Circuit Power Balance & Energy Conservation

Power balance turns a solved circuit back into a physical statement. With every element evaluated under the passive sign convention, positive power is absorbed and negative power is delivered. An ideal lumped interconnection stores no energy of its own, so the element powers must balance at every instant.

Instantaneous balance #

For all branches $k$:

$$\sum_k p_k(t)=\sum_k v_k(t)i_k(t)=0.$$

The result follows from KCL and KVL. In matrix language, branch voltages lie in the cut-space complement of branch currents, so their inner product vanishes.

For a DC resistor, $P_R=I^2R=V^2/R>0$. An independent source may have either sign: it delivers when current leaves its positive terminal and absorbs when current enters it.

Energy and storage #

Integrating the balance gives

$$\sum_k \Delta w_k=0.$$

Resistors convert electrical energy to heat. Capacitors and inductors change stored field energy. Sources exchange energy with non-electrical domains or with a model boundary. Over an interval, the energy delivered by sources equals dissipation plus net increase in stored energy.

Diagnostic procedure #

Create one row per element with its declared voltage, current, and $P=VI$. Keep signed values. Then check

$$r_P=\sum_k P_k.$$

Compare $r_P$ with the scale $\sum_k|P_k|$, not with one small branch:

$$\epsilon_P=\frac{|r_P|}{\sum_k|P_k|}.$$

A nonzero residual points to wrong current direction, wrong voltage polarity, inconsistent units, premature rounding, or an omitted branch. It does not identify which one, so pair the balance with KCL and KVL residuals.

For dynamic circuits, instantaneous power balance includes storage-rate terms. Over a periodic steady-state interval, the net change in stored energy is zero, but energy can move into and out of capacitors and inductors within the period.