Chapter One, the ideal transformer. transformers are commonly used in applications which require conversion of AC voltage from one voltage level to another. There are two broad categories of transformers, electronic Transformers which operate at a very low power and are used for consumer electronic equipment such as television set VCRs, CDs, personal computers, and many other devices to reduce the level of the voltage from 110 to 20, which is available at the AC mains to the desired level at which the device operates. The other category of transformer is power transformers which process thousands of watts of power. Power transformers are used in power stations transmission and distribution systems to raise and lower the level of the voltage To the desired levels. The basic principles of operation of both of these transport types of transformers are the same.

The transformer fundamentals we're going to talk about in the next few slides apply to both the electronic transformer and the power transformer. It was discovered quite some time ago that all objects are composed of extremely small building blocks known as atoms, and that these atoms are in turn composed of smaller components known as particles, protons, neutrons, and electrons. Whilst the majority of atoms have a combination of protons, neutrons and electrons, not all atoms have neutrons. Collectively, the protons and neutrons of an atom make up the nucleus or the center core of the Otto each electron has a negative charge of minus 1.602 times 10 to the negative 19. Cool ohms. And each proton in the nucleus has a positive charge of plus 1.602 times 10 to the minus 19.

Kudos, neutrons have no charge just the mass that's associated with it. Because of the opposite charge, there is some attraction force between the nucleus and the orbiting electrons. Electrons have relatively negligible mass compared to the mass of the nucleus. The mass of each proton and neutrons is 1840 times that the mass of an electron an atom becomes a positively charged ion when it loses an electron. And similarly an atom becomes negatively charged when it gains electron. I would now like to crudely demonstrate the magnitude and general situation of what's going on at the subatomic level and try to bring into reality.

What's going on in the electric circuit. Starting with one atom that has a proton, and one electron, as you can see, most of the atom is made up of free space, I'm going to assume that one electron is a free electron. That is, it's free to move from one end to another, concentrate on the electrons only, and start to zoom out and out and out and out to the point where you cannot collectively see all of the electrons and the circuit of at one time, so we just have to imagine what it is and know that The electric charge of one Kunal is made up of 6,250,000 with 15, zero electrons. That is 6.25 million, million, million, million electrons. That's a very, very, very, very, very large amount. Imagine if you would a million dollars in large denominational notes.

One person would have difficulty lifting that amount of money. Now multiply that by a million, that will require 1 million people to pick it up. Now multiply that by 1 million that would require 1 billion people to pick it up. Now multiply that by 1 million that require more people that are living on this planet. Now to pick it up. That is equivalent to in the order of magnitude of the number of electrons that are flowing in one amp.

Keeping this concept in mind, I would like to explore and compare the speed of electricity and the speed of electron flow. Looking at this example of an electric circuit made up of a battery and a resistor and a switch, if we close the switch, there seems to be an instantaneous movement of electrons at every point in the circuit. However, this is not quite instantaneous, but it is very close. It depends on a lot of parameters such as the wire size and the length of the wire and battery voltage and the temperature and a lot of things depend on exactly how fast the speed of electricity is. However, it is always very close to the speed of light, which is 186,000 miles per second. Now, if we open the switch the reaction speed is the same, the electricity will stop flowing almost instantaneously or close to the speed of light.

However, this is not the speed of electron flow, which is often referred to as drift velocity. And as I said, this is not the speed of an electron flow it is not an electron is not flowing at the speed of light. In fact, if we were to compare the speed of an electron flowing in a wire To that have a common garden snail a garden snail wood hands down beat the electron in a race because the garden snail moves at about half an inch per second, whereas an electron moves in the wire at about point 00 to eight inches per second. So, what's going on here? How can electricity seem to move at the speed of light? You know electron flow is slower than a garden snail.

In order to answer that question, I'm going to again zoom in to the subatomic level and consider what's going on in a wire. In this case in a two dimensional fashion three atoms by 10 atoms as you can see, there's a positive charge on the left hand side and a negative charge on the right hand side which would be indicative of connecting a battery with a positive end on the left and the negative end on the right. As you can see, electrons seem to be moving freely about. But if you watch for a while there seems to be a flow or a drift from right to left. Some electrons, however, seem to be moving left to right. But that is because they are bumping into each other.

