134 CHAPTER 3 NOOAL AND LOOP ANALYSIS TECHNIQ UES

N linea rl y independent equations must be written 10 solve for the loop currents.

• If currelll sources are present in the circuit , either of two techn iques can be employed. In the first case, one loop cur­rem is se lected 10 pass through one of the curren! sources.

The remaining loop currents are determined by opcn­circuiting the current sources in the ci rcuit and using this

modi lied circuit to select them. In the second case, a current is ass igned 10 each mesh in the circuit.

• Write a constraint equation for each current SOUfCC­

independent or dependent- in the circuit in terms of the

PROBLEMS

assigned loop currents using KCL. Each constraint equation represents one of the necessary linearly independent equa­

tions, and N/ current sources yield N/ linearly independent

equat ions. For each dependent current source, express the controlling variable for that source in tenns of the loop

currents. • Use KYL 10 formulate the remaining N - N/ linearl y inde­

pendell{ equations. Treat dependent voltage sources like

independent voltage sources when fonnulating the KYL

equmions. For each dependent vohage source, express the controlling variable in terms of the loop currents.

3 .1 Use nodal analysis to find VI in the circuit in Fig. P3. 1.

10 kn 5 kn

+

12 mA 4 kn 4 kn

Figure 1'3.1

3.2 Find both 10 and Vo in the network in Fig. P3.2 using nodal analysis.

2 mA 3 kn

I"

Figure 1'3.2

o 3·3 Use nodal analysis to find VI in the circuit in Fig. P3.3.

2kn

+

2kn 2 kn

Figure P3.3

6kn

4mA 12 kn

3.4 Find " I and V2 in the circuit in Fig. P3.4 using nodal analysis. Then solve the problem using MATLAB and compare your answers.

6mA

+ 4 kfl +

4mA 6 kn

Figure P3.4

o

o 3.6

0 3.5 Usc nodal analys is to fi nd both VI and V{} in the circuit i ll

Fig. P3.S. i

2 mA

VI V2 6kO 2 kO

12 mA 3kO 6 kO 1 kO

Figure P3.5

PROBLEM S 135

G +

Vo

0

Write the node equntions for the circuit in Fig. P3.6 in matri x form, and rind all the node voltages lIsing MATLAB.

3 .8 Find III in the c ircuit in Fig. P3.8 using nodal anal ysis.

8 kll 3 mA

.------{- } ----, 2 kll

1 mA t 1kO V2 3kll VI ~--~~~~~~--~ V3 3 kll

6 kll

2kO 6 mA 4 kll

Figure p).8

Figure p).6

o

~ 3.7 Find Y" in the network in Fig. P3.7. 3·9 Use nodal anal ys is 10 lind I" in the network in Fig. P3.9. 0

2 rnA

4 mA 2 kll

4 rnA t 4 k[J

1 k[J 1 kO +

1 k[J t 1 rnA 2 k[J

2 kf] 4 kO

6 rnA t 12 kO

10 Figure P).7

Figure P).9

136 CHAP TER 3 NODAL AND LOOP A N AL Y S IS TECHN IOUES

() 3.10 Use nodal analysis to find \1" in the circuit in Fig. P3. IO.

6 kfl 3 kfl

+

10V 12V

Figure 1'3.10

() 3.11 Find v" in the network in Fig. P3. 11 using nodal

analys is.

