Average power in the sinusoidal steady state

In a circuit with a sinusoidal generator at steady state (once transients have died out), the current and voltage oscillate at the same frequency but with a phase shift $\varphi$:

$\Delta V(t) = V_0\sin(\omega t), \qquad i(t) = I_0\sin(\omega t - \varphi)$

The instantaneous power $p(t) = \Delta V\cdot i$ oscillates, but its time average over a period $T = 2\pi/\omega$ is:

Average power

$\ev{\langle P \rangle = \frac{1}{2}\,V_0\,I_0\,\cos\varphi = V_\text{eff}\,I_\text{eff}\,\cos\varphi}$

where $V_\text{eff} = V_0/\sqrt{2}$ and $I_\text{eff} = I_0/\sqrt{2}$ are the RMS values, and $\cos\varphi$ is the power factor.

The power factor is crucial in practice: if $\varphi = 0$ (purely resistive circuit, or at resonance) the whole power $V_\text{eff}I_\text{eff}$ is actually consumed; if $\varphi \to 90^\circ$ (purely reactive circuit) the average power goes to zero, because the energy is only exchanged back and forth with the capacitor and inductor, without being dissipated.

Further detail — series RLC

For a series RLC circuit with sinusoidal EMF of amplitude $V_0$: $I_0 = \frac{V_0}{\sqrt{R^2 + (\omega L - 1/\omega C)^2}}, \qquad \tan\varphi = \frac{\omega L - 1/\omega C}{R}$ At resonance ($\omega = \omega_0 = 1/\sqrt{LC}$) we have $\varphi = 0$: minimum impedance, maximum current, maximum average power.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Alternating current · Power

Esercizi collegati: Problem — Reactances of an RLC as omega varies · Problem — Impedance and phase shift of a series RLC · Problem — Phasor diagram (XL greater than XC)