PHY.10:5.2 - Change a drive that makes storage and conduction coincide

A two-terminal element is driven with V(t)=V_0 exp(t/tau). One proposed account is an ideal resistor, I=G V. Another is an ideal capacitor, I=C dV/dt, prepared with charge C V_0 at the start of the recorded ramp.

Since dV/dt=V/tau, choosing C=G tau makes the complete current traces identical during this ramp. More accurate recording of that same drive cannot distinguish the accounts.

Use V_0=1 V, tau=2 s, G=1 mS and C=2 mF. Stop increasing the voltage when it reaches 2 V and hold it there. After the drive and readout have settled, the resistor predicts 2 mA and the ideal capacitor predicts zero current. The change removed dV/dt while retaining V.

The waiting interval must be interpreted physically. A source with finite output resistance and a detector with finite response can create a transient after the change. Include it or choose a readout time after its consequential effect. The ideal predictions also exclude a significant leakage path.

Now suppose the held-voltage current is 0.6 mA, while the earlier ramp still has I/V=1 mS. The ideal pair is inadequate for these records. A parallel conductance and capacitance give

I=G V+C dV/dt.

The hold gives G=0.3 mS. Substitution into the ramp relation G+C/tau=1 mS gives C=1.4 mF. The test has opened a physically different account in which conduction and storage coexist. These two records determine its two parameters under the stated idealization; they do not establish that this account suffices at every frequency or voltage.

The resulting distinction changes use. A continuously held voltage dissipates power through the conductance, while the capacitance stores charge and supplies a transient response. Subsequent pulse or frequency use can therefore ask a question that the original exponential ramp could not resolve.