r 6 kO 12 kfl

12 V 6 kfl + 6V

Figure 1'3.11

3.12 Usc nodal analysis to find Vo in the circuit in Fig. P3.12.

2 kfl 4 kfl

+

12V 2 kfl

Figure 1'3.12

3.13 Usc nodal analysis 10 lind v,J in the circuit in Fig. P3. 13.

2 kfl 6 kfl

12 kfl 3 kfl

5V +

6V

Figure 1'3.13

3.14 Use nodal analysis to fi nd Vo in the network in Fig. P3.1-t.

4 kfl 3mA

2 kfl

+

2 kfl 6V 6 kfl 1 kfl

L-------~------+-------~-----~l

Figure 1'3.14

3.15 Usc nodal analys is to find v" in the circuit in Fig. P3. IS.

4 kfl 6 kfl

6 kfl 2 kfl

12 V + Va -+ -

4 kfl

8kO 8 kfl

Figure 1'3.15

3.16 Find 10 in the ne lwork in Fig. P3. 16 using IlmJal tllli.tl y:-.i:-..

2 kfl

2mA 1 k!l

12V + 2kfl

Figure 1'3.16

3.17 Use nodal analys is to solve for the node vo lt ages in the circuit in Fig. P3.1 7. Also calcu late the power supplied by the I-A current source.

2A

,-----{-}-----,

120 6fl

48V 4fl 1 A

Figure 1'3.17

i 3.18 Find ~Q in the network in Fig. P3. IS using nodal equations.

1 kfl

2 mA 1 kfl -

6V + 1 kfl 1 kfl

+

Va

L-------~--____ ~ _____ ~o

Figure 1'].18

3.19 Find 10 in the network in Fig. P3. 19 using nodal analys is.

4 kfl

3 kfl 2 mA

12 V 6 kfl 2 kfl

Figure 1'].19

3.20 Use nodal analysis to find 10 in the circuit in Fig. P3.20.

2 kfl

2 mA 4 mA

12 V 2 kfl 2 kfl

Figure 1'].20

3.21 Find Vo in the circuit in Fig. P3.21 using nodal analysis.

6 mA 12 kfl

- r-------vVv---,

6 kfl 4 mA

+

12 V + 9V

Figure 1'].21

PROBLEMS 137

3.22 Use nodal analysis to find Vo in the network in

Fig. P3.22.

1 kfl 2 kfl 1 kfl

2kfl 1 kfl

12 V + t 2mA 1 kfl

Figure 1'].22

+

VO

3.23 Find 10 in the circuit in Fig. P3.23 using nodal analysis.

6kfl 6V

6 kfl +

6 kfl 12 kfl 12 kfl 6 kfl

Figure 1'].23

3.24 Usc nodal analysis to find If) in the circuit in Fig. P3.24.

3V

6V

r-~~--~~-+r-~--~~-. 6kfl 12 kfl

12 kfl 6 kfl

Figure 1'].24

3.25 Using nodal analysis, find \1;, in the network in Fig. P3.25.

3V

r-----~+-~----,

4 kfl 2 kfl ~--~~-4r-~~--~-------o

+

6V 2mA 6 kfl

Figure 1'].25

3V

138 CH A P T ER 3 NODA L AND L OO P ANALY S I S TEC H NIQUES

3.26 Find v" in the network in Fig. P3.26 lIsing nodal analysis. 3.30 Find Vo in the circuit in Fig. P3.30.

6V

3 kfl 12 V

r-~~~~~-+~~--NV--~ 4 kfl

4 kfl

Figure PJ.26

2 kll

3.27 Find \~) in the circuit in Fig. P3.27 using nodal analysis.

12 kn

6 kll 6V

~~NV--~~+-~~-----~O +

12V + 3 kll 6 kn Va

L-------~------~-----~o

Figure P3.27

3.28 Find v,J in the circuit in Fig. P3.28 using nodal analysis.

6V

r-----~-+~----,

2 mA 6 kn

+

12 kn 12 kn 4 kn

L-----~----~----~o

Figure P3 .28

4 kfl

6kfl 6V

2 kll +

6V r 12 kll 12 kll 4 kll

Figure P3.30

3.31 Find \1;, in the circuit in Fig. P3.31 using nodal analysis.

12V

r-----~+ -~----,

2 kn 1 kll

1 kfl 6V

--~O

+

1 kll

L----~----.-----o

Figure P3.31

3.32 Usc nodal analysis to fi nd v" in the circuit in Fig. P3.32.

+

Figure P3.32

o +

o

3.29 Use nodal analys is to fi nd Vo in the circuit in Fig. P3.29. 3.33 Use nodal analysis to fi nd Vo in the circuit in Fig. P3.33. 0

12 V

.-------{+ --'1-------,

6V 6 kn

- +

4 kll

Figure P3 .29

6kn

2 kll

(

+ 4 mA

2 kll Va

n

Figure P3.33

~ + 6V 1 kfl t 2mA

-+ 0 +

12 V

1 kll 1 kfl 1 kll Vo

e 3·34 Find v" in the network in Fig. P3.34.

