# A time-varying capacitor described by the constitutive relation is still a linear element, so that..

A time-varying capacitor described by the constitutive

relation

is still a linear element, so that superposition 3.pplies,

but-unlike a time-invariant capacitor-it is not usually an ideal energy-

storage element. Instead, a time-varying capacitor may supply power to or absorb

power from the circuit. If power is absorbed, it is not in general dissipated

but rather coupled or transduced into whatever form is producing the time

variations of the capacitance, such as a mechanical system changing the spacing

of the capacitor plates. Properly employed, a periodically varying capacitor

can produce parametric amplification. A simple circuit illustrating this effect

is shown below.

Exploitin linearity, we shall represent the steady-state

voltages and currents as complex exponentials; the actual sinusoidal waveforms

can be found by taking real parts. (The sinusoidal variations of the capacitor

must not, however, be represented as a complex exponential; circuit parameters

are different from dynamic variables.) The circuits Zl Uw) and Z2 Uw) are tuned

to the output frequency W_{l} + W_{o}. For simplicity we shall

idealize these impedances to be either open or short circuits as follows:

a) Exploiting the properties of Zl (jW) and Z2(jW), argue

that

Explain in particular why v(t) does not contain terms at the

frequencies W_{1} – W_{o} and 2wo + W1, which (as we shall see)

are contained in the capacitor current i(t).

b) Show that

c) Argue that the component of itt) at the frequency (WI +

WO) flows entirely through the load R. From this, solve for V_{2} to

obtain

d) The power supplied by the sinusoidal voltage source at

frequency Wl depends only on the component of il (t) at the frequency WI.

Writing this component in complex form as II i’w1 t, and recalling that the

average input power is

and the average output power is

show that

Taken together, (c) and (d) imply that V_{2} is

proportional to V_{i}; if V_{i} varies slowly, V_{2}

will vary slowly in the same pattern. On the other hand, the output power is

greater than the input power. The time-varying capacitor thus functions as an

amplifier.

In practice, the effect of a time-varying capacitor would

probably be achieved by using a large electrical voltage at frequency Wo

(called the pump voltage) to vary the operating point of a non-linear

capacitor. For generalizations of the parametric amplifier principle, see P. E.

Penfield, Frequency-Power Formulas (Cambridge, MA: Technology Press, 1 960) and

the fundamental paper by J. M. Manley and H. E. Rowe, Proe mE, 47, 7 (1956).