Chapter Four, three phase power transformers. three phase power has become the standard for power transmission and distribution today, three phase power generation transmission and distribution is advantageous over single phase power for the following reasons. three phase power distribution requires lesser amounts of copper or aluminum for transferring the same amount of power as compared to single phase. The size of three phase motor generator sets is smaller than the single phase motor generator of the same rating. three phase motors are self starting as they can produce a rotating magnetic field. The single phase motor requires special starting windings as it produces only a pulsing magnetic field.

Field. In the single phase motors the power transferred in motors is a function of instantaneous current which is constantly varying. Hence, single phase motors are more prone to vibrations whereas in three phase motors the power transfer is much smoother and and much more uniform throughout the cycle answers, fewer vibrations to be concerned about. mainly for these reasons, it's found that the generation transmission and distribution of electric power is more economical in three phase systems and in a single phase system. It is interesting to note that the development of three phase systems evolved starting with the War of the currents era or the Battle of the currents in the late 1800s, George Washington and Thomas Edison became adversaries due to Edison's promotion of direct current or DC for electrical power distribution against alternating current AC, advocated by several European companies and Westinghouse Electric based in pitch Pittsburgh, Pennsylvania, which had acquired many of the patents by Nikola Tesla.

Nikola Tesla went on to develop three phase power systems. In order to understand and work with three phase systems, we work with voltages and current vectors that are derived from single phase values that are sinusoidal and all sinusoidal waveforms Of course, are repetitive. In an AC power system, the power source will supply a voltage that is sinusoidal is a sinusoidal wave of one particular frequency, usually 60 cycles per second. The voltage starts at zero volts peaks at plus a volts travels through zero and peaks negatively at negative eight volts, then returns to zero. It repeats itself 60 times a second. When mathematically modeling a sine wave such as V is equal to A sine omega t, it can be considered to be directly related to a vector of length a, revolving, in a circle with an angular velocity of omega.

V can be represented by a revolving vector where its magnitude is a. its angular velocity is omega. And the angle of the vector at a particular time is given by omega t. When considering two sine waves that are not in phase one wave is often said to be leading or lagging the other. This terminology makes sense in the revolving vectors pictured here, the blue vector is said to be leading the red vector or conversely the red vector is lagging the blue vector, these vectors could be current or voltage or one of each. If the blue vector is voltage and the red vector is current, then the current is said to be leading the voltage or conversely, the voltage is lagging the current. In a utility power grid omega is 60 cycles per second or CPS in some countries, such as Europe, this is usually 50 cycles per second for the power grid there but Here in North America, it's 60 cycles per second, all voltages and currents are vectors all are rotating at the same angular velocity omega.

And all may be mathematically dealt with but must follow the rules for vector analysis. Because these vectors have this added property of angular velocity or rotation, they are renamed and known as phasers. When considering three phase power generation, you can assume that it's made up of three single phase generator connected together on one terminal. The generated voltage vectors or phasers are 120 degrees apart and rotating counterclockwise at 60 cycles a second. The loads can be connected in various configurations shown here is a Wye connected configured load. However, the load can be any configuration.

Three phase Transformers can be thought of as three single phase Transformers each consisting of a primary winding linked magnetically to a secondary winding. They become a three phase unit by virtue of their excitation voltage and how they are connected to one another. They may also share the same core, but they may be considered individually. Here they are energized by three voltage phasers, they're out of phase by 120 degrees, but are joined at one point or neutral and the phasers are considered rotating in a color clockwise direction. transformer and distribution stations with very few Exceptions handle three phase power and the transformers are either three phase units or three single phase units in a bank. The exception to this is where two transformers are operated in open delta to supply three phase power.

This is not a common situation is usually used in emergency situations. However, it does exist. The connections of a three phase circuit may be divided into two basic classes a star connection or a Delta Connection. The star connection has a neutral point at the junction of the three individual phases and single phase loads may be fed from any one of the phases to neutral. Such a connection is used at some transformer station to supply three phase power two motors and a single phase power to say the light system Delta circuits are mostly used for straight transformation of three phase power of medium voltage range, where single phase loads do not have to be provided for. When three single phase Transformers of equal capacity are connected star to star or delta delta, the capacity of the bank is equal to three times the capacity of each of the individual transformers.

Note that when the windings of a transformer are referred to as being primary or secondary windings, the winding regardless of the voltage level, the one connected to the source of the power is always considered the primary winding. three phase electric power is a common method used for the generation transmission and distribution of alternating current electric power. This polyface system is the most powerful common method used by electrical grids worldwide for the transfer of power. The three phase system was invented and developed in North America by Nikolai Tesla in the late 1800s. And I encourage you to go into YouTube and find some of the history of this fellow. He's very interesting.

A lot of what he developed is still in use today. And he's highly underrated inventor, and I think you'll find his life fairly interesting. So I encourage you to go there and have a look at it. As a result of this development, transformation of power between voltage level has given rise to various transformer connections or configurations, the most common of which we're about to look at in the next series of slides. Star stars sometimes called wind Why connected Transformers can deliver two voltage levels face to face or face to neutral on both the primary and secondary. Also the terminal bushings can be gradient insulated and hence less expensive to manufacture.

With this type of connection, one terminal of the primary terminals is connected to the system lines buses etc. The other terminals are connected to together to form a primary neutral. Similarly on the secondary side one of the terminals on the low voltage system is connected to the lines and buses the other terminals are connected together to form a secondary neutral. These neutrals primary and secondary are not necessarily connected together. And sometimes they're grounded or not grounded and sometimes one is and the other isn't. And sometimes they both are, but if they both are then of course they're Connect considered connected together.

The high voltage h1 terminals are connected to the individual phase conductors and the high voltage h2 terminals are connected together to form a neutral. The low voltage axon terminals are connected to the individual conductors and the low voltage x two terminals are connected together to form the neutral. The transformer having its h1 terminal connected to the read phase is referred to as the red phase transformer. Likewise, the transformer with the h1 terminal connected to the white phase is considered the white phase transformer and the transformer can with its h1 terminal connected to the blue phase is referred to as the blue phase transformer. The vectors or phasers look like this the read phase both primary and secondary disregarding the magnitudes are in face the white face on both primary and secondary are also in phase and the blue phase both primary and secondary are in phase.

The H two and x two terminals form a neutral for both the primary and secondary respectively. They are not necessarily connected together the phasers are 120 degrees apart and rotating counterclockwise Transformers for Star connection and solidly grounded neutrals may be made with only one terminal brought out in a bushing and the winding insulation graded so that less insulation is required to use towards the grounding. This results in considerable savings in costs in when constructing a transformer. As I have said the star star connected transformer can deliver two voltage levels at both the primary and secondary that is face to neutral or face to face. Let's look at the primary side only. For example, the secondary side is exactly the same only at different voltage levels.

