0:24

In the ideal model we're going to make a few assumptions that are never

completely true, but they allow us to get a decent idea of how a transformer is

going to behave without having to do a more complicated analysis.

So first of all we need to identify the assumptions that are used for

the ideal transformer model,

then use the ideal transformer model to do some basic circuit analysis.

Then we will describe the importance of transformers in power transmission.

First of all we need to define the Coefficient of Coupling.

So the Coefficient of Coupling is going to be represented by a lower case k.

And the Coefficient of Coupling is the value that

gives this relationship between mutual inductance M and

the self inductance is L1 and L2 in the transformer.

1:09

So it's always possible to find this k but

some k's are invalid in a physical sense and some k's are not.

K's basically have to be somewhere between 0 and 1, 0 meaning that there's a no

mutual inductance between the two coils, and they're completely independent of each

other, and 1 meaning that they're very tightly coupled.

But there's a physical limitation to how tightened this coupling can be.

1:33

So, to do an example problem, consider that L1 is 4 millihenries,

L2 is 9 millihenries, and M is 2mH, or we can simply use this equation to see

that k is equal to m divided by the square root of L1 times L2.

So that equals 2mH divided by 4mH times 9mH.

So that's 36mH squared.

Take the square root of that to give us 6 millihenries under 2 millihenries.

So that means that k is going to be equal to one-third.

This is quite simple to calculate the coefficient of coupling.

It's going to be very important, when we start talking about ideal transformers,

though, because we are going to make a certain assumption about what

this coefficient of coupling happens to be.

In the ideal transformer case, the coupling coefficient k is equal to 1,

which means that these two coils are very tightly coupled.

L2 and L1 are assumed to go to infinity, which means that so

is our mutual inductance.

Now, this is a limit.

It's not that it's equal to the value infinity.

But they're going to approach very, very large numbers.

And the reason that we make this assumption is that it allows the analysis

that leads to the transformer equations.

We're going to skip over the actual analysis.

It's available.

You can find it if you're interested, but we just need to make use of it.

But we clearly know that having this infinite inductance is not possible.

Finally we assume that losses from coil resistances are negligible.

Sometimes when we're doing analysis of transformers,

we're going to stick a resistor here.

And a resistor here to correspond to the resistance of

the two wires that are placed here and here.

It gives a better representation but

in the ideal transformer case we're going to assume that both of those go to zero.

Now the implications of this ideal transformer model are the following.

First of all, v1/N1 = v2/N2 where v1 and

v2 are the voltages across the primary and secondary coils respectively.

And then N1 and N2 are the number of rotations at the coil,

how many wraps of coil in the primary and the secondary coils respectively.

We also have the relationship N1i1 = N2i2.

Now these equations come from the Faraday's Law of Induction.

5:04

We have a phaser voltage here.

And I'm going to do all of my analysis for

this problem in terms of RMS of voltages and currents.

Because it's entirely possible to do that without any kind of confusion, but

I want to point that out.

So we have 120 kilovolts RMS here leading to 120 volts RMS here.

Now we don't know the number of coils because we don't have the number for

either of these.

But what we do know is the relationship between them.

So frequently with ideal transformers, we'll show something like this,

a ratio of the number of coils from one to the other.

Because the actual number of themselves don't really matter as much

as the relationship between them.

And in this case we see that there's 100 coils to one over here because if I take

120 and multiply it by 100 we get 12 kilovolts over here.

6:31

And so what we can then do is we can find the power that being consumed in each of

these devices.

So first of all we'll look at the resistor.

I2 is one-half amp rms and

the voltage across this is equal to 120.

And since this is a completely real device, P is going to be

equal to the apparent power which is one-half times 120.

The rms current and the rms voltage multiplied together.

So 60 watts, so this is basically a 60 watt lightbulb.

7:04

If I calculate the power that's being consumed here,

we'd find that P is equal to minus 60 watts based on the rough instructions and

the current and the voltage.

So this transformer is generating 60 watts of power, and that's because this side,

if we take this voltage and this current and

multiply them together, you'll find that that P is also equal to 60 watts.

7:43

And so we say that that's why we need to have this relationship between

voltage and current.

Because the power is being transferred from one side to the other.

And that's one of the nice things about ideal transformers,

is it's converting all of the power that's going over here,

pushing it to the other side to be used to do some type of work.

So essentially what we're doing is allowing ourselves to change the voltage

to operate at a different voltage.

So consequently transformers are often used in computers where you

have 120 volts coming out of the wall but you need 5 volt DC to run your components.

You need a transformer to change the voltage.

It's also used in power applications.

In this case this is what we're illustrating here.

This is like the power that's being sent by the power company.

This transformer is a big power transformer that's owned by the power

company that changes the voltage that they're transmitting to a smaller voltage.

The reason for this is that if you have high currents flowing through wires,

it leads to basically heaters.

And you lose a lot of power as heat lost in the wires.

Power companies don't want to lose all of that heat in their wires, so

they operate their transmission lines at very high voltages so

that the currents can remain small.

And they use transformers to go from those very high voltages

to smaller voltages that are used in residential homes.

8:58

So what are the implications?

Well transformers allow a change from one voltage to another voltage.

We take high-voltage and low-current power and transform it into,

using long-distance power distribution through transformers,

into something that is a low-voltage and high-current.

So that you can still get the power to all of the homes.

Now, it turns out that this is a very important phenomenon.

Before this event, it was required for

power stations to be placed very close to where the power was being consumed.

By being able to use transformers,

the power can now be sent over very long distances.

So things like the Niagara Falls power plant

can now send power all across northern the United States and Canada

without having to have a whole bunch of power stations all over the place.

9:44

So in summary, we showed the ideal transformer model and

used this model to solve an example system.

And identified that transformers are useful for power transmission because

they make possible sending power using very high voltages and low currents, to

be able to avoid getting high heat through high currents through transmission links.

In the next lesson, we'll be looking at a sensor that makes use of this concept of

mutual inductance, a linear-variable differential transformer, and

see places that they can find useful applications.

Until then.