With magnetic circuit analysis,
we take advantage of the duality between electric fields and
their equations in magnetic fields. What we're going to do is to develop
equivalent circuit models for magnetic elements that look
a lot like electric circuits and can be solved in a similar way. Let's consider here a magnetic core or
some other similar piece of magnetic material that has some flux phi
flowing down it through the material. The material has some dimensions,
so there's a length l and there's some area Ac. We think of this as being a magnetic
conductor, which is similar or analogous to an electrical conductor. So here instead of voltage, we have
magnetomotive force across this element. That magnetomotive force drives
a flux through the element, so these are the analogs of voltage and
current. And the element itself is
a conductor of magnetic field. From the previous lecture, we found
that with a uniform magnetic H field, H, down an element, there's a total magnetomotive force F
equal to H times the element length. More generally, it's the integral of H
down the element, but if H is constant, then we get this. We can also talk about the B field or
flux density in the material. It's related to the total flux phi by
integrating flux density over the area. So if we have some area Ac and
we have a constant flux density B, what we get is that B is the total
flux phi divided by the area. Finally, we have B is mu H or
H is B over mu. Let's substitute magnetomotive force, which is H times l where
the H is B over mu. So B over mu is H times the length. And then B is the flux,
total flux over the area, so we get total flux times length
over the area times mu. And finally, we can write this as that
magnetomotive force is equal to flux times this collection of constants. That's the length over the area times mu. This collection of constants is what
we call the reluctance of the material. It's a lot like the resistance
of a conductor. And so we have the magnetic analog of
Ohm's law, which is the magnetomotive force is the flux times the reluctance for
some magnetic element. What we will do is draw a magnetic circuit
element here that looks like a resistor, looks like an electrical circuit. It has a magnetomotive force F across it. We will represent the reluctance with
a symbol that looks like a resistor, and the current through it becomes
the flux through the element. So if we have some magnetic structure
that has different elements in it, let's say there's multiple windings, there's air gaps,
there are cores with different dimensions. We can represent each
element with a reluctance, such as the one on the previous slide. We found in the last lecture that windings
are sources of magnetomotive force from Ampere's law. So we can represent each
element in this way. And then what we get is a circuit model
that we can solve with standard circuit analysis. Okay, there are actually magnetic
analogs of Kirchoff's laws and they come from Maxwell's equations. The analog of Kirchoff's current laws,
where we have say some node and the node equation says the sum of
the currents into the node equals zero. The analog of that is the sum of
fluxes into a node equals zero. And the reason for this is that flux lines
are continuous and they can't just end. So one of Maxwell's equations
is divergence of B = 0, and that's what it says. So if we have a magnetic circuit,
we can have some node, and the sum of fluxes into the node equals
zero, just like Kirchhoff's current law. Kirchhoff's voltage law
comes from Ampere's law. Kirchhoff's voltage law says
the integral of H, dot, dl around a closed path is the total
current through the interior of the path. This is basically the sum of MMFs
around a closed loop, around a loop. And it adds up to zero as long as you view
the currents as being sources of MMF. So let's do an example. Let's say now we have an inductor that has
a core with uniform cross-sectional area, except it has an air gap right here. And so in the air gap,
the permeability is different, we have mu naught for permeability there. And then we have a winding
with n turns of wire, okay? So what is the magnetic
equivalent circuit for this? Well, we have the winding
is a source of MMF, there are n turns of
wire carrying current i. So the MMF here, which is the sum of the
currents flowing through the interior of the path, is n times i. And we have an MMF source, looks like a voltage source,
to represent the winding current. Then we have a core, and we'll take the core material, we'll assume
it's uniform all the way around the core from one side of the air gap
to the other around the core. And that has some reluctance. Okay, the reluctance is what? It's the length of the element. So let's take the length,
I'm going to define l core is the distance that the flux
takes going around the interior of the core from one side of
the air gap to the other. Okay, and then we divide by the area. Let's assume the core has some
