Power in Electric Circuits — Question 3

Question 3

Uniform charge delivery minimizes resistive loss over a fixed time

A nonnegative current waveform transports prescribed charge Q through constant resistance R during time T. No other constraint is imposed.

1. Use $\int_0^T[I(t)-Q/T]^2dt\ge0$ to derive the least possible heating.

2. Compare with delivery by a constant pulse lasting only fraction f of T, with zero current otherwise.

Question 3 - Solution

Uniform charge delivery minimizes resistive loss over a fixed time

(a) Method and calculation

$$\int_0^T I^2dt-2(Q/T)\int_0^T I\,dt+Q^2/T\ge0.$$$$W=R\int_0^T I^2dt\ge RQ^2/T,\quad I_{\mathrm{optimal}}=Q/T.$$

(b) Interpretation and further calculation

$$I_p=Q/(fT),\quad W_p=RI_p^2fT=RQ^2/(fT).$$$$W_p/W_{\min}=1/f\ge1.$$

Physical check

Equality requires constant current almost everywhere; concentrating transfer raises resistive loss.