Nodal Voltage Analysis:
Nodal Voltage Analysis complements the previous mesh analysis in that it is equally powerful and based on the same concepts of matrix analysis. As its name implies, Nodal Voltage Analysis uses the “Nodal” equations of Kirchhoff’s first law to find the voltage potentials around the circuit.
So by adding together all these nodal voltages the net result will be equal to zero. Then, if there are “n” nodes in the circuit there will be “n-1” independent nodal equations and these alone are sufficient to describe and hence solve the circuit.
At each node point write down Kirchhoff’s first law equation, that is: “the currents entering a node are exactly equal in value to the currents leaving the node” then express each current in terms of the voltage across the branch. For “n” nodes, one node will be used as the reference node and all the other voltages will be referenced or measured with respect to this common node.
Step of Nodal Voltage Analysis
- Find out Principal Node
- Find Reference and Non Reference Node
- Use KCL in Non Reference Node
- Principal Node: A node where three or more circuit elements are connected. It serves as a crucial point for applying Kirchhoff’s Current Law (KCL).
- Reference Node: A chosen node in the circuit that serves as a common ground or zero voltage point, against which all other node voltages are measured.
- Non-Reference Node: Any node in the circuit that is not the reference node. These nodes have unknown voltages that need to be determined using Nodal Voltage Analysis.
- Using KCL in Non-Reference Nodes: Apply Kirchhoff’s Current Law at each non-reference node by setting the sum of incoming currents equal to the sum of outgoing currents, then express these currents in terms of node voltages.
For example, consider the circuit from the previous section.
Nodal Voltage Calculation
Given a circuit with three nodes (Va, Vb, Vc) and node D as the reference node, we apply Kirchhoff’s Current Law (KCL) at node Vb.
Step 1: Write the Node Equation at Vb
(Va - Vb) / 10 + (Vc - Vb) / 20 = Vb / 40
Step 2: Substitute Given Values
Given Va = 10V and Vc = 20V, the equation becomes:
(10 - Vb) / 10 + (20 - Vb) / 20 = Vb / 40
Step 3: Simplify the Equation
(1 - Vb / 10) + (1 - Vb / 20) = Vb / 40
2 - (Vb / 10 + Vb / 20) = Vb / 40
Step 4: Factor Out Vb
2 = Vb (1 / 40 + 1 / 20 + 1 / 10)
2 = Vb (1 / 40 + 2 / 40 + 4 / 40)
2 = Vb * 7 / 40
Step 5: Solve for Vb
Vb = (2 * 40) / 7
Vb = 80 / 7 V
Step 6: Calculate I3
I3 = Vb / 40
I3 = (80 / 7) / 40
I3 = 2 / 7 A
I3 = 0.286 A
Final Answer
The nodal voltage at Vb is 80 / 7 V (≈ 11.43V), and I3 = 0.286 A
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