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

Resistors

The symbol for a resistor:

Real resistors:

Try wikipedia for more on resistors and for the resistor color codes.

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

$V=IR$

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:

$P=IV$

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

By plugging in different forms of V=IR, we can rewrite P=IV as:

$P=I^{2}R$

or

$P=\frac{V^2}{R}.$

Capacitors

The symbol for a capacitor:

or

The capacitor on the right is polarized. The potential on the straight side (with the plus sign) should always be higher than the potential on the curved side.

Real capacitors:

Notice that the capacitor on the far right is polarized; the negative terminal is marked on the can with white negative signs. The polarization is also indicated by the length of the leads: the short lead is negative, the long lead is positive.

A capacitor is a device that stores electric charges. The current through a capacitor can be changed instantly, but it takes time to change the voltage across a capacitor.

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:

$q(t)=Cv(t)$

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:

$i(t)=C\frac{dv}{dt}$

Rearranging and then integrating with respect to time give:

$v(t^{\prime })=\frac{1}{C}\underset{t_0}{\overset{t^{\prime }}{\int }}i(t)dt+v(t_0)$

If we assume that the charge, voltage, and current of the capacitor are zero at $t_0=-\mathrm{\infty }$, our equation reduces to:

$v(t^{\prime })=\frac{1}{C}\underset{-\mathrm{\infty }}{\overset{t^{\prime }}{\int }}i(t)dt$

The energy stored in a capacitor (in joules) is given by the equation:

$w_C(t)=\frac{1}{2}C{v}^2$

Inductors

The symbol for an inductor:

Real inductors (and items with inductance):

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. The voltage across an inductor can be changed instantly, but an inductor will resist a change in current.

Devices with coils of wire, such as motors or transformers, add inductance to a circuit. However, we generally don't purposefully add inductors to circuits.

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:

$v(t)=L\frac{di}{dt}$

If we multiply both sides by dt, we get:

$di=\frac{1}{L}vdt$

Integrating both sides from $t_0$ to $t^{\prime }$ gives:

$i(t^{\prime })-i(t_0)=\frac{1}{L}\underset{t_0}{\overset{t^{\prime }}{\int }}vdt$

which is equal to:

$i(t^{\prime })=\frac{1}{L}\underset{t_0}{\overset{t^{\prime }}{\int }}vdt+i(t_0)$

assuming that the voltage, current and energy of the inductor are all zero at $t=-\mathrm{\infty }$ reduces the equation to

$i(t^{\prime })=\frac{1}{L}\underset{-\mathrm{\infty }}{\overset{t^{\prime }}{\int }}vdt$

The energy stored in the inductor is given by:

$w_L(t)=\frac{1}{2}L{i}^2$

Elements in Series and Parallel

Resistors connected in series and parallel:

Two Elements in Series and Parallel

	Resistor	Capacitor	Inductor
Series	$R_{eq}=R_1+R_2$	$C_{eq}=\frac{C_1{C}_2}{C_1+C_2}$	$L_{eq}=L_1+L_2$
Parallel	$R_{eq}=\frac{R_1{R}_2}{R_1+R_2}$	$C_{eq}=C_1+C_2$	$L_{eq}=\frac{L_1{L}_2}{L_1+L_2}$

More than 2 Elements in series or parallel

Here we provide the equations for calculating the equivalant resistance of three or more resistors in parallel; the same form can be applied to the corresponding equations for capacitors and inductors. Of course, you can always just simplify a network of elements by combining two at a time using the equations above.

To find the combined resistance of resistors connected in series, simply add the resistances:

$R_{eq}=R_1+R_2+\cdot \cdot \cdot +R_n$

If the resistors are connected in parallel, the equation is:

$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdot \cdot \cdot +\frac{1}{R_n}$

Proof for Resistors in Parallel equation

Here we provide the derivation for the parallel resistors equation. The corresponding equations for capacitors and inductors can be derived with a similar method.

We can prove the equation for parallel resistors by using Kirchhoff's voltage and current laws:

$V_s=V_{R1}=V_{R2}$   (KVL)

$I_s=I_1+I_2$           (KCL)

Plugging in the constitutive law for resistors in the second equation yields:

$I_s=\frac{V_s}{R_1}+\frac{V_s}{R_2}=V_s(\frac{1}{R_1}+\frac{1}{R_2})=V_s\left(\frac{R_1+R_2}{R_1{R}_2}\right)$

$R_{eq}=\frac{V_s}{I_s}=\frac{R_1{R}_2}{R_1+R_2}$

References

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