Circuit Theory I Methods of Analysisfaraday.ee.emu.edu.tr/eeng223/Circuit Theory-1-C3-analysis...

Circuit Theory I Methods of Analysis Assistant Professor Suna BOLAT

Eastern Mediterranean University

Department of Electric and Electronic Eng.

Ref2: Anant Agarwaland Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare(http://ocw.mit.edu/), Massachusetts Institute of Technology 1

The road so far...

• Method 1: Basic KVL, KCL method of Circuit analysis

• Method 2: Apply element combination rules

• Method 3: Nodal & Mesh analysis – Particular application of KVL, KCL

2

Nodal & Mesh analysis

• Select reference node (ground) from which voltages are measured.

• Label voltages of remaining nodes with respect to ground.

• These are the primary unknowns.

• Write KCL for all but the ground node, substituting device laws and KVL.

• Solve for node voltages.

• Back solve for branch voltages and currents (i.e., the secondary unknowns)

3

Nodal analysis

• Steps to Determine Node Voltages:

1. Select a node as the reference node. Assign voltage v1,

v2, …vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.

2. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

4

Nodal analysis

• Steps to Determine Node Voltages:

reference node

5

(a) common ground, (b) ground, (c) chassis.

Nodal Analysis

• Typical circuit for nodal analysis

voltages v1 and v2 are assigned with respect to the reference node (i.e ground).

6 Reference Node

7

233

3

23

2122

2

212

111

1

11

or 0

)(or

or 0

vGiR

vi

vvGiR

vvi

vGiR

vi

1 2 1 2

2 2 3

I I i i

I i i

R

vvi

lowerhigher

Apply KCL to each non-reference node.

• Substituting element relations into KCL

8

3

2

2

212

2

21

1

121

R

v

R

vvI

R

vv

R

vII

232122

2121121

)(

)(

vGvvGI

vvGvGII

2

21

2

1

322

221

I

II

v

v

GGG

GGG

Example 3.1

• Calculate the node voltages in the circuit shown below.

9

• Calculate the node voltages in the circuit shown below.

Example

10

Determine the voltages at nodes 1, 2 and 3.

Example 3.2

11

Reminder: Cramer’s rule

12

• Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one.

• Case 2: The voltage source is connected between two nonreference nodes: a generalized node (supernode) is formed.

Nodal Analysis with Voltage Sources

13

• A circuit with a supernode.

14

• A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

• The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the voltage source.

Supernode

15

• Find the node voltages of the circuit below.

i1 i2

1 2

1 2 1 2

1 2

2 7 0

2 7 0 5 2 202 4 2 4

i i

v v v vv v

1 2 2v v

1 2

1 2

2 20

2

v v

v v

1 1

2

223 22 = 7.33V

3

22 16 +2= = 5.33V

3 3

v v

v

Example

16

• Find the node voltages of the circuit below.

Example

17

• Apply KCL to the two supernodes.

At supernode 1-2:

Example (continued...)

18

• Apply KCL to the two supernodes.

At supernode 3-4:


19

• Apply KVL around the loops:

Loop1:

Loop2:

Loop3:


20

• You can use Matlab to solve large matrix equations:

0

20

0

60

2103

0011

16524

2115

4

3

2

1

v

v

v

v

>> A=[5 1 -1 -2 4 2 -5 -16 1 -1 0 0 3 0 -1 -2]; >> B= [60 0 20 0]'; >> V=inv(A)*B V = 26.6667 6.6667 173.3333 -46.6667

V67.46

V33.173

V67.6

V67.26

4

3

2

1

v

v

v

v

We now use MATLAB to solve the matrix Equation. The Equation on the left can be written as AV =B → V= B/A= A-1B

Matlab Code


21

1. Mesh analysis: another procedure for analyzing circuits,

applicable to planar circuits. 2. A Mesh is a loop which does not contain any other loops

within it. 3. Nodal analysis applies KCL to find voltages in a given circuit,

while Mesh Analysis applies KVL to calculate unknown currents.

Mesh Analysis

22

• A circuit is planar if it can be drawn on a plane with no branches crossing one another. Otherwise it is nonplanar.

• The circuit in (a) is planar, because the same circuit that is redrawn(b) has no crossing branches.

Mesh Analysis

23

• A nonplanar circuit.

Mesh Analysis

24

• Steps to Determine Mesh Currents:

1. Assign mesh currents i1, i2, .., in to the n meshes.

2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.

3. Solve the resulting n simultaneous equations to get the mesh currents.

Mesh Analysis

25

• A circuit with two meshes.

Mesh Analysis

26

• A circuit with two meshes.

• Apply KVL to each mesh. For mesh 1,

• For mesh 2,

123131

213111

)(

0)(

ViRiRR

iiRiRV

223213

123222

)(

0)(

ViRRiR

iiRViR

Mesh Analysis

27

• Solve for the mesh currents.

• Use i for a mesh current and I for a branch current. It’s evident from the circuit that:

2

1

2

1

323

331

V

V

i

i

RRR

RRR

2132211 , , iiIiIiI

Mesh Analysis

28

• A circuit Find the branch currents I1, I2, and I3 using mesh analysis.

Example 3.5

29

• For mesh 1,

• For mesh 2,

• We can find i1 and i2 by substitution method or Cramer’s rule. Then,

123

010)(10515

21

211

ii

iii

12

010)(1046

21

1222

ii

iiii

2132211 , , iiIiIiI

Example 3.5

30

• Use mesh analysis to find the current Io in the circuit below.