And actually, they're not actually bumping into each other. They're coming close and being repelled into the opposite direction, which looks like they might be bumping into each other. But there is a general net flow from right to left. Now if we multiply that net effect by 6.25 million million million million, you would see that there is definitely one amp flowing in the circuit. But that still doesn't necessarily explain the instantaneous reaction. For that, let's move on.

Consider what is often referred to as Newton's cradle and ice. I'm sure you've all seen this at one time or another. A lot of people have been decorating their desk, you can see that you have a metal ball at one side. And as that ball swings in, it connects to the next one, which connects to the next one, which connects to the next one and pushes off the one on the end. And that reaction or that flow of action, if you would, seems to be taking place very quickly in fact, almost instantaneously, however, the ball is not quite moving from left all the way to right, but the action itself is moving from right to left. Now Let's take a look at and this again, I apologize for the crude animation.

But let's assume we're looking at a bunch of electrons in that wire. What is happening when you close the switch, you're actually pushing one electron into the bunch and almost instantaneously at the speed of light, an electron at the other end flows out. And what's happening is they are colliding as I said, but they are electrically charged. So you're actually getting a electrostatic wave going from right to left. And that is happening at the speed of light. The electron itself is not flowing down the line it is moving, but it's the actual action or the wave of electrostatics that's moving down the wire and not is what electric electricity is.

So I hope this Crude analysis helps you visualize what is actually going on in an electric circuit. Before diving into the inner workings of the transformer let's back up a couple of paces and review. electromagnetism first. Fundamentally transformers are governed by the modeling rules and observations involving electromagnetism, which involves the electromagnetic forces of physical interaction that occurs between electrically charged particles. These electromagnetic magnetic forces are modeled by making use of the existence of electromagnetic fields. A moving charged particle q one moving at a velocity v one will create a magnetic field which we will call B one and sometimes referred to as the beta rather than B but let's call it B one for now.

The magnetic field B one forms a closed loop or a ring around the moving particle. This magnetic field and ring actually has some direction, and we have arrows indicating that direction. But before going too much further in establishing that direction, I want to bring emphasis to a very subtle difference of reality and what is modeled here. In actual fact, there isn't a magnetic field as such, you can't see it, you can feel it. But what we're doing here is creating a model of what we call a magnetic field and we will just refer to it as a magnetic field from here on, but it isn't model and it what it is, it affects electrical charges or electricity. Because of the way we've modeled it.

Now we're going to continue to build this model. And this model will govern how things react inside a transformer later. But keep in mind, it is a model that we have created, but we're going to refer to it now as a magnetic field that forms a closed root loop with some direction. And this model of a magnetic field, as I said, has some direction. In order to find out the direction of that magnetic field. magnetic field we use what they call the right hand rule.

The field direction is identified by having using our right hand and having our thumb point in the direction of the charged velocity. The fingers of the right hand will point in the direction of the magnetic loop that is formed by the movement of the electric charge. So just repeating that, if you put your right hand and wrap your fingers around the velocity direction of the, of the charged particle and point your thumb in the direction of that philosophy, your fingers will point in the direction of the magnetic field. The field direction and strength depends on the type and the amount of charge. We've been looking at positive charges and if there are more than one charge or even many charges, the field strength will increase and is directly proportional to the amount of charges. Also, if the charges are negative, the field We'll be in the opposite direction to the motion of the negative charges and we're going to look at that when we start to talk about current.

However, for now, the field will be in the opposite direction if negative charges are moving, and just as in the case of positive charges, the more charges that are flowing, the stronger the field strength. This movement of electric charges can be extrapolated to the magnetic field that's produced by an electric current. Now, an electric current is movement of electrons. It is always oriented that is the magnetic field produced by the charged particles. The magnetic field is always oriented perpendicular to the direction of the flow. A simple method of showing this relationship again can be determined by the right hand Rule simply we can put the right hand around the wire with your thumb pointing in the direction of the current flowing and your fingers will point in the direction that is the magnetic field that is caused by the current flowing in the wire.

The magnetic field in circles this straight piece of wire carrying the current. And it should be noted that the magnetic flux here has no defined north or south pole at this particular time. However, it does have the lines of flux have a direction and that direction is indicated by the right hand rule. I've been using The terms flux and flux field rather loosely. So let's define what we're actually talking about here. When analyzing magnetic fields, we need to define them as being made up of flux lines, as you see here, and they have a direction and we assume the direction in this case is going from bottom left to top right.

