3
Simple Resistive Circuits
Assessment Problems
AP 3.1
Start from the right hand side of the circuit and make series and parallel
combinations of the resistors until one equivalent resistor remains. Begin by
combining the 6 Ω resistor and the 10 Ω resistor in series:
6 Ω + 10 Ω = 16 Ω
Now combine this 16 Ω resistor in parallel with the 64 Ω resistor:
16 Ωk64 Ω =
(16)(64)
1024
=
= 12.8 Ω
16 + 64
80
This equivalent 12.8 Ω resistor is in series with the 7.2 Ω resistor:
12.8 Ω + 7.2 Ω = 20 Ω
Finally, this equivalent 20 Ω resistor is in parallel with the 30 Ω resistor:
20 Ωk30 Ω =
600
(20)(30)
=
= 12 Ω
20 + 30
50
Thus, the simplified circuit is as shown:
3–1
3–2
CHAPTER 3. Simple Resistive Circuits
[a] With the simplified circuit we can use Ohm’s law to find the voltage across
both the current source and the 12 Ω equivalent resistor:
v = (12 Ω)(5 A) = 60 V
[b] Now that we know the value of the voltage drop across the current source,
we can use the formula p = −vi to find the power associated with the
source:
p = −(60 V)(5 A) = −300 W
Thus, the source delivers 300 W of power to the circuit.
[c] We now can return to the original circuit, shown in the first figure. In this
circuit, v = 60 V, as calculated in part (a). This is also the voltage drop
across the 30 Ω resistor, so we can use Ohm’s law to calculate the current
through this resistor:
60 V
=2A
30 Ω
Now write a KCL equation at the upper left node to find the current iB :
iA =
−5 A + iA + iB = 0
so
iB = 5 A − iA = 5 A − 2 A = 3 A
Next, write a KVL equation around the outer loop of the circuit, using
Ohm’s law to express the voltage drop across the resistors in terms of the
current through the resistors:
−v + 7.2iB + 6iC + 10iC = 0
So
Thus
16iC = v − 7.2iB = 60 V − (7.2)(3) = 38.4 V
iC =
38.4
= 2.4 A
16
Now that we have the current through the 10 Ω resistor we can use the
formula p = Ri2 to find the power:
p10 Ω = (10)(2.4)2 = 57.6 W
AP 3.2
Problems
3–3
[a] We can use voltage division to calculate the voltage vo across the 75 kΩ
resistor:
75,000
vo (no load) =
(200 V) = 150 V
75,000 + 25,000
[b] When we have a load resistance of 150 kΩ then the voltage vo is across the
parallel combination of the 75 kΩ resistor and the 150 kΩ resistor. First,
calculate the equivalent resistance of the parallel combination:
75 kΩk150 kΩ =
(75,000)(150,000)
= 50,000 Ω = 50 kΩ
75,000 + 150,000
Now use voltage division to find vo across this equivalent resistance:
50,000
vo =
(200 V) = 133.3 V
50,000 + 25,000
[c] If the load terminals are short-circuited, the 75 kΩ resistor is effectively
removed from the circuit, leaving only the voltage source and the 25 kΩ
resistor. We can calculate the current in the resistor using Ohm’s law:
200 V
i=
= 8 mA
25 kΩ
Now we can use the formula p = Ri2 to find the power dissipated in the
25 kΩ resistor:
p25k = (25,000)(0.008)2 = 1.6 W
[d] The power dissipated in the 75 kΩ resistor will be maximum at no load
since vo is maximum. In part (a) we determined that the no-load voltage
is 150 V, so be can use the formula p = v 2 /R to calculate the power:
p75k (max) =
(150)2
= 0.3 W
75,000
AP 3.3
[a] We will write a current division equation for the current throught the 80Ω
resistor and use this equation to solve for R:
R
(20 A) = 4 A
so
20R = 4(R + 120)
i80Ω =
R + 40 Ω + 80 Ω
480
Thus
16R = 480
and
R=
= 30 Ω
16
3–4
CHAPTER 3. Simple Resistive Circuits
[b] With R = 30 Ω we can calculate the current through R using current
division, and then use this current to find the power dissipated by R,
using the formula p = Ri2 :
40 + 80
(20 A) = 16 A
so
pR = (30)(16)2 = 7680 W
iR =
40 + 80 + 30
[c] Write a KVL equation around the outer loop to solve for the voltage v,
and then use the formula p = −vi to calculate the power delivered by the
current source:
−v + (60 Ω)(20 A) + (30 Ω)(16 A) = 0
Thus,
so
v = 1200 + 480 = 1680 V
psource = −(1680 V)(20 A) = −33,600 W
Thus, the current source generates 33,600 W of power.
AP 3.4
[a] First we need to determine the equivalent resistance to the right of the
40 Ω and 70 Ω resistors:
1
1
1
1
1
Req = 20 Ωk30 Ωk(50 Ω + 10 Ω)
so
+
+
=
=
Req
20 Ω 30 Ω 60 Ω
10 Ω
Thus,
Req = 10 Ω
Now we can use voltage division to find the voltage vo :
40
(60 V) = 20 V
vo =
40 + 10 + 70
[b] The current through the 40 Ω resistor can be found using Ohm’s law:
vo
20 V
=
= 0.5 A
40
40 Ω
This current flows from left to right through the 40 Ω resistor. To use
current division, we need to find the equivalent resistance of the two
parallel branches containing the 20 Ω resistor and the 50 Ω and 10 Ω
resistors:
(20)(60)
= 15 Ω
20 Ωk(50 Ω + 10 Ω) =
20 + 60
Now we use current division to find the current in the 30 Ω branch:
15
(0.5 A) = 0.16667 A = 166.67 mA
i30Ω =
15 + 30
i40Ω =
Problems
3–5
[c] We can find the power dissipated by the 50 Ω resistor if we can find the
current in this resistor. We can use current division to find this current
from the current in the 40 Ω resistor, but first we need to calculate the
equivalent resistance of the 20 Ω branch and the 30 Ω branch:
(20)(30)
= 12 Ω
20 + 30
Current division gives:
20 Ωk30 Ω =
i50Ω =
Thus,
12
(0.5 A) = 0.08333 A
12 + 50 + 10
p50Ω = (50)(0.08333)2 = 0.34722 W = 347.22 mW
AP 3.5 [a]
We can find the current i using Ohm’s law:
i=
1V
= 0.01 A = 10 mA
100 Ω
[b]
Rm = 50 Ωk5.555 Ω = 5 Ω
We can use the meter resistance to find the current using Ohm’s law:
imeas =
AP 3.6 [a]
1V
= 0.009524 = 9.524 mA
100 Ω + 5 Ω
3–6
CHAPTER 3. Simple Resistive Circuits
Use voltage division to find the voltage v:
v=
75,000
(60 V) = 50 V
75,000 + 15,000
[b]
The meter resistance is a series combination of resistances:
Rm = 149,950 + 50 = 150,000 Ω
We can use voltage division to find v, but first we must calculate the
equivalent resistance of the parallel combination of the 75 kΩ resistor and
the voltmeter:
(75,000)(150,000)
75,000 Ωk150,000 Ω =
= 50 kΩ
75,000 + 150,000
Thus,
vmeas =
50,000
(60 V) = 46.15 V
50,000 + 15,000
AP 3.7 [a] Using the condition for a balanced bridge, the products of the opposite
resistors must be equal. Therefore,
(1000)(150)
= 1500 Ω = 1.5 kΩ
100
[b] When the bridge is balanced, there is no current flowing through the
meter, so the meter acts like an open circuit. This places the following
branches in parallel: The branch with the voltage source, the branch with
the series combination R1 and R3 and the branch with the series
combination of R2 and Rx . We can find the current in the latter two
branches using Ohm’s law:
100Rx = (1000)(150)
so
Rx =
5V
5V
= 20 mA;
iR2 ,Rx =
= 2 mA
100 Ω + 150 Ω
1000 + 1500
We can calculate the power dissipated by each resistor using the formula
p = Ri2 :
iR1 ,R3 =
p100Ω = (100 Ω)(0.02 A)2 = 40 mW
p150Ω = (150 Ω)(0.02 A)2 = 60 mW
Problems
3–7
p1000Ω = (1000 Ω)(0.002 A)2 = 4 mW
p1500Ω = (1500 Ω)(0.002 A)2 = 6 mW
Since none of the power dissipation values exceeds 250 mW, the bridge
can be left in the balanced state without exceeding the power-dissipating
capacity of the resistors.
