Nodal analysis
Nodal analysis is a systematic way to solve circuits by finding node voltages. It turns any circuit into a set of linear equations based on Kirchhoff's current law.
The idea
Choose one node as the reference, ground, at 0 V. Assign unknown voltages to every other node. Then write KCL at each non-reference node, expressing every branch current through Ohm's law in terms of node voltages. Solving the resulting equations gives all node voltages, from which every current and power follows.
Branch currents from node voltages
The current through a resistor R from node a to node b is \(\frac{v_a - v_b}{R}\), the voltage difference divided by resistance. Writing every current leaving a node in this form and setting their sum to zero (or equal to any source current entering) produces one equation per node, linear in the unknown voltages.
Choosing the reference
Any node can be the reference, but choosing the one with the most connections, often the negative terminal of the main source, usually gives the simplest equations. All other node voltages are then measured relative to it, matching how engineers probe circuits with a meter's black lead on ground.
Current sources
Independent current sources simply contribute known currents to the KCL equations at their nodes. A circuit with only resistors and current sources gives equations of the form G v = i, where G is the conductance matrix, v the vector of node voltages, and i the vector of source currents into each node.
Voltage sources and supernodes
A voltage source connected between a node and the reference fixes that node's voltage directly, reducing the number of unknowns. A voltage source between two non-reference nodes forms a supernode: write one KCL equation for the combined region enclosing both nodes, plus the constraint equation stating that their difference equals the source voltage.
Dependent sources
Dependent sources are handled like independent ones, with an extra equation expressing the controlling variable, such as a current through some resistor, in terms of node voltages. This keeps the system solvable, though the conductance matrix may no longer be symmetric.
Solving the equations
Small systems can be solved by substitution or elimination; larger ones by matrix methods such as Gaussian elimination or Cramer's rule, or by software. Circuit simulators such as SPICE, developed at the University of California, Berkeley in the early 1970s, use a generalised form of nodal analysis called modified nodal analysis.
Worked example
A 10 V source connects through \(R_1 = 2\) \(\Omega\) to node A; A connects to ground through \(R_2 = 4\) \(\Omega\) and through \(R_3 = 4\) \(\Omega\). KCL at A: \(\frac{v_A - 10}{2} + \frac{v_A}{4} + \frac{v_A}{4} = 0\). Multiplying by 4: \(2v_A - 20 + v_A + v_A = 0\), so \(v_A = 5\) V. The source current is \(\frac{10 - 5}{2} = 2.5\) A, splitting into 1.25 A through each 4 \(\Omega\) resistor.
Key takeaways
Nodal analysis picks a reference node and solves for the other node voltages using KCL. Each resistor current is the difference of its node voltages divided by R. Choose the most connected node as reference. Current sources add known terms, giving G v = i. Voltage sources to ground fix node voltages; floating voltage sources form supernodes with an extra constraint. Dependent sources need an equation for their controlling variable. Solve by elimination, matrices, or software; SPICE uses modified nodal analysis.
Flashcard
Solve for node voltages using KCL
Check your understanding
In nodal analysis, how many unknown node voltages does a circuit with 5 nodes have (with no voltage sources)?