We can use the phase to neutral voltages as seen here or we can use the face to face voltage. The red to white voltage, for example is a difference between the red phase voltage and the white phase voltage or red minus white, which is the red phase vector plus the negative white phase vector which gives us the red to white phaser. Similarly, we can find the white to blue phase two phase voltage and the blue to red phase voltage. As we know in a balanced a star star system, the face to neutral voltages are 120 degrees apart the face to face voltages then as can be seen form an equal lateral triangle, the interior angles of which are equal and equal to 60 degrees. As can be seen the two phase two neutral voltages and the phase two phase voltage form two adjacent jacent 60 3090 degree triangles, each of which have the sides in the ratio of one, two and the square root of three.

Therefore, it can be seen that the magnitude of the phase two neutral voltage and the magnitude of the phase two phase voltage are in the ratio of two over two times the root of three or the phase two phase voltages equal root three times the phase two neutral voltages and this is magnitude only, of course As can be seen, the phasor of the red to white voltage leads the phasor of the red to neutral voltage by 30 degrees. Or generally speaking, the phase two phase values are vectors lead the face to neutral vectors or phasers by 30 degrees. Although I've been demonstrating this using the primary voltages of the three phase transformer, generally speaking, this is usually referred to as the phase relationship in a three phase system. I now want to take a closer look at this phase relationship in regard to the primary and secondary of this transformer.

The primary and secondary phase two phase voltage relationship are both the same with respect to their respective face to neutral voltages. The primary phase two phase voltage leads the primary face to neutral voltage by 30 degrees and the magnitude is root three bigger similarity, the secondary phase two phase leaves the secondary face in neutral by 30 degrees and the magnitude is root three bigger. the turns ratio of this transformer is given by the face to neutral primary over the face to neutral secondary voltages. Because the phase relationship the line to neutral voltages can be written in terms of the line to line voltage, which as you can see, can also describe the turns ratio When analyzing these transformer connections for the most part, if if not all of the time we are considering this system to be balanced that is, we have balanced voltages that are equal in magnitude and 120 degrees apart.

So, if we wanted to develop the per phase analysis of a transformer that was why why connected the per phase equivalent of this transformer would look like this. We would be using the line to neutral voltages both for the primary and secondary and if we were to convert the secondary line to neutral voltage to the primary line to neutral voltage, we would have to multiply by the turns ratio which is done needed as a here. And if there was current flowing in this single phase equivalent, then the ratios of the line currents would be the inverse of the turns ratio. In other words, if we wanted to calculate the secondary line current, we'd have to multiply by the turns ratio. And it goes without saying that there are two voltage levels to be looking at a primary and a secondary voltage levels. And we can describe them in terms of the line to neutral voltages.

And if we adopted these two voltages as the base values when considering a per unit equivalent circuit, then our per unit equivalent circuit would look like this. Which is says was just essentially is just two wires. neglecting a course the turns ratio of the transformer because we're dealing in per unit values here now, we have used the base voltages and we can either use line to neutral base voltages or line to line base voltages. However, the per unit magnitudes of the voltages are the same, because we're dealing in per unit values now, as are the per unit values for the current. In this slide, I made a quick reference to the per unit equivalent circuit. And in each one of the configurations as we go through them, I will be referring to a per unit equivalent circuit.

If this seems very mysterious to you, I'd encourage you to have a look at the course that we offer on per unit analysis. Because per unit analysis is a whole whole other course in itself. However, I will make this reference if you don't have an interest in per unit values and you can disregard the last little portion referred to. However, as I said, I would encourage you to have a look at the course on per unit analysis and that will remove the mystery of the per unit equivalent circuit. harmonic voltages and currents in an electrical power system are the result of nonlinear electric loads. harmonic frequencies in the power grid are a frequent cause of power quality problems.

These in turn could result in an increased heating of the equipment and conductors or the misfiring and variable speed drives and torque pulsation motors. When a nonlinear loads such as a rectifier is connected to the system And it draws a current that is not necessarily sinusoidal the current waveform can become quite complex depending on the type of load and its intersection or interaction that is with other components in the system. All the way forms no matter how complex as long as they are repetitive can be broken down into the sum of the fundamental sinusoidal wave plus multiples or harmonics of the fundamental frequency and order to analyze this we have a system of doing that called Fourier analysis. However, just an understanding that complex waveforms if repetitive, are made up of the fundamental frequency plus integral harmonics or multiples of the fundamental component harmonics. One of the major effects of public Power System harmonics is the unwanted increase in system third harmonic currents.

Third harmonic currents are often the result of nonlinear transformation caused by the history effect of the iron core. This can cause a sharp increase in the zero sequence current and therefore increase the current in the neutral conductor. This effect can require a special consideration in the design of the electric system. We are about to look at one of those solutions but first let's take a closer look at that third harmonic generated by the nonlinear transformation. If the fundamental frequency is a sine omega t, then the third harmonic is a frequency that is three times that of the fundamental frequency and can be written signed three times omega t. So in a three phase system if we place the read phase, fundamental frequency as our reference Phase are at zero degrees, then the sinusoidal wave formula is given by a is equal to A sine omega t plus zero degrees.

And I've plotted it on the bottom of the slide, the blue phase fundamental frequency is leading the red phase by 120 degrees, then the blue phase sinusoidal wave formula is given by b is equal to b sine omega t plus 120 degrees. And the white phase formula frequency is lagging the red phase by 120 degrees making the sinusoidal wave form formula w is equal to W sine omega t minus 120 degrees. Now, keep in mind, we're talking about a balanced system here. So the magnitudes of the A B and W phasers are equal, but I'm going to leave them as a B and W for now. And the following third harmonic magnitudes a third harmonic, B, third harmonic and W third harmonic are also equal in magnitude. Now the read phase third harmonic frequency is given by a magnitude a three h times sine three times omega t plus zero degrees and if we take the three times inside the frequency brackets, it's going to read a three h sine three omega t plus three times zero degrees which is equal to a three h sine three omega t. Looking at the blue phase, third harmonic, it can be written B three h sine three times omega t plus 120 degrees.

Taking the three inside the bracket, we get b, three h sine three omega t plus 360. And 360 degrees is the same as zero degrees. So the result will be B three h sine three omega t. The white phase is given by W three h sine three times omega t minus 120 degrees. Bringing the three times inside the frequency bracket gives us w three h sine three omega t minus 360. Well, three 60 is still zero degrees. So if you subtract zero degrees, you're left with w three h sine three omega t, and all three of the A B and C third harmonics you can see are equal and in face.