AP 3.8 Convert the three Y-connected resistors, 20 Ω, 10 Ω, and 5 Ω to three
∆-connected resistors Ra , Rb , and Rc . To assist you the figure below has both
the Y-connected resistors and the ∆-connected resistors
(5)(10) + (5)(20) + (10)(20)
= 17.5 Ω
20
(5)(10) + (5)(20) + (10)(20)
= 35 Ω
Rb =
10
(5)(10) + (5)(20) + (10)(20)
= 70 Ω
Rc =
5
Ra =
The circuit with these new ∆-connected resistors is shown below:
From this circuit we see that the 70 Ω resistor is parallel to the 28 Ω resistor:
70 Ωk28 Ω =
(70)(28)
= 20 Ω
70 + 28
Also, the 17.5 Ω resistor is parallel to the 105 Ω resistor:
17.5 Ωk105 Ω =
(17.5)(105)
= 15 Ω
17.5 + 105
3–8
CHAPTER 3. Simple Resistive Circuits
Once the parallel combinations are made, we can see that the equivalent 20 Ω
resistor is in series with the equivalent 15 Ω resistor, giving an equivalent
resistance of 20 Ω + 15 Ω = 35 Ω. Finally, this equivalent 35 Ω resistor is in
parallel with the other 35 Ω resistor:
35 Ωk35 Ω =
(35)(35)
= 17.5 Ω
35 + 35
Thus, the resistance seen by the 2 A source is 17.5 Ω, and the voltage can be
calculated using Ohm’s law:
v = (17.5 Ω)(2 A) = 35 V
Problems
Problems
P 3.1
[a] From Ex. 3-1: i1 = 4 A,
i2 = 8 A,
is = 12 A
at node b: −12 + 4 + 8 = 0,
at node d: 12 − 4 − 8 = 0
[b] v1
=
4is = 48 V
v3 = 3i2 = 24 V
v2 = 18i1 = 72 V
v4 = 6i2 = 48 V
loop abda: −120 + 48 + 72 = 0,
loop bcdb: −72 + 24 + 48 = 0,
loop abcda: −120 + 48 + 24 + 48 = 0
P 3.2
[a] p4Ω
=
i2s 4 = (12)2 4 = 576 W
p18Ω = (4)2 18 = 288 W
p3Ω
=
(8)2 3 = 192 W
p6Ω = (8)2 6 = 384 W
[b] p120V (delivered) = 120is = 120(12) = 1440 W
[c] pdiss = 576 + 288 + 192 + 384 = 1440 W
P 3.3
[a] 3 kΩ and 4 kΩ resistors are in series. The simplified circuit is shown
below:
[b] The 1000 Ω and 1500 Ω resistors are in series, as are the 200 Ω & 100 Ω
resistors. The simplified circuit is shown below:
3–9
3–10
CHAPTER 3. Simple Resistive Circuits
[c] The 25 Ω, 10 Ω, and 15 Ω are in series, as are the 18 Ω and 22 Ω resistors.
The simplified circuit is shown below:
[d] The 60 Ω and 55 Ω resistors are in series, as are the 65 Ω and 40 Ω
resistors. The simplified circuit is shown below:
P 3.4
[a] The 40 Ω and 20 Ω resistors are in parallel. The simplified circuit is shown
below:
[b] The 100 Ω and 140 Ω resistors are in parallel, as are the 240 Ω and 300 Ω
resistors. The simplified circuit is shown below:
[c] The 150 kΩ, 160 kΩ, and 80 kΩ are in parallel, as are the 80 kΩ and
60 kΩ resistors. The simplified circuit is shown below:
[d] The 700 Ω and 600 Ω resistors are in parallel, as are the 2 kΩ and 6 kΩ
resistors. The simplified circuit is shown below:
Problems
P 3.5
3–11
Always work from the side of the circuit furthest from the source. Remember
that the current in all parallel-connected circuits is the same, and that the
voltage drop across all series-connected resistors is the same.
[a] Circuit in Fig. P3.3(a):
Req = [(3 k + 4 k)k7 k] + 6 k = [7 kk7 k] + 6 k = 9.5 kΩ
Circuit in Fig. P3.3(b):
Req = [600k2500] + 200 + 100 = 783.87 Ω
Circuit in Fig. P3.3(c):
Req = (25 + 10 + 15)k(18 + 22) = 50k40 = 22.22 Ω
Circuit in Fig. P3.3(d):
Req = ([(65 + 40)k50] + 60 + 55)k200 = [(105k50) + 60 + 55]k200
= 148.8k200 = 85.36 Ω
Vs2
202
[b] P =
= 42.1 mW
=
Req
9.5 × 103
P =
302
= 1148.1 mW
783.87
P =
702
= 220.5 W
22.22
2
P = I 2 R = 20 × (10−3 ) × 85.36 = 34.14 mW
P 3.6
Always work from the side of the circuit furthest from the source. Remember
that the current in all series-connected circuits is the same, and that the
voltage drop across all parallel-connected resistors is the same.
[a] Circuit in Fig. P3.4(a):
Req = (40k20) + 30 = 13.33 + 30 = 43.33 Ω
Circuit in Fig. P3.4(b):
Req = 100k140k[(240k300) + 200] = 100k140k(133.3 + 200)
= 100k140k333.33 = 49.6 Ω
3–12
CHAPTER 3. Simple Resistive Circuits
Circuit in Fig. P3.4(c):
Req = (80 kk60 k) + (150 kk160 kk80 k) + 100 k
= 34.28 k + 39.34 k + 100 k = 173.6 kΩ
Circuit in Fig. P3.4(d):
Req = [(650 + 800)k700k600] + (2 kk6 k) + 4 k = (1450k700k600) + 1.5 k + 4 k
= 264.2 + 1.5 k + 4 k = 5.76 kΩ
V2
202
=
= 9.2 W
R
43.33
For the circuit in Fig. P3.4(b):
[b] P =
2
P = I 2 (Req ) = 40 × (10−3 ) × 49.6 = 79.36 mW
For the circuit in Fig. P3.4(c):
P =
802
V2
= 36.86 mW
=
Req
173.6 × 103
For the circuit in Fig. P3.4(d):
P =
P 3.7
V2
702
= 850.6 mW
=
Req
5.76 × 103
[a] Circuit in Fig. P3.7(a):
Req = ([(15k60) + (30k45) + 20]k50) + 25 + 10 = [(12 + 18 + 20)k50] + 25 + 10
= (50k50) + 25 + 10 = 25 + 25 + 10 = 60 Ω
Circuit in Fig. P3.7(b) – begin by simplifying the 75 Ω resistor and all
resistors to its right:
[(18 + 12)k60 + 30]k75 = (30k60 + 30)k75 = (20 + 30)k75 = 50k75 = 30 Ω
Now simplify the remainder of the circuit:
Req = ([(30 + 20)k50] + (20k60))k40 = [(50k50) + 15]k40 = (25 + 15)k40
= 40k40 = 20 Ω
Circuit in Fig. P3.7(c) – begin by simplifying the left and right sides of
the circuit:
Rleft = [(1800 + 1200)k2000] + 300 = (3000k2000) + 300 = 1200 + 300 = 1500 Ω
Rright = [(500 + 2500)k1000] + 750 = (3000k1000) + 750 = 750 + 750 = 1500 Ω
Now find the equivalent resistance seen by the source:
Req = (Rleft kRright ) + 250 + 3000 = (1500k1500) + 250 + 3000
Problems
3–13
= 750 + 250 + 3000 = 4000 = 4 kΩ
Circuit in Fig. P3.7(d):
Req = ([(750 + 250)k1000] + 100)k([(150 + 600)k500] + 300)
= [(1000k1000) + 100]k[(750k500) + 300] = (500 + 100)k(300 + 300)
= 600k600 = 300 Ω
[b] Note that in every case, the power delivered by the source must equal the
power absorbed by the equivalent resistance in the circuit. For the circuit
in Fig. P3.7(a):
P =
302
Vs2
= 15 W
=
Req
60
For the circuit in Fig. P3.7(b):
P = Is2 (Req ) = (0.08)2 (20) = 0.128 = 128 mW
For the circuit in Fig. P3.7(c):
P =
Vs2
202
=
= 0.1 = 100 mW
Req
4000
For the circuit in Fig. P3.7(d):
P = Is2 (Req ) = (0.05)2 (300) = 0.75 = 750 mW
P 3.8
[a] Rab = 28 + (70k80) + 10 = 28 + 37.33 + 10 = 75.3 Ω
[b] Rab = [(6 k + 10 k + 4 k)k9 k] + 6.8 k
= (20 kk9 k) + 6.8 k = 6.2 k + 6.8 k = 13 kΩ
[c] Rab = 1000k800k(430 + 560) = 1000k800k990 = 444.4k900 = 297.5 Ω
P 3.9
Write an expression for the resistors in series and parallel from the right side
of the circuit to the left. Then simplify the resulting expression from left to
right to find the equivalent resistance.
