Relation between the Stored and the Dissipated Energies of a Circuit Composed of Linear Capacitors, Linear/Nonlinear Resistors and dc Voltage Sources

Recently the reduction of power consumption is one of the most important issues in the design of LSI circuits. Circuit designers quite commonly evaluate the power consumption on the basis of the “fact” that when we charge a capacitor C by a voltage source E or when we discharge the capacitor with the voltage E we consume the energy of 12CE . This “fact” can be verified typically for the simple RC circuit in Fig. 1 composed of a linear capacitor, a linear resistor, and a dc voltage source. That is, let the total energy dissipated at the resistor during the time interval [0, t] be WR(t) and the energy stored at the capacitor at the time t be WC(t), respectively. Then WR(t) > WC(t) and WR(∞) = WC(∞) = 12CE hold independently of the value of the resistor R if the capacitor has no initial charge. Nishi [1] and Nishi and Kawane [2] showed that similar relations hold for most general linear passive RC circuits in which steady state currents vanish. We may conjecture from the above “fact” that when we store some amount of energy in capacitors, the same amount of energy must inevitably be dissipated at resistors for general (linear/nonlinear) circuits. However we easily see that the above relation does not hold for the following three cases: Case 1) where capacitors have nonzero initial charges, Case 2) where variable voltage sources are applied instead of dc voltage sources and Case 3) where LC resonances are used to charge a capacitor. We can easily verify Case 2) even for the circuit in Fig. 1 by replacing the dc voltage source with a variable voltage source and by gradually increasing the source