0:22

We will look at losses associated with each of these phenomena in detail.

But, in summary hysteresis loss is because of

the energy required to rotate the magnetic domains in the core material.

And the eddy current loss is a resistive loss, or an ie squared

r loss because of the induced Eddy currents that flow in the core.

0:47

Let's consider Hysteresis loss first.

The magnetic course we typically use are made of ferromagnetic material.

Ferromagnetic materials are made up of tiny magnetic domains.

You can think of a magnetic domain as a tiny magnet that can rotate

when a magnetic field is applied to it.

Sort of like a compass rotates when in the Earth's magnetic field.

1:30

Just like a magnet, the magnetic domains produce magnetic flux

because of their internal electron spins.

Under normal conditions, without an externally applied magnetic field,

the magnetic domains are randomly oriented and therefore produce no net flux.

However, when a magnetic field is applied to them, the magnetic domains

start to rotate to align themselves with the applied magnetic field.

Hence resulting in much larger magnetic flux than

would have been predicted by simply mu naught times h.

2:09

This amplification of the b field is the reason why the apparent permeability

of cores is much higher than the permeability of free space.

However, it takes energy to rotate these magnetic domains.

And if the applied magnetic fields is varying and possibly

changing direction then the magnetic domains also have to rotate along with it.

For example if you are applying a sinusoidal current to a winding wrapped

around the core, the h field in the core is also varying sinusoidally.

As a result the magnetic domains will also have to

rotate their alignment at the frequency of the applied current.

However, to rotate the magnetic domains have to overcome frictional forces and

that requires energy.

Hence there are losses because of this effect.

This effect is what causes the BH loop to be not a single-valued function.

And in fact creates a hysteresis in the bh gov which is why

this effect is also known as hysteresis.

And the losses associated with it are known as hysteresis losses.

3:30

Let's try and develop a model for

this hysteresis loss in terms of the physical parameters of our core.

To do this, let's do the following thought experiment.

Let's assume that all of the losses

associated with our magnetic device are due to hysteresis loss.

In that case, if you would to measure the energy going into the winding

of our magnetic structure over one complete cycle then if there was

any net energy, that energy must be associated with the hysteresis loss.

4:09

Note that if there was no loss in our magnetic device,

then the energy that goes into our magnetic device and part of the cycle

would be equal to the energy that flows out in the rest of the cycle.

So if there is any loss that loss must be measurable by simply measuring

the average energy going into our magnetic device over a complete cycle.

4:41

Lets consider the core shown here with a winding, with N turns.

The core itself has a cross section area of Ac and

it has a mean length in the core through which the flux flows

of lc and a permeability of mu c.

The flux that flows in the core is v and

this flux flows and the result of the current i flowing in the winding.

5:45

Recall that the voltage of the terminals is simply equal to d

lambda dt, where lambda is the flux linkage.

And the flux linkage in this case is simply n times phi.

So we can write this as n d phi dd.

Phi in turn is just

the flux density b times the cross sectional area Ac.

So this can be further written

as N times Ac times dB dt.

6:26

Similarly, we can find an expression for i.

Recall from Amperes law that the integral of H

around this loop which will be H times lc must equal

the total current cutting the surface of this loop which is just N times i.

So Hl is nothing but Ni and

therefore i is can be written as H times lc over N.

7:49

We can also change this to an integral in dB by canceling out the dt.

And so this expression can be rewritten

as simply W Equal

to Ac times lc since they're both constants,

I can pull that out of the integral and so,

the integral is again over 1 cycle.

And now, it's just an integral of H with respect to B.

9:08

To integrate H with respect to B over one cycle, we simply

have to integrate H as a function of B as we transcend this BH curve.

Let's say we have some starting value which corresponded

to some minimum value of B, let's say Bmin.

This has some corresponding value of H and so,

to find the integral of H with respect to B, we integrate which

is essentially the area under this curve as we move along this curve.

And note, we're moving along the curve to the right,

since that's the curve to follow when H or B are increasing.

So, we move along this curve up, until we reach the maximum value of P,

let's say that's some value B max.

10:26

To complete the cycle, we have to integrate all the way back to Bmin.

Since this time, the value of B and H is decreasing,

we have to follow the curve to the left and so,

we go down on this spot until we reach Bmin.

And this time, the area is the area between the dB axis and

this line, so that's this shaded part.

So what is, then, the value of this integral?

HdB over one cycle.

Well, using that example, we can rewrite

this as an integral, which goes as follows.

So, we integrate from Bmin to Bmax,

H, and the H we follow is the one to the right,

so we'll just call that HrdB.

Now, we also have integrate

on the way back and that's starting from Bmax coming back to Bmin.

So, that's an integral

from Bmax to Bmin of HDB, but

this is now the H to the left,

so we'll call it HlDB.