[a] Rab = [(26 + 10)k18 + 6]k36 = (36k18 + 6)k36 = (12 + 6)k36 = 18k36 = 12 Ω
[b] Rab = [(12 + 18)k10k15k20 + 16]k30 + 4 + 14 = (30k10k15k20 + 16)k30 + 4 + 14
= (4 + 16)k30 + 4 + 14 = 20k30 + 4 + 14 = 12 + 4 + 14 = 30 Ω
[c] Rab = (500k1500k750 + 250)k2000 + 1000 = (250 + 250)k2000 + 1000
= 500k2000 + 1000 = 400 + 1000 = 1400 Ω
[d] Note that the wire on the far right of the circuit effectively removes the
60 Ω resistor!
3–14
CHAPTER 3. Simple Resistive Circuits
Rab = [([(30 + 18)k16 + 28]k40 + 20)k24 + 25 + 10]k50
= ([(48k16 + 28)k40 + 20]k24 + 25 + 10)k50
= ([(12 + 28)k40 + 20]k24 + 25 + 10)k50 = [(40k40 + 20)k24 + 25 + 10]k50
= [(20 + 20)k24 + 25 + 10]k50 = (40k24 + 25 + 10)k50 = (15 + 25 + 10)k50
= 50k50 = 25 Ω
P 3.10
[a] R + R = 2R
[b] R + R + R + · · · + R = nR
[c] R + R = 2R = 3000 so R = 1500 = 1.5 kΩ
This is a resistor from Appendix H.
[d] nR = 4000;
so if n = 4, R = 1 kΩ
This is a resistor from Appendix H.
P 3.11
[a] Req = RkR =
[b] Req
=
=
=
R2
R
=
2R
2
RkRkRk · · · kR
(n R’s)
R
Rk
n−1
R2
R
R2 /(n − 1)
=
=
R + R/(n − 1)
nR
n
R
= 5000 so R = 10 kΩ
2
This is a resistor from Appendix H.
R
[d]
= 4000 so R = 4000n
n
If n = 3
r = 4000(3) = 12 kΩ
This is a resistor from Appendix H. So put three 12k resistors in parallel
to get 4kΩ.
[c]
P 3.12
150 × 2 k
vR2
=
= 41.66 V
5.2 k + 2 k
R1 + R2
[b] i = 150/7200 = 20.8 mA
[a] vo =
[c] PT =
P 3.13
v0 =
1002
10
v2
=
=
=5W
3
R
2 × 10
2
10 × R2
100
10 × 10 Ω
=
= 2.5 V
=
R1 + R2
30 + 10
40
v0 when Rt = 30 kΩ
v0 =
10 × 30 × 103
= 9.99 V
10 + 30 × 103
Problems
P 3.14
[a] vo =
40R2
=8
R1 + R2
so
R1 = 4R2
R2 RL
R2 + RL
Let Re = R2 kRL =
vo =
3–15
40Re
= 7.5
R1 + Re
so
Then, 4R2 = 4.33Re =
Thus, R2 = 300 Ω
R1 = 4.33Re
4.33(3600R2 )
3600 + R2
and
R1 = 4(300) = 1200 Ω
[b] The resistor that must dissipate the most power is R1 , as it has the largest
resistance and carries the same current as the parallel combination of R2
and the load resistor. The power dissipated in R1 will be maximum when
the voltage across R1 is maximum. This will occur when the voltage
divider has a resistive load. Thus,
vR1 = 40 − 7.5 = 32.5 V
32.52
= 880.2 m W
1200
Thus the minimum power rating for all resistors should be 1 W.
p R1 =
P 3.15
Refer to the solution to Problem 3.16. The voltage divider will reach the
maximum power it can safely dissipate when the power dissipated in R1 equals
1 W. Thus,
vR2 1
=1
1200
so
vR1 = 34.64 V
vo = 40 − 34.64 = 5.36 V
So,
40Re
= 5.36
1200 + Re
Thus,
and
(300)RL
= 185.68
300 + RL
Re = 185.68 Ω
and
RL = 487.26 Ω
The minimum value for RL from Appendix H is 560 Ω.
P 3.16
Req = 9k[4 + 2k(7 + 14)] = 9k[4 + 2k21] = 9k[4 + 1.82] = 9k5.82 = 3.53 Ω
v8A = v9Ω = 8 × 3.53 = 28.27 V
Using voltage division:
v14Ω =
4.6k2
14k(7k2)
× 28.27 =
× 28.27 = 7.33 V
4 + (14k7k2)
4 + 4.6k2
3–16
P 3.17
CHAPTER 3. Simple Resistive Circuits
[a] io =
v
R
i8 =
3 × 42
= 2.08 A
42 + 18.71
i = 3 − 2.08 = 0.92 A
[b] vo = 30 × I = 30 × 0.92 = 27.6 V
12
= 0.233 A
[c] I100 = I8 ×
112
P1 = I 2 R = (0.23)2 × 100 = 4.97 W
P 3.18
Begin by using KCL at the top node to relate the branch currents to the
current supplied by the source. Then use the relationships among the branch
currents to express every term in the KCL equation using just i2 :
0.05 = i1 + i2 + i3 + i4 = 0.6i2 + i2 + 2i2 + 4i1 = 0.6i2 + i2 + 2i2 + 4(0.6i2 ) = 6i2
Therefore,
i2 = 0.05/6 = 0.00833 = 8.33 mA
Find the remaining currents using the value of i2 :
i1 = 0.6i2 = 0.6(0.00833) = 0.005 = 5 mA
i3 = 2i2 = 2(0.00833) = 0.01667 = 16.67 mA
i4 = 4i1 = 4(0.005) = 0.02 = 20 mA
Since the resistors are in parallel, the same voltage, 25 V, appears across each
of them. We know the current and the voltage for every resistor so we can use
Ohm’s law to calculate the values of the resistors:
R1 = 25/i1 = 25/0.005 = 5000 = 5 kΩ
R2 = 25/i2 = 25/0.00833 = 3000 = 3 kΩ
R3 = 25/i3 = 25/0.01667 = 1500 = 1.5 kΩ
R4 = 25/i4 = 25/0.02 = 1250 = 1.25 kΩ
The resulting circuit is shown below:
Problems
P 3.19
(24)2
= 80,
R1 + R2 + R3
Therefore, R1 + R2 + R3 = 7.2 Ω
(R1 + R2 )24
= 12
(R1 + R2 + R3 )
Therefore, 2(R1 + R2 ) = R1 + R2 + R3
Thus, R1 + R2 = R3 ;
2R3 = 7.2;
R3 = 3.6 Ω
R2 (24)
=5
R1 + R2 + R3
4.8R2 = R1 + R2 + 3.6 = 7.2
Thus, R2 = 1.5 Ω;
P 3.20
R1 = 7.2 − R2 − R3 = 2.1 Ω
[a]
20 kΩ + 40 kΩ = 60 kΩ
30 kΩk60 kΩ = 20 kΩ
vo1 =
vo =
20,000
(180) = 120 V
(10,000 + 20,000)
40,000
(vo1 ) = 80 V
60,000
[b]
i=
180
= 4.5 mA
40,000
30,000i = 135 V
vo =
40,000
(135) = 90 V
60,000
3–17
3–18
CHAPTER 3. Simple Resistive Circuits
[c] It removes the loading effect of the second voltage divider on the first
voltage divider. Observe that the open circuit voltage of the first divider
is
30,000
′
(180) = 135 V
vo1
=
40,000
Now note this is the input voltage to the second voltage divider when the
current-controlled voltage source is used.
