Inductors

So far we have seen electrical dampers - resistors, and electrical springs - capacitors. You might be wondering if there is an equivalent of the mass in the electrical domain. Here is a summary so far:
element Mechanical Domain Electrical Domain
damper
spring
mass ?????
Note that in each constitutive law, the relationships in each domain are conisistent in how they relate the effort variable (F, P, V) to the flow variable (v, F, i). Thus, to have an analogy to a mass in electrical systems, we need an element relating effort to the time derivative of the flow variable: V = constant * i'. There is indeed such an element, and it is called an inductor. These are pictures of some real inductors:

Inductors are two-ended elements consisting of a coil of wire, sometimes wound around a ferromagnetic core. They create a magnetic field in their vicinity proportional to the current in the wire, the number of turns in the coil, and some other things (e.g. the core material). There is energy stored in the field.

Because the field contains energy, you can't instantaneously increase the current through the inductor: to do so would produce an instantaneous change in magnetic field, and where would the energy come from to accomplish that? (Power is iV; so to accumulate additional energy requires power over a period of time.)

Interestingly, you can't instantaneously decrease the current flowing in an inductor, either. You may ask, what if you just yank it out of a circuit, so that the current instantly must be zero? In principle the current, desperate to continue, will arc through the air if you do this! To arc through the air requires a lot of voltage, and indeed the inductor can develop a large voltage in this circumstance.

An ideal inductor carrying a constant current has no voltage across it. If you change the current (slowly) the inductor will develop a voltage, with a polarity such as to discourage you from making that change in current. Its constitutive law describes that behavior:
(1)
The voltage is proportional to the inductance L (measured in henries) and to the rate of change of current, i' = di / dt.

Continuing the analogy to a mass: Once there is a current (velocity) in the inductor (mass), it takes a voltage (force) for a period of time to effect a change in that current (velocity). In other words, the integral of the constitutive law of the inductor is analogous to that for a mass:

      is similar to    
Here is an inductor in a piece of a circuit with the voltmeter shown explicitly:

Note that the polarities of the voltage across an inductor and the current through an inductor are measured in the usual way. According to the constitutive law, if we want to increase the current through this inductor, we must make the left side positive with respect to the right side. And we must do this over a period of time.

Here is a picture of a circuit board from a cell phone. Can you find the inductor in this picture? Click on the appropriate location on the image!