Now, remember what I said that the magnitudes of the third harmonics are all equal to so that the third harmonic frequencies are all in phase and you can see them plotted one on top of the other in the time domain graph at the bottom of the page. A solution to this problem is to provide a path for these third harmonics. This is accomplished with a tertiary winding connected in a way that allows the circulating currents to flow the provision of this extra winding increases the cost of a transformer considerably, but it may be used to supply power to an extra load such as station service. This consists of a third winding on each of the transformer legs, they are connected to form a Delta Connection. The fundamental voltage vectors look like this and thus the main transformation may be 230 to 110 with a with a circuit of 13 kV being fed from the tertiary winding for local use to tie in the synchronous condensers or supply local power.

The third harmonic currents however, are rotating at three times 60 cycles a second are in phase with each other and will circulate in the Delta as shown here. Now, let's look at a delta delta configuration. In this configuration, one terminal of the primary terminals are connected to the system lines or buses, the other terminals are connected to the adjacent primary terminal. Similarly, one terminal of the secondary terminals are connected to the system lines or buses and the other terminals are connected to the adjacent secondary terminal. That is to say, the h1 terminals are connected to individual face conductors and the high voltage h two terminals are connected to the adjacent primary h one terminal. Similarly, the x one terminal of the secondaries are connected to the individual phase conductors while the low voltage acts to turn nodes are connected to the adjacent secondary terminal x one.

The transformer having its h1 terminal connected to the RVs or the read phase is referred to as the read phase transformer and likewise a transformer with the h1 terminal connected to the white phase. line or bus is and is green in our diagram of course, is referred to as the white phase transformer, and the transformer with its h1 terminal connected to the blue phase is referred to as the blue phase transformer. The vectors or phasers look like this both primary and secondary will put them both up at the same time because they're connected the same way or similar way. The red to white phasers face to face are showing here. The white to blue face is showing like this, and the blue to red phasor. Both the primary and secondary are showing like this.

All of these phasers, of course, are rotating counterclockwise. You'll notice that there is no neutral connection in a Delta Delta Connection. no difficulty due to third harmonic circulating currents is encountered since the Delta connected windings provide a path for them. The third harmonic current vectors of each phase are in phase and will circulate in the Delta as shown. The Delta Connection is used for transformation in a medium voltage ranges where a neutral point is not necessary for load purposes. Transformers with a Delta two Delta operation must have the entire world winding and all the bushings insulated for the full line to line voltage of the circuit to which they are connected.

Therefore construction is more expensive. Start to SAR or delta delta connections do not introduce any phase shift into the circuit from primary to secondary. Thus, all related primary voltages will be in phase with the related secondary voltages. I'm going to move everything slightly here and shrink it down so that I can fit some equations on the slide while looking at the diagram at the same time, and I've rotated the phasers slightly which I'm allowed to do as long as I maintain the relativity. For a delta delta connected transformer the turns ratio is given by the primary phase two phase voltage over the secondary Phase two phase voltage. That is because the phase two phase voltages bridge the entire coil of each of the transformer legs.

In coming up with a per phase equivalent circuit for this type of transformer The first step in the procedure for a PR phase analysis is to convert all Delta loads and sources in the transformer is a source in this case to their equivalent y connections. The neutral of course, is hypothetical because there is no neutral in this connection. However, it can be calculated by referring to the three phase phase relationships because we're talking about a balanced condition here. So, again we can use the phase relationships to come up with a turns ratio using the hypothetical face to neutral voltages, then, right That phase two phase turns ratio in terms of a line to neutral voltage. We can now construct the per phase equivalent circuit using the hypothetical phase to neutral voltages. The phase two neutral primary voltage can be calculated from the face to neutral secondary voltage by multiplying by the turns ratio or if current is flowing in this single phase transformer, the secondary line current can be found by multiplying the primary line current by the turns ratio.

There are two voltage levels that be used for per unit analysis and the voltage basis that are required for that per unit analysis. And this is what the per unit equivalent circuit would look like for a delta delta connected transformer. And of course, there are two zones, voltage zones for that per unit equivalent circuit. And we'd have to use whichever v base we use to either the line to neutral or line to line. However, the per unit voltages on in either zone are equal in magnitude as are the currents in each of the the base souls. For a star Delta connected to transformer, we have already gone through the exercise of seeing how the high side of a star connected transformer is as well as we've gone through the exercise of connecting the low side in adult connection.

However I'm going to go through the process now because this is the first time we're looking at connecting why across to a delta. One terminal of the primary terminals are connected to the system lines or buses and the other terminals are connected together to form a primary neutral. The in the secondary of the transformer and one terminal of the secondary terminals are connected to the system lines buses etc. The other terminals are connected to the adjacent secondary terminals. The high voltage h2 terminals are connected together to form a neutral and the h1 terminals are connected to the individual phase conductors. The transformer heavy The h1 terminal connected to the red phase boss is referred to as the red phase transformer.

Likewise the transformer with the h1 terminal connected to the white phase. It is referred to as the white phase transformer and the transformer with his h1 terminal connected to the blue phase bus or line is referred to as the blue phase transformer. The x one terminal of the transformer are connected to the individual conductors and the low voltage x two terminals are connected to the adjacent secondary x one terminal. before going any further, I want to explain the nomenclature that I'm going to be using in describing the voltages of the transformer primary and secondary as we go from You've already seen that I'm separating out things in color to make things a little bit easier to understand. When I'm talking about the primary side voltages I'm going to use uppercase letters or capital letters are W and B, for red, white, and blue. And in for if I'm describing the voltages in the secondary side of the transformer I'm going to use the lowercase letters are W and B.

If we're talking about face to face values, I will use both letters. In other words, if the voltage is red to white, I will say our W and if it's white to blue, I'll say WB and cetera. I will use color and they should make sense but don't necessarily. The thing that's most important is the letters themselves. The R in The w not necessarily the color, although I will try to maintain some sense in regard to the color. But as you can see, it gets a little bit confusing when we're talking face to face values.

Because we're talking about two colors. However, I think it'll be pretty obvious as we go along. And if I'm talking about face to face values in the secondary, I'm going to be adopting the same standard, except that it's going to be lowercase letters. Now when I talk about the face to neutral voltages, they are and they should always be two designation or two points because that's what describes a potential difference. So in the case of the primary side, I'm going to be using capital letters again, red to neutral, white to neutral blue to neutral in other words are in WNB n and in the in the secondary I'll be using lowercase letters are n, w, n and b. And now, I'm going to make one exception and that is because we are going to be trying to get everything condensed onto one slide in cases, I will be taking the liberty of dropping the end or the neutral.

So a lot of times I will be referring to the phase to neutral value with just the upper or lower case letter. Now in a wide Delta transformer Of course there is no neutral, but we will be talking at times about a hypothetical neutral, in which case I will just be referring to phase two neutral value with just the one letter. This could lead to some countries fusion as you go forward, but I'm going to make sure that things are pretty clear as we go along. And even though I should be having the end in there, you will see it's clearer and easier to understand if I do leave it out during the analysis. So the connections are fairly straightforward because this is the second time we've gone through them. The thing that is different though is the phasor relationships between the primary and the secondary.