P 3.21
[a] At no load:
vo = kvs =
At full load:
vo = αvs =
Therefore k
=
α
=
Thus
1−α
α
and
and
where Re =
Ro R2
Ro + R2
(1 − k)
R2
k
(1 − α)
Re
R1 =
α
R1 =
(1 − k)
R2 Ro
=
R2
Ro + R2
k
R2 =
(k − α)
Ro
α(1 − k)
(k − α)
(1 − k)
R2 . · .
R1 =
Ro
k
αk
0.05
Ro = 2.5 kΩ
0.68
0.05
Ro = 14.167 kΩ
0.12
R1 =
[b] R1
=
R2
=
[c ]
Re
vs ,
R1 + Re
R2
R1 + R2
Re
R1 + Re
Solving for R2 yields
Also,
R2
vs .
R1 + R2
Maximum dissipation in R2 occurs at no load, therefore,
PR2 (max) =
[(60)(0.85)]2
= 183.6 mW
14,167
Maximum dissipation in R1 occurs at full load.
PR1 (max) =
[60 − 0.80(60)]2
= 57.60 mW
2500
Problems
3–19
[d ]
P 3.22
PR1
=
PR2
=
(60)2
= 1.44 W = 1440 mW
2500
(0)2
=0W
14,167
[a] Let vo be the voltage across the parallel branches, positive at the upper
terminal, then
ig = vo G1 + vo G2 + · · · + vo GN = vo (G1 + G2 + · · · + GN )
It follows that
vo =
ig
(G1 + G2 + · · · + GN )
The current in the k th branch is
ik =
[b] i5 =
P 3.23
ik = v o G k ;
Thus,
ig G k
[G1 + G2 + · · · + GN ]
40(0.2)
= 3.2 A
2 + 0.2 + 0.125 + 0.1 + 0.05 + 0.025
[a] 7 kk(3 k + 4 k) = 7 kk7 k = 3.5 kΩ
Using voltage division,
v7k =
3.5 × 103
× (20) = 7.3 V
(6 × 103 ) + (3.5 × 103 )
v4k =
3.5 × 103
× (20) = 0.538 × 20 = 10.7 V
(3 × 103 ) + (3.5 × 103 )
[b] i =
V
10.7
=
= 2.675 mA
R
4 × 103
v3k =
3 × 103
× 7.3 = 3.128 V
(3 × 103 ) + (4 × 103 )
3–20
CHAPTER 3. Simple Resistive Circuits
P 3.24
Applying kVL
−15I1 − 10i1 − 25i1 + 70 = 0
−50i1 + 70 = 0
i1 =
70
= 1.4 A
50
−18i2 − 22i2 − 70 = 0
−40i2 − 70 = 0
i2 =
−70
= −1.75 A
40
Voltage across 25 Ω resistor
V25 = −I1 × 25 = −35 V
P 3.25
[a] The equivalent resistance of the circuit to the right of, and including, the
50 Ω resistor is
[(60k15) + (45k30) + 20]k50 = 25 Ω
Thus by voltage division,
25
(30) = 12.5 V
25 + 25 + 10
[b] The current in the 25 Ω resistor can be found from its voltage using Ohm’s
law:
12.5
= 0.5 A
i25 =
25
[c] The current in the 25 Ω resistor divides between two branches – one
containing 50 Ω and one containing (45k30) + (15k60) + 20 = 50 Ω. Using
current division,
v25 =
i50 =
50k50
25
(i25 ) = (0.5) = 0.25 A
50
50
Problems
3–21
[d] The voltage drop across the 50 Ω resistor can be found using Ohm’s law:
v50 = 50i50 = 50(0.25) = 12.5 V
[e] The voltage v50 divides across the equivalent resistance (45k30) Ω, the
equivalent resistance (15k60) Ω, and the 20 Ω resistor. Using voltage
division,
v60 = v15k60 =
P 3.26
12
15k60
(12.5) =
(12.5) = 3 V
(15k60) + (30k45) + 20
12 + 18 + 20
[a] The equivalent resistance to the right of the 36 Ω resistor is
6 + [18k(26 + 10)] = 18 Ω
By current division,
36k18
(0.45) = 0.15 = 150 mA
36
[b] Using Ohm’s law,
i36 =
v36 = 36i36 = 36(0.15) = 5.4 V
[c] Before using voltage division, find the equivalent resistance of the 18 Ω
resistor and the resistors to its right:
18k(26 + 10) = 12 Ω
Now use voltage division:
12
(5.4) = 3.6 V
v18 =
12 + 6
10
[d] v10 =
(3.6) = 1 V
10 + 26
P 3.27
[a] Begin by finding the equivalent resistance of the 30 Ω resistor and all
resistors to its right:
([(12 + 18)k10k15k20] + 16)k30 = 12 Ω
Now use voltage division to find the voltage across the 4 Ω resistor:
4
(6) = 0.8 V
v4 =
4 + 12 + 14
[b] Use Ohm’s law to find the current in the 4 Ω resistor:
i4 = v4 /4 = 0.8/4 = 0.2 A
[c] Begin by finding the equivalent resistance of all resistors to the right of the
30 Ω resistor:
[(12 + 18)k10k15k20] + 16 = 20 Ω
Now use current division:
30k20
(0.2) = 0.12 = 120 mA
i16 =
20
3–22
CHAPTER 3. Simple Resistive Circuits
[d] Note that the current in the 16 Ω resistor divides among four branches –
20 Ω, 15 Ω, 10 Ω, and (12 + 18) Ω:
20k15k10k(12 + 18)
(0.12) = 0.048 = 48 mA
10
[e] Use Ohm’s law to find the voltage across the 10 Ω resistor:
i10 =
v10 = 10i10 = 10(0.048) = 0.48 V
P 3.28
[f ] v18 =
18
(0.48) = 0.288 = 288 mV
12 + 18
[a] v6k =
6
(18) = 13.5 V
6+2
v3k =
3
(18) = 4.5 V
3+9
vx = v6k − v3k = 13.5 − 4.5 = 9 V
6
[b] v6k = (Vs ) = 0.75Vs
8
v3k =
3
(Vs ) = 0.25Vs
12
vx = (0.75Vs ) − (0.25Vs ) = 0.5Vs
P 3.29
i=
=
(8 k + 12 k)k(2 k + 14 k)
× 20 × 10−3
8 k + 12 k
20 kk16 k
8.8 k
× 20 × 10−3 =
× 20 × 10−3 = 8.8 mA
20 k
20 k
Use Ohm’s law to find the voltage drop across the 12 kΩ resistor, positive at
the top:
v12k = −(8.8 × 10−3 )(22 × 103 ) = −105.6 V
Find the current in the branch containing the 2 k and 14 k resistors, from
bottom to top
i10k+15k =
(8 k + 12 k)k(2 k + 14 k)
8.8 k
× 20 × 10−3 =
× 20 × 10−3 = 11 mA
2 k + 14 k
16 k
Use Ohm’s law to find the voltage drop across the 14 k resistor, positive at the
top:
v14k = −(14 k)(11 × 10−3 ) = −154 V
vo = v14k − v12k = −1056 − (−154) = −1056 + 154 = 48.4 V
Problems
P 3.30
3–23
Req = [(120k60) + 30]k(20 + 90) = (40 + 30)k(110) = 70k110 = 42.77 Ω
Use voltage division to find the voltage drop between the top and bottom
nodes:
vReq =
42.7
(5) = 1.5 V
42.7 + 100
v1 =
40
120k60
(1.5) =
(1.5) = 0.85 V
120k60 + 30
40 + 30
v2 =
20
(1.5) = 0.27 V
90 + 20
1.52
v2
=
= 112.3 mW
P =
R
20
P 3.31
Find the equivalent resistance of all the resistors except the 2 Ω:
5 Ωk20 Ω = 4 Ω;
4 Ω + 6 Ω = 10 Ω;
10k(15 + 12 + 13) = 8 Ω = Req
Use Ohm’s law to find the current ig :
ig =
125
125
= 12.5 A
=
2 + Req
2+8
Use current division to find the current in the 6 Ω resistor:
i6Ω =
8
(12.5) = 10 A
6+4
Use current division again to find io :
io =
P 3.32
5k20
5k20
i6Ω =
(10) = 2 A
20
20
Use current division to find the current in the 8 Ω resistor. Begin by finding
the equivalent resistance of the 8 Ω resistor and all resistors to its right:
Req = ([(10k60) + 8]k40) + 6 = (8.57 + 8)k(40 + 6) = 17.7 Ω
i6Ω =
50kReq
13.07
50k17.7
(0.3) =
(0.3) = 221.5 mA
(0.3) =
Req
17.7
17.7
Use current division to find i8Ω from i8 :
i8Ω =
i20Ω =
40k16.57
11.71
40k[8 + (60k10)]
× (0.220) =
× 0.221 =
× 0.221 = 156.2 mA
8 + (60k10)
16.57
16.57
60k16.57
× (0.156) = 122.2 mA
16.57
3–24
P 3.33
CHAPTER 3. Simple Resistive Circuits
The current in the shunt resistor at full-scale deflection is
iA = ifullscale − 3 × 10−3 A. The voltage across RA at full-scale deflection is
always 150 mV; therefore,
150 × 10−3
150
RA =
=
−3
ifullscale − 3 × 10
1000ifullscale − 3
150
= 30.018 mΩ
5000 − 3
[b] Let Rm be the equivalent ammeter resistance:
[a] RA =
0.15
= 0.03 = 30 mΩ
5
150
= 1.546 Ω
[c] RA =
100 − 3
0.15
[d] Rm =
= 1.5 Ω
0.1
Rm =
P 3.34
Original meter:
Re =
60 × 10−3
= 7.5 mΩ
8
Modified meter:
Re =
(0.03)(7.5 × 10−3 )
= 6 mΩ
(0.0375)
.·. (Ifs )(6 × 10−3 ) = 60 × 10−3
Ifs =
P 3.35
60 × 10−3
= 10 A
6 × 10−3
At full scale the voltage across the shunt resistor will be 200 mV; therefore the
power dissipated will be
PA =
(200 × 10−3 )2
RA
Therefore RA ≥
(200 × 10−3 )2
= 40 mΩ
1.0
Otherwise the power dissipated in RA will exceed its power rating of 1 W
Problems
3–25
When RA = 40 mΩ, the shunt current will be
iA =
200 × 10−3
=5A
40 × 10−3
The measured current will be imeas = 5 + 0.002 = 5.002 A
.·. Full-scale reading for practical purposes is 5 A.