~

2 kfl + 6V 12V 1 kfl

2 kfl + - 0

+ 2V

2 kfl + 4V 1 kfl 1 kfl Vo

Figure 1'J.34

o 3·35 Find 10 in the circuit in Fig. P3.35 using nodal

analysis.

12 kfl

12 kfl 12 mA

Figure 1'J.35

12 kfl

10

3.36 Find Vo in the circuit in Fig. P3.36 using nodal analys is.

12V

r-~~--~~-+·~~·----~O

Vo + 2

1 kfl

2kfl

+

1 k!l

L-------~ __ ------_+·-------O

Figure 1'J.36

0 3.37 Find Vo in the circuit in Fig. P3.37.

iii -- + 1 kD

Vx 1 kD

Figure 1'J.37

2V, 1000

0 +

1 k!l Vo

0

PROB L EMS 139

3.38 Use nodal analys is to find v" in the circuit in Fig. P3.3S. i 10 V,

+

12 mA 6 kfl -

+ +

12 kfl 4 kfl V, 8 kfl

-

o Figure 1'J.38

3.39 Use noda l ana lysis to find v" in the circuit in Fig. P3.39. e In add iti on, tind all branch currents and eheck your

answers using KCL at every node.

- + 12 kfl

20001,. 2 kfl +

6V 4 kfl 2 mA 4 kfl Vo

1, ------4-----~~----~---~r

Figure 1'J.39

3.40 Determine v" in the network in Fig. P3.40 using nodal

analys is.

21, j 1 kfl j 2 mA

Ix 1 kfl 0

1 kfl +

1 kfl 12V 1kfl Vo

0

Figure 1'J.40

3.41 Determine Vo in the network in Fig. P3.4!.

1 k!l 2 kfl 2V 1 kfl

2kfl 12V

-+ +

4mA

6mA 6V 2 kfl t L-----4-----~ ______ ~----~O

Figure 1'J.41

140 CHAPTER 3 NODA L AND L OOP ANALYSIS TECHNIQ U ES

~ 3.42 Find 1" in the circuit in Fig. P3.42.

5 kO

~--~+-~--------~~--~

3000 I,

4 kO 2 kO 6 mA

+

6 kO 2 kO V, 6 kO

Figure P3.42

~ 3.43 Use nodal analys is to solve for fA in the network in Fig. P3.43.

v, 30 V5

5 fA 120 t 2A

18 V

V, V, +- V3

80

40 ~ SA 50

fA

Figure PJ.43

~ 3.44 Use nodal analysis to find VI, Vz• V). and V4 in the circu it in Fig. P3.44.

12 V

+-

V, V, - VA + V3

60 80 100

3A ~ 40 V, 10

+ 2 VA

Figure PJ.44

t 4A

3.45 Use nodal analysis to find v" V" V3, and V, in the network in Fig. P3.45

+ -)-..... - -N'I-----, 50

20

V, 80 V, 100

40

Figure P3.45

3.46 Find 10 in rhe network in Fig. P3.46 using mesh analysis.

6 kO 4 kO

6V 6 kO 24 V

3.47 Use mesh analys is lO find Va in the circuit in

Fig. P3.47.

6 kO

12V +

Figure P3.47

4 kO

8 kO

---0 +

+ 24V

o

C) 3.48 Find If) in the circui t in Fig. P3.48.

0 3.49

i

4 kl1 4 kl1

2 kl1

24V + 2 kl1

+ 6V

Figure 1'3.48

Find \1;, in the network in Fig. P3.49 using mesh equations.

