CHAPTER

3

Resistive Network Analysis

Figure 3.2

Figure 3.2

3-1

Branch current formulation in nodal analysis

Use of KCL in nodal analysis

Figure 3.3

3-2

Network Analysis

- Determine each of the unknown branch currents and node voltages

Branch voltages

Vs = Va - Vd = Vs

VR1 =Va – Vb

VR2 = Vb – Vd = Vb

VR3 = Vb – Vc

VR4 = Vc – Vd = Vc

R1 R3b c

d

R2

+VR2

- Vs

+−

+VR4

-

a+ VR3 -+ VR1 -

Identify branch and node voltage

Node Voltages

Va = Vs

Vb = VR2

Vc = VR4

Vd = 0 (ref)

Node Voltage Method-Based on defining the voltage at each node as independent variable

-Select one node as reference (V=0)

-Once each node voltage is defined, apply Ohm’s Law to determine current in each branch

-Branch current is expressed in terms of one or more node voltages.

-Apply KCL at each node

+ V -i

R

a bVa Vb I = Va – Vb = VR

R R

Figure 3.4

Figure 3.4

3-3

Illustration of nodal analysis

Figure 3.6

Figure 3.6

3-4

Summary of nodal analysis method

1) Select a ref node

2) Define remaining n-1 node voltage

3) Apply KCL at each node

4) Solve linear equations

Figure 3.11

Figure 3.11

3-5

Circuit for Example 3.6

Ex:

i

I

R1

R2

R3 R4

− +Va Vb

Vc

i1

i4 i3

i2

VUse nodal analysis to find i

R1 = 2 Ω, R2 = 2 Ω

R3 = 4 Ω, R4 = 3 Ω

I = 2A, V = 3V

Ex: R2

ia ibR3R1 R4

ia = 1 mA

ib = 2 mA

R1 = 1 kΩ

R2 = 500 Ω

R3 = 2.2 kΩ

R4 = 4.7 kΩ

Figure 3.12

3-6

Basic principle of mesh analysisFigure 3.12

Figure 3.13Use of KVL in mesh analysis

Figure 3.13

3-7

A two-mesh circuitFigure 3.14

3-8

Figure 3.15

Figure 3.15

Assignment of currents and voltages around mesh 1

3-9

Figure 3.17, 3.18

Figure 3.17

3-10

Figure 3.18

Figure 3.21

3-11

Circuit used to demonstrate mesh analysis with current sourcesFigure 3.21

Figure 3.25

3-12

Figure 3.25

Summary

1) Define each mesh current

2) In a circuit with n meshes and m current sources, n-m independent eq’s will result

3) Apply KVL

4) Solve eq’s

Ex:

Ex:

i1

R3

R4R2

R1

i2

+

−

+

−

+

−

V1 V2 V3

Find mesh eq’s

V1 = 10 V, V2 = 9 V, V3 = 1 V

R1 = 5 Ω, R2 = 10 Ω, R3 = 5 Ω, R4 = 5 Ω

I

R1

R2

R3

R4

− + i1

i3

i2

V

Find mesh current

I = 0.5 A, V = 6 V,

R1 = 3 Ω, R2 = 8 Ω,

R3 = 6 Ω, R4 = 4 Ω

Figure 3.27

3-13

The principle of superposition Figure 3.27

In a linear cct containing N sources, each branch voltage and current is the sum of N voltages and currents, each of which may be computed by setting all but one source equal to zero and solving the cct containing that single source

Net current = (VB1 + VB2)/R = VB1/R + VB2/R

= iB1 + iB2

Figure 3.28

3-14

Zeroing voltage and current sources Figure 3.28

Ex: Use superposition principle

Solution:

a) Set Vs = 0,

b) Set Is = 0

R3

R2

+

−

Is

Vs

R1

+

V3

Find V3

Is = 12 A, Vs = 12 V

R1 = 1 Ω, R2 = 0.3 Ω, R3 = 0.23 Ω

R3R2

IsR1

+

V3a

R3R2

Vs

R1

+

V3b

+

Thevenin and Norton Equivalent Circuit

• Thevenin Theorem – When viewed from the load, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal voltage source VT in series with an equivalent resistance RT

• Norton Theorem - When viewed from the load, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal current source IT in parallel with an equivalent resistance RN

Figure 3.34, 3.35

Illustration of Thévenin theorem

Illustration of Norton theorem

Determination of Thevenin or Norton Eq Resistance

• Remove the load

• Zero all independent voltage and current sources

• Compute total resistance between load terminals with the load removed

Figure 3.36, 3.37

3-16

Computation of Thévenin resistance Figure 3.36

Figure 3.37

Equivalent resistance seen by the load

Computing Thevenin Voltage

• Defined as – the eq (Thevenin) source voltage is equal to the open cct voltage present at the load terminal (with the load removed)

• Method1. Remove the load2. Define open cct voltage Voc3. Apply analysis method to solve for Voc

4. Thevenin voltage is VT = Voc

Figure 3.43

3-17

Equivalence of open-circuit and Thévenin voltage Figure 3.43

Figure 3.47

3-18

A circuit and its Thévenin equivalent Figure 3.47

Figure 3.54

3-19

Computation of Norton current Figure 3.54

Figure 3.67

3-20

Measurement of open-circuit voltage and short-circuit current Figure 3.67

3-21

Power transfer between source and load

Figure 3.70

Graphical representation of maximum power transfer

Figure 3.69

Figure 3.73, 3.74

3-22

The i-v characteristic of exponential resistor Figure 3.73 Figure 3.74

Representation of nonlinear element in a linear circuit

Figure 3.75, 3.76

3-23

Load line Figure 3.75

Figure 3.76

Graphical solution of equations 3.44 and 3.45