# Power Triangle Challenge

#### xptpcrewx

##### Power System Engineer
A challenge for the electrical engineers…

From a physics standpoint the only true concept behind electrical power is the instantaneous power p(t) = v(t)•i(t). Real, Reactive and Apparent power are made up engineering concepts…

Given this fact, how many of you have actually questioned where the “power triangle” came from and it’s connection to physics? Do you understand this relationship and more importantly how it came to be? Why are the components in quadrature? What does this even mean in terms of instantaneous power? Can you provide a derivation or rigorous proof of its validity and use? If not, can you truly consider yourself an electrical engineer?

Have fun… the results may surprise you.

#### bwat

##### EE
If not, can you truly consider yourself an electrical engineer?
lol at this part

I'm all for people understanding things at a fundamental level. The more you understand something at a ground/root cause scenario, the less you have to memorize because you can always derive the subsequent outcomes from the root cause physics. I feel like there's an Einstein quote about this... That being said, somebody not being able to provide "derivation or rigorous proof of its validity and use" of the power triangle is nowhere close to a good metric to determine whether somebody can call themselves an EE. There's a chance that being able to do that would help them 0% over the course of their EE career.

Unless someone hangs out it in the academia world, there are very few scenarios that such knowledge would come into play. I'm not bashing academia here. I have both my bachelors and masters in EE. I'm just pointing out what everybody largely agrees; everything that you learn in class doesn't always find a way to be useful in the real world. e.g. I'm still waiting for the first time to use my laplace transforms knowledge that was drilled into my head years ago.

It is good to refresh this type of thing though and have these thought experiments, so I appreciate the questions. I would have agreed with your quoted question if maybe it said something like "If not, do you think it could help you be a better EE or Electrician if you did know?" .... and I'd probably say yes to that.

#### Besoeker3

##### Senior Member
lol at this part

I'm all for people understanding things at a fundamental level. The more you understand something at a ground/root cause scenario, the less you have to memorize because you can always derive the subsequent outcomes from the root cause physics. I feel like there's an Einstein quote about this... That being said, somebody not being able to provide "derivation or rigorous proof of its validity and use" of the power triangle is nowhere close to a good metric to determine whether somebody can call themselves an EE. There's a chance that being able to do that would help them 0% over the course of their EE career.

Unless someone hangs out it in the academia world, there are very few scenarios that such knowledge would come into play. I'm not bashing academia here. I have both my bachelors and masters in EE. I'm just pointing out what everybody largely agrees; everything that you learn in class doesn't always find a way to be useful in the real world. e.g. I'm still waiting for the first time to use my laplace transforms knowledge that was drilled into my head years ago.

It is good to refresh this type of thing though and have these thought experiments, so I appreciate the questions. I would have agreed with your quoted question if maybe it said something like "If not, do you think it could help you be a better EE or Electrician if you did know?" .... and I'd probably say yes to that.
Good post if I may say so. And I take your point about Laplace and Nyquist too for that matter. But I suppose many of us ended in a particular field - specialised. It was was me - power electronics.

#### xptpcrewx

##### Power System Engineer
lol at this part

I'm all for people understanding things at a fundamental level. The more you understand something at a ground/root cause scenario, the less you have to memorize because you can always derive the subsequent outcomes from the root cause physics. I feel like there's an Einstein quote about this... That being said, somebody not being able to provide "derivation or rigorous proof of its validity and use" of the power triangle is nowhere close to a good metric to determine whether somebody can call themselves an EE. There's a chance that being able to do that would help them 0% over the course of their EE career.

Unless someone hangs out it in the academia world, there are very few scenarios that such knowledge would come into play. I'm not bashing academia here. I have both my bachelors and masters in EE. I'm just pointing out what everybody largely agrees; everything that you learn in class doesn't always find a way to be useful in the real world. e.g. I'm still waiting for the first time to use my laplace transforms knowledge that was drilled into my head years ago.

It is good to refresh this type of thing though and have these thought experiments, so I appreciate the questions. I would have agreed with your quoted question if maybe it said something like "If not, do you think it could help you be a better EE or Electrician if you did know?" .... and I'd probably say yes to that.

The point is it’s such a fundamental concept right up there with ohms law, that it’s just embarrassing if you can’t do it. It’s not an academic thing or some difficult proof, It’s just basic trigonometry. Where is this exercise useful? How about when explaining what reactive power is… you cannot claim to understand real and reactive power without knowing what it means mathematically.