And if we group them together, they form what is known as a magnetic flux field, symbolized by the Greek letter phi, and it's a measure of the quantity of magnetism. And it's measured in terms of Weber's Short Form W b. It takes into account the strength and the extent of the magnetic field. If we consider a flat surface area, such as indicated here, we'll call it a the area that the magnetic field As through then we can define the magnetic flux density as the amount of magnetic flux in an area that is perpendicular to the direction of the magnetic flux. Or in other words, the magnetic flux density is the amount of magnetic flux perpendicular to a flat surface area divided by that area. In the SI system, this flux density is measured in Tesla, but mostly we just talked about Weber's per square meter.

And in summary, we have this chart which is magnetic flux is indicated by the Greek letter phi and the units are Weber's and the magnetic flux density is B or beta, which is measured in terms of a Tesla or More simply Weber's per square meter to create a stronger magnetic field force and consequently more field flux. With the same amount of electric current we can wrap the wire into a coil shape, where the circling magnetic fields around the wire will join to create a larger field with defined magnetic north and south polarity. The amount of magnetic field force generated by a coil wire is proportional to the current through the wire multiplied by the number of turns or wraps of the wire in the coil. This field force is called Magnum motive force or mmf and it's measured in ampair turns. Now this is a formula we'll be returning to Quite often, and it's a very important formula and that is the magnetic flux is proportional to the current.

The right hand rule comes into play here again, the pole of a magnetic field of a solenoid can be determined again by the right hand rule. Imagine your right hand gripping the coil of the solenoid such the fingers are pointing in the same way as the current is flowing in the coil. Your thumb then points in the direction of the field that is pointing towards the north the way the arrows are coming out of the field. Since the magnetic field line is always coming out of the North Pole, therefore the thumb points towards the north pole here we are have wrapped our coil of wire around the block of iron, which I'll refer to as core material. The nature of this material is such that the magnetic lines of flux will want to travel through it rather than the air. In fact, it will boost the amount of flux that is produced by the current in the coil.

In this example, the magnetic flux will still vary as the current in the coil, but at a greater extent due to the core material. The magnitude of the magnetic flux will also depend on the number of terms of the coil. As I said, the magnitude of the magnetic flux will also depend on the material through which the flux will travel. If an iron core is substituted for an air core, in that given coil, the magnitude of the magnetic field is greater and it's greatly increased. All materials have a property defined as permeability, which is the measure of the ability of the material to support the formation of a magnetic field within itself. Hence, it is the degree of magnetization that the material obtains in response to an applied magnetic field.

Magnetic permeability is typically represented by the Greek letter mew. permeability also called magnetic permeability is constant. And it's a constant constant proportionality that exists between magnetic induction due to current flow and magnetic magnetic field intensity. The amount of flux produced this constant is equal to approximately 1.257 times 10 to the minus six Henry's per meter, in free space or a vacuum. In other materials it can be much different, often substantially greater than the free space value, which is symbolized by the Greek letter mew with a subscript zero. materials that cause the lines of flux to move further apart, resulting in a decrease in magnetic flux density compared to a vacuum are called diamagnetic materials.

Materials that concentrate magnetic flux by a factor of more than one but less than or equal to 10 called paramagnetic. And magnetic materials that concentrate the flux by a factor of more than 10 are called federal magnetic. The permeability factors of some substances change with rising or falling temperatures or with the intensity of the applied magnetic field. In engineering applications permeability is often often expressed in relative rather than absolute terms. If mewn not represents the permeability of free space, and mu represents the permeability of a substance in question also specified in Henry's per meter. Then the relative permeability is mew subscript R is given by mew times 7.958 times 10 to the fifth Power, although there is an ability to increase magnetic flux due to the magnetic material, the material is also said to have a reluctance or a resistance to the ability to support a magnetic field.

The reluctance of magnetic materials proportional to the mean length and the it's inversely proportional to the product of the cross sectional area and the permeability of the magnetic material where r is the reluctance magnetic resistance of the material and mew is the magnetic permeability coefficient. l, which is measured in meters is the length of the material and a is the cross sectional area the mean of the material in meter squared Therefore, ferromagnetic material is said to have a reluctance r to the flux. The flux current and reluctance are related by the equation and i is equal to phi times R and I of course is known as the mag motive force or mmf Magnum motive force or mmf of course, is measured in terms of M pair turns. It is sometimes helpful to draw an analogy between the electric circuit and the magnetic circuit where mmf is related to EMF current is related to flux Resistance is related to reluctance.