4V 12 V

1--.----o +

4 kl1 4 kl1 4 kl1

L-------~------~-----_o

Figure 1'3.49

PROBLEMS 141

3.52 Find V;, in the circuit in Fig. P3.52 using mesh analysis.

~ __ -{_ + 6V

1 kl1 1 kl1

1 kl1 12 V

Figure 1'3.52

3.53 Solve Problem 3.27 using loop analys is.

3.54 Solve Problem 3.32 using loop analysis.

3.55 Use loop analysis to find Vf) in the network in Fig. P3.55.

2 kl1 2 kl1

1 kl1 2 mA

+

L-------~--------------4------0

Figure 1'3.55

3.56 Use mesh analysi s to find Vo in the network in o 3.50 Find Vf) in the network in Fig. P3.50 using mesh equations. Fig. P3.56.

12V 4 V

-+-~~~+-~~------~ +

2 kO 2 kl1 2 kl1

L----~~---~---~o

Figure 1'3.50

3.51 Find '" in the circuit in Fig. P3.51.

3V

6V

~~~--~~-+.l--+---NV-_ 6 kfl 12 kfl

12 kfl 6 kfl

Figure 1'3.51

3V

r-~~--~~~--~------o 3 kfl 2 kfl +

12 V 6 kfl

L----~----.----o

Figure P3.56

3.57 Find 10 in the circuit in Fig. P3.5 7 using mesh analysis .

3 kfl

+ 3V

2 mA t 3 kfl

6 kfl

Figure 1'3.57

o

o

142 CHAPTER 3 NODAL AND L OOP ANAL Y SIS TE CHNIQUES

i 3.58 Find~) in the ci rcuit in Fig. P3.58 using mesh

analys Is.

6 kn

2kn

12V +

Figure 1'3.58

3kn

---0 +

~ 8mA

3.59 Usc mesh ana lys is to find Vo in the circuit in Fig. P3.59.

2 kn

6V 4 kn

4kn

+ V o

8 kn

8 kn

3.62 Use loop analysis to calculate Ihe power supplied by the 20-V voltage source in the circui t in Fig. P3.62 .

sn

sn lOn

20n + 20V

Figure 1'3.62

3.63 Find /" in the network in Fig. P3.63 using loop analysis.

1 kn

1 kn

12V

t----I'./II'-- t---{+ -1 kn

1 kn t 6mA

Figure 1'3.59 Figure 1'3.63

3 .64 Use loop analysis to find It] and I, in the network in

3.60 Usc mesh analysis 10 find I" in the network in Fig. P3.60. Fig. P3.64.

3 kn

12V

-+~~--~~--~--~~-, 3kn 2 kn

2 kn

Figure 1'3.60

1 kn

o 3.61 Use loop analysis to find Vo in the circuit in Fig. P3.61.

1 kn 6V

-+

1 kn 2kn 0 +

1 kn

2 mA t 1 kn VO 12V +

0

Figure 1'3.61

6 kn

3 kn

12 mA 4 kn + 12V 2 kn

10 I , L-____ ~----~----~

Figure 1'3.64

3.65 Find /0 in the network in Fig. P3.65 using loop analysis. Then solve the problem using MATLAB and compare your answers.

6 kn

6kn 6kn

BV

Figure 1'3.65

o 3.66 Use loop analys is to find Vo in the circui t ill Fig. P3.66.

r-----~. __ ~----_.

6 rnA

12kil 6kil

12V + 12 kil ~ 2rnA

Figure P3.66

o 3.67 Using loop analysis, find Vo in the Network in Fig. P3.67.

2 kil t 4 rnA 1 kil

6V

+ - )--+----., - +)-4 -------<:

2 kil

12V

1 kil

+

1 kil

L-------~------~-------o

Figure P3.67

o 3.68 Find 10 in the circuit in Fig. P3.68 .

10

6V + 2 kil 4 kil 2 rnA

