Nodal, Mesh & Admittance Methods for AC Networks

At one angular frequency, a linear RLC network becomes a complex linear system. The circuit topology does not change: KCL still sums currents and KVL still sums voltages. Only the element coefficients become frequency-dependent complex numbers.

Phasor-domain nodal analysis #

For a branch admittance $Y_{ab}$:

$$\underline I_{ab}=Y_{ab}(\underline V_a-\underline V_b).$$

The nodal system is

$$Y(\omega)\underline V=\underline I_s.$$

The diagonal stamp is the sum of incident admittances and each off-diagonal stamp is the negative admittance between the node pair. Voltage sources create the same supernode constraints as in DC.

The "method of conductances" used in older course material is this admittance formulation. It is especially compact for parallel branches because admittances add directly.

Phasor-domain mesh analysis #

The mesh system is

$$Z(\omega)\underline I=\underline V_s.$$

Each diagonal contains the sum of impedances around a mesh; mutual entries are the negative shared impedance. Coupled inductors add signed mutual terms and must follow the dot convention.

Workflow #

1. choose cosine or sine reference and convert every source consistently;
2. state whether phasors are RMS or peak;
3. compute all $Z(\omega)$ or $Y(\omega)$ at the given frequency;
4. solve one complex linear system;
5. convert results to both rectangular and polar form when interpretation needs both;
6. verify KCL/KVL and complex power before returning to time-domain waveforms.

The choice between nodal and mesh analysis follows the same unknown-count and source-type tradeoff as in DC. Complex arithmetic does not make one method more physically correct than the other.

When component values are specified rather than reactances, calculate $Z_L$ and $Z_C$ at the stated frequency once and record it beside the solve. Reusing a result at another frequency without recomputing impedances is a model change disguised as arithmetic reuse.