P 3.36
[a] The model of the ammeter is an ideal ammeter in parallel with a resistor
whose resistance is given by
100 mV
= 50 Ω.
2 mA
We can calculate the current through the real meter using current
division:
25
1
(25/12)
(imeas ) =
(imeas ) = imeas
im =
50 + (25/12)
625
25
Rm =
[b] At full scale, imeas = 5 A and im = 2 mA so 5 − 0.002 = 4998 mA flows
throught the resistor RA :
RA =
im =
100
100 mV
=
Ω
4998 mA
4998
(100/4998)
1
(imeas ) =
(imeas )
50 + (100/4998)
2500
[c] Yes
P 3.37
For all full-scale readings the total resistance is
RV + Rmovement =
full-scale reading
10−3
We can calculate the resistance of the movement as follows:
Rmovement =
Therefore,
20 mV
= 20 Ω
1 mA
RV = 1000 (full-scale reading) − 20
[a] RV = 1000(50) − 20 = 49, 980 Ω
[b] RV = 1000(5) − 20 = 4980 Ω
[c] RV = 1000(0.25) − 20 = 230 Ω
[d] RV = 1000(0.025) − 20 = 5 Ω
3–26
P 3.38
CHAPTER 3. Simple Resistive Circuits
[a] vmeas = (50 × 10−3 )[10k50k(4980 + 20)] = 50 × 10−3 [8.33k5000] = 0.415 V
[b] vtrue = (50 × 10−3 )(8.33) = 0.416 V
0.415
− 1 × 100 = −0.2403%
% error =
0.416
P 3.39
The measured value is
ig =
60
= 1.69 A;
(15.5 + 20)
The true value is
ig =
70k20 = 15.55 Ω.
imeas =
70
(1.69) = 1.47 A
80.1
70k30 = 21 Ω.
60
= 1.46 A;
(21 + 20)
itrue =
70
(1.46) = 1.27 A
80
1.47
− 1 × 100 = −15.21%
%error =
1.27
P 3.40
itrue =
10
(50 × 10−3 ) = 8.33 mA
10 + 50
imeas =
10
(50 × 10−3 ) = 8.30 mA
10 + 50 + 0.2
% error =
P 3.41
8.30
− 1 100 = −0.36%
8.33
[a ]
20 × 103 i1 + 80 × 103 (i1 − iB ) = 7.5
80 × 103 (i1 − iB ) = 0.6 + 40iB (0.2 × 103 )
. ·.
100i1 − 80iB = 7.5 × 10−3
80i1 − 88iB = 0.6 × 10−3
Calculator solution yields iB = 225 µA
Problems
3–27
[b] With the insertion of the ammeter the equations become
100i1 − 80iB = 7.5 × 10−3
(no change)
80 × 103 (i1 − iB ) = 103 iB + 0.6 + 40iB (200)
80i1 − 89iB = 0.6 × 10−3
Calculator solution yields iB = 216 µA
216
[c] % error =
− 1 100 = −4%
225
P 3.42
[a] Since the unknown voltage is greater than either voltmeter’s maximum
reading, the only possible way to use the voltmeters would be to connect
them in series.
[b ]
Rm1 = (300)(900) = 270 kΩ;
Rm2 = (150)(1200) = 180 kΩ
.·. Rm1 + Rm2 = 450 kΩ
i1
max
=
300
× 10−3 = 1.11 mA;
270
i2
max
=
150
× 10−3 = 0.833 mA
180
.·. imax = 0.833 mA since meters are in series
vmax = (0.833 × 10−3 )(270 + 180)103 = 375 V
Thus the meters can be used to measure the voltage.
320
[c] im =
= 0.711 mA
450 × 103
vm1 = (0.711)(270) = 192 V;
P 3.43
vm2 = (0.711)(180) = 128 V
The current in the series-connected voltmeters is
im =
130
200
=
= 7.3 × 10−4 A
270,000
180,000
v60 kΩ = (7.3 × 10−4 )(60 × 103 ) = 43.8 V
Vpower
supply
= 200 + 130 + 43.8 = 373.8 V
3–28
P 3.44
CHAPTER 3. Simple Resistive Circuits
Rmeter = Rm + Rmovement =
600 V
= 1200 kΩ
0.5 × 10−3
vmeas = (60 kk280 kk1200 k)(10 × 10−3 ) = (49.4 kk200 k)(10 × 10−3 ) = 396.18 V
vtrue = (60 kk280 k)(10 × 10−3 ) = 494.11 V
396.18
% error =
− 1 100 = −19.81%
494.11
P 3.45
[a] vmeter = 180 V
[b] Rmeter = (100)(200) = 20 kΩ
20k70 = 15.555556 kΩ
vmeter =
180
× 15.555556 = 78.75 V
35.555556
[c] 20k20 = 10 kΩ
vmeter =
[d] vmeter
a
180
(10) = 22.5 V
80
= 180 V
vmeter b + vmeter c = 101.26 V
No, because of the loading effect.
