 + ((1 + 5)||(3))))
= 4 + ((36)||(10 + ((6)||(3))))
1
= 4 + ((36)||(10 + (2)))
= 4 + ((36)||(12))
= 4 + 9
= 13 mH
(b)
Clearly, the given circuit has 1.2 mH and 0.6 mH in parallel, and this com-
bination is in series with 2.4 mH, the equivalent of which is in parallel with
7 mH. Thus
Leq = 7||(2.4 + (1.2||0.6))
= 7||(2.4 + 0.4)
= 7||2.8
= 2 mH
Problem 41
(a)
Remember that the equivalent capacitance expressions for series and par-
allel connections are opposite of that used for resistances. The equivalent
capacitance can be written as
Ceq = C1||(C2 + (C3||C4))
= 4 +6 × 3
9= 6 µF
vs(t) =1
Ceq
∫ t
−∞
is(τ) dτ
=1
6sin(104
t) V
(b)
Ceq = C1||(C2 + (C3||C4) + C5)
2
(C2 + (C3||C4) + C5) =(
1
18+
1
54+
1
10.8
)
−1
= 6 µF
⇒ Ceq = 6 + 60
= 66 µF
vs(t) =1
Ceq
∫ t
−∞
is(τ) dτ
=1
660(1 − cos(105
t)) V
3
8-18(a)
8-18(b)
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
18
20
t(s)
Vc(t
)(V
)
0 2 4 6 8 10 12 14 160
1
2
3
4
5
6
7
8
9
10
t(s)
Vc(t
)(V
)
8-19(a)
8-19(b)
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
t(s)
i L(t)(
A)
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.05
-0.045
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
t(s)
i L(t)(
A)
8-20(b)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-10
-5
0
5
10
15
20
t(s)
Vc(t
)(V
)
0 0.1 0.2 0.3 0.4
-5
0
5
10
15
20
8-20 (d) Vc
I(t)
t (s)
VcI(t
) (V
)
0 0.1 0.2 0.3 0.4
-5
0
5
10
15
20
8-20 (d) Vc
II(t)
t (s)
VcII(t
) (V
)
0 0.1 0.2 0.3 0.4
-5
0
5
10
15
20
8-20 (d) Vc(t)
t (s)
Vc(t
) (V
)
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
8-20 (d) Ic
I(t)
t (s)
I cI(t
) (A
)
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
8-20 (d) Ic
II(t)
t (s)
I cII(t
) (A
)
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
8-20 (d) Ic(t)
t (s)
I c(t
) (A
)
8-18
8-19
0 5 10 1510
11
12
13
14
15
16
17
18
19
20
8-18 (c) VC
(t)
t (s)
VC
(t)
(V)
0 5 10 15-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
8-18 (d) VC
(t)
t (s)
VC
(t)
(V)
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-3 8-18 (e) IC
(t)
t (s)
I C(t
) (A
)
0 0.005 0.01
0
0.05
0.1
0.15
0.2
8-19 (c) IL(t)
t (s)
I L(t
) (A
)
0 0.005 0.01
-0.22
-0.2
-0.18
-0.16
-0.14
-0.12
-0.1
8-19 (d) IL(t)
t (s)
I L(t
) (A
)
0 0.005 0.010
5
10
15
20
25
8-19 (e) VL(t)
t (s)
VL(t
) (V
)
ECE 201 Spring 2010
Homework 21 Solutions
Problem 31
(a)
Using voltage division,
vC(0−) =20R
50RV0
= 0.4V0
= vC(0+)
(b)
The Thevenin equivalent is given by
Voc = V0
R3
R3 + (R1||R2)
= 0.8V0
Rth = R1||R2||R3
= 4R
(c)
vC(∞) = 0.8V0
vC(0+) = 0.4V0
⇒ vC(t) = vC(∞) + [vC(t0+) − vC(∞)]e−
t−t0
RthC
= 0.8V0 − 0.4V0e−
t
4RC
1
(d)
vC(T−) = vC(T+) = 0.8V0 − 0.4V0e−1.5
= 0.71075V0
(e)
τ = RthC
= 4RC
(f)
Using the same relation as in part (c), with t0 = T ,
vC(t) = 0.71075V0e−
t−T
4RC
(g)
For simplicity, let all quantities be normalized to 1. Then the plot for vC(t)
would look as shown on the next page.
