A circuit has 4 nodes. One is designated as ground. How many independent KCL equations does nodal analysis require to find all unknown voltages?
A4 — one equation per node, including the ground node
B3 — one equation per non-reference node
CAs many as there are resistors in the circuit
DAs many as there are independent current sources
The ground node has a fixed voltage of 0 by definition — it provides no unknown and needs no equation. Each of the remaining 3 nodes has one unknown voltage, so exactly 3 KCL equations are needed. This 'one equation per non-reference node' rule is the key to nodal analysis: the number of equations always equals the number of unknowns, giving a uniquely solvable system.
Question 2 Multiple Choice
When writing a KCL equation at node Vᵢ in nodal analysis, how is the current through resistor R connecting node i to node j expressed?
AVⱼ / R — the neighbor's voltage divided by the resistance
B(Vᵢ + Vⱼ) / R — the sum of the two node voltages divided by the resistance
C(Vᵢ − Vⱼ) / R — the voltage difference divided by resistance, representing current leaving node i toward node j
D1 / (R · Vᵢ) — conductance times the inverse of the node voltage
Ohm's law for a branch: current = voltage across the branch / resistance. The voltage across a resistor connecting node i to node j, measured in the direction of assumed current flow (leaving node i), is Vᵢ − Vⱼ. So current leaving node i through R is (Vᵢ − Vⱼ)/R. This sign convention — expressing currents as leaving the node being analyzed — is what makes the KCL equation consistent: sum of all leaving currents = 0.
Question 3 True / False
The choice of which node to designate as ground affects the values of branch voltages (voltage differences across circuit elements) in the final solution.
TTrue
FFalse
Answer: False
Branch voltages are differences between node voltages: V_AB = V_A − V_B. If you shift the reference (choose a different ground), all individual node voltages shift by the same constant, but every difference remains unchanged. Only the individual node voltages (measured relative to the reference) change. The physics of the circuit — currents, branch voltages, power dissipation — is independent of the choice of reference node.
Question 4 True / False
A supernode arises when a voltage source connects two non-reference nodes, because the current through an ideal voltage source cannot be expressed directly as a function of the node voltages.
TTrue
FFalse
Answer: True
An ideal voltage source enforces a fixed voltage difference between its terminals, but the current through it is determined by the rest of the circuit — it is not a function of the node voltages alone. This makes it impossible to write separate KCL equations at each terminal node in the usual way. The supernode technique treats both nodes as a combined region: write KCL around the outside of the supernode (the source current becomes internal and disappears) and add the constraint V_A − V_B = V_source as a supplementary equation.
Question 5 Short Answer
Why is the ground (reference) node essential to nodal analysis? What mathematical problem does it solve?
Think about your answer, then reveal below.
Model answer: Node voltages are defined as potential differences relative to a reference. Without fixing one node to a known value (0 V), the system of KCL equations is underdetermined — there are infinite solutions related by adding a constant to every node voltage. The ground node removes this degree of freedom by anchoring the solution to a specific reference potential.
Mathematically, the KCL equations alone constrain the differences between node voltages but not their absolute values. The circuit physics only determines relative potentials (voltage drops), not absolute ones. Choosing a ground node adds one constraint (V_ground = 0) that makes the system square and uniquely solvable. This is analogous to needing a boundary condition to solve a differential equation — without it, the solution family has a free parameter.