INC 111 Basic Circuit Analysis

Week 3

Nodal Analysis

Systematic Methods for Analyzing a Circuit

• Nodal Analysis

• Mesh Analysis (Loop analysis)

NodeNode = Point that connected together on the same wire

10V

4K

6K VExample

3 nodes

Nodal AnalysisProcedure

1. Count the number of nodes = N

2. Choose one of the nodes to be a reference node (0V) when there is no ground

3. Give the voltage for other nodes to be V1, V2, V3, …

4. Use Kirchoff’s current law on each node to form N-1 equations

5. Solve the equations to find V1, V2, V3, …

Example

1Ω2Ω3A

5Ω

2A

0V (reference)

V1 V2

1Ω2Ω3A

5Ω

2A

0V (reference)

V1 V2

302217

0)2212(1530

05

21

2

013

VV

VVV

VVV

Equation 1

1Ω2Ω3A

5Ω

2A

0V (reference)

V1 V2

10261

01025)12(

021

02

5

12

VV

VVV

VVV

Equation 2

Solve the Equations

1Ω2Ω3A

5Ω

2A

V1 V2

V1 = 5 VV2 = 2.5V

Example

1Ω

2Ω

-8A 5Ω-25A

3Ω

4Ω

-3A

V1 V2 V3

0V

1Ω

2Ω

-8A 5Ω-25A

3Ω

4Ω

-3A

V1 V2 V3

0V

132332417

0241436331396

03

213

4

318

VVV

VVVV

VVVV

Equation 1

1Ω

2Ω

-8A 5Ω-25A

3Ω

4Ω

-3A

V1 V2 V3

0V

183321112

0263323181222

01

02

2

323

3

12

VVV

VVVVV

VVVVV

Equation 2

1Ω

2Ω

-8A 5Ω-25A

3Ω

4Ω

-3A

V1 V2 V3

0V

50031921015

0345001535210310

05

0325

4

13

2

23

VVV

VVVVV

VVVVV

Equation 3

Solve the Equations

50031921015

183321112

132332417

VVV

VVV

VVV

Cramer’s Rule (Optional)

50031921015

183321112

132332417

VVV

VVV

VVV

956.0816

780

19105

3112

347

1910500

31118

34132

1

V

576.10816

8628

19105

3112

347

195005

3182

31327

2

V

132.32816

26220

19105

3112

347

500105

18112

13247

3

V

50031921015

183321112

132332417

VVV

VVV

VVV

Supernode

When there is a voltage source in the circuit, direct KCL cannot be used because we do not know the current in the voltage source.

We will use the idea of supernode.

Supernode is the method that combines 2 nodes together when using KCL. It will include the voltage source within the circle when using KCL.

Example

1Ω-8A 5Ω-25A

3Ω

4Ω

-3A

1V

V1 V2 V3

0V

1Ω-8A 5Ω-25A

3Ω

4Ω

-3A

1V

V1 V2 V3

0V

132332417

0241436331396

03

213

4

318

VVV

VVVV

VVVV

Equation 1

1Ω-8A 5Ω-25A

3Ω

4Ω

-3A

1V

V1 V2 V3

0V

132

1680327280135

02603121500115315180120220

01

02

5

0325

4

133

3

12

VV

VVV

VVVVVV

VVVVVV

Equation 2

supernode

Equation 3

132

1680327280135

132332417

VV

VVV

VVV

V1 = -4.952 VV2 = 14.333 VV3 = 13.333 V

Example

1Ω2Ω

5Ω

2A3V

0V

V1 V2

6

132

01

022

5

12

31

V

VVV

V

Sanwa YX360TRF Multimeter

Sanwa YX360TRF Multimeter

RangeSelector

Dial ResistanceZero Adjust

Resistance Measurement

Read this Range

Select these ranges

Voltage Measurement

Read this Range

Select these ranges

Current Measurement

Read this Range

Select these ranges

Measure the Resistance ofAmmeter and Voltmeter

Adjust a multimeter to measure volt and use another multimeter to measure its resistance.

Adjust a multimeter to measure amp and use another multimeter to measure its resistance.

Resistor Code

Standard Resistor Value

1R0 10R 100R 1K0 10K 100K 1M0 10M

1R2 12R 120R 1K2 12K 120K 1M2 n/a

1R5 15R 150R 1K5 15K 150K 1M5 n/a

1R8 18R 180R 1K8 18K 180K 1M8 n/a

2R2 22R 220R 2K2 22K 220K 2M2 n/a

2R7 27R 270R 2K7 27K 270K 2M7 n/a

3R3 33R 330R 3K3 33K 330K 3M3 n/a

3R9 39R 390R 3K9 39K 390K 3M9 n/a

4R7 47R 470R 4K7 47K 470K 4M7 n/a

5R6 56R 560R 5K6 56K 560K 5M6 n/a

6R8 68R 680R 6K8 68K 680K 6M8 n/a

8R2 82R 820R 8K2 82K 820K 8M2 n/a

E12 -Standard

Protoboard