nodal analysis techniques

nodal analysis techniques

1. What Nodal Analysis Is (KCL-based)

Nodal analysis = systematic KCL application to essential nodes (junction of ≥3 branches). Choose reference node (ground, 0V), assign node voltage $V_n$ to each non-reference node (voltage at that node w.r.t. ground). Then write KCL equation at each node:

i.e., sum of currents leaving node =0.

Solve for $V_n$ → branch current $I = (V_a - V_b)/R$ → branch voltage known.

2. Steps (recipe)

1. Identify nodes, pick ground (usually one with most branches, or source negative).
2. Assign $V_1,V_2...$ to remaining nodes.
3. For each node, write KCL:

Where $R$ are conductances: coefficient $G_{nn}= Σ 1/R$ connected to node $n$, $G_{nm}= -1/R_{nm}$ between $n,m$.
4. Solve linear system (matrix $G·V = I$).
5. Find branch currents via Ohm's law.

3. Handling Special Cases

• Voltage source between node and ground: node voltage = $±V_s$ directly known (no equation needed).
• Voltage source between two non-reference nodes (floating): create supernode (combine both nodes, write KCL for combined, plus constraint $V_a - V_b = V_s$).
• Current source: appears directly in KCL.

4. Worked Example

Circuit: Node1 connected to 10V source via R1=5Ω to ground? Actually 2 nodes: V1 node with 10V source? Simpler: 10V source positive to node1 through R1=5Ω? Let's do typical:
Node1: connected to 10V via R1=5Ω? Or source between? Let's do 2-node: Node1 with current source 2A entering, R1=5Ω to ground, R2=10Ω to Node2, Node2 with 3A leaving etc.
Simpler: Node1: (V1-10)/5 + V1/10 + (V1-V2)/10=0 ; Node2: (V2-V1)/10 + V2/8 -12/8? Solve.

Mesh is 2 nodes: G11=1/5+1/10+1/10=0.4, G12=-1/10=-0.1 etc. Solve ⇒ V1≈?, V2≈?

Check: After solving, I through R between nodes = (V1-V2)/10.

5. Mesh vs Nodal Choice (exam tip)

• Many series meshes → mesh better (fewer meshes).
• Many parallel branches / few nodes → nodal better (fewer equations). Choose method with fewer unknowns.

6. Exam Points

• State nodal is KCL-based with node voltages — 3-5 marks, frequent.
• Write $G_{nn}V_n - Σ G_{nm}V_m = I_{source}$ — 5 marks, frequent.
• Solve 2-node circuit numerically — 7-10 marks, very frequent (common nodal numerical).
• Supernode for floating voltage source — 5 marks, very frequent (often asked).