2 kil -6 kil t 1 rnA 2 kil

Figure PJ.68

o 3.69 Use loop analysis to find 10 in the network in

Fig. P3.69.

1 kil

12V 1 kil ~ 2 rnA

1 kil 1 kil

10

1 kil 4 rnA 1 kil

Figure P3.69

PROB L EMS 143

3·70 Use loop analysis to find 10 in the network in Fig. P3.70.

1 kfl

12 V + 1 kil ~ 2 rnA

10

1 kil 1 kn

1 kil t 4 rnA 1 kfl

Figure P3.70

3.71 Using loop analysis, fi nd \1;, in the c ircui t in Fig. P3.71. ()

6 rnA t 1 kil 2 kil 12 V

t---~~~--4-+)-~·--------O +

6V + 1 kil 2 kil

L-------~------~-------o

Figure P3.71

3.72 Using loop analysis, fi nd v" in the network in Fig. P3.72. 0 (i)

2 kil + 6V 1 kil t 2 rnA

-+ 0 +

4 rnA 12 V

2 kil 1 kil 1 kil 1 kil Vo

0

Figure P3.72

3.73 Using loop analysis, find I" in the c ircuit in Fig. P3.73. o 1 kil t 4 rnA 1 kil 1 kn

1 kil 12 V 1 kil -+

1 kil 1 kil ~ 2 rnA 1 kil

10

Figure P3.73

144 CHAPTER 3 NODA L AND LOOP ANALYSIS TECHNIQUES

() 3.74 Use MATLAB to find the mesh cun'ents in the network in Fig. P3.74.

2 kfl

0 1kfl0 2 mA t + 12 V 12

0 2 kfl 2 kfl 1 kfl

6V +

0 0 1 kfl

Figure P3.74

o 3·75 Use loop analysis to find V" in the network in Fig. P3.75.

1 kfl

I,. 4 kfl r-------~~~----~- +r_~-----O

12V

2 kfl 2 kfl

Figure PJ.75

o 3.76 Find V" in the circuit in Fig. P3.76 using nodal analysis.

r-~~--~-{+-~~-----~O 1 kfl

12 V +

1 kfl 2 kfl

Figure P3.76

o 3·77 Use nodal analysis 10 find Y" in Fig. P3.77.

10 kfl 10 kfl

Ix ~------~~------~·------~O

Figure P3.77

3.78 Use nodal analysis 10 find ~J in the network in Fig. P3.78.

1 kfl

+

t 1 kfl 1 kfl 1 kfl

Figure P3.78

3 .79 Find the power suppl ied by the 2-A current source in the 0 network in Fig. P3.79 using loop analys is.

4fl

10V 4fl 5fl 2A

Figure PJ.80

3.81 Find Vo in the circuit in Fig. P3.81 using loop analysis. 0 Then solve the problem using MATLAB and compare your answers.

r-----~~-+r_~~~--~-----Q

12V 1 kfl +

i, + 1 kfl 1 kfl 1 kfl

Figure PJ.81

3 .82 Find Vo in the network in Fig. P3.82 using nodal analys is. 0

10 kfl +

40001, 10 kfl 4 mA '0 kfl

Figure P3.82

() 3.83 Find V" in the circuit in Fig. P3.83 using nodal analys is.

~ 2 ki ,

- +)-----,

1k0 1k0

1 kO 12 V

i ,.

1 kO

L----~---~---~o

Figure 1'3.83

() 3.84 Use mesh analysis to find Vo ill (he circuit in

fi1 Fig. P3.84. - 6 V,

r---~+-->---~

6mA B kO

+ +

12 kO Vx 12kO

4 kO

3.85 Fi nd Vr in the circu it in Fig. P3.85.

2 kO

10 mA t + 2 Vx 4 kO

6kO

Figure 1'3.85

Using mesh analysis, find Vo in the circuit in Fig. P3.86.

Vx 4000 t

6 mA t

Figure 1'3.86

4kO

2 kO

4 kO +

Vx 4 kO

+

Vo

0