Comparing the phase two phase voltages, starting with the red to white phases, both the primary and secondary. Looking at the primary side, the phase relationship tells us that the phase two phase voltages root three times the phase two neutral voltage and the phase two phase voltage leads the Phase two neutral voltage by 30 degrees. And because the phase two neutral primary is magnetically linked to the phase two phase secondary, let me repeat that, because the phase two neutral primary is magnetically linked on each transformer to the phase two phase secondary. As you can see from the phasor diagrams, the phase two phase primary leads the phase two phase secondary by 30 degrees and the turns ratio is given by the magnitude of the primary phase to neutral voltage over the magnitude of the secondary phase to phase voltage which means the secondary red to white voltage is equal to the primary Read to neutral voltage divided by the turns ratio.

Or mathematically, you can also say that the primary red to neutral voltage is equal to the turns ratio times the secondary red to white voltage. We're talking about magnitudes here of course, and it relates the primary and the secondary phase to phase voltage magnitudes. Since we already know that the secondaries lags the primary by 30 degrees then we can calculate the primary and secondary phasor relationships for a y to delta transformer. Let's look at the currents now in this transformer and in order to look at the current Of course, we have to add a load and I've chosen that delta load, I could have just as easily chosen a y load, it's not going to make that much difference to the flow of the secondary or the primary currents for that matter. And I've chosen a resistive load because I don't want any phase shifting going on due to the load itself.

Certainly, it will add to the shifting if there is any. But that isn't what we're looking at right now. So let's just assume that the load is purely resistive. I've also drawn in the voltage phasers in miniature near the title there, just so we can keep track of them. Now, I've adopted the same standard pretty much that I did with the voltages. But keep in mind now we're looking at currents.

So all the currents that are associated with what we're going to look at On the primary side, I'm gonna use capital letters or upper uppercase letters. And on the secondary side, I will be using the lowercase letters. The line Currents, I'm going to title them as I blind, our primary, eyeline white primary and eyeline blue primary. You'll notice that I've capitalized the R prime and the w prime and the B prime to indicate that that is the primary side that we're talking about. And I'm going to do a similar thing on the secondary. The lines coming off of the transformer, I'm going to call a line with lowercase r secondary and a lowercase w secondary and I'll lowercase blue secondary The transformer primary winding current I'm going to refer to with a capital R and the current coming in the line is going to flow down and into the spot of that red transformer.

And all of the current flowing through the red face coil or winding of the transformer is going to be equal to eyeline are primed in a wide Delta connected transformer. Similarly, the current flowing in the winding of the primary white transformer is going to flow into the spot on that transformer from the eyeline w prime. And the same thing is it's going to be equal to the line current. And similarly, the blue phase is going to Be The boy is going to be designated B. And it's going to be equal to the line current on the blue face line coming in. That is because we have a Wye connected primary.

Now on the secondary winding of the transformer, I'm going to call the current flowing in that winding as lowercase r. And in the case of a lowercase r, and the current flowing in this transformer because it's flow, the current is flowing into a spot on the primary, it's going to flow out of the spot on the secondary. And similarly, the white phase secondary current coil is going to flow out of the spot and the blue phase current coil is going to flow out of a spot. Now we would like to find Don't what the current is flowing in the eyeline are secondary or our sec, W sec and B sack and we're going to have a look at that right now. This transformer is connected to a balanced system. And let's assume that it's connected to a bus. And it's gonna have the red white and blue phase currents coming in from the primary side.

And it's a balanced system. So those currents coming in will be the equal in magnitude and 120 degrees apart. And the current that's flowing in the red phase transformer is going to flow into the spot primary. And it's going to be I'm going to assume that it's straight up and down or it's pointing straight up. It is magnetically With the secondary, so the current flowing out of the spot on the secondary is going to be in phase with a current flowing into the spot on the primary. And I've indicated the phasers for both currents in the primary and the secondary windings of the transformer.

Similarly, the white phase current is 120 degrees a lagging the red phase and that's going to be flowing into the spot on the white phase transformer primary. And because it is also magnetically linked with the secondary, the current will be flowing out of the spot on the secondary. So it will be in phase with the current flowing in the primary side. However, the x one terminal of the red light phase is connected to the x two terminal of the red face. So the head of the arrow of the white phase will be connected to the tunnel. Have the arrow of the red face.

Similarly, the blue phase current is going to be 120 degrees lagging the white phase current because the system again is balanced. So the current flowing into the spot on the second on the primary side of the transformer is going to be 120 degrees from the white phase and I've indicated it there on the slide. Now, the secondary winding of the Transformers magnetically linked to the primary. So the current flowing in the secondary it has to be flowing out of the spot but it has to be in phase with the current flowing in the primary. But it's x one terminal is connected to the white phase, x two terminal and it's x two terminal that is of the blue phases connected to the x one terminal The red face so the tail of the blue phase arrow will be on the head of the red face, and the head of the blue face will be on the tail of the white face as indicated on the slide.

I would now like to figure out what the currents are that are flowing in the line to the load. And I'm going to start with the red phase line secondary current. And if you go back to the transformer, you'll see it's connected into places. It's connected to the x one terminal of the red transformer and is connected to the x two terminal of the blue phase transformer. What that means is the current that's flowing in the line is going to be made up of the red phase current. But the current we have to subtract the current is flowing into the blue phase because As it's flowing out of the line, so we have to subtract the blue phase current.

And we know what the plus blue phase current is, the minus blue phase current is just 180 degrees from that. So these are the two phasers that make up dough line current. So all we need to do is connect them and that will be the current that's flowing in the secondary of the line of the red face. The Delta connected secondary currents form a triangle of three equal sides which is an equilateral triangle and that contained angles are 60 degrees. So in trying to come up with the angle between the red phase and the minus blue phase if we move them As blue phase over, it forms a straight line with a plus blue phase. And if we subtract 60 degrees from 180 degrees, which the straight line forms, then we're left with 120 degrees.

And that is that contained angle between the red phase and the minus blue phase current. Now the triangle formed by the red and minus blue phase currents and the eye line our secondary current. It is in a saucer least triangle. So the two sides that are equal, we'll share the 60 degrees that are left over when you subtract 120 from 180. And we're left with 30 degrees. So that's just a quick breeze of trigonometry there.

Just to prove that we have a triangle with two contained angles of 30 degrees. And one of 120 degrees. If we move I line our secondary up to compare it to the primary side of the transformer. We know that the red phase primary current flowing in the primary side of the transformer is equal to the wind current coming in. So we can see that the eye line our secondary will lie in the eyeline our primary by 30 degrees or in generally speaking, we'll say that the secondary lags the primary by 30 degrees because we've just done the investigation for just one line me. I line our are secondary.

