Phase Angle and Power Factor

Phase Angle

Impedance is an AC (alternating current) analogue to resistance in a DC circuit. As we studied in a previously Atom ("Impedance"), current, voltage and impedance in an RC circuit are related by an AC version of Ohm's law: $I = \frac{V}{Z}$, where I and V are peak current and peak voltage respectively, and Z is the impedance of the circuit.

In a series RC circuit connected to an AC voltage source as shown in , conservation of charge requires current be the same in each part of the circuit at all times. Therefore we can say: the currents in the resistor and capacitor are equal and in phase. (We will represent instantaneous current as i(t). )

Series RC Circuit

Series RC circuit.

On the other hand, because the total voltage should be equal to the sum of voltages on the resistor and capacitor, so we have:

$\begin{aligned} v(t)& = v_R(t)+v_C(t) \\ &= i(t)R + i(t)\frac{1}{j \omega C} \\ &= i(t) (R+\frac{1}{j\omega C}) \end{aligned}$,

where $\omega$ is the angular frequency of the AC voltage source and j is the imaginary unit; j²=-1. Since the complex number $Z = R+\frac{1}{j\omega C} = \sqrt{R^2+(\frac{1}{\omega C})^2} e^{j\phi}$ has a phase angle $\phi$ that satisfies $cos\phi = \frac{R}{\sqrt{R^2+(\frac{1}{\omega C})^2}}$,

we notice that voltage $v(t)$ and current $i(t)$ has a phase difference of $\phi$.

For R=0, $\phi = 90^{\circ}$. As learned from the preceding series of Atoms—the voltage across the capacitor V_C follows the current by one-fourth of a cycle (or 90º).

Power Factor

Because voltage and current are out of phase, power dissipated by the circuit is not equal to: (peak voltage) times (peak current). The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is I_rmsV_rmscosϕ, where I_rms and V_rms are the root mean square (rms) averages of the current and voltage, respectively. For this reason, cosϕ is called the power factor, which can range from 0 to 1.