Phasor & Topographic Voltage Diagrams

A phasor diagram is the complex plane made visible. Length represents RMS magnitude, angle represents phase relative to a declared reference, and vector addition represents a KCL or KVL equation. A topographic diagram adds circuit order: branch voltage phasors are placed head-to-tail along the chosen path.

Choose one reference #

Set one nonzero phasor angle to zero, commonly the source voltage or a series current. Every other angle is relative to it. Rotating the entire diagram changes no physical phase difference.

For a series RLC branch with current reference $\underline I$:

$$\underline V_R=R\underline I,$$

$$\underline V_L=j\omega L\underline I,$$

$$\underline V_C=\frac{1}{j\omega C}\underline I=-j\frac{1}{\omega C}\underline I.$$

$V_R$ is aligned with current, $V_L$ leads it by $90^\circ$, and $V_C$ lags it by $90^\circ$. The source is their vector sum, not the sum of magnitudes.

Topographic construction #

1. draw the first node at an arbitrary origin;
2. traverse the circuit in the declared order;
3. add each signed branch voltage head-to-tail;
4. label every intermediate point with its node;
5. require a closed-loop path to return exactly to its starting point.

For a branched circuit, different paths between the same pair of nodes must end at the same phasor point.

What the diagram catches #

A topographic diagram uses the same vectors but anchors each head to the next circuit node. Two different paths to one node must end at the same complex coordinate; failure to close localizes a sign or branch-order error.