However, we can go through the same exercise from The white phase and blue phase. And we'll come up with the same thing that the line currents have the secondary lag the line currents of the primary by 30 degrees. And that is not necessarily too difficult to think about because we already saw that the line two line voltages, lag the primary line to line voltages or phase two phase voltages by 30 degrees and we just added a resistance load so the current should be lagging by 30 degrees as well. However, what we haven't discovered yet we're going to go on to see that in the next few slides is the magnitudes of these currents that are flowing in the primary and the secondary. Looking closer at this triangle, which is made up of The red phase current and the minus blue phase current in the eye line of the red face secondary.

We'll call them our minus B and I line it is an N A Sausalito triangle. And we've seen this exercise before, the triangle is made up of 260 3090 triangles, which means the length of the arrow eyeline are secondary, or the magnitude eyeline our secondary is made up of the ratios of the red magnitude, root three over two plus the minus b magnitude root three all over two. And because we're dealing with just magnitudes, we can say that the Read magnitude is equal to the minus blue magnitude. So we can replace the minus b magnitude in our equation with the, our magnitude. rewriting the equation, we get root three times two times the R magnitude all over two. And the twos in the numerator and denominator cancel out, leaving us with eyeline our secondary is equal to root three times our magnitude.

Now I'm just going to switch back a little bit for before progressing. And I'm going to recall the voltages of this transformer we did this in a couple of slides before the turns ratio of the voltage of the transformer is given. By the voltages of the read phase magnitude, the voltage across the coil or the windings of the read phase over the voltage drop across the windings of the red to white secondary terminals because red to white is connected delta. So, the turns ratio when we discovered this before is given by the magnitude are all over the magnitude are W and the currents then if you look at our diagram, the currents in the secondary windings over the currents in the primary windings the magnitudes also for the turns ratio of the transformer and just to recall the theory I'm going to look back at the slide that that that discovered this, that the turns ratio can be described in either the voltages or the currents, but they are the inverse of each other.

So going back to our equations, and looking at the current formula that we just developed, the we can rewrite the, or sorry, we can replace the uppercase r in the primary because all of the current flowing in the eyeline primary flows through the windings of the read phase transformer. So I'm going to replace the capital R with the magnitude eyeline r prime and rewriting the equation so that I have the lowercase r magnitude on the left hand side of that equation because now I Want to go across to the other side of this slide and replace the lowercase r magnitude in the last formula that we developed. And I'm going to replace that with a eyeline are primed, which gives us the relationship between the line current in the secondary on the red phase and the line current in the primary on the red face. And I'm going to just do something to get rid of the two terms root three times a, just to make it a little bit more concise.

I'm going to say let a primed equal A times the root three. And I can rewrite that equation now and instead of just looking at the red face, because we could analyze each one of the red, white and blue phases and come up with the same thing So, we can say that the magnitude of the secondary lying current is equal to A primed times, eyeline primary. And these are magnitudes. But we also in a previous slide discovered the phase angle or the phase shift between the primary and secondary. So, I can rewrite this equation placing in the phase shift, and I can drop the magnitude signs and say these are the actual phasor relationships now, where I line secondary current is equal to A prime times eyeline primary current minus 30 degrees. I'm switching back now to the voltage diagrams for this transformer because I want to go forward Develop a per phase and a per unit equivalent circuit for this transformer.

The phase relationship on the primary side we've already discovered that the line two line primary voltage is equal to root three times the line to neutral voltage on the secondary and that the primary line to line voltage will lead the secondary line to line voltage by 30 degrees, giving us the equation there and I can rewrite that equation, just taking the line to neutral voltage primary on the left hand side and putting it all equal to the voltage line to line all over root three at 30 degrees. Now in order to find out what the turns ratio is, I could take the line to neutral primary voltage and put it over the line to line voltage on the secondary. And that would give us the turns ratio of each of these legs are these transformers. And I'm going to rewrite that equation such that the voltage line to neutral on the primary is equal to a times the voltage line to line on the secondary.

Now, as I said, I want to develop a per phase equivalent circuit. So in starting to do a per phase analysis, we have to work with why connected sources and loads so we have to convert all of our delta sources and loads to a Wye connection. So I'm going to develop a hypothetical neutral on the secondary side because there isn't one, but because it is a balanced system, I can develop a hypothetical not neutral for the secondary. And there is a phase relationship associated with the secondary and this hypothetical neutral such that the line to line voltage of the secondary is going to be root three times of line to neutral secondary and it is going to be leading the line to neutral voltage by 30 degrees. This is a phase relationship if there is a three phase system with a neutral, which we've hypothetically developed now, I rewriting the equation, bringing the voltage line to neutral over to the left hand side.

That's all I've just done here, and bringing all the rest of the terms to the right hand side of the equation, giving us this final equation here. Now in developing the phase per phase equivalent circuit, I'm going to take these two equations, and I'm going to divide one by the other. And I can do that mathematically as long as I do the same with the left hand side of the equation as I do with the right hand side of the equation. So if I bring the left hand side of the equation down, I've got the voltage line to neutral all over the voltage line to neutral primary to secondary. Now I have to bring down the right hand sides of the equation and do the same thing with them. Voltage line to line all over the fraction that you see there voltage line to line secondary, and that is all over root three times 30 degrees.

Well, I can make this a little bit more simply simple by bringing the denominator, the denominator up onto the numerator, which you can do math quickly and that will get rid of at least one of the division signs and make the equation a little bit easier to to work with. Now you can see that we have a voltage line to line secondary in both the private though new numerator and denominator so they can cancel out. And again, I'm going to replace the a root three with a primed and that gives us the equation, the primary voltage line to neutral over the secondary line to neutral, a prime at 30 degrees. And I'm going to rewrite that equation right away because I want to use it in this other fashion, the voltage primary line to neutral is equal to A prime the voltage line to neutral secondary at 30 degrees.

And that tells us again that the secondary lags the primary by 30 degrees, which we already knew. Now, I want to bring down a couple of terms and equate them to the line to neutral voltage, but I want to do it using the line to line values. And I can do it by this fraction. And you might say well, where did We get the numerator and denominator from for that fraction. Well, these two equations tell us that the primary line to neutral is equal to the numerator and the voltage line to neutral is the denominator, we've already developed that. So I just brought them down and put them one over the other, which is equivalent to the voltage line to neutral over the voltage line to neutral primary to secondary.

