S A R I A T I B I N T I D A L I B

NODAL ANALYSIS

Kirchoff’s Current Law (KCL)

DEFINITION:

The total current leaving each branch is equal to zero.

OR

Step to determine Nodal Voltages:

1. Identify every node in the circuit. 2. Label each node with a node voltage (V1, V2, etc). The node with the

highest number of branches connected should be labeled as the ground node (reference node) having zero potential.

3. At a particular node of interest (except ground), use Ohm’s law to express the current through any branch connected to that node as the difference between the two node voltages at both end of that branch divided by the branch impedance. The voltage at the node of interest is always considered to be at higher potential than the rest of the node voltages.

4. Apply KCL to sum all currents at that node of interest. The resulting algebraic equation (called nodal equation) has all node voltages as its unknowns.

5. Solve the resulting simultaneous nodal equations to obtain the values of the unknown node voltages. Use the values of node voltages above to find voltages and/or currents throughout the rest of the circuit.

EXAMPLE 1

Q: Refer to the circuit below, determine V1 value using the Nodal Voltage Method

Solution:

+

–

4–j3

010 V

1+j2 2+j3

06 V

–

+

V1 I3

I2

I1

1. Identify every node in the circuit.- Only One node 2. Label each node with a node voltage : V1

Label each current leaving the node : I1 ,I2 and I3

Step3: Apply KCL to sum all currents at V1

𝐼1 + 𝐼2 + 𝐼3 = 0

𝑉1 − 1000

2 + 𝑗3+

𝑉1

4 − 𝑗3+

𝑉1 − −6∠0°

1 + 𝑗2= 0

𝑉1

1

2 + 𝑗3+

1

4 − 𝑗3+

1

1 + 𝑗2−

1000

(2 + 𝑗3)+

6∠0°

(1 + 𝑗2)= 0

Step 4: Solve the resulting simultaneous nodal equations to obtain the values of the unknown node voltages

𝑉1 0.725 −44.830 =1000

(2 + 𝑗3)−

6∠0°

1 + 𝑗2

𝑉1 =0.3515.26°

0.72− 44.83°

= 0.48V60.08°

Q: Using the Nodal Voltage Method to find the voltage of Vo

Example 2

Solution:

1. Identify every node in the circuit.- Only One node 2. Label each node with a node voltage : V0

Label each current leaving the node : I1 ,I2 and I3

Step3: Apply KCL to sum all currents at Vo

𝐼1 + 𝐼2 + 𝐼3 = 0

𝑉𝑜 − 1000

𝑗6+

𝑉𝑜

3+ 6.5−300= 0

𝑉𝑜1

𝑗6+

1

3= −6.5 −300 +

1000

𝑗6

Step 4: Solve the resulting simultaneous nodal equations to obtain Vo

𝑉𝑜1

𝑗6+

1

3= −6.5 −300 +

1000

𝑗6

𝑉𝑜3 + 𝑗6

𝑗18= −5.63 + 𝑗3.25 +

10 < 00

6 < 900

𝑉𝑜 = 5.85163.370𝑗18

3 + 𝑗6

= 15.7𝑉− 170.06V

Exercise 1:

Q : Refer to the circuit below 1. How many nodes in the circuit below. 2. Write the KCL equations for every nodes

Exercise 2

+

–

2–j3

3+j2 5+j3

–

+

I2

Q: 1. Write the equation for I1 , I2 and I3 2. Using the Nodal Voltage Method to find the voltage of V1

V1

I1 I3

END