Problem 33
(a)
iL(0+) = iL(0−) =−10
80 + 20= −0.1 A
2
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
v C(t
)
Plot of vC
(t) for 0<=t<=4T
Figure 1: Plot of vC(t) for problem 31
(b)
Here we have to consider two intervals, 0 ≤ t < 80 ms and 80 ms ≤ t <
160 ms. For 0 ≤ t < 80 ms,
iL(t) = iL(∞) + [iL(0+) − iL(∞)]e−tR/L
=20
100+ [−0.1 − 0.2]e−25t
= 0.2 − 0.3e−25t
iL(80−) = iL(80+) = 0.2 − 0.3e−2
= 0.1594 A
Again, for the interval 80 ms ≤ t < 160 ms,
iL(t) = iL(∞) + [iL(80+) − iL(∞)]e−(t−80)R/L
= −0.1 + [0.1594 + 0.1]e−25(t−80)
= −0.1 + 0.2594e−25(t−80)
We can write vout(t) as the following,
vout(t) = vin(t) − R1iL(t)
3
= 4 + 24e−25t, 0 ≤ t < 80 ms
= −2 − 20.752e−25(t−80ms), 80 ms ≤ t < 160 ms
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16−30
−20
−10
0
10
20
30
t(seconds)
v out(t
)(vo
lts)
Plot of vout
(t) for 0<=t<=160ms
Figure 2: Plot of vout(t) for problem 33
4
9-16(b)
0 1 2 3 4 5 6 7 8-1
-0.5
0
0.5
1
1.5
2
t(s)
Vc(t
) (V
)
9-18(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9
10
t(s)
Vc(t
)(V
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
t(s)
i L(t)(
A)
9-18(b)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
12
t(s)
Vc(t
)(V
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
t(s)
i L(t)(
A)
ECE 201 Spring 2010
Homework 25 Solutions
Problem 43
(a)
To find the initial conditions at t=0-, we can write the following KVL equa-
tions,
20 + 40iL(0−) − 60(0.1 − iL(0−)) = 0
⇒ iL(0−) = iL(0+) = iC(0+) = −0.14 A
⇒ vC(0+) = vC(0−) = 20 − 0.14 × 40
= 14.4 V
After t=0, the circuit is a series RLC circuit with the following characteristic
equation,
s2 +
R
Ls +
1
LC= 0
s2 + 15s + 50 = 0
⇒ s1 = −5, s2 = −10
⇒ vC(t) = c1e−5t + c2e
−10t
⇒ c1 + c2 = 14.4
(5 × 10−3)(5c1 + 10c2) = 0.14
⇒ c1 = 23.2, c2 = −8.8
⇒ vC(t) = (23.2e−5t− 8.8e−10t) V
The plot of vC(t) for 0 < t ≤ 1 is shown on the next page.
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
t(seconds)
v C(t
)(V
olts
)
Plot of vC
(t) (part (a)) for 0<t<=1s
(b)
In this case, the situation after t=0 can be visualized as a series RLC circuit
with a constant input voltage source (after source transformation) which will
have the value 0.5× 60 = 30 V . The initial conditions and the characteristic
equation do not change. However, the expression for vC(t) will now have a
term due to the constant input also. Thus,
vC(t) = c1e−5t + c2e
−10t + 30
⇒ c1 + c2 + 30 = 14.4
5c1 + 10c2 = 28
⇒ vC(t) = (−36.8e−5t + 21.2e−10t + 30) V
The plot of vC(t) for 0 < t ≤ 1 follows.
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 114
16
18
20
22
24
26
28
30
t(seconds)
v C(t
)(V
olts
)
Plot of vC
(t) (part (b)) for 0<t<=1s
Problem 49
(a)
At t=0-, the capacitor is open circuit and the inductor is short circuit. At
t=0+, the current source is off and the voltage source is still operating. Thus
we can write the following equations,
vC(0−) = vC(0+) = 10 V
iL(0−) + 0.005 =10
1000⇒ iL(0+) = iL(0−) = 0.005 A
10 − vL(0+) − vC(0+) = 0
⇒ vL(0+) = 0
iC(0+) +10
1000= 0.005
⇒ iC(0+) = −0.005 A
3
(b)
For t > 0, we can write the following KVL and KCL equations,
10 − LdiL
dt− vC = 0
iL = CdvC
dt+
vC
R
⇒d2vC
dt2+
1
RC
dvC
dt+
vC
LC=
10
LC
Thus the characteristic equation is
s2 + 2000s + 1.087 × 107 = 0
⇒ s1,2 = −1000 ± j1000π
⇒ vC(t) = e−1000t[A cos(1000πt) + B sin(1000πt)] + 10
A = 0, B = −10
π(Using initial conditions)
⇒ vC(t) = 10 −10
πe−1000t sin(1000πt) V
iL(t) = CdvC
dt+
vC
R
= 0.005[
2 − e−1000t
1
πsin(1000πt) + cos(1000πt)
]
A
(c)
vC(t) is computed above in part (b).