constant cross-sectional area A sub c, and then we divide by the mu of the core. And finally, we have an air gap, okay? So the air gap is a, we can treat as
another element in the magnetic circuit. It has its reluctance,
Define that as our gap. The reluctance of the gap is a similar
formula, it's the length of the gap, so this distance,
divided by the cross-sectional area. Let's assume the cross-sectional area
is the same as the area of the core. Actually fringing flux will make it
effective, the area a little bit larger, but let's ignore that for now. And then the mu of the air is mu naught,
the permeability of free space. So we have an equivalent circuit, and we can directly write
the elements just like this. Finally, the flux in the core, I'll label
that is the flux around this circuit. And what happens is the magnetomotive
force, or ni source, drives flux through the reluctances
of the core in the air gap. Here's the model I just
derived with the reluctances. And finally,
what we can do is solve the circuit. So what we have from just solving
the circuit is the voltage over the resistance,
or MMF over reluctance, equals the flux phi, basically Ohm's law. Finally, let's solve. Here's what we had from
the last two slides. We've solved for
the relationship between current and flux. The last thing we need to do to get
the terminal equations is to find the relationship between the voltage and
the current, the winding. So we have current here in this equation,
we need to get voltage. And if you recall from
the previous lecture, the relationship between voltage and
flux is given by Faraday's law, that the voltage induced in each turn
here for Faraday's law is d phi/dt. So the total voltage v
is d phi/dt in one turn. There are n turns and each have the same
flux phi in them, so each turn will have the same voltage, and the total voltage
of the winding will be n times d phi/dt. So we can plug that into here, or really solve this for
phi and plug it into here, what phi would be ni over
the sum of the reluctances. Plug that into here, so v is n times d phi/dt, the derivative of this. We have another n, so n squared times
di/dt, and we divide by the reluctances. Okay, so we have an equation of the form v is some
collection of constants times di/dt. We can recognize this as the form
of the equation of an inductor, and apparently this is the inductance,
so v is di/dt. And there's what the inductance is, it's n
squared over the sum of the reluctances. Okay, it's interesting now to
look at what the air gap does. People put air gaps in inductors. Why would you do that? So if we look at our equation for inductance right here,
what is the effect of adding the air gap? Well, it adds additional
reluctance to the denominator, which would reduce the inductance. Well, why would you want to do that? So there are several reasons for this. One reason is that it makes the inductance
less dependent on the core reluctance. And one thing about magnetic cores
is that the mu of magnetic cores often is a strong function of things like
temperature and can have variations. Whereas the mu of air,
mu naught, is a fixed number. So it's possible here to make more
stable inductors, whose value doesn't change as much with things like
temperature, by adding an air gap. So we sacrifice some inductance, but what we end up getting is a more
stable value of inductance. There's a second reason for
this as well, which is saturation. So last time we saw that cores can
saturate, we have a saturation flux density B sat of core material, and so
there's a flux that corresponds to that. So there is a value of total flux
that makes the inductor saturate. Okay, let's plug this phi sat into here to find the current at
which the core saturates. So we'll call it i sat would be equal to phi sat times the sum of the reluctances. And we can simplify this and
what we get is this equation, which gives the value of winding current
that makes the inductor saturate. And what you can see is that
by adding air gap reluctance, you make the value of i sat increase. So for a given core that has
a given reluctance, has a given cross-sectional area, B sat is a function
of the core material that is given. What we can do by adding the air
gap is to increase the current at which the inductor saturates. So we're trading off inductance for
saturation. Often you'll see plots here
that look like the BH loop. We've actually multiplied B by
cross-sectional area to get flux. And H is proportional to ni. So when we plot in ni versus phi, we get a curve that has a shape similar
to that of the BH loop of the material. And what we're doing is we're reducing
the slope by adding in reluctance, but we're increasing the saturation
current of the inductor. So to summarize,
the magnetic circuit approach is a pretty simple way to model more
complex magnetic structures and to see the effects of things
such as air gaps and inductors. And we can extend this further to more
complicated systems such as coupled inductors or transformers,
which we'll do in the near future.