Note, that the second integral is integral from Bmax to Bmin, so

to put it back in the same form as the first integral.

I can simply swap the sign here and

rewrite this integral as simply minus

the integral Bmin to Bmax of HldB.

Therefore, our original integral HdB, or

one cycle Is nothing but the difference between the area

of the first integral which is the area under the H curve

to the right minus the area of the H curve to the left.

On this figure, that simply

the area between the two curves.

So, this area here is simply

our integral HdB over one cycle.

If the range of flux density B over which we moved along this curve

was narrower, then the area would be smaller and our loop would be smaller.

If on the other hand, the values of B over which we moved extended to a much

larger region of this PH loop, then the Hysteresis Loss would certainly be bigger.

13:46

Therefore, to summarize, a Hysteresis Loss is given by

the area enclosed by the BH loop over the values of B and

H that we move over, multiplied by the volume of the core.

So the bigger the core, the bigger the loss.

Now, this is the energy lost in one cycle.

To determine what is our average power lost,

you must multiply it by the number of cycles we transcend in the second,

which is essentially an operating frequency.

So to determine the average power loss,

you must multiply the result By the operating frequency f.

And so this expression then gives us the expression for

the average power loss due to Hysteresis.

15:01

Note that this full BH loop that's shown here, is only a snapshot

of the BH loop that you may actually go over and it depends on the range of B and

H values over which you move, as well as the actual operating frequency.

If we assume tha the shape of the BH loop does not depend on the operating

frequency, then we can come up with an approximate model for

the average power loss due to Hysteresis that has a linear dependence on frequency.

We can also roughly model the area of the BH loop that we surround

by a parameter delta b which is the flux swing.

The flux swing, Delta B is simply

the average to peak value of the flux swing we have.

So, if our swing in B is from some

Bmin to some Bmax So

that we are essentially covering the loop as shown.

16:37

In theory, you could try and find their values by fitting this expression

to measure data, but in practice, measuring hysteresis loss by itself

is extremely difficult unless you have no other losses in your core.

Let's turn our attention then to the other type of losses that occur in cores

17:29

Therefore, when we have changing flux in the magnetic core

we will also have induced Eddy currents.

By Maxwell's equations, eddy currents flow so as to oppose a change in flux.

Let's see how eddy currents will flow in this example structure.

Here we have some time varying current applied to a winding around a core.

This current produces a flux that goes through the core.

And this flux is also changing with time.

The Eddy Currents will flow in a plain perpendicular

to the direction of the flux.

Now here, the flux generated by the winding current,

both flow from right to left as shown.

18:59

You can also see that in this picture right here,

we have the original flux is flowing from left to right.

And the eddy currents are flowing in a direction,

such that, they are producing a field which is

going in the opposite direction and trying to cancel the original field.

Mathematically, this comes straight from Maxwell's equation or

from Faraday's law in particular.

You can see when you have a changing magnetic field,

it's going to result in an electric field and that electric field in turn,

will generate a current depending on the conductivity of

the material in which the electric field is present.

20:06

In addition, they redistribute how the flux is flowing in the core.

Note that in the center of the core, they're cancelling the original flux.

However, near the surface of the core,

they're actually reinforcing the original flux,

because if you draw the full flux for this loop formed by the Eddy current,

the flux lines will actually look something as shown.

So near the surface, the flux lines created by the eddy current

are actually adding to the original flux lines, while in the middle of the core,

they're cancelling them.

As a result, the flux distribution is nonuniform inside the core.

And the flux density increases around the surface of the core and

is essentially zero in the middle.

So essentially, the cross-sectional area available for

the flux to flow on the core is substantially reduced, because now,

the same amount of flux is essentially flowing just near the surface of the core.

So if you design your magnetic device assuming a certain cross sectional area of

available for flux flow, that is probably no longer going to be true.

21:52

To determine the average power loss,

you will simply average this expression over one time period.

To determine the functional dependence of Eddy current losses on the physical

properties of the core, let's see what the eddy currents are proportional to.

In a material whose impedance is purely resistive, the strength of the Eddy

currents will be proportional to the voltage that's driving it.

From Faraday's law, this driving voltage itself

22:36

Which can in turn be written simply as, A c,

which is the cross of the core,

times db dt would be is the flux density in the core.

Let's assume that our driving current is sinusoidal

in which case our flux density will also be changing sinusoidally.

Let's say B is equal to some B-naught

times some sin of 2 pi

23:35

Therefore the voltage, VE which is driving the Eddy current,

is proportionate to the frequency as well as

the amplitude of the flux density B naught.

Since the current ie is proportional to ve,

we also have that the current ie is proportional to the frequency f and

proportional to the magnitude of the flux density.

And since power loss is proportional to the square of the current,

then the power loss is also proportional to the square of the frequency and

the square of the magnitude of the flux density.

