Phasors describe a sinusoidal steady state

One frequency, settled response

A phasor represents the amplitude and phase of a sinusoid at one angular frequency $\omega=2\pi f$. It is useful after transient effects have settled in a linear circuit. It does not directly describe startup, a clipped waveform, or a mixture of frequencies as one number. Analyze different sinusoidal frequencies separately and combine them only when linear superposition applies.

Use a consistent amplitude convention. Here phasor amplitudes are RMS. For a sine wave, RMS is peak divided by √2, and peak-to-peak is twice peak. A 1 V RMS sine has 1.414214 V peak and 2.828427 V peak-to-peak. Mixing a peak input with an RMS output produces a false gain even when every other calculation is correct.

Complex impedances

An ideal resistor has impedance R, a capacitor $1/(j\omega C)$, and an inductor $j\omega L$. At 1 kHz, a 100 nF capacitor has reactance magnitude 1591.549 Ω and phase −90°. A 10 mH inductor has reactance 62.8319 Ω and phase +90°. The sign encodes the voltage-current phase relation.

Drive a series 1 kΩ resistor and 100 nF capacitor with 1 V RMS at 1 kHz. Total impedance is $1000-j1591.549$ Ω, with magnitude 1879.635 Ω. Current magnitude is 0.532018 mA, and current leads the source voltage by approximately 57.86°. The resistor voltage is 0.532018 V RMS. Capacitor voltage magnitude is approximately 0.846733 V RMS.

Those two voltage magnitudes do not add arithmetically to 1 V because their phases differ. Their complex sum does equal the source. This is the AC version of KVL, using phasors instead of unsigned magnitudes.

Keep the interpretation

For average power in the resistor, use RMS current squared times R, giving approximately 0.283043 mW here. The ideal capacitor has zero average dissipation, although it stores and returns energy. A real capacitor may dissipate power through its losses; that requires additional model terms.