P 3.46
[a] R1
=
(100/2)103 = 50 kΩ
R2
=
(10/2)103 = 5 kΩ
R3
=
(1/2)103 = 500 Ω
[b] Let ia
=
actual current in the movement
id
=
design current in the movement
Then % error =
ia
− 1 100
id
For the 100 V scale:
100
100
ia =
=
,
50,000 + 25
50,025
50,000
ia
=
= 0.9995
id
50,025
id =
100
50,000
% error = (0.9995 − 1)100 = −0.05%
Problems
For the 10 V scale:
5000
ia
= 0.995
=
id
5025
% error = (0.995 − 1.0)100 = −0.4975%
For the 1 V scale:
ia
500
= 0.9524
=
id
525
P 3.47
% error = (0.9524 − 1.0)100 = −4.76%
From the problem statement we have
Vs (10)
(1) Vs in mV; Rs in MΩ
50 =
10 + Rs
48.75 =
Vs (6)
6 + Rs
[a] From Eq (1)
(2)
10 + Rs = 0.2Vs
.·. Rs = 0.2Vs − 10
Substituting into Eq (2) yields
48.75 =
6Vs
0.2Vs − 4
or
Vs = 52 mV
[b] From Eq (1)
50 =
520
10 + Rs
or
50Rs = 20
So Rs = 400 kΩ
P 3.48
[a] Rmovement = 50 Ω
R1 + Rmovement =
30
= 30 kΩ
1 × 10−3
R2 + R1 + Rmovement =
150
= 150 kΩ
1 × 10−3
R3 + R2 + R1 + Rmovement =
.·. R3 = 150 kΩ
.·. R1 = 29,950 Ω
.·. R2 = 120 kΩ
300
= 300 kΩ
1 × 10−3
3–29
3–30
CHAPTER 3. Simple Resistive Circuits
[b]
v1 = (0.96 m)(150 k) = 144 V
imove =
i1 =
144
= 0.96 mA
120 + 29.95 + 0.05
144
= 0.192 mA
750 k
i2 = imove + i1 = 0.96 m + 0.192 m = 1.152 mA
vmeas = vx = 144 + 150i2 = 316.8 V
[c] v1 = 150 V;
i2 = 1 m + 0.20 m = 1.20 mA
i1 = 150/750,000 = 0.20 mA
.·. vmeas = vx = 150 + (150 k)(1.20 m) = 330 V
P 3.49
[a] Rmeter = 300 kΩ + 600 kΩk200 kΩ = 450 kΩ
450k360 = 200 kΩ
Vmeter =
200
(600) = 500 V
240
[b] What is the percent error in the measured voltage?
True value =
360
(600) = 540 V
400
500
% error =
− 1 100 = −7.41%
540
P 3.50
is = i1 + i2 + i3 + i4
Problems
3–31
v s = i1 R1 = i2 R2 = i3 R3 = i4 R4
i1 =
vs
,
R1
is = v s
i2 =
vs
,
R2
i3 =
vs
,
R3
1
1
1
1
+
+
+
R1 R2 R3 R4
i4 =
vs
R4
is
1
1
1
1
1
=
=
+
+
+
vs
Req
R1 R2 R3 R4
XR 1
1
1
1
1
= i=1
=
+
+ ··· +
Req
R1
R1 R2
Rk
(1)
Note that resistance of all equivalent resistor is always (less) smaller than the
resistance of the smallest resistor in parallel connection.
Sometimes, using conductance when dealing with resistors connected in
parallel is more convenient. In that case, Eq. (1) becomes
Geq =
Xk
i=1
Gi = G1 + G2 + · · · + Gk .
1
1
R2 + R1
1
=
+
=
Req
R1 R2
R1 R2
Req =
R1 R2
R1 + R2
3–32
P 3.51
CHAPTER 3. Simple Resistive Circuits
[a]
The condition for a balanced bridge is that the product of the opposite
resistors must be equal:
(1000)(750)
= 1500 Ω
500
[b] The source current is the sum of the two branch currents. Each branch
current can be determined using Ohm’s law, since the resistors in each
branch are in series and the voltage drop across each branch is 24 V:
(500)(Rx ) = (1000)(750)
is =
so
Rx =
24 V
24 V
+
= 28.8 mA
500 Ω + 750 Ω 1000 Ω + 1500 Ω
[c] We can use Ohm’s law to find the current in each branch:
ileft =
24
= 19.2 mA
500 + 750
iright =
24
= 9.6 mA
1000 + 1500
Now we can use the formula p = Ri2 to find the power dissipated by each
resistor:
p500 = (500)(0.0192)2 = 184.32 mW
p1000 = (1000)(0.0096)2 = 92.16 mW
p750 = (750)(0.0192)2 = 276.18 mW
p1500 = (1500)(0.0096)2 = 138.24 mW
Thus, the 750 Ω resistor absorbs the most power; it absorbs 276.48 mW
of power.
[d] From the analysis in part (c), the 1000 Ω resistor absorbs the least power;
it absorbs 92.16 mW of power.
P 3.52
Note the bridge structure is balanced, i.e. (10 × 6) k = 20 × 2, hence there is
no current in the 7 kΩ resistor. It follows that the equivalent resistance of the
circuit is
Req = 800 + (10 k + 2 k)k(20 k + 6 k) = 800 + (12 kk26 k) = 9.01 kΩ
The source current is 150/(9.0 × 103 ) = 16.64 mA.
Problems
3–33
The current down through the branch containing the 10 kΩ and 2 kΩ
resistors is
11,250
(16.64 m) = 15.3 mA
12 × 103
i7k =
P 3.53
Redraw the circuit, replacing the detector branch with a short circuit.
8 kk20 k = 5.71 kΩ
8 k × Rx = 20 k × 10 k
Rx =
20 k × 10 k
= 25 kΩ
8k
→ ix =
100
= 3.25 × 10−3 A
30.71
v1 = 5.71 k(3.25 × 10−3 ) = 18.55 V
v2 = (3.25 × 10−3 )(25 kΩ) = 81.25 V
i=
V
v1
18.55
⇒ i1 =
= 2.31 mA
=
R
R1
8 × 103
i2 =
81.25
v2
=
= 8.125 mA
R2
10 × 103
vd = v2 − v1 = 81.25 − 18.55 = 62.7 V
ld = l2 − l1 = (8.125 × 10−3 ) − (2.31 × 10−3 ) = 5.815 mA
Rx =
P 3.54
vd
62.7
=
= 10.78 × 103 Ω
id
5.815 × 10−3
In order that all four decades (1, 10, 100, 1000) that are used to set R3
contribute to the balance of the bridge, the ratio R2 /R1 should be set to 0.001.
3–34
P 3.55
CHAPTER 3. Simple Resistive Circuits
Use the figure below to transform the ∆ to an equivalent Y:
R1 =
(40)(25)
= 9.756 Ω
40 + 25 + 37.5
R2 =
(25)(37.5)
= 9.1463 Ω
40 + 25 + 37.5
R3 =
(40)(37.5)
= 14.634 Ω
40 + 25 + 37.5
Replace the ∆ with its equivalent Y in the circuit to get the figure below:
Find the equivalent resistance to the right of the 5 Ω resistor:
(100 + 9.756)k(125 + 9.1463) + 14.634 = 75 Ω
The equivalent resistance seen by the source is thus 5 + 75 = 80 Ω. Use Ohm’s
law to find the current provided by the source:
is =
40
= 0.5 A
80
Thus, the power associated with the source is
Ps = −(40)(0.5) = −20 W
Problems
P 3.56
3–35
Use the figure below to transform the Y to an equivalent ∆:
Ra =
(25)(100) + (25)(40) + (40)(100)
7500
=
= 300 Ω
25
25
Rb =
(25)(100) + (25)(40) + (40)(100)
7500
=
= 187.5 Ω
40
40
Rc =
7500
(25)(100) + (25)(40) + (40)(100)
=
= 75 Ω
100
100
Replace the Y with its equivalent ∆ in the circuit to get the figure below:
Find the equivalent resistance to the right of the 5 Ω resistor:
300k[(125k187.5) + (37.5k75)] = 75 Ω
The equivalent resistance seen by the source is thus 5 + 75 = 80 Ω. Use Ohm’s
law to find the current provided by the source:
is =
40
= 0.5 A
80
Thus, the power associated with the source is
Ps = −(40)(0.5) = −20 W
3–36
P 3.57
CHAPTER 3. Simple Resistive Circuits
Use the figure below to transform the Y to an equivalent ∆:
Ra =
(25)(125) + (25)(37.5) + (37.5)(125)
8750
=
= 233.33 Ω
37.5
37.5
Rb =
(25)(125) + (25)(37.5) + (37.5)(125)
8750
=
= 350 Ω
25
25
Rc =
8750
(25)(125) + (25)(37.5) + (37.5)(125)
=
= 70 Ω
125
125
Replace the Y with its equivalent ∆ in the circuit to get the figure below:
Find the equivalent resistance to the right of the 5 Ω resistor:
350k[(100k233.33) + (40k70)] = 75 Ω
The equivalent resistance seen by the source is thus 5 + 75 = 80 Ω. Use Ohm’s
law to find the current provided by the source:
is =
40
= 0.5 A
80
Thus, the power associated with the source is
Ps = −(40)(0.5) = −20 W
Problems
P 3.58
3–37
[a] Use the figure below to transform the Y to an equivalent ∆:
Ra =
3500
(25)(30) + (25)(50) + (30)(50)
=
= 116.67 Ω
30
30
Rb =
(25)(30) + (25)(50) + (30)(50)
3500
=
= 70 Ω
50
50
Rc =
(25)(30) + (25)(50) + (30)(50)
3500
=
= 140 Ω
25
25
Replace the Y with its equivalent ∆ in the circuit to get the figure below:
Find the equivalent resistance to the right of the 13 Ω and 7 Ω resistors:
70k[(50k116.67) + (20k140)] = 30 Ω
Thus, the equivalent resistance seen from the terminals a-b is:
Rab = 13 + 30 + 7 = 50 Ω
[b] Use the figure below to transform the ∆ to an equivalent Y:
R1 =
(50)(20)
= 10 Ω
50 + 20 + 30
3–38
CHAPTER 3. Simple Resistive Circuits
R2 =
(50)(30)
= 15 Ω
50 + 20 + 30
R3 =
(20)(30)
= 6Ω
50 + 20 + 30
Replace the ∆ with its equivalent Y in the circuit to get the figure below:
Find the equivalent resistance to the right of the 13 Ω and 7 Ω resistors:
(50 + 10)k(25 + 15) + 6 = 30 Ω
Thus, the equivalent resistance seen from the terminals a-b is:
Rab = 13 + 30 + 7 = 50 Ω
[c] Convert the delta connection R1 —R2 —R3 to its equivalent wye.