And I'm just going to make that fraction simpler by moving the denominators around and I'm left with this equation which has a common element in both the numerator and denominator now, which we can cancel out, and that leaves us with the primary line to line voltage over the secondary line to line voltage is equal to the voltage Lying to neutral, primary all over the voltage line to neutral secondary. And both of those are equal to A prying at 30 degrees. And I can rewrite the equation again. And this time it's the voltage line to line primary is equal to A prime and the voltage line to line secondary at 30 degrees. Now you might have been wondering why in our black box that I dropped down and say the secondary lags of primary by 30 degrees. And you might be wondering why I left off the voltages and currents or whether we're talking line to line or line to neutral because it doesn't matter whether we're talking about the voltages, line to line or the voltages line to fictitious neutral light into neutral, or the currents in the primary secondary, they all like the primary by 30 degrees.

If it's a y Delta Connection, we now have enough equations to establish a per phase equivalent circuit. And we use the formulas for voltage line to neutral primary line to neutral secondary as our per phase because we're gonna we're talking about a balanced system here. So, we can look at one face, do the analysis and extrapolate it back to a three phase quantity. So our single phase equivalent circuit would look like what we have there on the left hand side of the slide and the primary voltage is lying to neutral voltage and equate it's equal to A primed times a line to neutral secondary voltage, but it's leaving it by 30 degrees. So our equivalent circuit tells us that if we wanted to calculate the line to neutral primary voltage using the line to neutral secondary voltage, we have two operators that have to operate on that voltage.

One is a phase shift of 30 degrees, we'd have to add 30 degrees to it. And the second operator is the turns ratio that we'd have to operate on the line to neutral secondary voltage. So we'd have to multiply by a prime. And we have established what a primed is it's the actual turns ratio of the transformer times the root of three. Now we've already established this equation for the Secondary line current in terms of the primary line current and these are phasor values. So if we wanted to look at the primary line current in terms of the secondary line current, we'd have to divide by the operator and we'd have to add 30 degrees.

So our operators would look like this. In going from the primary side to the secondary side, we would have to multiply the primary voltage by the turns ratio a prime. However, the phase shift doesn't change, we still going from the primary to the secondary, we would have to subtract 30 degrees because the operator has not changed. In other words, if we are going from the secondary to the primary, we would have to add 30 degrees. So you remember our black box says the voltages and currents are shifted the same by 30 degrees. So there's no difference between the shifting of the phase of the voltage or the current.

However, the turns ratio is inversed. We can now move on to the building of our per unit equivalent circuit. And in doing so, we have to be aware of the two voltage levels that are that are available on the transformer. And we establish the voltage basis of the base one and the base two for our voltage zones. And the per unit equivalent circuit would look like this. We have to still carry the 30 degree phase shift with operator with our per unit equivalent circuit The per unit equivalent circuit nicely gets rid of the turns ratio for us.

However, it does not in a three phase system, get rid of the 30 degree phase shift. So we have to be aware of the fact that we're talking about a Y delta transformer there is a 30 degree phase shift from one side of the transformer to the other regardless if we're talking about per unit values, or actual value, so we have to put the operator in there to remind us we got to do the phase shift. We still have the two voltage levels, which can either be a V base that is based on the line to line voltages, or the line to neutral voltages. It doesn't matter which one we pick as long as we use it consistently for establishing the per unit values and then the actual values going the other direction. The What that means is the magnitudes of the voltages in per unit values is the same to pet doesn't matter when you on the primary secondary, you don't have to worry about the turns ratio, the magnitude of the currents sorry, the voltages are the same as well as the magnitudes of the current are the same in a per unit equivalent circuit.

However, we have to be aware of that 30 degree phase shift for both the currents and the voltages. So if we're going from the secondary to the primary, we have to add 30 degrees. For a delta to Wye transformer, one terminal of the primary terminals are connected to the system lines or buses. The other terminals of the high side are connected to the adjacent primary terminal. On the secondary one terminal of the secondaries are connected to the low voltage system lines or bus as well. The other terminals are connected together to form a neutral which means that The h1 terminals are connected to the individual phase conductors, the H, two terminals are connected to the adjacent h one terminal.

On the secondary side the x one terminals of the secondary are connected to the individual phase conductors. While the low voltage x two terminals are connected together to form a neutral on the secondary side. The phasers would look like this. You have the h1, h2 forming the red to white voltage, you have the h1 to h2, forming the white to blue voltage and you have the h1 to h2 forming the blue to red voltage. You'll notice that the x two terminals are jumpered to the actual one terminal of the adjacent phase so that you form that recognizable Delta Connection. On the secondary side, you have the red, white and blue phases.

The x one, two x two are showing there. The x one, two x two on the white phase are showing there with the x two connected have the white face connected to the x two of the red phase and the x one of the blue phase connected to the x two of the white phase and the red face which together because they are all connected together, the two terminals form a neutral on the secondary side. Again, it's an important thing to notice that the primary Red White voltage is in phase with a secondary red to neutral voltage, as well as the primary white to blue voltage is in phase with a secondary white to neutral voltage and the blue to red voltage is in phase with a secondary blue to neutral voltage. Or in other words of saying this is that the primary phase two phase voltages are in phase with the secondary phase to neutral voltages.

I'm now going to compare the phase two phase voltages are the primary and secondary of this transformer. And I'm only going to look at the red phase transformer because the the white and the blue are virtually the same because of the phase relationship, and I'm looking only at the secondary side of the transformer, we know that the phase two phase voltage will lead the phase to neutral voltage by 30 degrees. And we know that the red phase secondary is magnetically linked to the red to white phase primary of the red phase transformer. So that means Is that the phase two phase secondary voltage must lead the phase to phase voltage of the primary. And in general terms, we might say that the secondary leads the primary by 30 degrees. Let's look at the magnitudes of the voltages now.

And we know from the phase relationship of the secondary side of the transformer that the face to face secondary voltage is root three times the phase two neutral secondary voltage. And because the phase two phase primary is magnetically linked to the phase two neutral secondary, the turns ratio may be given by the magnitude of the primary phase to phase voltage over the magnitude of the phase two neutral voltage on the secondary which can also be written the magnitude secondary read to neutral voltage is equal to the magnitude primary red to white voltage divided by the turns ratio. We can now replace the red two neutral voltage in the secondary phase relationship equation, which gives us the magnitude of the secondary red to white voltage is equal to root three times the magnitude of the red to white primary voltage all over the turns ratio. So, having looked at the voltages as far as the phase relationship and the magnitude relationship is concerned, I would like to now look at the currents of this transformer and we're going to have to add a load of course to make the current flow In the transformer and as I said, I'm going to add a load to this transformer.

And it's going to be connected in a y configuration, I could have just as well connected in delta, but we have to choose something. So I chose a y. And I'm going to maintain a balanced system here. And I'm going to also say that the load is purely resistive, because I want to maintain that any phase shifting that's going on is because of the transformer and not the load. I'm going to designate the current flowing in the secondary read phase as I align with a subscript r dash sec or secondary and that is the same current because of the Wye connection on the secondary as the current that is flowing in the secondary read phase winding of the transformer. And because it's magnetically linked with the primary side, it is in phase with a current flowing in the primary winding of the transformer.

