Electric circuits and the hydraulic analogy: inductor behavior.An inductor is
a magnetic flux storage device. Now you may be wondering: What is magnetic flux,
and why do you want to store it? We'll get to that. Let's start by looking at
the hydraulic analogy. A hydraulic inductor is a water wheel connected
through a rigid axle to a heavy stone flywheel. When water flows from left to
right, it's directed in a clockwise direction. Here I've drawn the same thing
in a cross-section side view. The flywheel momentum is proportional to the
flow rate in liters per second. For example, if we have one liter per second
flowing through, there are 20 momentum units; 2 liters per second, 40 momentum
units; 3 liters per second, 60 momentum units, and so on. The momentum is
proportional to the flow rate, and the constant of proportionality is called
the inductance. For example here we have 60 momentum units equals the inductance,
20 units, times 3 liters per second. Now let's take a look at the electronic
inductor, a coil of wire wrapped around a cylindrical core. The magnetic flux is
proportional to the flow rate in amps. For example, if we have 1 amp flowing
through this inductor, we have 20 webers of magnetic flux; if we have 2 amps, 40
webers; 3 amps, 60 webers. The flux is proportional to the current, and the
constant of proportionality is called the inductance. For example, here we have
60 webers equals 20 inductance units times 3 amps. The units of inductance are
webers per amp, and there's another name for that, henrys. Henrys means webers
per amp
Now back to the question, what is magnetic flux? It is a number of lines of
magnetic field that pierce the cross-sectional area of the core, and,
there's a mathematical description of that but I'm not going to get into that.
In fact, this definition is too complicated. For our purposes, we can
think of magnetic flux as the momentum of an invisible flywheel that tends to
keep the same current flowing and resists change. So when you see a picture
of lines of magnetic force going through an inductor, use your vivid imagination
to visualize a turning flywheel that keeps the current flowing. All right, what about
the pressure and changes in flow rate? There is no pressure drop with a
constant flow. The pressure will be the same on the two sides when there's a
constant flow. There's a pressure drop only when the flow changes. For example,
let's connect a pump that maintains a pressure of 20 kiloPascals. We turn on
the pump. The flywheel starts turning faster and faster, and over three seconds
the flow rate changed from 0 to 3 liters per second. Now what happens when
we reduce the flow? We build up pressure on the other side; that's a negative
pressure. Higher pressure on this side than this side is a negative pressure
drop and the momentum of the flywheel becomes less. So we saw before, momentum
equals inductance times flow rate. That means the change in momentum is equal to
the inductance times the change in flow rate. Now the change in momentum is equal
to the pressure applied to the inductor. This notation dF/dt means change in flow
rate with a change in time. When I attached the pump here and pumped up the pressure
to 20 kPa, the flow rate increased at a rate of 1 liter
per second. When I remove thed pump and there was a constant flow rate, there was
no pressure drop across here and the flow rate was constant; no change in the flow
rate. When I blocked the flow on this side, there was a buildup of pressure on
this side. That's a negative pressure drop from here to here; and the flow rate
decreased at 1 liter per second for each second. This is very much like a
revolving door. If people are in a hurry, they'll press on the doors as they go
through and cause the momentum to increase. If they're relaxed and walk through
at the same speed that the door is already turning, the momentum of the
revolving door stays constant. As some very slow people go through the
door, the door will catch up to them and push them through; that's negative
pressure and the momentum of the revolving door will decrease. Let's watch
it again: turn on the pump: 1, 2, 3. Three liters per
second, no pressure difference on these two sides, and a constant flow rate. And
now, a negative pressure
a negative drop from here to here, and the flow rate decreased at1 liter per
second per second. Now let's look at the electronic inductor. There's no voltage
drop with a constant current. There is a voltage drop only when the current
changes. For example, let's insert a battery in this circuit: 1, 2, 3, and now we
have 3 amps flowing through this circuit with a 0 voltage drop a constant 3 amps
and constant 60 webers. This is like the momentum of a flywheel keeping the flow
constant. Mow let's put a battery in the reverse direction and apply a negative
voltage: 1, 2, 3. We're back to where we started. We saw
earlier, flux is inductance times current. That
means the change in flux with time equals inductance times the change in
current with time. The change in flux per unit of time is the voltage applied; you
could deduce that from Maxwell's equations. And the notation di/dt means
the change in current with the change in time. For example, when we applied 20
volts, the current increased at a rate of 1 amp per second. When we removed the
battery, there was zero volts around this loop and the flow was constant; there was
no change in the current. And when we put the battery in backwards, it applied a
negative voltage; the current decreased by one amp per second. Let's watch that
again: go, 1, 2, 3, constant current. We're at this stage now: zero volts zero change.
And now let's put the battery in backwards: go, 1, 2, 3. OK! A larger device has
more inductance. For example, for this hydraulic inductor, we extend this axle
and put another heavy flywheel on, it doubles the
inductance. It's twice as hard to get the flywheels moving and twice as hard to
stop. Similarly for the electronic inductor, twice as many windings around
the core doubles the inductance. That's twice as hard to develop flux and twice
as hard for the flux to disappear. You may recall from your freshman physics
that kinetic energy is proportional to the velocity squared. And so it is with
the inductor. The energy stored by this turning flywheel is one-half the
inductance times the flow rate squared, in this case 10 joules. And for this
electronic inductor, the energy stored in this magnetic flux is 1/2 times the
inductance times the flow rate squared, and 10 joules.
So in summary, comparing the hydraulic and electric models, the flow rate is
like the current, the momentum of the flywheel is like the magnetic flux of
the electric inductor. Inductance has these units in the hydraulic system and
henrys in the electric system and the energy is in joules: these units in the
hydraulic system and these units in the electric system.