INC 111 Basic Circuit Analysis
Week 3
Nodal Analysis
Systematic Methods for
Analyzing a Circuit
• Nodal Analysis
• Mesh Analysis (Loop analysis)
Node
Node = Point that connected together on the same wire
Example
6K
V
10V
4K
3 nodes
Nodal Analysis
Procedure
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
V1
V2
5Ω
3A
2Ω
0V (reference)
1Ω
2A
V1
V2
5Ω
3A
2Ω
1Ω
0V (reference)
V1 0 V1V 2
3

0
2
5
 30  5V 1  (2V 1  2V 2)  0
7V 1  2V 2  30
Equation 1
2A
V1
V2
5Ω
3A
2Ω
1Ω
0V (reference)
V 2 V1 V 2  0

20
5
1
(V 2  V 1)  5V 2  10  0
 V 1  6V 2  10
Equation 2
2A
Solve the Equations
V1
V2
5Ω
3A
2Ω
1Ω
V1 = 5 V
V2 = 2.5V
2A
Example
4Ω
-3A
3Ω
V1
V2
1Ω
-8A
0V
2Ω
V3
5Ω
-25A
4Ω
-3A
3Ω
V1
V2
2Ω
1Ω
-8A
V3
5Ω
0V
V1V 3
V1V 2
3
0
4
3
96  3V 1  3V 3  36  4V 1  4V 2  0
8
7V 1  4V 2  3V 3  132
Equation 1
-25A
4Ω
-3A
3Ω
V1
V2
2Ω
1Ω
-8A
V3
5Ω
0V
V 2 V1
V 2 V 3 V 2  0
3

0
3
2
1
2V 2  2V 1  18  3V 2  3V 3  6V 2  0
 2V 1  11V 2  3V 3  18
Equation 2
-25A
4Ω
-3A
3Ω
V1
V2
2Ω
1Ω
-8A
V3
5Ω
0V
V 3 V 2 V 3 V1
V30

 25 
0
2
4
5
10V 3  10V 2  5V 3  5V 1  500 4V 3  0
 5V 1  10V 2  19V 3  500
Equation 3
-25A
Solve the Equations
7V 1  4V 2  3V 3  132
 2V 1  11V 2  3V 3  18
 5V 1  10V 2  19V 3  500
Cramer’s Rule (Optional)
7V 1  4V 2  3V 3  132
 2V 1  11V 2  3V 3  18
 5V 1  10V 2  19V 3  500
 132  4
18
11
3
3
500  10 19 780
V1 

 0.956
7
4 3
816
 2 11  3
 5  10 19
7
2
 132  3
18
3
5
V2
7
7V 1  4V 2  3V 3  132
 2V 1  11V 2  3V 3  18
 5V 1  10V 2  19V 3  500
500 19 8628

 10.576
4 3
816
 2 11  3
 5  10 19
7
4
 132
2
11
18
 5  10 500
26220
V3 

 32.132
7
4 3
816
 2 11  3
 5  10 19
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
4Ω
-3A
3Ω
V1
1Ω
-8A
0V
V3
V2
1V
5Ω
-25A
4Ω
-3A
V1
3Ω
1Ω
-8A
V3
V2
1V
-25A
5Ω
0V
V1V 3
V1V 2
8
3
0
4
3
96  3V 1  3V 3  36  4V 1  4V 2  0
7V 1  4V 2  3V 3  132
Equation 1
4Ω
-3A
3Ω
V1
supernode
1Ω
-8A
V3
V2
1V
5Ω
-25A
0V
V 2 V1
V 3 V1
V30 V 20
3
 25 

0
3
4
5
1
20V 2  20V 1  180 15V 3  15V 1  1500 12V 3  60V 2  0
 35V 1  80V 2  27V 3  1680
V 2 V 3  1
Equation 3
Equation 2
7V 1  4V 2  3V 3  132
 35V 1  80V 2  27V 3  1680
V 2 V 3  1
V1 = -4.952 V
V2 = 14.333 V
V3 = 13.333 V
Example
V1
V2
5Ω
2Ω
3V
1Ω
0V
V1  3
V 2  V1
V20
2
0
5
1
13
V2 
6
2A
Sanwa YX360TRF Multimeter
Sanwa YX360TRF Multimeter
Dial
Range
Selector
Resistance
Zero 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 of
Ammeter 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
E12 -Standard
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
Protoboard