Now, a couple of things to make note of here that the current is going to be coming for the power is going to be coming from the primary side connection, because the load is on the secondary. So, the current on the read phase as well as the other phases are going to be flowing into the spot of the primary winding and if the current is flowing into the spot of the primary, it has to be flowing out of the spot on the secondary and a similar relationship holds true for the other two phases. before going any further you have to remember That the voltage on the secondary side of the transformer we've already gone through the process is made up of the phasers of a red white and blue balanced system and those voltages are 120 degrees apart. So the current on the secondary side that's flowing into the load is going to be in phase with the voltages because the load is resistive.

So the currents on the secondary are going to be 120 degrees apart and balanced Of course, so that the current flowing in the transformer windings as you see there are going to be at 120 degrees apart, which means the current flowing in the windings of the primary are also going to be 120 degrees apart. So, once the load is connected to the secondary it will draw current from the primary lines. Looking at the read phase line for example, it will be made up of the current that will be flowing in the read phase transformer winding minus the current flowing in the blue phase transformer. Adding those two vectors or phasers together gives us a total current flowing in the red face primary line which forms and I saw Sully's triangle which we should recognize whose contained angles are 120 and two at 30 degrees. Comparing it to the line currents of the secondary it can be seen that the secondary will lead the project Primary by 30 degrees that is like secondary line currents, we'll be leaving the primary line current by 30 degrees.

I now want to look at the magnitudes of the currents. The familiar Sausalito triangle makes the magnitudes of the red and blue phasers equated in this way. Remembering that we are dealing with magnitudes only here, then the magnitude of red will be equal to the magnitude of minus blue. Which means we can write our magnitude of our line current or red phase line current in terms of the current As flowing in the winding of the red phase only, such that the line current of the red phase is equal to root three times the quantity, the magnitude r plus the magnitude are all over two, which equals root three, two times the magnitude of r over two. And of course, the twos will cancel out, leaving us with the equation that the current flowing in the red face primary line is equal to root three times the current that's flowing in the industry or into the red face transformer winding.

You can say it in general terms also that the current flowing in the primary line, depending on which phase you're talking about, is equal to root three times the current is flowing in the winding of the associated transformer league. And I call that I subscript capital line to line. So we know we're talking about the primary side. And at the risk of repeating myself, but I'll try to put some visuals up for it. The current that's flowing in the individual windings of the transformer are going to be equated to the current that's flowing in the individual windings of the transformer such that the magnitude of the line current is equal to root three. The current that's flowing in the individual transformer winding on the primary side Looking at the turns ratio of the individual legs of the transformer, we can define the turns ratio by looking at the ratio of the voltages of the line to line primary voltage over the line to neutral secondary voltage.

And that's of course will equate that to a, which is the turns ratio of the individual leg of the transformer. And we know that if we were to had current flowing in that transformer, that the turns ratio would be defined by just the inverse of that which is the current that's flowing in the secondary line all over the current that's flowing in the primary line to line now we're talking magnitudes here because the ratio is magnitude. So we got these two fractions that define the turns ratio of the transformer, we can rewrite that second one. And you can rewrite it again such that the magnitude of the current that's flowing in the windings of the primary side transformer is equal to the magnitude of the secondary line current all over the turns ratio, which now allows us to make a substitute in the previous equation that we developed over on the right hand side of the slide.

Which means we can now define the primary line current in terms of the secondary line current and that is the primary line current is equal to root three times the secondary line current All over the turns ratio of the transformer. And now I'd like to define the term one over a prime as the ratio of root three all over the turns ratio. That's just to make our equation a little bit easier to look at. So we can rewrite that equation such that the primary line current is equal to one over a primed times the secondary line current. And those are magnitudes. However, we can change them into phasers by simply adding the angle of the phasor that which which we've already developed and we know that the secondary leaves the primary by 30 degrees.

So in phasor terms, eyeline primary is equal to one over a primed, eyeline secondary, minus 30 degrees. I now want to develop a PR phase in a per unit equivalent circuit for this type of a connection. So I'm going to return to my voltage phasers here momentarily, and I'm going to shift everything up to the upper left to make some room for my equations. And we'll see where that takes us. Looking at the secondary side of the transformer to start with, we can see from the phase relationship, the relationship between the phase two phase and phase two neutral voltages are given by these equations and the per phase turns ratio is given by the line to line voltage all over the line to neutral voltage. Now, in order to look at the phase relationship of the primary, of course, there is no neutral, so we have to come up with a hypothetical neutral, but we can do that because it's a balanced equation.

And the procedure for a per phase analysis, the step one procedure is to convert all Delta loads and sources to their y equivalents. So we can do that, as I said by having a hypothetical neutral, and we can come up with the equation that aligned to neutral voltage on the primary is equal to the line to line voltage minus root three all over root three I'm going to take the two equations that we have just developed here and do some mathematical manipulations with them. I'm going to look at these two equations. And I'm going to divide the line to neutral primary voltage by the line to neutral secondary voltage. And then whatever I do with the left hand side of those equations, I have to do with the right hand side. So that equates to the line two line voltage minus 30 degrees over root three, all over the line to line voltage divided by the turns ratio.

And I can make that equation simpler by getting rid of two of the fraction signs, which is just again a mathematical manipulation which says that the line to neutral primary voltage over the line to neutral secondary voltage is equal to the turns ratio times a line to line voltage at minus 30 degrees, all over root three, line two line primary voltage. I am going to make the equation even simpler by cancelling out the common term in both the numerator and denominator, namely the line to line primary voltage. And I'm going to establish the A primed ratio which is the turns ratio all over root three. So, our equation now becomes the line to neutral primary voltage over the line to neutral secondary voltage is equal to A primed at minus 30 degrees. Now, there is no primary neutral so I'd like to reestablish this equation using line to line voltages.

And we've got that conversion there of line to line or line to neutral voltages by establishing our phase relationship earlier on, and we can now say that we can replace the line to neutral values of this equation by their equivalent line to line values. And then I'll simplify the equation as I've already done in the past by getting rid of fractions, side signs and the common terms. And so we're left with the voltage primary line to line all over the secondary voltage line to line is equal to the ratio of the line to neutral voltage primary over the line to neutral voltage secondary which is equal to A prime at minus 30 degrees. This last equation is actually made up of two equations, which can be rewritten with the primary voltages on the left hand side of the two equations and the secondary voltages on everything else on the right hand side of the equation, which will help us establish our our per phase equivalent circuit a little bit easier because in a per phase equivalent circuit, we're dealing with line to neutral values and this is a balanced system.