(d)
vL(t) = 10 − vC(t)
=10
πe−1000t sin(1000πt) V
(e)
iC(t) = CdvC
dt
=0.005
πe−1000t[sin(1000πt) − π cos(1000πt)] A
The plots for the respective quantities follow.
4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
5
6
7
8
9
10
11
12x 10
−3
t(seconds)
i L(t)(
A)
Plot of iL(t) for 0<t<=5ms
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
7.5
8
8.5
9
9.5
10
10.5
11
t(seconds)
v C(t
)(V
)
Plot of vC
(t) for 0<t<=5ms
5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
−1
−0.5
0
0.5
1
1.5
2
2.5
t(seconds)
v L(t)(
V)
Plot of vL(t) for 0<t<=5ms
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
−5
−4
−3
−2
−1
0
1
2
3x 10
−3
t(seconds)
i C(t
)(A
)
Plot of iC
(t) for 0<t<=5ms
6
ECE 201 Spring 2010
Homework 29 Solutions
Problem 40
(a)
The following differential equation can be written using KCL at the inverting
terminal of the op amp and using the virtual ground concept,
vs(t)
R1
+vout(t)
R2
+ Cdvout(t)
dt= 0
vs(t) = −100u(t)
= −100, t ≥ 0
⇒ vout(t) =100R2
R1
(1 − e−t/R2C)
= 400(1 − e−2t) mV, t ≥ 0
Problem not from the book
If we denote the voltage at the non-inverting terminal of the op amp as
vL(t), then using the virtual ground concept, the output voltage is given
by Vout(t) = 10vL(t). Also, vL(t) = VIN(t) − 1000iL(t). The problem then
reduces to finding iL(t). Due to the square wave shape of VIN(t), the graph
for iL(t) will also be periodic with maximum and minimum values of x and
−x (say). The task is to find the value of x, which can be done using the
following equation,
iL(t) = iL(∞) + [iL(0) − iL(∞)]e−t/τ
τ =L
R
= 2 µs
x = 2 × 10−4 + [−x − 2 × 10−4]e−0.5
⇒ x = 0.489 × 10−4
1
The plot of Vout(t) for 0 ≤ t ≤ 2µs is drawn below. Note that Vout(t) will be
periodic with period 2µs.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−6
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
t(seconds)
Vou
t(t)(
Vol
ts)
Plot of Vout
(t) for 0<=t<=2us. Note that it is periodic with period 2us
2
ECE 201 Spring 2010
Homework 33 Solutions
Problem 30
(a)
Yin(jω) =1
R+
1
jωL
= 0.05 −j0.25
ω
Zin(jω) =1
Yin(jω)
=j20ω
5 + jω
(b)
IL =IinZin(jω)
jωL
= 10 ×10(i + j)
j20
= 5√
2e−jπ/4
⇒ iL(t) = 10 cos(5t − π/4) mA
Problem 40
Let the impedances of R1, C and R2, L combinations be Z1 and Z2 respec-
tively.
Z1 =[
1
500+ j(400 × 5 × 10−6)
]
−1
= 250(1 − j)
Z2 =
[
1
100+
1
j(400 × 0.125)
]
−1
= 20(1 + 2j)
1
Let the phasor current through the source be I.
