Power in Alternating-Current Circuits — Question 9

Question 9

Harmonic current lowers true power factor even with an in-phase fundamental

A load has sinusoidal voltage $v=V_p\cos\omega t$ but current $i=I_1\cos(\omega t-\phi)+I_3\cos3\omega t$. The harmonic amplitudes are positive.

1. Find real power and true power factor using total rms current.

2. Evaluate power factor when phi=0 and I3=I1.

Question 9 - Solution

Harmonic current lowers true power factor even with an in-phase fundamental

(a) Method and calculation

$$P=\langle vi\rangle=V_pI_1\cos\phi/2,\quad I_{\mathrm{rms}}=\sqrt{(I_1^2+I_3^2)/2}.$$$$\boxed{\mathrm{pf}_{\mathrm{true}}=\frac{P}{V_{\mathrm{rms}}I_{\mathrm{rms}}}=\frac{I_1\cos\phi}{\sqrt{I_1^2+I_3^2}}}.$$

(b) Interpretation and further calculation

$$\phi=0,\quad I_3=I_1\Rightarrow P=V_pI_1/2.$$$$\boxed{\mathrm{pf}_{\mathrm{true}}=1/\sqrt2=.707107}.$$

Physical check

Correcting fundamental phase alone cannot eliminate distortion current.