Convert the wye connection R1 —R3 —R4 to its equivalent delta.
P 3.59
R1 =
30 × 12
= 32 Ω;
30 + 12 + 70
= 18.75 Ω
R2 =
12 × 70
= 75 Ω;
30 + 12 + 70
R3 =
70 × 30
70 + 30 + 12
Problems
3–39
The transformed circuit.
The equivalent resistance seen by the 30 V source can be calculated by making
series and parallel combination of the resistors to the right of the 30 V source.
Req = (20 + 7.5)k(2 + 3.2) + 18.75 = 27.5k5.2 + 18.75 = 23.1 Ω
.·.
i=
the current i in the 30 V source is given by
30
= 1.2 A
25.1
i1 =
3.2
3.2
(i) =
× 1.2 = 0.13 A,
7.5 + 20
27.5
i2 = i − i1 = 1.2 − 0.13 = 1.06 A
Now use KVL & Ohm’s law to calculate v1 .
v1 = 3.2i2 + 18.75i = 3.2(1.06 A) + 18.75(1.2 A)
v1 = 25.89 V
Now use KVL and Ohm’s law to calculate v2 . v2 is the sum of the voltage
drop across the 75 Ω resistor, 7.5i1 , & the voltage drop across the 18.75 Ω
resistor, 18.75i:
v2 = 7.5i1 + 18.75i = 7.5(0.13) + 18.75(1.2) = 23.47 V
P 3.60
[a] Convert the upper delta to a wye.
R1 =
(50)(50)
= 12.5 Ω
200
R2 =
(50)(100)
= 25 Ω
200
R3 =
(100)(50)
= 25 Ω
200
Convert the lower delta to a wye.
R4 =
(60)(80)
= 24 Ω
200
3–40
CHAPTER 3. Simple Resistive Circuits
R5 =
(60)(60)
= 18 Ω
200
(80)(60)
= 24 Ω
200
Now redraw the circuit using the wye equivalents.
R6 =
Rab = 1.5 + 12.5 +
(120)(80)
+ 18 = 14 + 48 + 18 = 80 Ω
200
[b] When vab = 400 V
400
=5A
ig =
80
48
i31 = (5) = 3 A
80
p31Ω = (31)(3)2 = 279 W
P 3.61
[a] After the 20 Ω—100 Ω—50 Ω wye is replaced by its equivalent delta, the
circuit reduces to
Now the circuit can be reduced to
i=
96
(1000) = 240 mA
400
Problems
io =
400
(240) = 96 mA
1000
80
(240) = 48 mA
400
[c] Now that io and i1 are known return to the original circuit
[b] i1 =
v2 = (50)(0.048) + (600)(0.096) = 60 V
i2 =
60
v2
=
= 600 mA
100
100
[d] vg = v2 + 20(0.6 + 0.048) = 60 + 12.96 = 72.96 V
pg = −(vg )(1) = −72.96 W
Thus the current source delivers 72.96 W.
P 3.62
8 + 12 = 20 Ω
20k60 = 15 Ω
15 + 20 = 35 Ω
35k140 = 28 Ω
28 + 22 = 50 Ω
50k75 = 30 Ω
30 + 10 = 40 Ω
ig = 240/40 = 6 A
io = (6)(50)/125 = 2.4 A
i140Ω = (6 − 2.4)(35)/175 = 0.72 A
p140Ω = (0.72)2 (140) = 72.576 W
3–41
3–42
P 3.63
CHAPTER 3. Simple Resistive Circuits
[a] Replace the 60—120—20 Ω delta with a wye equivalent to get
is =
750
750
=
= 10 A
5 + (24 + 36)k(14 + 6) + 12 + 43
75
i1 =
(24 + 36)k(14 + 6)
15
(10) = (10) = 2.5 A
24 + 36
60
[b] io = 10 − 2.5 = 7.5 A
v = 36i1 − 6io = 36(2.5) − 6(7.5) = 45 V
v
45
= 7.5 +
= 8.25 A
60
60
= (750)(10) = 7500 W
[c] i2 = io +
[d] Psupplied
P 3.64
Ga Gc
Gb Gc
(1), G2 =
GT
GT
Ga + Gb + Gc
G1
=
GT
=
G1 G2
=
Ga Gb G2c
G2T
G1 G3
=
Ga G2b Gc
G2T
G2 G3
=
G2a Gb Gc
G2T
(2),
G3 =
Ga Gb
GT
(3)
sum of these three equation is
Ga Gb G2c + Ga G2b Gc + G2a Gb Gc
G1 G2 + G1 G3 + G2 G3 =
G2T
=
divide (4) by (1)
Ga Gb Gc (Ga Gb Gc )
G2T
(4)
G1 G2 + G2 G3 + G1 G3
G1
=
(G
Ga G
b Gc T)
G
G
T
b Gc
Ga
=
G1 G2 + G2 G3 + G1 G3
G1
Similar manipulations generate the expressions for Gb and Gc .
Problems
P 3.65
3–43
[a] Subtracting Eq. 3.42 from Eq. 3.43 gives
R1 − R2 = (Rc Rb − Rc Ra )/(Ra + Rb + Rc ).
Adding this expression to Eq. 3.41 and solving for R1 gives
R1 = Rc Rb /(Ra + Rb + Rc ).
To find R2 , subtract Eq. 3.43 from Eq. 3.41 and add this result to
Eq. 3.42. To find R3 , subtract Eq. 3.41 from Eq. 3.42 and add this result
to Eq. 3.43.
[b] Using the hint, Eq. 3.43 becomes
R1 + R3 =
Rb [(R2 /R3 )Rb + (R2 /R1 )Rb ]
Rb (R1 + R3 )R2
=
(R2 /R1 )Rb + Rb + (R2 /R3 )Rb
(R1 R2 + R2 R3 + R3 R1 )
Solving for Rb gives Rb = (R1 R2 + R2 R3 + R3 R1 )/R2 . To find Ra : First
use Eqs. 3.44–3.46 to obtain the ratios (R1 /R3 ) = (Rc /Ra ) or
Rc = (R1 /R3 )Ra and (R1 /R2 ) = (Rb /Ra ) or Rb = (R1 /R2 )Ra . Now use
these relationships to eliminate Rb and Rc from Eq. 3.42. To find Rc , use
Eqs. 3.44–3.46 to obtain the ratios Rb = (R3 /R2 )Rc and
Ra = (R3 /R1 )Rc . Now use the relationships to eliminate Rb and Ra from
Eq. 3.41.
P 3.66
[a] Rab = 2R1 +
Therefore
Thus
R2 (2R1 + RL )
= RL
2R1 + R2 + RL
2R1 − RL +
R2 (2R1 + RL )
=0
2R1 + R2 + RL
RL2 = 4R12 + 4R1 R2 = 4R1 (R1 + R2 )
When Rab = RL , the current into terminal a of the attenuator will be
vi /RL .