Example 3.6

31

• Apply KVL to each mesh. For mesh 1,

• For mesh 2,

126511

0)(12)(1024

321

3121

iii

iiii

02195

0)(10)(424

321

12322

iii

iiiii

Example 3.6

32

• For mesh 3,

• In matrix from:

• we can calculate i1, i2 and i3 by Cramer’s rule, and find Io.

02

0)(4)(12)(4

, A, nodeAt

0)(4)(124

321

231321

210

23130

iii

iiiiii

iII

iiiiI

0

0

12

211

2195

6511

3

2

1

i

i

i

Example 3.6

33

• Consider the following circuit with a current source.

Mesh analysis with current source

34

• Possibe cases of having a current source.

Case 1

• Current source exist only in one mesh

mesh 2:

mesh 1:

• One mesh variable is reduced

Case 2

• Current source exists between two meshes, a super-mesh is obtained.


35

• Possibe cases of having a current source.

• A supermesh is considerred when two meshes have a (dependent/independent) current source in common.


36

• Applying KVL in the supermesh below:

• KVL in the supermesh above: • Apply KCL at node 0 above:


37

• Properties of a Supermesh

1. The current is not completely ignored

• provides the constraint equation necessary to solve for the mesh current.

2. A supermesh has no current of its own.

3. Several current sources in adjacency form a bigger supermesh.

4. A supermesh requires the application of both KVL and KCL.


38

• Apply KVL in the supermesh (mesh 1 + mesh 2 + mesh 3):

• Apply KVL in mesh 4:

• Apply KCL at node P:

• Apply KCL at node Q:

0

5

5

0

3110

0011

5400

4631

4

3

2

1

i

i

i

i

• 4 equations for 4 variables. Using Cramer’s Rule


39

• The analysis equations can be obtained by direct inspection

a) For circuits with only resistors and

independent current sources. (nodal analysis)

b) For planar circuits with only resistors and independent voltage sources. (mesh analysis)

Nodal & Mesh analysis

40

a) For circuits with only resistors and independent current sources. (nodal analysis).

• The circuit has two nonreference nodes and the node equations:

• In Matrix form:

• Consider the following example:

41

• In general, if a circuit with independent current sources has N nonreference nodes, the node-voltage equations can be written in terms of conductances as:

• G is called the conductance matrix, v is the output vector, and i is the input vector. 42

b) For planar circuits with only resistors and independent voltage sources. (mesh analysis)

• The circuit has two meshes and the mesh equations :

• In Matrix form:

• Consider the following example:

43

• In general, if a circuit has N meshes, mesh-current equations can be expressed in terms of resistances as:

• R is called the resistance matrix, i is the output vector, and v is the input vector. 44

Write the node-voltage matrix equations in the following circuit.

Example 3.8

45

625.11

1

2

1

8

1 ,5.0

4

1

8

1

8

1

325.11

1

8

1

5

1 ,3.0

10

1

5

1

4433

2211

GG

GG

125.0 ,1 ,0

125.0 ,125.0 ,0

11

1 ,125.0

8

1 ,2.0

0 ,2.05

1

434241

343231

242321

141312

GGG

GGG

GGG

GGG

• The circuit has 4 nonreference nodes, so the diagonal terms of G are:

• The off-diagonal terms of G are:

Example 3.8

46

• The input current vector i in amperes:

642 ,0 ,321 ,3 4321 iiii

6

0

3

3

.6251 0.125 1 0

0.125 .50 0.125 0

1 0.125 .3251 0.2

0 0 0.2 .30

4

3

2

1

v

v

v

v

• The node-voltage equations can be calculated by:

Example 3.8

47

Write the mesh-current equations in in the following circuit.

Example 3.9

48

6 ,0 ,6612

,6410 ,4

543

21

vvv

vv

6

0

6

6

4

4 3 0 1 0

3 8 0 1 0

0 0 9 4 2

1 1 4 01 2

0 0 2 2 9

5

4

3

2

1

i

i

i

i

i

• The input voltage vector v in volts :

• The mesh-current equations are:

Example 3.9

49

• Both nodal and mesh analyses provide a systematic way of analyzing a complex network.

• The choice of the better method is dictated by two factors.

First factor: nature of the particular network.

The key is to select the method that results in the smaller number of equations.

Second factor: information required.

Nodal vs mesh analysis

50

• Both nodal and mesh analysis provide a systematic way of analyzing a complex network.

• The choice of the better method is dictated by two factors.

1. Nodal analysis: Apply KCL at the nonreference nodes. (The circuit with fewer node equations)

2. A supernode: Voltage source between two nonreference nodes.

3. Mesh analysis: Apply KVL for each mesh. (The circuit with fewer mesh equations)

4. A supermesh: Current source between two meshes.

Summary

51

(a)An npn transistor, (b) dc equivalent model.

Application: DC Transistor Circuits: BJT Circuit Models

52

For the BJT circuit in the figure =150 and VBE = 0.7 V. Find v0.

Example 3.13

53

• Use mesh analysis or nodal analysis

Example 3.13

54

Circuit Theory I Methods of Analysisfaraday.ee.emu.edu.tr/eeng223/Circuit Theory-1-C3-analysis...

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