Circuit Theory/Capacitors

Capacitors are passive circuit elements that can be used to store energy in the form of an electric field. In the simplest case, a capacitor is a set of parallel metal plates separated by a dielectric substance. Dielectric used can be air or any other substance.



Electric charges build up on the opposite plates as a voltage is applied to the capacitor.

With a constant voltage across the capacitor there will be no change in electric charge and thus the steady-state current becomes zero. Stored energy can be discharged from a capacitor by connecting its terminals with a wire. Big capacitors can create lightening bolts, so tape the wire to the end of a long wooden stick.

Capacitance
Capacitance is defined as the capability of a capacitor to store the charge of a voltage. Capacitance is measured in units called "Farads," abbreviated by an "F".

The ratio of charge over voltage gives a value of capacitance
 * $$C = \frac{q}{v}$$

Capacitor Terminal Relation
The relationship between the current and the voltage of a capacitor is:


 * $$i = C\frac{dv}{dt}$$

The real world disturbs a capacitor by changing the voltage across it rapidly. This instantly turns the capacitor into a short. The capacitor eventually catches up to the changed condition, the voltage stops changing, the current drops to zero and the capacitor acts like an open.

Capacitor Safety
The bigger a capacitor is, the more dangerous it is.

A capacitor can continue storing its energy after the circuit is turned off. A capacitor the size of a soda bottle can throw a person across the room and/or damage nerves. Touching a capacitor the size of a kitchen trash destroyed a persons arms to the point they were cut off at the shoulders. Touching a capacitor the size of an oil drum has killed people. Super capacitors are packing more power into smaller, lighter objects.

If capacitor is connected to a circuit wrong, it can explode youtube video.

A leyden jar is a capacitor. Aluminum foil of about 200 mm by 450 mm placed on either side of glass can store charge. If charged with a static electricity generator such as a Wimshurst machine can cause the glass, at the edges of the aluminum foil to shatter into little shards like a grenade exploding.

Example
Problem: Given a current of 10 amps flowing through a 1uF capacitor for 2 us, what is the voltage across the capacitor?

Solution: The problem appears to be a capacitor hit by a pulse of current: $$ i(t) = \begin{cases} 0, & \text{for } t {\leq} 0\\ 10, & \text{for } 0 < t {\leq} .000002\\ 0, & \text{for } .000002 < t \end{cases}$$

The initial voltage across the capacitor is not given. Assume it is zero for time less than 0. While current is flowing into the capacitor it is going to charge. When the current stops, the capacitor will hold it's charge and the voltage will remain constant.

The charging period requires transforming $$i = C\frac{dv}{dt}$$ into the integral $$v(t) = \frac{\int\limits_{0}^{t}i({\tau})\,d{\tau}}{C}$$ The solution involves an integral: $$ v(t) = \begin{cases} 0, & \text{for } t {\leq} 0\\ \frac{\int\limits_{0}^{t}10\,d{\tau}}{.000001}, & \text{for } 0 < t {\leq} .000002\\ v(.000002), & \text{for } .000002 < t \end{cases}$$

To calculate the answer, the integral needs to be evaluated twice (links are to wolfram alpha):
 * once with the limits 0,t
 * second with the limits 0,0.000002

This results in the solution:$$ v(t) = \begin{cases} 0 \text{  volts}, & \text{for } t {\leq} 0 \text{  seconds}\\ 10^7t \text{ volts}, & \text{for } 0 < t {\leq} .000002 \text{  seconds}\\ 20 \text{ volts}, & \text{for } .000002 < t \text{  seconds} \end{cases}$$