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#### petersonra

##### Senior Member
I would have to study up to deal with Laplace or Nyquist. Just never used it after school. I went through a BSEET program so it was a little different than an EE program. Mostly the same math though. An engineer I ran across in a textile plant when we were trying to get a DC drive working called me a "5 Volt engineer". He pretty much had it right. I was well prepared for the 5 Volt world coming out of school but somehow ended up never working in that world.

I would be hard pressed to even work simple calculus problems these days. However, for some reason youtube has been recommending math videos in Spanish for me. Most are linear algebra, exponents, trig, geometry, and log type problems. I have been watching some of them even though I cannot understand a word of Spanish. I can follow the math though, and more often than not solve the problem in my head before the professor does. I had forgotten how much fun math can be. However, I have some bad math habits that have not gone away, like trying to jump ahead two or three steps to save time, which gets me now and then.

I guess youtube decided I am weird enough and lately has been suggesting electronics videos to me. Mostly mesh equations (KVL). Most, but not all are in Spanish for some reason. I guess youtube thinks I speak Spanish. However, I was truly surprised when in one of the English language videos the professor told the students not to bother to solve the linear algebra problem the mesh equations give you but to just type it into your calculator. That seems to be the practice of most of the linear algebra problems I saw that ended up needing to be solved with matrices.

#### bwat

##### EE
The point is it’s such a fundamental concept right up there with ohms law, that it’s just embarrassing if you can’t do it. It’s not an academic thing or some difficult proof, It’s just basic trigonometry. Where is this exercise useful? How about when explaining what reactive power is… you cannot claim to understand real and reactive power without knowing what it means mathematically.
You can certainly explain what real and reactive power is without providing a detailed proof of how we got to the power triangle.

I would wholeheartedly agree that it's a fundamental concept up there with ohm's law. However, you're taking it a step further. It would be analogous to saying that you need to be able to provide a proof for ohm's law.

#### xptpcrewx

##### Power System Engineer
I would have agreed with your quoted question if maybe it said something like "If not, do you think it could help you be a better EE or Electrician if you did know?" .... and I'd probably say yes to that.
Let's say we rephrase it the way you want. Are you more inclined to accept the challenge?

It seems a power engineer not knowing the power triangle, is similar to a lawyer arguing a case but not knowing the principle behind the law, or a brain surgeon who can follow a surgical procedure but not really know how the brain works, or an electrician who does installs but never cared to learn the code. In each of these cases, I think something is missing.

You can certainly explain what real and reactive power is without providing a detailed proof of how we got to the power triangle.
Go for it. Tell us what they represent and how to quantify them. Also if you can, explain why real and reactive power are in quadrature and a pythagorean identity applies.

I would wholeheartedly agree that it's a fundamental concept up there with ohm's law. However, you're taking it a step further. It would be analogous to saying that you need to be able to provide a proof for ohm's law.
Ohms law is an empirical law derived experimentally so I disagree. The power triangle on the other hand is an arbitrary engineering construct.

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#### jim dungar

##### Moderator
Staff member
The point is it’s such a fundamental concept right up there with ohms law, that it’s just embarrassing if you can’t do it. It’s not an academic thing or some difficult proof, It’s just basic trigonometry. Where is this exercise useful? How about when explaining what reactive power is… you cannot claim to understand real and reactive power without knowing what it means mathematically.
It has probably been 43 years since I last proved an engineering formula or theorem. I had people that worked for me that could and had no problem following their work.

I can't remember being embarrassed by not doing this and I am proud call myself a power systems engineer.

#### wwhitney

##### Senior Member
The function space of sinewaves of a fixed period T is two dimensional and forms an inner product space under <X,Y> = \integral_0^T X * Y. For any such sinewave X(t), the shifted sinewave X'(t) = X(t - T/4) is orthogonal to X, that is <X,X'> = 0, so X and X' form a basis for the space.

For current I and voltage V, the instantaneous power transferred is P(t) = I(t)*V(t), so the power transferred over one period is just <I,V>. For non zero V, let I1 = (|I|/|V|)*V and I2 = I1'. Then I1 and I2 are a basis, with |I1| = |I2| = |I|, and we may write I = a * I1 + b * I2 for some a, b with a^2 + b^2 = 1, since it is an orthogonal basis.

Now note <I,V> = a <I1, V> = a |I| |V|. That is the real power transferred in a cycle, and a is the power factor. |I| |V| is called the apparent power, and b |I| |V| is called the reactive power. The power triangle is the statement that (|I| |V|)^2 = (a |I| |V|)^2 + (b |I| |V|)^2.

