Power in Alternating-Current Circuits — Question 10

Question 10

Maximizing resistor power by changing resistance differs from tuning resonance

An ideal fixed rms voltage V drives a series impedance with adjustable positive R and fixed nonzero net reactance X. Frequency and reactive components do not change.

1. Find the R that maximizes real resistor power.

2. Find the maximum power and power factor at this optimum.

Question 10 - Solution

Maximizing resistor power by changing resistance differs from tuning resonance

(a) Method and calculation

$$P(R)=\frac{V^2R}{R^2+X^2},\quad\frac{dP}{dR}=V^2\frac{X^2-R^2}{(R^2+X^2)^2}.$$$$\boxed{R_{\mathrm{optimum}}=|X|}.$$

(b) Interpretation and further calculation

$$P_{\max}=\frac{V^2|X|}{2X^2}=V^2/(2|X|).$$$$\boxed{\mathrm{pf}=R/\sqrt{R^2+X^2}=1/\sqrt2}.$$

Physical check

With reactance held nonzero, very small resistance draws mostly reactive current and dissipates little real power.