ELECTENG291 · Fundamentals of Electrical Engineering
Nodal & Mesh Analysis
This Module 1 chapter of University of Auckland ELECTENG 291 turns circuit-solving into a system: the node-voltage (nodal) method writes KCL at each essential node in terms of node voltages, and the mesh-current method writes KVL around each loop in terms of mesh currents. It covers choosing between them and the special cases — the supernode (a voltage source between two non-reference nodes) and the supermesh. Nodal analysis is a course staple that recurs in the tests and the final exam; mesh is flagged as extra reading but appears as an alternative method.
What this chapter covers
- 01Node-voltage method: pick a reference node, write KCL at each essential node using node voltages and conductances, solve
- 02Mesh-current method: define loop currents, write KVL around each mesh, solve (the dual of nodal)
- 03Choosing a method: nodal suits many parallel branches / current sources; mesh suits series loops / voltage sources
- 04Supernode: a voltage source between two non-reference nodes — enclose both, write one KCL for the supernode plus a KVL constraint (e.g. V_A = V_B + 15 V)
- 05Supermesh: a current source shared by two meshes — combine them and add the current-source constraint
- 06Handling dependent sources: keep them active and add the controlling-variable relation as an extra equation
- 07Cross-checking a node/mesh answer against superposition or a Thévenin reduction
Node-voltage solve with a supernode
- +1Identify the supernode: the 15 V source sits between A and B, so enclose both nodes and treat them as one supernode for KCL.
- +1Write the supernode KCL (currents leaving to ground through the resistors balance the injected 5 mA): V_A/3k + V_B/2k = 5 mA.
- +1Write the source constraint linking the two node voltages: V_A = V_B + 15 (the 15 V source raises A above B by 15 V).
- +1Substitute and clear fractions (multiply through by 6 kΩ): 2(V_B + 15) + 3V_B = 30 → 2V_B + 30 + 3V_B = 30 → 5V_B = 0.
- +1Solve: V_B = 0 V, so V_o = 0 V (and V_A = V_B + 15 = 15 V). The supernode plus its KVL constraint gave exactly the two equations needed for two unknowns.
Key terms
- Node-voltage (nodal) analysis
- A systematic method: choose a reference node, define each other node's voltage relative to it, write KCL at each essential node in terms of node voltages and conductances, and solve the resulting linear system.
- Mesh-current analysis
- The dual method: define a circulating current in each independent loop (mesh), write KVL around each mesh in terms of those currents, and solve. Best suited to planar circuits dominated by series voltage sources.
- Reference node (ground)
- The node chosen as the zero-voltage datum against which all node voltages are measured. Choosing the node with the most connections usually simplifies the algebra.
- Supernode
- The region formed when a voltage source connects two non-reference nodes. You write a single KCL for the enclosed pair together with the KVL constraint V_A − V_B = source voltage, because the current through an ideal voltage source is unknown.
- Supermesh
- The mesh-analysis counterpart: when a current source is shared by two meshes, the two meshes are combined into one KVL loop and the current-source value is added as a constraint equation.
- Essential node
- A node where three or more circuit elements meet. Writing KCL only at essential nodes (relative to the reference) gives the minimum set of node-voltage equations.
Nodal & Mesh Analysis FAQ
When should I use nodal versus mesh analysis?
Count what you would have to solve. Nodal analysis needs one equation per essential node (minus the reference) and handles current sources and many parallel branches naturally. Mesh analysis needs one equation per independent loop and handles series voltage sources naturally, but it only applies cleanly to planar circuits. Pick whichever gives fewer unknowns; both are valid and should agree.
What is a supernode and how do I handle it?
A supernode appears when a voltage source connects two non-reference nodes. Because you can't write an ordinary KCL through an ideal voltage source (its current is unknown), you enclose both nodes, write one KCL for the whole supernode, and add the source's voltage as a constraint linking the two node voltages (e.g. V_A = V_B + 15 V). That gives you exactly the equations you need.
How do dependent sources change nodal or mesh analysis?
Keep the dependent source active and write its value in terms of the node voltages or mesh currents you are already solving for, then add the relation between the controlling variable and those unknowns as an extra equation. The system stays linear; you just have one more equation tying the controlled source to the circuit.
Can Sia help me set up nodal and mesh equations in ELECTENG 291?
Yes, as a study aid. Sia can pick a reference node, write the KCL/KVL system (supernodes and supermeshes included), and check your algebra, then cross-check the answer against another method. It explains and drills; it does not do graded assessment for you, and University of Auckland academic-integrity rules apply — confirm what is permitted on Canvas.
Exam move
Adopt a fixed nodal routine: choose the reference node (usually the most-connected one), label every other node voltage, write KCL at each essential node as currents-out = currents-in, then solve. Train the supernode and supermesh cases specifically — they are where marks are lost — by always pairing the enclosed KCL with the source constraint. Learn to pick the lighter method: fewer essential nodes → nodal, fewer meshes → mesh. Keep dependent sources active and write their controlling variable in terms of your unknowns. Because nodal, mesh, superposition and Thévenin must all give the same answer, use a second method as a self-check whenever time allows. Show the KCL/KVL equations explicitly — the method marks live there. Confirm assessment details on Canvas.
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