Looking back, at a coil of wire in air or a vacuum with current flowing in at the current produces an mmf or a field intensity that is directly proportional to the current which produces a magnetic field which measured over an area is known as magnetic flux density. If we were to plot this flux density as the current or field intensity increases, it would be linear or a straight line. This is known as a magnetization curve or a bH curve, even though it's not much of a curve, but the reason we call it a curve will be evident when we see the next few slides. Now, if that coil of wire wrapped around a ring of magnetic material, the current flowing in that coil also produces an mmf or field intensity that is greatly enhanced due to the properties of the material and will also increase with the current.

Again just as in a coil of wire in air, that current produces a magnetic field which measured over an area is known as flux density. However, this time if we were to plot this flux density, as the current or field intensity increases, it would not entirely be linear but would curve and start to flatten at some point depending on the type of material that it is inside the coil. This flattening of the curve is known As saturation, and is characteristic of what happens in Transformers that are wound on a magnetic material. Most transformers are designed to operate in the linear region of the bH curve, but at times they are pushed into the nonlinear regions of the curve which could produce and be problems that we'd have to deal with. And we'll see those a little bit later. Another quirk to confound our analysis of magnetic flux versus force is the phenomenon of magnetic hysteresis.

As a general term history says means a lag between the input and the output in a system upon the change of directions much the same as what you might have experienced in an old car whose steering is very, very sloppy. And as you steer you have to oversteer in order to To bring it back in the other direction and once going in one direction. In a magnetic system hysteresis is seen in effect in a ferromagnetic material that tends to stay magnetized after an applied field force has been removed if the force is and if the forces are in reverse direction. So let's see how that works. Let's use the same graph again, only extending the axis in the negative direction. First, we'll apply an increasing field force current through the coils of our electromagnet.

We should see the flux density increase go up into the right according to the normal magnetization curve. Next, we'll stop the current going through the coil of the electromagnet and see what happens to the flux leaving The first curve still on the graph. Due to the relativity of the material, we still have a magnetic flux with no applied force no current through the coil, our electromagnetic magnetic core is actual actually become a permanent magnet at this point. Now we will slowly apply the same amount of magnetic field force in the opposite direction. The flux diversity has now reached a point equivalent to what it was in the full positive value of field intensity, except it's in the negative direction or in the opposite direction. That's stop the current going through the coil again and see how much flux remains.

Once again, due to the nature of the relativity of the material. It will hold a magnetic field With no power applied to the coil, except this time, it's in the direction opposite to that of the last time when we stopped the current if we reapply power in the positive direction again, we should see the flux density reach its prior peak in the upper right hand corner of the graph again. This s shaped curve trace by the steps form what is called the history says curve of a ferromagnetic material for a given set of field and intensity extremes. When a ferromagnetic material approaches magnetic flux saturation, disproportionate levels of magnetic field force mmf are required to deliver equal increases in the magnetic field flux phi As you can see from the graph, a large increase of mmf is required to supply the needed increases in flux which result in a large increase in coil current.

Thus, coil current increases dramatically at the peaks in order to maintain the flux waveform that isn't distorted. This is why transformers are designed to operate in the linear region of the magnetic material. And that's why problems can occur when transformers are driven into saturation, especially in the cases of instrument Transformers that rely on linearity to accurately measure current and voltages. Our modern world would be impossible without electromagnetic induction, the phenomenon that underlies the operation of many devices including transformers. qualitatively here is how electromagnetic induction works. As the lines of flux cut across a coil or coils of wire, the number of magnetic lines of flux cutting the coil will be increasing or decreasing as it increases and decreases a voltage will be induced in the coils.

Quantitatively Faraday's law of electromagnetic induction relates the magnitude of an induced voltage to the rate of change of the magnetic flux linking a circuit. Faraday's law states that the magnitude of the inductor or induced voltage in a circuit is equal to the magnitude of the rate of change of the magnetic flux linking the circuit can be described by this formula or in terms of calculus, the magnitude of the dose voltage in a circuit is equal to n d phi by dt. The terms d phi by dt means the same thing as delta phi by delta t only in terms of calculus, the d phi by dt nd indicates an infantile massively small change in the flux and the time the course is instantaneous induced voltage, phi is a magnetic flux in Weber's d phi is the change of magnetic flux also in Weber's and dt is a change of time in seconds.

