Electric Circuit - Lecture 04

Electric Circuit Analysis

Lec4: Nodal analysis

Ahsan KhawajaAhsan_khawaja@comsats.edu.pkLecturerRoom 102Department of Electrical Engineering

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis.

1. Reference Node

The reference node is called the ground node where V = 0

+

–

V 500W

500W

1kW

500W

500WI1 I2

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

2. Node Voltages

V1, V2, and V3 are unknowns for which we solve using KCL

500W

500W

1kW

500W

500WI1 I2

1 2 3

V1 V2 V3

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

Currents and Node Voltages

500W

V1

500WV1 V2

50021 VV

5001V

3. KCL at Node 1

500W

500WI1

V1 V2

500500

1211

VVVI

3. KCL at Node 2

500W

1kW

500W V2 V3V1

0500k1500

32212

VVVVV

3. KCL at Node 3

2323

500500I

VVV

500W

500W

I2

V2 V3

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

Typical circuit for nodal analysis

• Find the node voltages in the circuit shown below.

• At node 1

2

0

45 121

321

vvv

iii

• At node 2

6

0

45 212

5142

vvv

iiii

Practice

• Determine the voltage at the nodes in Fig. below

• At node 1,

243

3

2131

1

vvvv

ii x

• At node 2

4

0

8223221

32

vvvvv

iiix

• At node 3

2

)(2

84

2

213231

21

vvvvvv

iii x

• In matrix form:

0

0

3

8

3

8

9

4

38

1

8

7

2

14

1

2

1

4

3

3

2

1

v

v

v

Spot quiz…

Example:Use nodal analysis to find the voltage at each node of this circuit.

• Step 1:Identify and label, each node in the circuit. Ground has been chosen for you.

• Step 2:Write the Nodal equation for each node identified in Step 1. • Node 1:Because the voltage at node 1 is known with respect to our

reference point, the equation is:V1 = 71 volt

• Node 2:At node 2, assume all currents are leaving the node as shown in Figure 5.

All currents are assumed to be leaving node V2,

so all terms are positive.

Node 2 I (1) + I (2) + I (3) = 0

Ohm's Law I (1) = (V2 - V1)/2

I (2) = (V2 – 0)/11

I (3) = (V2 - V3)/10

Substituting: (V2 - V1)/2 + V2/11 + (V2 - V3)/10 = 0

• I (4) and I (5) are assumed to be leaving node V3, so these terms are positive. But Is (the 2A source) is entering, so it is assigned a negative sign.

Node 3 -Is + I (4) + I (5) = 0

Ohm's Law I (4) = (V3 - V2)/10

I (5) = V3/5

Substituting:-2 + (V3 - V2)/10 + V3/5 = 0

• The three equations are shown below:

Node 1: V1 = 71v

Node 2 : (V2 - V1)/2 + V2/11 + (V2 - V3)/10 = 0

Node 3: -2 + (V3 - V2)/10 + V3/5 = 0

• After solving we find that the node voltages are:

V1 = 71v

V2 = 55v

V3 = 25v

Example:Use nodal analysis to find the voltage at each node of this circuit.