So we can bring up the per phase equivalent circuit and the per phase equivalent circuit would look like this. And we have two operators that we have to be concerned about and one is the turns ratio for the per phase transformer in we have to establish what We do to the secondary voltage to come up with a primary voltage. In other words, if I'm given the secondary voltage, I have to multiply by the turns ratio a prime in order to get the primary voltage and that's what's indicated with the yellow arrow there. Now, remember the A primed is a ratio of the regular turns ratio of the individual transformer windings divided by root three. And the second operator that we have to be concerned about is the phase shift that happens with a voltage going through the transformer. And if we take the secondary voltage and move it in and want to know what the primary voltage is, not only do we have to be concerned about the turns ratio, we have to be concerned about the 30 degree phase shift.

In other words, we have to subtract 30 degrees from the secondary voltage in order to get the primary voltage in That's indicated there with the burgundy or purple arrow that you see on the diagram. Now we know and we've already talked about this before, that if the turns ratio of the transformer is one thing involving the voltage, it's the inverse if we're looking at the currents, so in this case, the current would have the primary current line current is given by one over a prime times the secondary line current minus 30 degrees. So we have to in going from the secondary side to the primary side, otherwise, we get the we have the secondary current and we want to know what the primary current is we have to divide by the turns ratio a primed however, the phase shift remains unchanged. It's the same as the voltage the primary voltage will lag the secondary voltage by 30 degrees, so we have to use that operator to come up with the primary voltage the same way as we do with the voltage.

Before going any further, I just like to point out a fact that it stands to reason that there will be the same phase shift for the current as the voltage when going from the primary to the secondary. However, there will be a change in value of the current. In other words, it will be the inverse of the voltage for the magnitudes because if we added which we have purely resistive load to the transformer, it wouldn't be make any sense if the current was at a different phase shift than the voltage because it doesn't change from the voltage. So, really the operator This is just establishes the fact that the phase shift operator in this per phase equivalent circuit is doing the right thing. Because we add a resistive load, the current would be in phase with the voltage. So now I'd like to establish the per unit equivalent circuit.

And establishing the per unit equivalent circuit of course, we have to pick up our two base voltages. And the two base voltages are determined by the turns ratio of the transformer. And I've outlined them they're in in zones with the red, green and blue box. Now the equivalent circuit or per unit equivalent circuit is just going to be two straight lines because the turns ratio disappears when establishing a per unit equivalent circuit. However, the operator for phase shift has to be kept with the per unit equivalent circuit. This is one of the anomalies that makes working with the per unit transformer equivalent circuit a little bit difficult because in a Delta two why or a y to delta transformer, we have to keep track of the phase shift going through the transformer.

Whereas, a Delta two Delta or a y two y type transformer, you don't have to worry about a phase shift and you wouldn't have to worry about turns ratio. So it's a little bit easier. However, if you consider the fact that we have two operators, one is the turns ratio and the other is the phase shift. Then you just have to keep track of the phase shift and going from one zone to the other using your per unit equivalent circuit. The two zones are still there, you can have the phase shift operator in any one of the zones doesn't matter which one you pick. Because the voltage, the magnitude of the voltages going from the primary to secondary is the voltages are going to be still the same.

They're equal, as is the magnitudes of the currents. It's just the voltage phase shift that you have to worry about. Before moving on, let's show firstly the need for the next configuration type connection. In order to do that, let's look at a hypothetical system growth. We see here an existing station with two y two Delta transformers, ultimately feeding a low voltage distribution grid as a system grows a why why transformer station is built and its primary is fed from the same source as the other old existing system with the two wide Delta transformers. As the system grows, at some point, it would meet on the low voltage feeds and for logistical reasons reasons it might seem practical to parallel the systems.

Of course, the problem here is the 30 degree phase shift difference on the secondaries of the transformers and their connecting fields. There is a solution to this and it is it is accomplished with the aid of the star zigzag or another name for it is the interconnected star transformer A transformer that is used to supply unbalanced single line to neutral and three phase loads for three phase four wire systems. The following advantages are achieved, the transformer can supply single phase line to neutral loads while maintaining a stable secondary neutral. The transformer is relatively free of third harmonic residues. Therefore, there is no need for a tertiary winding as required with a star star transformer. Stars zigzag connected Transformers can be paralleled with Star Delta transformers.

The phase relationship of the phase two phase voltage and we're looking at the red to white in this case is the same as a star Delta transformer. That is, there's a 30 degree phase shift from primary to secondary, yet a neutral connection is available for relaying and other single phase loads. So let's look at the connections now that make up this star zigzag transformer. The transfer in this transformer each phase is made up of a single primary winding and two secondary windings. One terminal of the primary terminals are connected to the system lines or buses, the other terminals are connected together and form a primary neutral. The phasers look like this which of course should not be a surprise because we've already seen this before.

The secondary of the transformer is made up of two windings on each phase one set of these windings are connected the same as the previously studied secondary star connections. One terminal of the secondary terminals are connected to the low voltage system the other terminal of that winding are connected together to form a secondary neutral and not surprising. The transformer phasers on the secondary on those windings look like this. The other set of these windings are connected like this. The red unspotted terminal of the secondary is connected to the red system lines, buses etc. The other red spotted terminal is connected to the spotted terminal of the white terminal of the other A secondary whitening added to the existing vectors would look like this.

The white unspotted terminal of the secondary is connected to the white system lines or buses. The other terminal is connected to the spotted terminal of the blue terminal of the other secondary winding. When added to the existing vectors would look like this. The blue unspotted terminal of the secondary is connected to the blue system lines and buses. The other blue on the Heather blue spotted terminal is connected to the spotted terminal of the red terminal of the other secondary winding completing the vectors to look like this. The H one terminals are connected to the individual phase conductors and the high voltage h two terminals are connected together to form a primary neutral.

The transformer having its h1 terminal connected to the read phase is referred to as a red phase transformer and likewise the h1 terminal connected to the white phase is referred to the white phase transformer and the h1 terminal of the blue phase is connected to the what we call the blue phase transformer. The secondary terminals are brought out to form x one red X one white and x one blue The red white face to face secondary voltage is in phase with the primary red to neutral voltage. The white to blue face to face secondary voltage is in phase with the primary white to neutral voltage and the blue to red face to face secondary voltage is in phase with the primary blue to neutral voltage. Now, let's revisit our hypothetical growth system and we still have the existing station with two y two Delta transformer ultimately feeding needle voltage distribution grid.

This time however, it grows and it builds into a y zigzag transformer station. As the system continues to grow at some point it would meet for logistical reasons again It might seem practical to parallel the systems. This time. The feeders are in phase with a much happier outcome. And this ends the chapter and I would like to give credit for these contributors of some of the features of this particular chapter.