PROBLEMS 145

3.87 Find 10 in the circuit in Fig. 1'3.87.

B kO 12V +

6 kO

j 20 i ,

5 kO B kO 5V

4 kO

io

Figure 1'3.87

3.88 Write mesh equations for the circuit in Fig. 1'3.88 using 0 the ass igned currents.

2A

-40

0 _ Vx + 40

50

t 0.sY.,. 20

61 C0 12 V +

Figure 1'3.88

3.89 Find v" in the network in Fig. P3.89.

Vx

1 kO

1 kO 12mA

1k0 1k0

Vx 1000

Figure 1'3.89

1 kO

+

1 kO

4mA

20V

+

146 CHAP TE R 3 N O D AL AN D LOOP ANA LY S I S TECHN I O UE S

Using loop analysis, fi nd ~, in the circu it in Fig. P3.90.

2 kO

12V

1 kO

2 kO

+

1 kO V,

~--VV~--~--i--r-~~------~O +

2 mA 1 kO

L-------+-------~-----~O

Figure P3.90

3.91 Solve for the assigned mesh currents in the network in Fig. P3.9 1.

80 60 50 411

Figure P3.91

~ 3.92 Using the assigned mesh currents shown in Fig. P3.92, solve for the power supplied by the dependent voltage source.

411

30

1203 t 2A

18 V

t--~iVV'--+--++ -811

3 j 5 <~) 511

f A L--__ --+--__ ----'

Figure P3.92

Using loop analysis, find Vo in the circuit in

Fig. P3.93.

2I , 1 kO j 2mA

1 kO 12V 1 kO

L-------+-------~-----~O

Figure P3.93

3.94 Using loop analysis, fi nd Vo in the network in

Fig. P3.94.

1 kO t 4 mA 1 kll

12 V

-+ 1 kO

1 kO 2kll

Figure PJ.94

3.95 Using loop analysis, tind ~, in the circuit in

Fig. P3.9S.

1 kll 1 kO 1 kO

2Vx 12V

-+ -+ + 2kO

4 mA t 2 kf! 2 kO V,

Figure PJ.95

3.96 Using loop analysis, find 10 in the network in

Fig. P3.96.

0 +

Vo

0

1 kO

1 kf!

0 +

Vo

0

1 kll 1 kf! + 2Vx 1 kll

I , 1 kf! 1 kO

1 kO Io +

t 2Ix + 12 V 1 kll Vx

Figure P3.96

o

(} 3-97 Using loop analysis, find 10 in the circuit in

Fig. P3.97.

1 kl1 12 V 1 kl1 t SmA

10 1 kl1

1 kl1 1 kl1 +

2V, + 1 kl1 2I, 1 kl1 V,

I ,

Figure P3.97

Use mesh analysis to determine the power delivered by the independent 3-V source in the network in Fig. P3.98.

10011

200 11 S V,.

300 11

Figure P).98

C 3·99 Use mesh analys is to find the power delivered by the current-contro lled voltage source in the circu it in

Fig. P3.99.

SI1 V .. ~ 3A

+ 15 I ..

t-----+--(-+

V, 8 t

Figure P).99

3211

SI1

I,

111

PROBLEMS 147

3.100 Usc both nodal ancl loop analyses to determine J" in the circui t in Fig. P3. IOO.

1 kl1 1 kl1 2Vx 1 kl1

I , 1 kl1 1 kl1

1 kl1 10 +

2 mA ~ 2/ , + 12V 1 kO V,

Figure P3.1OO

3.101 Use both nodal and loop analyses to find v" in the circuit in Fig. P3. 1 0 I.

Figure P).101

3.102 Find 10 in the network in Fig. P3.1 02 using nodal analysis.

1 kl1 12 V 1 kl1 t 10 1 kl1

1 kl1 1 kl1 +

SmA

2V\· + 1 kl1 t 2/x 1 kn Vx

I,

Figure P).102