Node & Loop Analysis#

When series/parallel simplification can’t reduce a circuit to something obvious, systematic methods are needed. Node analysis (nodal analysis) and loop analysis (mesh analysis) are the two workhorses. Both are mechanical — follow the procedure, get the answer. The skill is choosing the right one and interpreting the results.

Node Analysis (Nodal Analysis)#

Write KCL at each unknown node. Express all currents in terms of node voltages using Ohm’s law (I = V/R or I = V × Y where Y = 1/R). Solve the resulting system of equations.

Procedure#

1. Choose a reference node (ground) — Pick the node with the most connections. This simplifies the equations by giving one node a voltage of zero
2. Label the remaining node voltages — V1, V2, etc. These are the unknowns
3. Write KCL at each unknown node — Sum of currents leaving the node = 0 (or sum of currents in = sum of currents out)
4. Express each current in terms of node voltages — Current through a resistor from node A to node B is (V_A - V_B) / R
5. Include known voltages — Voltage sources fix relationships between nodes. A voltage source between nodes sets V_A - V_B = V_source
6. Solve the system — Algebra, substitution, or matrices

When to Use Node Analysis#

• Circuits with many parallel branches
• When node voltages are needed (which is most of the time)
• Circuits with current sources (they directly give a current entering a node)

Dealing with Voltage Sources#

A voltage source between two nodes constrains their voltages but doesn’t directly indicate the current through it. The supernode technique handles this: draw a boundary around the voltage source and its two nodes, write KCL for the boundary, and add the constraint equation V_A - V_B = V_source.

Loop Analysis (Mesh Analysis)#

Write KVL around each independent loop. Express all voltages in terms of loop (mesh) currents using Ohm’s law (V = IR). Solve the resulting system of equations.

Procedure#

1. Identify independent loops (meshes) — In a planar circuit, meshes are the “windowpanes” — the loops that don’t contain other loops inside them
2. Assign a mesh current to each mesh — All clockwise (or all counterclockwise) by convention
3. Write KVL around each mesh — Sum of voltage drops = sum of voltage rises
4. Express each voltage drop in terms of mesh currents — A resistor shared by two meshes carries the difference of their mesh currents
5. Include known currents — Current sources fix mesh currents or differences between them
6. Solve the system

When to Use Loop Analysis#

• Circuits with many series elements
• When branch currents are needed
• Circuits with voltage sources (they directly give a voltage in a loop equation)

Dealing with Current Sources#

A current source in a mesh fixes that mesh current (if it’s in only one mesh) or constrains the difference between two mesh currents. The supermesh technique handles this: combine the two meshes sharing the current source into one KVL equation, and add the constraint that the current source current equals the difference of the two mesh currents.

Choosing Between Them#

In practice, node analysis is more commonly used because voltages are what’s usually needed and what can be directly measured with probes.

Tips#

• Count unknowns before starting — node analysis needs (N-1) equations for N nodes; mesh analysis needs M equations for M independent meshes
• Use supernode/supermesh techniques to handle sources cleanly
• Matrix methods (Cramer’s rule, Gaussian elimination) systematize the solution process for complex circuits

Caveats#

• Reference direction consistency — The most common source of errors. Pick a convention and stick with it through the entire analysis. If I is defined as flowing left to right, the voltage drop across R is IR with the + on the left
• Dependent sources — Transistor models and op-amp models include dependent sources (voltage or current controlled by another voltage or current). These add equations but don’t add unknowns. Include their controlling variable in the equations
• Non-planar circuits — Mesh analysis only works directly for planar circuits (circuits that can be drawn without crossing wires). Non-planar circuits need node analysis or generalized loop analysis
• Overcounting or undercounting equations — For node analysis: (N-1) equations are needed for N nodes. For mesh analysis: M equations are needed for M independent meshes. Getting the count wrong means an unsolvable or redundant system

In Practice#

• Calculated node voltages that don’t match measured values indicate a modeling error, component fault, or measurement issue
• A node voltage of exactly 0 V where a non-zero value is expected suggests a short to ground
• A node voltage equal to the supply where it shouldn’t be suggests an open circuit in the path to ground
• Sanity-checking results against bench measurements validates the analysis and catches errors in both