Chapter 4

Methods of Analysis of Resistive Circuits

• Node voltage analysis - with independent source : current source / voltage source - with dependent source

• Mesh current analysis - with independent source : voltage source / current source

- with dependent source

Resistive Circuit Analysis

Figure 4.2-1(a) A circuit with three nodes.(b) The circuit after the nodes have been labeled and a reference node has been selected and marked.(c) Using voltmeters to measure the node voltages.

Node Voltage Analysis – with current sources

To write a set of node equations, we do two things:

1. Express element currents as functions of the node voltages.

2. Apply KCL at each of the nodes of the circuits, except for the

reference node.

Figure 4.2-2Node voltages, v1 and v2 , and element voltage, va, of a circuit element.

Figure 4.2-3 Node voltages, v1 and v2, and element voltage, v1 - v2, of a (a) generic circuit element, (b) voltage source, and (c) resistor.

21 vvva

Node Voltage Analysis – with current sources

Figure 4.2-4(a) A circuit with three resistors.(b) The resistor voltages expressed as functions of the node voltages.(c) The resistor currents expressed as functions of the node voltages.

Node Voltage Analysis – with current sources

12 R

vv

R

vi baaS

KCL

31 R

v

R

vv bba

If

15.04 baa vvv

5.01bba vvv

AiRRR S 4,5.0,1 321 node (a)

node (b)

Example 4.2-1

Node Voltage Analysis – with current sources

node (a)

node (b)

0510

4

baa vvv

045

R

vvv bba

Example 4.2-2

Example 4.2-3

Node Voltage Analysis – with current sources

05

22

11

R

vvi

R

vvi

R

vv bacacanode (a)

node (b)

node (c)

03435

2

i

R

v

R

vv

R

vvi bcbba

21215521

111111iiv

RRv

Rv

RRR cba

2335435

11111iiv

Rv

RRRv

R cba

16321321

1111111iv

RRRRv

Rv

RR cba

Exercise 4.2-1

Exercise 4.2-2

Node Voltage Analysis – with current sources

32

1

2

1

4

1

ba vv

43

1

2

1

2

1

ba vv

node (a)

node (b)

Figure 4.3-1Circuit with an independent voltage sourceand an independent current sources.

Figure 4.3-2Circuit with a supernode that incorporate va and vb.

Node Voltage Analysis – with current and voltage sources

A supernode consists of two nodes connected by an

independent

or a dependent voltage source

Sa vv 23 R

vv

R

vi abbS

23 R

vv

R

vi SbbS

32

332

RR

vRiRRv SSb

Sba vvv bSa vvv

SbbS iR

v

R

vv

21

21

221

RR

vRiRRv SSb

Example 4.3-1

Example 4.3-2

Node Voltage Analysis – with current and voltage sources

Method #1

Apply KCL to the Supernode

Method #2

Vva 4

Vvv cb 8

212126

cbab vvvv

8 cb vv

Method #1

Method #2 : apply KCL to the Supernode

Vva 1212 ab vv

65.1 avi

Vvb 0

03

5.3 bvi

635.35.1 ab vv

360.2 ba vv

35.3

65.1 ba vv

36

0.2 ba vv

Example 4.3-3

Exercise 4.3-1 Exercise 4.3-2

Node Voltage Analysis – with current and voltage sources

Let Supernode be the 10V source

Apply KCL to the Supernode

524010

bcba vvvv

Vvb 12

10 ca vv

530

220

ba vv10 ba vv

10

12

403

ab vv8 ba vv

Example 4.4-1

Example 4.4-2

Example 4.4-3

Node Voltage Analysis – with dependent sources

6ba

x

vvi

24

6

833 bb

xc

vviv

KCL at node (b)

32

6

8 cbb vvv

Vvb 7 Vvc 5.0

aaxba vvvvv 444

KCL at supernode

aaaba vvvvv

4

3

10

5

41043

Vva 4 Vvv ab 205

ab vv 5

2010

6 baa

a vvi

v

KCL at node (b)

ab iv

520

0

20

12 ba

vi

KCL at node

(a)

Vvb 10

Exercise 4.4-1

Exercise 4.4-2

Node Voltage Analysis – with dependent sources

12

4

128

6 aaa

baa

a ivi

vvi

v

Vva 0

8

9ai

KCL at node (a)

Viv ab 5.44

015

4

20

6

aaa vvv

Figure 4.5-1Nonplanar circuit with a crossover.

Figure 4.5-2Circuit with four meshes.Each mesh is identified by dashed lines.

- Loop

- Mesh : 다른 loop 를 포함하고 있지 않은 loop (cross over 가 없는 circuit 에만 적용

가능 )

Mesh Current Analysis – with independent voltage source

Figure 4.5-3(a) A circuit with two meshes(b) Inserting ammeters to measure the mesh currents.

Figure 4.5-4 Mesh currents, i1 and i2,and element current, i1 – i2, of a(a) generic circuit element,(b) current source, and (c) resistor.

321 ii

Mesh Current Analysis – with independent voltage source

To write a set of mesh equations, we do two things:

1. Express element voltages as functions of the mesh

currents.

