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Resistors and Inductors Together
A circuit containing resistance and inductance.

When we combine a resistor and an inductor in a single circuit, some interesting things can happen. The circuit we will use to explore this behavior is shown to the right. Note that this time we are using a three-position switch, rather than the two-position switch we used with the RC circuit. This is because an inductor, unlike a capacitor, will not simply hold its stored energy if the circuit is opened. Therefore, we must consider what will happen if the circuit should open.

In this circuit, we begin with switch S in position 3. It has been there for a long time, so no current flows through R and L, and there is no voltage across either. There is also no magnetic field around L at this time. Thus, we start with a quiescent circuit.



The voltage across L as it charges.

Now, we move switch S to position 2. The current through an inductor cannot change instantaneously, so at this instant there is still zero current through R, and the coil develops a counter-EMF equal to the battery voltage, E.

Note that for this circuit we are using a time constant of L/R. The curve itself plots the function:

vL  = ε(-Rt/L)

E

As time passes, the current through this circuit increases to a maximum value of E/R, and the voltage across L falls to zero. Once that has happened, the current no longer changes and the magnetic field around L stops expanding.

Incidentally, the L/R time constant meets the requirements of dimensional analysis as follows:

    Inductance

    Inductance is the capacity to store energy in a magnetic field. In so doing, it will develop a counter-EMF to oppose any change in current through itself. An inductance of 1 Henry will develop a counter-EMF of 1 volt if its current changes by 1 ampere/second. 

    Resistance

    Resistance opposes the flow of current through a circuit. By Ohm's Law, R = E/I. Thus, 1 ohm may also be expressed as 1 volt/ampere. 

L  =  volts  =   volts × seconds   = seconds


 amperes/seconds  amperes



R volts volts


amperes amperes


The voltage across L as it discharges.

When we move switch S from position 2 to position 3, we still have a circuit, containing just L and R. We are assuming an ideal switch (perhaps an electronic switch) that will make one connection at the same instant that it breaks the other.

Prior to this moment, current has been flowing through R and L, with electrons moving upwards from bottom to top. At the moment we move S to position 3, current cannot change instantaneously, so it must still be flowing through R and L, with the current through L still moving from bottom to top. This requires that L generate an EMF equal to the battery voltage, E, but opposite in polarity. This will decay exponentially, as shown in the graph to the right.

The energy to keep this current flowing comes from the magnetic field around the coil as it collapses; since the field is shrinking as it gives up energy, it is moving with respect to the coils of wire, and therefore induces a voltage in the coil which will tend to keep current flowing in the same direction as before.



A circuit containing resistance and inductance.

We have repeated our RL test circuit to the right. Now, let's consider what happens if we leave the switch in position 2 long enough to allow the magnetic field to fully charge as before, and then move S to position 1. We now have an open circuit, and yet L will attempt to maintain the flow of current through itself. What will happen?

The answer is found in Ohm's Law, with due notice taken of the amount of energy stored in the magnetic field. An open circuit is effectively an infinite resistance. For the same current to flow through such a resistance, the coil must develop an infinite voltage across itself.

In practice, this doesn't happen, of course. However, the coil does develop a high enough voltage to arc across the switch contacts as they open. This immediately discharges most of the energy in the magnetic field. The remaining energy dissipates rapidly as the magnetic field collapses.



A variation on the open circuit discharge idea is to use a higher value of R for discharge, than was used to charge the magnetic field. The inductor will develop a voltage according to the new value of R, thus increasing the voltage beyond the value of the battery, E. Of course, the higher value of R must also be used for the discharge time constant, and discharge will be more rapid than charging the magnetic field.


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