I =V
270 − j210
⇒ VC = I × 250(1 − j)
=120√
2(1.0256 − j0.1282)
⇒ vC(t) = 124.032 cos(400t− 7.125) V
IL =
(
V
270 − j210
)(
R2
R2 + jωL
)
=V × 100
(270 − j210)(100 + j50)
=240√
2(0.00128205 + j0.0002564)
⇒ iL(t) = 0.3138 cos(400t + 11.3099) A
2
0 0.5 1 1.5 2 2.5 3-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.0111-2(a) Instantaneous Current i(t)
time (s)
curr
ent
(A)
Instantaneous Current
0 0.5 1 1.5 2 2.5 30
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
11-2(a) Instantaneous Power p(t) and Average Power pave
time (s)
pow
er
(W)
Instantaneous Power
Average Power
0 0.5 1 1.5 2 2.5 3-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.0111-2(b) Instantaneous Current i(t)
time (s)
curr
ent
(A)
Instantaneous Current
0 0.5 1 1.5 2 2.5 30
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
11-2(b) Instantaneous Power p(t) and Average Power pave
time (s)
pow
er
(W)
Instantaneous Power
Average Power
0 0.5 1 1.5 2 2.5 3-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.0111-2(c) Instantaneous Current i(t)
time (s)
curr
ent
(A)
Instantaneous Current
0 0.5 1 1.5 2 2.5 30
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
11-2(c) Instantaneous Power p(t) and Average Power pave
time (s)
pow
er
(W)
Instantaneous Power
Average Power
ECE 201 Spring 2010
Homework 37 Solutions
Problem 5
(a)
For Figure (a),
I2
eff =1
T
∫ T
0
i2(t) dt
=1
9
[∫
3
0
9 dt +∫
6
3
16 dt
]
=25
3
⇒ Ieff = 5/√
3
For Figure (b),
I2
eff =1
4
[∫
1
0
9 dt +∫
3
2
16 dt
]
= 25/4
⇒ Ieff = 2.5
(b)
Current through RL is given by i(t) × 60/(30 + 60) = (2/3)i(t). Power
absorbed by RL is then given by the square of the effective current through
it times the resistance. Thus
P =4
9×
25
3× 30
=1000
9W
1
(c)
P =4
9×
25
4× 30
=250
3W
Problem 7
In all parts of the problem, the following identities will be used,
∫
2π/m
0
cos(mnx) dx = 0
∫
2π/m
0
sin(mnx) dx = 0
Here m and n are integers.
(a)
V2
1eff =1
T
∫ T
0
v2
1(t) dt
=1
T
∫ T
0
[102 + 40 cos(20t) + 2 cos(40t)] dt
= 102
⇒ V1eff = 10.0995
(b)
V2
2eff =1
T
∫ T
0
v2
2(t) dt
=1
T
∫ T
0
[501 + cos(4t) + 12.51 + cos(8t) + 50cos(6t) + cos(2t)] dt
= 62.5
⇒ V2eff = 7.9057
(c)
V2
3eff =1
T
∫ T
0
v2
3(t) dt
=1
T
∫ T
0
[501 + cos(4t) + 36.42771 + cos(8t) + 6.24991 − cos(8t) + . . .] dt
= 92.6776
⇒ V3eff = 9.6269
2
Problem 9
(a)
ZL =[
1
5+ j(30 × 5 × 10−3)
]
−1
= 0.8(4 − j3)
⇒ VL = Iin × ZL
=4√
2(4 − j3)
⇒ vL(t) = 20 cos(30t − 36.87) V
(b)
Pinst(t) = iin(t)vL(t)
= 100 cos(30t) cos(30t − 36.87) W
Paverage =VmIm
2cos(θv − θi)
= 50 cos 36.87
= 40 W
Problem 10
(a)
Is =50 6 − 90
6 + j12 − j4
= 5 6 − 143.13 (magnitude = 5)
(b)
Paverage = VeffIeff cos(θv − θi)
= 5 × 50 cos(−53.13)
= 150 W
3
(c)
Paverage = R|Is|2
= 6 × 25
= 150 W (same as part (b))
(d)
Is =50 6 − 90
30 + j50 − j10
= 1 6 − 143.13 (magnitude = 1)
Paverage = VeffIeff cos(θv − θi)
= 1 × 50 cos(−53.13)
= 30 W
4
ECE 201 Spring 2010
Homework 40 Solutions
Problem 30
(a)
Zth = (R1||R2) − j/ωC
= (20 − j10) Ω
Voc = Vs
R2
R1 + R2
= 20 Vrms
For maximum power transfer, ZL = Z∗
th = (20 + j10) Ω.
(b)
Pavg =V 2
oc
4RL
= 5 W
Problem 37
(a)
Here we cannot apply the maximum power transfer theorem because RL is
fixed. The expression for average power across RL is given by
Pavg =V 2
s RL
(R + RL)2 + (ωL − 1/ωC)2
Clearly, the average power is maximum when the denominator is minimum,
which happens when the following two conditions hold,
R = 0, ωL =1
ωC
1
C =1
ω2L
= 2.5 mF
Pavgmax=
V 2
s
RL
=502
5= 500 W
(b)
Pavg =V 2
s RL
(R + RL)2 + (ωL − 1/ωC)2
For maximum Pavg ,
dPavg
dRL
= 0
⇒ R2
L = R2 + (ωL − 1/ωC)2
⇒ RL = 4.011 Ω
2