Notice the minus sign this becomes significant if the circuit is complete and current flows but for now, we are considering it an open circuit Let's close the loop now and we'll do that using a resistor. In order to limit the amount of current that will be flowing, because of Faraday's law a voltage will be induced in the coil. And since we have closed the loop of the circuit, the coil current will flow through that resistor. This brings us to another significant law of electromagnetic induction called lenses law. lenses law goes along with Faraday's law in that it states when an emf is generated by a change of magnetic flux. According to Faraday's law.

The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change will which produces it. Let's read that again and follow it step by step. an emf is generated by a change of magnetic flux. That would be induced in the voltage drop across the resistor R. And according to lenses law, the induced current will produce a magnetic field. Remember now that anytime current flows, there's a proportional amount of magnetic field induced and that it opposes the change which produces it as indicated by the green dashed line. In other words, the induced magnetic field is always opposite to the magnetic flux which produced it.

This is the reason for the negative sign in the Faraday's law equation. So, in order to check this out, we will apply the right hand rule, the fingers must point in the direction of the magnetic field right to left, the thumb will point in the direction of the current, which is up and over which flows through the resistor indicating the polarity of the voltage drop as it flows through the resistors or the resistor. This is all happening for one direction of the permanent magnet movement, which is a net increasing of magnetic flux. When the permanent magnet stops, the voltage will go to zero and when it moves in the other direction, the magnetic lines of flux linking the coil is decreasing causing a reversal of the flow of current as well as the polarity of the voltage drop across the resistor. Faraday's law and lenses law are very important when analyzing a transformer as you will see in the next series of slides.

If two coils of wire are brought into close proximity to each other, so that the magnetic field from one link to the other, a voltage will be generated in the second coil as long as there is a changing magnetic field. This is called mutual inductance. When changing voltage impressed upon one coil induces a voltage in the other. The key here is Faraday's law, the induced coil they experience a change in flux over a period of time in other words, a circuit of wire a coil must experience a change in the flux delta phi over time delta t. Faraday's law states that if a coil of wire experiences a change in magnetic flux, a voltage will be induced in that coil such that a voltage will be induced when the flux is increasing, then returns to zero when the flux ceases to change, current is steady, then, a voltage will be induced in the other direction.

When the flux is decreasing to zero then returns to zero when flux is at zero, and ceases to change. lenses law does not play a role in this situation as there is no current flowing in the second coil at this time, because the volt meter has a very high impedance and is measuring only the open circuit voltage So let's repeat that. A voltage will be induced when the flux is increasing, then returns to zero when the flux ceases to change that is, the current is steady, then a voltage will be induced in the other direction when the flux is decreasing to zero, then returns to zero when that flux is at zero and is ceasing to change. Before proceeding I want to emphasize an important concept to remember and that is, flux and flux density are related to the current such that the current is always proportional to the flux and the flux density.

That is, regardless of whether that current is AC or DC Which means that when flux and flux density are zero, the related current must be zero. When the flux and flux density are at a maximum or a positive maximum, the related current is at a positive maximum. And when the flux and flux density are at a minimum or at a minus maximum, the related current is at a minimum or a minus maximum. In other words, flux and flux density are always proportional to the current that's flowing that causes the flux or flux density. And you can also look at the inverse and that is if the flux or flux density is the thing that is driving the current then the current is always proportion To that flux and flux density, regardless of whether it's positive, negative or zero. If we wind two coils on a steel core, we can cause almost all of the flux to link both coils and we can further hypothesize the case in which 100% of the flux is linked by both coils.

In this ideal case, it is called an ideal transformer. Now, let's input a sinusoidal AC voltage on the red coil on the left. This is known as the primary winding. This AC voltage will cause a small current to flow in the primary coil. The amount of current flowing is limited by the reactance of the primary coil. In an ideal transformer this reactance is 100% inductance.