25:20

Although we have separately modeled, hysteresis loss and

Eddy Current Loss using approximate models.

In practice, it is very difficult to measure hysteresis loss and

Eddy Current Loss separately.

So it is near impossible to determine the fitting parameters for

our approximate models.

In practice, if you measure the core loss by measuring the energy that's going into

the core over one cycle.

What you would really be measuring is the sum of the hysteresis loss and

the Eddy Current Loss.

In practice, we use a single expression to model the combined effect of

hysteresis loss and Eddy Current Loss and just simply call it the Total Core Loss.

This Total Core Loss is modeled by the well known

Steinmetz equation that is shown here.

In the Steinmetz equation, the Total Core Loss is written as a function

of the operating frequency F and the flux swing Delta B.

And the broadened core loss is also proportional to the volume of the core.

27:28

With power loss density is the power loss per unit volume of the core essentially

equal to, The actual power loss in the core,

Pc, divided by the volume of the core, Vc, which would be equal to.

If we were to use the second version of the Steinmetz equation,

K1 times delta B raise to the power of Beta.

28:14

The power loss density in this chart is given as a function of

the flux swing delta-b for a range of values of delta-b.

And also for a range of different operating frequencies

from 20 kilohertz down here, all the way up to 1 megahertz.

So, if you want to know what your power loss density is at a delta b of,

say, 0.03 tesla, which would be here,

then you would simply go up to the frequency of interest.

Let's say that was 500 kilohertz so that's this first line here and

then you can read off what your power loss density would be for

that delta B and 500 kilohertz operating frequency.

Now, you would have to multiply the power loss density by the actual volume of your

core to figure out what your actual power loss in the core is.

29:41

and let's say these are the two points, then for each point you can read

off the value of delta B and the corresponding value for power loss.

That gives you one equation, which has two unknowns and

then from the other point, you get another equation, which also has two unknowns.

And by solving these two equations simultaneously you can find the values for

K1 and beta that would be suitable for 200 kilohertz operation.

You can repeat the process and get different values of K1 and beta for

500 kilohertz or 1 megahertz or whatever frequency you are interested in

30:39

The only things you could do is reduce your operating frequency and

reduce your flux swing, if that's a possibility.

For Eddy current losses, we can do one more thing, and

that is to use laminations.

So instead of using a solid core, we'll slice up the core

Into thin laminations as I'm showing here.

Essentially the core is built through thin slices

of core material that are stacked together with insulation between them.

31:34

When the core is sliced up like this with insulation between each

of our slices, It blocks the natural flow of the eddy current and

the eddy currents are forced to flow in much smaller loops.

We can see that pictorially here where the top picture here shows

the original core without any laminations.

And in this case, You have the flux going down the center of the core and

the Eddy currents are flowing in a way so as to oppose that flux.

Since the Eddy currents can flow in large loops in this case,

corresponding to a large amount of flux enclosed by them.

Then by Faraday's law, we can see that we have essentially the integral

of flux density over a large cross sectional area.

Which results in a large electric field being generated which,

subsequently means, a large Eddy current being produced.

On the other hand, if we slice up the core, as shown in the pictorial below,

then the Eddy currents flow in much tighter loops,

corresponding to this area becoming small.

And so, the corresponding electric fuse becomes small and so

do the generated Eddy currents.

You can also see this from a more visual perspective by seeing that as you

slice the core into thinner and thinner slices.

The Eddy currents are essentially being forced to flow back and

forth on top of one another.

Hence, there is not much effective eddy current flowing at all.

33:48

At low frequencies such as at 50 or 60 hertz,

commonly used core materials are laminated iron alloys.

The most popular amongst these is silicon steel.

Iron alloys are good because they have high saturation flux density.

As high as two tesla but the problem is that they have relatively high

core losses especially because their high conductivity and high eddy currents.

34:41

At even higher frequencies, especially frequencies from 20 kHz up to 20 MHz,

the material of choice are ferrites.

Ferrites have even lower saturation flux density, but

they have the lowest core loss.

Ferrites are essentially a ceramic material containing iron oxide and

other metals.

35:04

There are very hard resistivities, and therefore,

resent very low Eddy Current Loss until you get to very high frequencies.

Commonly used ferrites are generally classified into manganese-zinc ferrites or

nickel-zinc ferries.

The manganese-zinc ferrites have relatively high permeability, but

the resistivity relative to nickel-zinc ferrites.

Is lower and therefore,

they're useful up to frequencies in megahertz or a few megahertz.

Above a few megahertz of operating frequency,

Nickel-Zinc ferrites are the preferred choice.

Once you get to frequencies much above 20 megahertz, the losses in the available

ferrites also become too high, and you're best off using air core magnetics.

That is to say that you have no magnetic core at all and typically,

you wind your winding either on some on non-magnetic plastic material, or

they're just made in the PCB itself.