2. Apply KVL to each of the meshes of the circuits.

Figure 4.5-5

Mesh Current Analysis – with independent voltage source

21311 iiRiRvs

22213 iRiiR

Figure 4.5-6

sviRiRR 24141 )(

0)( 35252414 iRiRRRiR

gviRRiR 35325 )(

Mesh #1

Mesh #2

Mesh #1

Mesh #2

Mesh #3

21411 iiRiRvs

012432522 iiRiiRiR

0)( 23533 gviiRiR

)8(123)36( 21 ii

8)63(3 21 ii

Exercise 4.5-1

Mesh Current Analysis – with two independent voltage sources

812336 211 iii

06338 212 iii

Figure 4.6-2Circuit with an independent current sourcecommon to both meshes.

Figure 4.6-1Circuit with an independent voltage source and an independent current source.

sii 2

sab vviR 11

abviRR 232 )(

12 iiis

Mesh Current Analysis – with current and voltage source

Mesh #1

Mesh #2

Mesh #1 sviRiRR 22121 )(

21

21 RR

iRvi ss

sviRRiR 23211

ss viiRRiR 13211

Example 4.6-1

Figure 4.6-4 Circuit with a supermesh that incorporates mesh 1 and mesh 2, indicated by the dashed line.

Mesh Current Analysis – with current and voltage source

10121 abviiR

0)( 33132 abviRiiR

41 i

Mesh #2

Mesh #3

532 ii

A Supermesh is one larger mesh created from two meshes that have an independent or dependent current source in common.

SupermeshMesh #3

102)(3)(1 23231 iiiii

Current source

10451 321 iii

0631 321 iii

521 ii

0)(32)(1 23313 iiiii

SupermeshMesh #3

Example 4.6-2

Mesh Current Analysis – with current and voltage source

0129 1 vi

5.121 ii

Method #1

Method #2

063 22 vii

1299 21 ii 012639 221 iii

1299 21 ii012639 221 iii

Exercise 4.6-1

Exercise 4.6-2

Mesh Current Analysis – with current and voltage source

Ai4

31

04)23(9 122 iii

Ai3

4

9

122

Vv 433

4

0)3(6315 ii

Example 4.7-1

Mesh Current Analysis – with dependent source

21 iiia

KVL to mesh #1

02432 1 i

12 4

5ii

032 2 mviKVL to mesh #2

25 iia

Ai4

31 Ai

16

152

Vivm 3016

153232 2

Example 4.7-2

Mesh Current Analysis – with dependent source

VAia 2.72.7

21 iiia

KVL to mesh #1

KVL to mesh #2

03610 1 i Ai 6.31

Ai 8.12 02.74 2 i

AVii

Ai

i

AiA a

a

a /48.16.3

2.7

21

Example 4.8-1

Figure 4.8-1

Node Voltage & Mesh Current Analysis – comparison

Figure 4.8-2

0122 23313 iiRiii

Ai 11

2R

Ai 32

Ai 5.03

KVL to mesh #3

Example 4.8-2

Figure 4.8-4

Node Voltage & Mesh Current Analysis – comparison

Figure 4.8-3

R

vvv 331 22

Vv 161

8R

Vvv 1821

KCL at node 3

Vv 22

Vv 163

Mesh Current Analysis – MATLAB

0)( 12131411 viiRiRiR 0)( 21322225 iiRviRiR

1231341 viRiRRR 2232513 viRRRiR

123131 viRiRaRR p

223213 1 viRRRaiR p

Node Voltage and Mesh Current Analysis – PSpice

Example 4.10-1

Example 4.11-1

Example 4.11-2

Node Voltage and Mesh Current Analysis – MATLAB

Figure 4.12-1Proposed circuit for measuring and displaying the angular position of the potentiometer shaft.

Potentiometer Angle Display

Figure 4.12-2Circuit diagram containing models of the power supplies, voltmeter, and potentiometer.

Figure 4.12-3 The redrawn circuit showing the mode v1.

kvo /1.0 Vk ov101.0ovwhere

Angle varies from to

180 180Potentiometer varies from to

0 1

Potentiometer Angle Display

Figure 4.12-3 The redrawn circuit showing the mode v1.

0)1(

)15(15

2 21

p

i

p

ii

RaR

v

aRR

v

M

v

RRR 21

ppp

pi RRRMRaRaRR

RRaRMv

2121

21

2)1(

15)12(2

Assumption MRp 2 KR 20

Then, ppp RRMRaRaRR 22)1(

pp

i RR

aRv

2

15)12(

180

15

2

V

RR

Rv

p

pi

180

5.7

180

15

2

1 VVviLet kRkR p 10,5

4.2)1.0(5.7

180b

180

5.7 Vbbvv io

ree

voltVb

deg1.0

180

5.7

Figure 4.12-4 The final designed circuit.

Potentiometer Angle Display

ppp

pi RRRMRaRaRR

RRaRMv

2121

21

2)1(

15)12(2

150 9167.02

1

360

150

aSuppose

24.6

1052210)9167.01(5109167.05

15)19167.02(102

kkMkkkk

kMvi

98.14)24.6()4.2( ov8.14910 ov

Problems P4.2-4 / P4.2-7 / P4.3-5 / P4.3-11 /

P4.4-7 / P4.4-10 / P4.4-17 / P4.5-6

/

P4.6-10 / P4.7-10 / P4.8-2

Homework #4