Resistors (Ohm's Law), Capacitors, and Inductors

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Resistors

The symbol for a resistor:

File:Resistor symbol.jpg

The relationship between the current through a conductor with resistance and the voltage across the same conductor is described by Ohm's law:

where V is the voltage across the conductor, I is the current through the conductor, and R is the resistance of the conductor.

The power dissipated by the resistor is equal to the voltage multiplied by the current:

If is measured in amps and in volts, then the power is in watts.

By plugging in different forms of , we can rewrite as:

or

Capacitors

The symbol for a capacitor:

File:Capacitor symbol.jpg

A capacitor is a device that stores electric charges. (Do not confuse this with a battery; a battery creates an electric potential difference, but doesn't store charge. A battery is analogous to a water pump, while a capacitor is analogous to a water tank.) At steady state, a capacitor has a constant voltage across it and zero current through it. In this state, the capacitor acts like an open circuit.

The unit of measurement for the capacitance of a capacitor is the farad, which is equal to 1 coulomb per volt.

The charge(q), voltage (v), and capacitance(C) of a capacitor are related as follows:

where q(t) and v(t) are the values for charge and voltage, expressed as a function of time.

Differentiating both sides with respect to time gives:

Rearranging and then integrating with respect to time give:

If we assume that the charge, voltage, and current of the capacitor are zero at , our equation reduces to:

or

,

which is equivalent to the first equation.

Assuming the same zero initial states as before, the energy stored in a capacitor (in joules) is given by the equation:

Inductors

The symbol for an inductor:

File:Inductor symbol.jpg

An inductor stores energy in the form of a magnetic field, usually by means of a coil of wire. An inductor resists change in the current flowing through it; an inductor at steady state has a constant current flowing through it and no voltage across it. In this state, the inductor acts like a short circuit.

The relationship between the voltage across the inductor is linearly related by a factor L, the inductance, to the time rate of change of the current through the inductor. The unit for inductance is the henry, and is equal to a volt-second per ampere.

The relationship between the voltage and the current is as follows:

If we multiply both sides by dt, we get:

Integrating both sides from to gives:

which is equal to:

assuming that the voltage, current and energy of the inductor are all zero at reduces the equation to

The energy stored in the inductor is given by:

References

Hayt, William H. Jr., Jack E. Kemmerly, and Steven M. Durbin. Engineering Circuit Analysis. 6th ed. New York:McGraw-Hill, 2002.