Sinusoidal Waveforms, RMS Values & Phase

Sinusoidal steady state is the regime reached by a stable linear circuit after transients have decayed while the excitation remains sinusoidal. Every voltage and current then shares the source angular frequency; only magnitude and phase differ. That closure is why differential equations become complex algebra.

Parameterizing the waveform #

Write

$$x(t)=X_m\cos(\omega t+\phi),$$

where $X_m$ is peak amplitude, $\omega=2\pi f$ is angular frequency, and $\phi$ is phase relative to a declared reference. The period is $T=1/f$.

Phase has meaning only as a difference. If

$$v(t)=V_m\cos(\omega t+\phi_v),\qquad i(t)=I_m\cos(\omega t+\phi_i),$$

then the physically relevant angle is $\phi=\phi_v-\phi_i$. Changing the time origin shifts both phases equally and leaves $\phi$ unchanged.

RMS is a power-equivalent value #

For any periodic waveform,

$$X_{rms}=\sqrt{\frac{1}{T}\int_0^T x^2(t)\,dt}.$$

For a sinusoid, $X_{rms}=X_m/\sqrt2$. A resistor driven by a sinusoidal voltage dissipates the same average power as it would under DC voltage $V_{rms}$:

$$P=\frac{V_{rms}^2}{R}.$$

Peak, peak-to-peak, average, and RMS are different quantities. A zero-mean sinusoid has average value zero but nonzero RMS.

Phase as time displacement #

A phase difference corresponds to time shift

$$\Delta t=\frac{\Delta\phi}{\omega}.$$

The same time shift produces a different phase angle at a different frequency. This is why a constant propagation delay is not a constant phase shift over bandwidth.

The phasor method applies one frequency at a time. A nonsinusoidal periodic waveform must be decomposed into harmonics, analyzed frequency by frequency in a linear circuit, and recombined.

RMS must be computed over a stated interval. For a periodic waveform, use an integer number of periods; otherwise the value can depend on the window position even when the underlying signal is unchanged.