# Short Circuit Calculations – Per Unit and Symmetrical Components

Per Unit System and Symmetrical Components are used for many short circuit calculations. Electric Utility Data is also often provided in terms of Per Unit and Symmetrical Components. Watch this video to see Jim Phillips explain how to use the positive, negative and zero sequence impedances in a per unit format to calculate the three phase and line-to-ground short circuit current. To learn more, take Jim’s Short Circuit Analysis on-demand class.

*About Jim Phillips:** Electrical Power and Arc Flash Training Programs – For over 30 years, Jim Phillips has been helping tens of thousands of people around the world, understand electrical power system design, analysis, arc flash and electrical safety.*

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### Video Transcript

How do you perform short circuit calculations using a method known as symmetrical components and also the per unit system? Well, the actual calculation process is pretty simple. To calculate the three phase short circuit current, you just take the voltage, and you divide by what I’m showing here is Z 1 plus Z f. Now, what Z 1 is, that’s defined as the positive sequence impedance. And it’s basically just the impedance of your circuit, the impedance of the conductors, the transformers, the utility company.

Z f is defined as the impedance for the fault. So if you have something that’s actually causing the fault, like a tree or something like that, you can introduce Z with the subscript f. However, most people just assume Z sub f is 0. They assume that it’s a bolted short circuit, and there’s no additional impedance involved.

The line-to-ground short circuit calculation is done by taking 3 times the voltage, divided by several different impedances. Z 1, the positive sequence impedance, plus Z 2 which is the negative sequence impedance, plus Z 0 the 0 sequence impedance, plus 3 times the fault impedance. So again if you have a tree or something that’s in the way.

Now, what does Z 1, Z 2, and Z 0 actually mean? These values are based on a theory known as symmetrical components. And Z 1, as I mentioned, that that’s just the impedance of your circuit elements, the conductors, transformers, the utility system. Z 2, the good news is Z 2 in almost every case, is the same as Z 1. So if you know the impedance of your conductors, your transformers, and so forth, you have both Z 1 and Z 2.

Z 0, however, 0 sequence impedance, that’s the impedance that’s the result of current flowing on something other than a face path. So Z 0 is going to be based on a return path other than phase conductors, and it’s also going to be affected significantly by grounding, such as how transformers are grounded, and how generators are grounded. So I want to walk you through a simple problem in performing short circuit calculations when you have the symmetrical component values positive and 0 sequence impedance. But also bases, base values, and I actually talk about the base values and per unit in another video.

So you may look at this example and see that I have Z 1, I have Z 0, but I’m not showing Z 2, the negative sequence impedance. So you may be thinking, well, yeah, how do we handle this if Z 2 isn’t given. Well, as I mention, Z 2 in almost every case is the same as Z 1. And most of the time you won’t be given Z 2, or the negative sequence impedance. So what you do is you just use Z 1 twice. You assume that Z 2 and Z 1 are both the same value.

So looking at this problem, the first thing that I do is I calculate what is known as the base current. We’re going to calculate the 3-phase short circuit current and the line-to-ground short circuit current in terms of per unit. And the per unit is simply a decimal equivalent, or a percentage, of some frame of reference known as the base current.

And so to calculate the base current, you take the MVA base, and someone has to define what the MVA base is. Typically, it’s 100 MVA. But you’ll need to obtain that from your source of data. And you multiply times 1000, that converts MVA to kVA, divide by the square root of 3, and the kV base.

And so for this example, we were given that we have a 100 MVA base, multiplied times 1,000, that’s 100,000 kVA. We divide by the square root of 3. We divide by the voltage that we were provided, 13.8 kV, which was also defined as the base. And our base current is 4,183.7 amps. And that’s going to be our frame of reference. So when we actually have our calculated per unit value, we’ll take that value and we multiply times 4,183.7.

So to calculate the 3-phase short circuit current we take the voltage divided by the positive sequence impedance Z 1, and the voltage that I’m showing is just simply 1. And you might think, where did that come from, the 1. The voltage of 1, that means 1 per unit. And actually, you should be factoring in angles, but I’m just showing just the basic format of how to perform the calculations.

The 1, that assumes that the actual operating voltage is the same as the base voltage. And since we were given a base voltage of 13.8 kV, the assumption is, then we’ll say the system is operating at 13.8 kV. And that’s the normal assumption that you make. And unless you’re given something else, normally you assume that the system is operating at whatever the base, or the nominal voltage is, and therefore the per unit voltage is 1 per unit.

So we take 1 per unit voltage, we divide by the 0.99 per unit positive sequence impedance that we were provided, and that’s 1.010 per unit, 3-phase short circuit current. So we take this value and we multiply times the base current that we calculated 4,183.7 Amps. And our 3-phase short circuit current in this case is 4,225 Amps.

To calculate the line-to-ground short circuit current, it’s not a whole lot more difficult. We have a few more numbers that we need to include, but the process is actually pretty simple. We take the equation that I showed you a moment ago. We take 3 times the voltage. And in this case it’s 3 times 1. And this time we have to divide by Z 1, Z 2 and Z 0, the positive, negative, and 0 sequence impedances.

Well, remember, we were provided Z 1 and Z 0 but we weren’t provided Z 2. So in this case, we just assume that Z 2, the negative sequence impedance, is the same as the positive sequence impedance. So 3 times our 1 per unit voltage, that’s 3.

We divide by 0.99, that’s the positive sequence impedance again, plus 0.99, that’s the negative sequence impedance Z 2, plus 3.12 that we were provided. And, this gives us a per unit line-to-ground short circuit current of 0.588 per unit. And if we take the 0.588 per unit, times our base current 4,183.7 Amps. Our line-to-ground short circuit current is 2,460 Amps.

So obviously, there is more to the theory of symmetrical components than I’m showing here. But as far as just the calculations, the calculations are actually quite simple.