So, the current will lag the voltage by 90 degrees. This current is known as the magnetization current. Remember that anytime a current flows, it will produce a magnetic flux proportional to it which means it to is sinusoidal and in phase with the current and since we are dealing with an ideal transformer, all of that flux flows in the iron and Lynx both coils considering the primary coil, we will measure a voltage drop. This voltage drop will call the one across its terminals and it will be equal to the applied voltage V AC because it's directly connected to the applied voltage In coil one, the flux produced by the generator is related to the voltage V one by Faraday's law, which involves the changing flux times the number of turns in the primary coil. In the coil to the secondary coil, the voltage produced by that same flux by mutual inductance is also given by Faraday's law, which involves the same changing flux, but times the turns in the secondary coil.

That voltage is either larger or smaller than v one depending on and one and two the terms of the primary and secondary coils but it is in phase with the applied voltage the AC Mathematically, we can rewrite the two farraday law equations for both the primary and secondary coils keeping only the associated voltages and coil turns numbers on the right hand side of the equations. As you can see, both the right hand sides are equal to the same changing flux minus d phi by dt. Therefore, we can write minus d phi by dt is equal to v one divided by n one is equal to v two divided by n two, which means v one all over n one is equal to v two all over into or rewriting it we can see that it's v1 over v2 is equal to n one all over in two these two ratios v1 to v2 In one two and two are known as the turns ratio of the transformer and sometimes it is designated with the letter A.

Now, let's add some impedance to the secondary coil of the transformer. We know that the voltage drop across the impedance will be the two and that it is in phase with the applied voltage V AC. By virtue of ohms law, a current will flow in said to whose magnitude is given by v2 Oliver Zed two and depending on the impedance Zed to the current will either lead or lag or be in phase with the voltage, the two and the AC. The current in Zed two will produce a Magnum motive force m m f two which is equal to n two times I two and cause a flux in the core limited only by the reluctance of the core material are depending on the direction that the coil is wrapped in, and it may be in this direction we'll assume that for now, this magnet motive force mmf two links the primary coil resulting in an induced current itu which must equate to m m f two, but because there are n one turns in the primary coil mm f two must equal and one times I one, since the Magnum motive force is the same in both coils then in one times I one must equal into times I two.

Now remember lenses law which states the induced currents, magnetic fields oppose the change which produces it or the induced magnetic field acts to keep the magnetic flux in the loop constant. So, if the increase in current I too will try to increase the magnetic flux or mmf which I've indicated by the mmf, I to arrow then I one will counteract the attempt to increase the flux or mmf by producing a counter flux or mmf in the opposite direction. This is lenses law the net effect We'll be that the magnetic flux in the core will not change due to the current flowing in the secondary coil, but will remain the same that being the magnetization flux only. We can now express the turns ratio and one two and two in terms of the primary and secondary currents. That is the turns ratio is equal to i two all over I one.

That is of course, if we are loading the transformer in there is current flowing in the secondary side. In comparison, this is the inverse of the voltage ratio, which is equal to v1 over v2. Now, let's look at the power flow through the transformer. If the transformer is loaded, as is the case by adding Zed to To the secondary, we can measure the power flow into the primary by measuring the voltage V one and the current i one and multiplying them together to give us this equation, we can measure the power flow out of the secondary similarly by measuring v2 and I two and that will give us the power flow out of the secondary. Now, we can take the power flow into the transformer equation and converting I one and v one to I two and v2 using the turns ratio, we get this equation and clearly we can cancel out the turns numbers.

In other words, the N one in the numerator cancels out with and then one in the denominator. And similarly, the two n twos can be cancelled out as well, leaving us with v one v two I two, which is equal to the power out. This means that the power out is equal to the power in, which should be no big surprise because a transformer can't add power to the system. It's neither created nor destroyed, it just flows through the transformer. Let's take another look at this transformer set up with a load on the secondary of the transformer. And there's going to be current flowing in the primary as we have calculated and there is a voltage drop across the primary coil.

So the power supply sees an impedance because it's reflected due to the impedance on the secondary and we can calculate what that reflected impedance is by measuring the voltage drop across the primary coil and the current into that coil. And that would give us the reflected impedance. In other words, v one all over AI one is going to give us the reflected impedance, we can substitute for the one by using the turns ratio and the two, and we can substitute for I one using AI two and the turns ratio of the transformer to give us this equation, which is n one all over in two times v two, all over and two over n one times I two. Well, the two all over I two is just Zed two. And, and one all over in to all over into overhead. In one is n one over N two squared.

So Zed one is equal to Zed two, multiplied by the square of the turns ratio, which is the same as a squared times Zed to this ends chapter one.