Capacitance and Inductance

Short Topical Videos

• The Capacitor (by MAKE magazine)
• The Inductor (by MAKE magazine)

Reference Material

• Horowitz & Hill, The Art of Electronics, 2nd Ed., Ch. 1
• Capacitor (wikipedia)
• Inductor (wikipedia)

Capacitors

Some typical capacitors

Capacitors collect and hold charge. The larger the capacitor, the more electrons it can hold for a given voltage:

$C={\frac {Q}{V}},\,\!$

where $C$ is capacitance, $Q$ is charge, and $V$ is voltage. Since current, $I$, is the flow of charge versus time, we may alternately express capacitance as relating current to a voltage change versus time:

$I=C{\frac {dV}{dt}}\,\!$

A useful way of thinking of a capacitor is as two parallel plates with a wire connected to each side. Indeed, many capacitors are actually built this way. Imagine that we have a positive charge $Q$ and put it on one side of the capacitor. Electrons, being naturally attracted to a negative charge, will collect at the other plate, if they are free to move, because that’s the closest they can get to positive charge that is attracted them. If these electrons are not given another path to the opposite plate, they will collect until their charges sum to $Q$, whereupon the repulsion of like charges offsets the attraction of the positive charge, and we will have achieved equilibrium. In the steady state, the charge difference between the two plates means there is a potential difference, or voltage $V$, between them. For a given separation between them, larger plates make for bigger capacitors, as measured by capacitance $C$. Capacitance is measured in Farads ($F$), with typical capacitors ranging from hundreds of $pF$ (slightly more than the capacitance between nearby wires) to $F$ (will weld metal, and maybe explode, if you touch the leads together).

Capacitors in Series

Capacitors in series

Capacitors in series add reciprocally. You can maybe see this by considering the total voltage drop across two capacitors in series. Clearly, each capacitor must individually see only a fraction of that voltage drop, and so will not collect as much charge. Since capacitance is the charge collected for a given voltage, the total capacitance must have decreased.:

$C={\frac {1}{{\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\dots +{\frac {1}{C_{n}}}}}\,\!$

Capacitors in Parallel

Capacitors in parallel

Capacitors in parallel add, as should be apparent from examining the parallel plate capacitor discussed above. Two pairs of parallel plates behave just as one larger pair of plates whose area is the sum of the two original plates:

$C=C_{1}+C_{2}+\dots +C_{n}\,\!$

Inductors

Some typical inductors

Inductors are like crotchety curmudgeons. They hate change. Specifically, changes in current:

$V=L{\frac {dI}{dt}},\,\!$

where $V$ is voltage, $L$ is inductance, and $I$ is current.

At their simplest, inductors are coils of wire. As current begins to flow, inductors sap a bit of energy and store it in a magnetic field that they set up. As long as the current keeps flowing, inductors behave very much like wires. But if current stops flowing, the magnetic field starts to collapse. The energy stored in the field has to go somewhere, and it goes into pushing electrons in the direction they were formerly flowing (this is because a changing magnetic field induces an electromotive force, or voltage, around a loop). The larger the inductor (e.g. the more loops in the coil of wire), the larger the magnetic field that gets set up for a given current, and therefore, the larger the voltage that is induced when the magnetic field collapses. Inductors are measured in Henrys ($H$), and typically range from $\mu H$ (a bit more than the inductance in a wire) to 100 $H$ (a powerful electromagnet).

A large inductance by itself isn’t dangerous, but a large current flowing through even a moderate inductor can be extremely unsafe to disconnect without bleeding it off through a resistor. Relatedly, the energy stored in the magnetic field of an inductor is given by:

$E={\frac {1}{2}}LI^{2}\,\!$

Inductors in Series

Inductors in series

Inductors in series add:

$L=L_{1}+L_{2}+\dots +L_{n}\,\!$

Inductors in Parallel

Inductors in parallel

Inductors in parallel add reciprocally:

$L={\frac {1}{{\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\dots +{\frac {1}{L_{n}}}}}\,\!$