
Capacitors
A
capacitor can be made physically using two parallel conducting
plates which are held close together (but not touching). Electric
charge can be stored in a capacitor by applying a voltage across the
plates.

The defining equation of a capacitor

is
 |
(E.2) |
where

denotes the capacitor's
charge in
Coulombs,

is the
capacitance in
Farads, and

is the
voltage drop across the capacitor in volts. Differentiating with
respect to time gives
where

is now the
current in
Amperes. Note that, by convention, the current is taken to be
positive when flowing from plus to minus across the capacitor (see the
arrow in Fig.
E.1 which indicates the direction of current
flow--there is only one current

flowing clockwise around the
loop formed by the voltage source, resistor, and capacitor when an
external voltage

is applied).
Taking the
Laplace transform of both sides gives
by the
differentiation theorem for Laplace transforms (§
D.4.2).
Assuming a zero initial voltage across the capacitor at time 0, we have
We call this the
driving-point impedance of the capacitor. The
driving-point
impedance facilitates
steady state analysis (zero
initial conditions) by allowing the capacitor to be analyzed like a
simple resistor, with value
Ohms.
Mechanical Equivalent of a Capacitor is a Spring
The mechanical analog of a capacitor is the
compliance of a
spring. The voltage

across a capacitor

corresponds to the
force 
used to
displace a spring. The charge

stored in
the capacitor corresponds to the
displacement 
of the spring.
Thus, Eq.

(
E.2) corresponds to
Hooke's law for ideal springs:
where

is called the
spring constant or
spring stiffness.
Note that
Hooke's law is usually written as

. The quantity

is called the
spring compliance.
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