# 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

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A time-varying capacitor described by the constitutive relation is still a linear element, so that..
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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 Wl + Wo. 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 W1 – Wo 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 V2 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 V2 is
proportional to Vi; if Vi varies slowly, V2
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).