Welcome back to our class on linear 
circuits. 
Today we're going to be talking about the 
ideal transformer model. 
So we're going to use the ideal 
transformer model to analyze basic 
transformer circuits and then see the 
relationships between voltages and 
currents. 
So in the previous lesson we looked at 
the linear model. 
It's a fairly complicated model that 
makes use of phasor analysis and mutual 
inductance to do these calculations. 
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 
ideal of how a transformer's going to 
behave without having to do more 
complicated analysis. 
So first of all, we need to identify the 
asusmptions that are used for the ideal 
transformer model, and 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 the mutual inductance, M, and the 
self inductance is L1 and L2 in the 
transformer. 
So, it's always possible to find this k, 
but some k's are valid in a physical 
sense and some k's are not. 
K's basically have to be similar between 
0 and 1, 0 meaning that there's no mutual 
inductance between the two coils, and 
they're completely independent of each 
other. 
And one, meaning that they're very 
tightly coupled. 
There's a limit to, a physical limitation 
to how tightly this coupling can be. 
So to do an example problem, consider 
that L1 is 4mH, L2 is 9mH 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 it's 36mH squared. 
Take the square root of that to get 6mH 
under 2mH. 
So that means that k is going to be equal 
to one third. 
So it is quite simple to calculate 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, that their 
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, to make use of it. 
But we clearly know that having this 
infinite in-, inductance is not possible. 
Finally, we assume that losses from coil 
[UNKNOWN] 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 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 over N1 is equal to v2 
over 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 of the coil, or how many reps 
of the coil in the primary and the 
secondary coils respectively. 
We also have the relationship that N1 
times i1 is equal to N2 times i2. 
And these equations come from the 
uh,Faraday's law of induction so what we 
have is we're making a circle like this. 
Then we have a changing B field through 
that surface. 
But as I wrap coils around again, and 
again, and again, that corresponds to 
additional surfaces. 
And so every time we have another wrap 
it's just another one of these surfaces. 
And so we take the number of these wraps 
and multiple it by the voltage that's 
induced, by the changing magnetic flux, 
in each of these sections. 
Which is why we get this, N [INAUDIBLE] 
behavior. 
But this equation is very simple to work 
with. 
They are quite easy to, to use for 
analysis. 
And so that's the most important thing 
that we're going to need to take out of 
this. 
So let's do an example. 
Here we're going to start by writing down 
our two equations. 
That v1 over N1 is equal to v2 over N2, 
and that N1 times i1 is equal to N2 times 
i2. 
We have a phasor voltage here, and I'm 
going to do all of my analysis for this 
problem in terms of RMS 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 kilo Volts rms here, 
leading to 120 Volts rms here. 
Now we don't know the number of coils 
because we don't have a number for either 
of these, but what we do know is the 
relationship between them. 
So frequently, with ideal transformers 
will show something like this, a ratio of 
the number of coils from one to the 
other. 
Because the actual number themselves 
don't really matter 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 kilo-volts, over here. 
And if I want to find the current, it's 
going through over here, we had 120 volt 
rms, 240 ohms. 
So that means that i2 is going to be 
equal to half an amp with phase angle of 
zero. 
On this side i1, we're going to have the 
same relationship using this i2 and i1, 
that means that i1 is going to be equal 
to 5mA, yeah, 5mA rms. 
And so what we can then do is we can find 
the power that's 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 it crosses 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 in the rms voltage 
together, so 60 watts. 
So this is basically 60 watts. 
If I calculate the power that's being 
consumed here we find that p is equal to 
minus 60 watts based on the reference 
structure 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. 
We find that that p is also equal to 60 
watts. 
So we'll call this p of the secondary. 
And we'll call this p of the primary. 
And this is p of the resistor. 
And that's because this source is going 
to be generating 60 watts as well. 
So we see 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 transformer is it's converting all 
of the power that's going in over here, 
pushing it to the other side to be used 
to do some type of work. 
So essentially what we're doing is we're 
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 volts 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 being sent by 
the power company, this transformer is a 
big power transformer that's owned by the 
power company that changes the voltage 
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. 
So what are the implications? 
Well transformers allow change from one 
voltage to another voltage. 
With a 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 you can still get the 
power to all of the homes. 
And 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 the United States and Canada 
without having to have a whole bunch of 
power stations all over the place. 
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 
for 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, linear-variable 
differential transformer. 
And see places that they can find useful 
applications. 
Until then.