Using current division, the current in the RL branch will be
vi
R2
·
RL 2R1 + R2 + RL
Therefore
and
vo =
R2
vi
·
RL
RL 2R1 + R2 + RL
vo
R2
=
vi
2R1 + R2 + RL
[b] (300)2 = 4(R1 + R2 )R1
22,500 = R12 + R1 R2
R2
vo
= 0.5 =
vi
2R1 + R2 + 300
3–44
CHAPTER 3. Simple Resistive Circuits
.·. R1 + 0.5R2 + 150 = R2
0.5R2 = R1 + 150
R2 = 2R1 + 300
.·. 22,500 = R12 + R1 (2R1 + 300) = 3R12 + 300R1
.·. R12 + 100R1 − 7500 = 0
Solving,
R1 = 50 Ω
R2 = 2(50) + 300 = 400 Ω
[c] From Appendix H, choose R1 = 47 Ω and R2 = 390 Ω. For these values,
Rab 6= RL , so the equations given in part (a) cannot be used. Instead
Rab = 2R1 + [R2 k(2R1 + RL )] = 2(47) + 390k(2(47) + 300)
= 94 + 390k394 = 290 Ω
290
% error =
− 1 100 = −3.33%
300
Now calculate the ratio of the output voltage to the input voltage. Begin
by finding the current through the top left R1 resistor, called ia :
vi
ia =
Rab
Now use current division to find the current through the RL resistor,
called iL :
iL =
R2
ia
R2 + 2R1 + RL
Therefore, the output voltage, vo , is equal to RL iL :
vo =
R2 RL v i
Rab (R2 + 2R1 + RL )
Thus,
390(300)
R2 RL
vo
=
= 0.5146
=
vi
Rab (R2 + 2R1 + RL )
290(390 + 2(47) + 300)
% error =
0.5146
− 1 100 = 2.92%
0.5
Problems
P 3.67
[a] After making the Y-to-∆ transformation, the circuit reduces to
Combining the parallel resistors reduces the circuit to
Now note:
Therefore
0.75R +
Rab
2.25R2 + 3.75RRL
3R
3R + RL
3R(3R + 5RL )
! =
=
2
15R + 9RL
2.25R + 3.75RRL
3R +
3R + RL
If R = RL , we have
Therefore
2.25R2 + 3.75RRL
3RRL
=
3R + RL
3R + RL
!
Rab =
3RL (8RL )
= RL
24RL
Rab = RL
[b] When R = RL , the circuit reduces to
io =
ii (3RL )
1 vi
1
ii =
=
,
4.5RL
1.5
1.5 RL
Therefore
P 3.68
vo
= 0.5
vi
[a] 3.5(3R − RL ) = 3R + RL
10.5R − 1050 = 3R + 300
1
vo = 0.75RL io = vi ,
2
3–45
3–46
CHAPTER 3. Simple Resistive Circuits
7.5R = 1350,
R = 180 Ω
2(180)(300)2
= 4500 Ω
R2 =
3(180)2 − (300)2
[b ]
vo =
42
vi
=
= 12 V
3.5
3.5
io =
12
= 40 mA
300
i1 =
30
42 − 12
=
= 6.67 mA
4500
4500
ig =
42
= 140 mA
300
i2 = 140 − 6.67 = 133.33 mA
i3 = 40 − 6.67 = 33.33 mA
i4 = 133.33 − 33.33 = 100 mA
p4500
top
= (6.67 × 10−3 )2 (4500) = 0.2 W
= (133.33 × 10−3 )2 (180) = 3.2 W
p180
left
p180
right
p180
vertical
p300
load
= (33.33 × 10−3 )2 (180) = 0.2 W
= (100 × 10−3 )2 (180) = 0.48 W
= (40 × 10−3 )2 (300) = 0.48 W
The 180 Ω resistor carrying i2
[c] p180
left
= 3.2 W
[d] Two resistors dissipate minimum power – the 4500 Ω resistor and the 180
Ω resistor carrying i3 .
[e] They both dissipate 0.2 W.
Problems
P 3.69
[a ]
va =
vin R4
Ro + R4 + ∆R
vb =
R3
vin
R2 + R3
vo = va − vb =
R3
R4 vin
−
vin
Ro + R4 + ∆R R2 + R3
When the bridge is balanced,
R4
R3
vin =
vin
Ro + R4
R2 + R3
.·.
R3
R4
=
Ro + R4
R2 + R3
Thus,
vo
=
=
=
≈
R4 vin
R4 vin
−
Ro + R4 + ∆R Ro + R4
1
1
R4 vin
−
Ro + R4 + ∆R Ro + R4
R4 vin (−∆R)
(Ro + R4 + ∆R)(Ro + R4 )
−(∆R)R4 vin
,
since ∆R << R4
(Ro + R4 )2
[b] ∆R = 0.03Ro
Ro =
R2 R4
(1000)(5000)
= 10,000 Ω
=
R3
500
∆R = (0.03)(104 ) = 300 Ω
. · . vo ≈
[c] vo
=
=
=
−300(5000)(6)
= −40 mV
(15,000)2
−(∆R)R4 vin
(Ro + R4 + ∆R)(Ro + R4 )
−300(5000)(6)
(15,300)(15,000)
−39.2157 mV
3–47
3–48
P 3.70
CHAPTER 3. Simple Resistive Circuits
−(∆R)R4 vin
(Ro + R4 )2
[a] approx value =
true value =
.·.
−(∆R)R4 vin
(Ro + R4 + ∆R)(Ro + R4 )
approx value
(Ro + R4 + ∆R)
=
true value
(Ro + R4 )
.·. % error =
−∆R
Ro + R4
− 1 × 100 =
× 100
Ro + R4 + ∆R
Ro + R4
Note that in the above expression, we take the ratio of the true value to
the approximate value because both values are negative.
But Ro =
R2 R4
R3
.·. % error =
[b] % error =
P 3.71
−R3 ∆R
R4 (R2 + R3 )
−(500)(300)
× 100 = −2%
(5000)(1500)
∆R(R3 )(100)
= 0.5
(R2 + R3 )R4
∆R(500)(100)
= 0.5
(1500)(5000)
.·. ∆R = 75 Ω
% change =
P 3.72
75
× 100 = 0.75%
10,000
[a] Using the equation for voltage division,
Vy =
βRy
βRy
VS =
VS = βVS
βRy + (1 − β)Ry
Ry
[b] Since β represents the touch point with respect to the bottom of the
screen, (1 − β) represents the location of the touch point with respect to
the top of the screen. Therefore, the y-coordinate of the pixel
corresponding to the touch point is
y = (1 − β)py
Remember that the value of y is capped at (py − 1).
Problems
P 3.73
3–49
[a] Use the equations developed in the Practical Perspective and in Problem
3.72:
1
Vx
= = 0.2
Vx = αVS
so
α=
VS
5
Vy
3.75
Vy = βVS
so
β=
= 0.75
=
VS
5
[b] Use the equations developed in the Practical Perspective and in Problem
3.72:
x = (1 − α)px = (1 − 0.2)(480) = 384
y = (1 − β)py = (1 − 0.75)(800) = 200
Therefore, the touch occurred in the upper right corner of the screen.
P 3.74
Use the equations developed in the Practical Perspective and in Problem 3.72:
x = (1 − α)px
so
α=1−
x
480
=1−
= 0.25
px
640
Vx = αVS = (0.25)(8) = 2 V
y = (1 − β)py
so
β =1−
192
y
= 0.8125
=1−
py
1024
Vy = βVS = (0.8125)(8) = 6.5 V
P 3.75
From the results of Problem 3.74, the voltages corresponding to the touch
point (480, 192) are
Vx1 = 2 V;
Vy1 = 6.5 V
Now calculate the voltages corresponding to the touch point (240, 384):
x = (1 − α)px
so
α=1−
x
240
= 0.625
=1−
px
640
Vx2 = αVS = (0.625)(8) = 5 V
y = (1 − β)py
so
β =1−
384
y
= 0.625
=1−
py
1024
Vy2 = βVS = (0.625)(8) = 5 V
When the screen is touched at two points simultaneously, only the smaller of the
two voltages in the x direction is sensed. The same is true in the y direction.
Therefore, the voltages actually sensed are
Vx = 2 V;
Vy = 5 V
These two voltages identify the touch point as (480, 384), which does not correspond
to either of the points actually touched! Therefore, the resistive touch screen is
appropriate only for single point touches.