Other Single-Loop Circuits — Question 10

Question 10

A constant-power load admits two operating currents below a source limit

A source with emf E and internal resistance r drives an ideal load that absorbs fixed positive power P. Its terminal voltage is positive and $P=VI$. Assume a static operating point, without specifying load dynamics.

1. Find the possible currents and the condition for their existence.

2. For $E=10.0$ V, $r=1.00\,\Omega$ and $P=16.0$ W, find both currents and terminal voltages. State what static equations cannot decide.

Question 10 - Solution

A constant-power load admits two operating currents below a source limit

(a) Method and calculation

$$V=E-rI,\quad P=I(E-rI)\Rightarrow rI^2-EI+P=0.$$$$I_\pm=\frac{E\pm\sqrt{E^2-4rP}}{2r},\quad P\le E^2/(4r).$$

(b) Interpretation and further calculation

$$I_-=2.00\,\mathrm A,\ V_-=8.00\,\mathrm V;\qquad I_+=8.00\,\mathrm A,\ V_+=2.00\,\mathrm V.$$$$I_-V_-=I_+V_+=16.0\,\mathrm W.$$Static equations establish possible states; their stability requires a dynamic model.

Physical check

No real operating point exists above the discriminant limit in this source model.