Cheers, Wayne

#### xptpcrewx

##### Power System Engineer
The function space of sinewaves of a fixed period T is two dimensional and forms an inner product space under <X,Y> = \integral_0^T X * Y. For any such sinewave X(t), the shifted sinewave X'(t) = X(t - T/4) is orthogonal to X, that is <X,X'> = 0, so X and X' form a basis for the space.

For current I and voltage V, the instantaneous power transferred is P(t) = I(t)*V(t), so the power transferred over one period is just <I,V>. For non zero V, let I1 = (|I|/|V|)*V and I2 = I1'. Then I1 and I2 are a basis, with |I1| = |I2| = |I|, and we may write I = a * I1 + b * I2 for some a, b with a^2 + b^2 = 1, since it is an orthogonal basis.

Now note <I,V> = a <I1, V> = a |I| |V|. That is the real power transferred in a cycle, and a is the power factor. |I| |V| is called the apparent power, and b |I| |V| is called the reactive power. The power triangle is the statement that (|I| |V|)^2 = (a |I| |V|)^2 + (b |I| |V|)^2.

Cheers, Wayne
Interesting approach. Anyone else?

#### wwhitney

##### Senior Member
I don't think practicing engineers need to prove they can do the mathematics underlying their work at the drop of a hat. They've done it before, and they know they can do it again if necessary.

I'm not a practicing engineer, so I was happy to respond to your challenge.

Cheers, Wayne

#### xptpcrewx

##### Power System Engineer
I don't think practicing engineers need to prove they can do the mathematics underlying their work at the drop of a hat. They've done it before, and they know they can do it again if necessary.

I'm not a practicing engineer, so I was happy to respond to your challenge.

Cheers, Wayne
I appreciate it. And I agree. I don't expect anyone to do it at the drop of a hat - theres no time limit. It actually took me a couple days to prove it to myself, but once I did it, I gained some critical insight in the process.

The main idea behind this thread is to revive some basic concepts we take for granted. I think a key difference between an engineer and a technician is that a technician knows how to apply a formula, whereas an engineer knows how to derive it.

#### xptpcrewx

##### Power System Engineer
It has probably been 43 years since I last proved an engineering formula or theorem. I had people that worked for me that could and had no problem following their work.
Im sure you are more than capable and have the tools to solve it even after 43 years.
I can't remember being embarrassed by not doing this and I am proud call myself a power systems engineer.
Maybe it's just me. I like to beat myself up over these things...

#### bwat

##### EE
If not, can you truly consider yourself an electrical engineer?
The point is it’s such a fundamental concept right up there with ohms law, that it’s just embarrassing if you can’t do it.
These are similar statements and what I was objecting to. The "challenge" in and of itself is a good one, as I mentioned in my first post (#2).

Similar to Jim's post, I could have done this with ease probably 15 or so years ago. But I'd have to refresh a few things to be able to do it now. And I'm sure it's not surprising to hear that I would say it's laughable to compare my ability as an EE right now than with where I was 15 years ago. Hence, my objection to saying it is a requirement and/or embarrassing if you can't do it. There isn't an ounce of me that would be embarrassed to say I'd have to look some things up, or that would say I was somehow a better EE 15 years ago. I didn't really know anything other than the underlying math.

but once I did it, I gained some critical insight in the process.
I'm curious. What'd you discover?

#### xptpcrewx

##### Power System Engineer
I'm curious. What'd you discover?
I'll post my version of the derivation, but would like to give members a chance to try it out first. We will all have different approaches but it would be interesting to see if we can arrive at a similar result or conclusion. Let me add an interesting rhetorical question and possible hint...

The portion of instantaneous power averaged over a complete cycle results in a net transfer of energy in one direction. This time average is known as the active power or real power. The same idea does not apply to reactive power because no net energy transfer takes place (energy oscillates between the source and load, so the average power is zero), yet somehow reactive power is still represented as a component of the same right triangle... Therefore any non-zero value used to quantify reactive power must represent something different altogether. It gets a little abstract...

#### xptpcrewx

##### Power System Engineer
Headache because you find it difficult to understand? (its supposed to be a challenge)

#### petersonra

##### Senior Member
Headache because you find it difficult to understand? (its supposed to be a challenge)
I don't care all that much about E=MC^2 either. That would give me a headache if I thought about it too much too.

#### xptpcrewx

##### Power System Engineer
I don't care all that much about E=MC^2 either. That would give me a headache if I thought about it too much too.
To each their own but why mention anything then?

#### Besoeker3

##### Senior Member
Interesting approach. Anyone else?
Another approach. Just a brief section of it.
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