# Trouble with use of Symmetrical components

I see some trouble with how symmetrical components are treated. In lots of texts, I have heard something along these lines
When the current in a 3 phase motor is unbalanced, we can resolve it into three sets of balanced currents.
1. Positive sequence currents.
These produce rotating magnetic fields in usual direction and produces +ve torque and power
2. Negative sequence currents
There produce rotating magnetic fields in opposite direction and produce -ve torque
3. Zero sequence currents
These produce stationary and pulsating magnetic fields and produce no torque.

Fine upto now. the net torque is sum of all torques.
I fully agree that mathematic ally, 3 phase unbalanced set of currents (or any phasors) can be thought off as sum of 3 sequence components.

What I disagree is when people talk about the effects of these currents.
For example I don't think we can say the losses in stator winding = (I_positive^2*r + I_negative^2*r + I_zero^2*r).
Mostly, I find texts talking about negative sequence currents.
"Since negative sequence currents produce rotating magnetic fields in opposite direction, it will have large relative velocity with rotor (nearly 2*Ns , Ns is synchronous speed). It will create large induced currents in rotor and huge eddy losses."
But I don't think we are allowed to find out the induced currents and eddy losses due to the 3 sequence currents individually, just like we can't find the I^2*R losses individually.

For example, consider that a DC current of 2A is flowing in one of the winding of a motor. Or even better, lets consider a motor with no current in any winding.
We can think of 0 current as 0 = 100*Sin(wt) + (-100*Sin(wt)) , as composed of two AC currents. Now Clearly, we can't individually calculate the losses (eddy and I^2*R) due to each current and Add.

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## Answers and Replies

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jim hardy
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i've heard of Fortescue's theorem, but never studied it. I heard of sequence analysis from some genuine power engineers who were in relay department at same plant i worked.. So i am very foggy on it.

But - aren't you just saying the sum of the squares doesn't equal the square of the sum?

Near the end of this link, a kindly professor speaks to power associated with sequence components. Each sequence gets its power calculated independently and they are summed.

http://www.elect.mrt.ac.lk/EE201_3phase_sym_comp.pdf

Thus P = 3 (Va0 Ia0 cos φ0 + Va1 Ia1 cos φ1 + Va2 cos φ2)

I hope to learn along with you on this one.

old jim

dlgoff
Gold Member
... I don't think we are allowed to find out the induced currents and eddy losses due to the 3 sequence currents individually, just like we can't find the I^2*R losses individually.
From the Wikipedia page Fault (power engineering) (bold by me),

To determine the currents resulting from an asymmetrical fault, one must first know the per-unit zero-, positive-, and negative-sequence impedances of the transmission lines, generators, and transformers involved. Three separate circuits are then constructed using these impedances. The individual circuits are then connected together in a particular arrangement that depends upon the type of fault being studied (this can be found in most power systems textbooks). Once the sequence circuits are properly connected, the network can then be analyzed using classical circuit analysis techniques. The solution results in voltages and currents that exist as symmetrical components; these must be transformed back into phase values by using the A matrix.
where the A matrix is

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The impedance of a synchronous generator for each sequence can be measured or calculated. And they are not always equal to stator and rotor resistance and Xd, Xq etc. So in your first example, the stator resistance might not be; r=r0=r1=r2. And hence the equation is wrong.

The transformation is power invariant, so power calculated in abc sequence should equal the 012 sequence. Thus using the correct measured or calculated resistance for each sequence should give the same answer as using normal quantities.

Disclaimer: This is just my quick reasoning, so I could be wrong.

i've heard of Fortescue's theorem, but never studied it. I heard of sequence analysis from some genuine power engineers who were in relay department at same plant i worked.. So i am very foggy on it.

But - aren't you just saying the sum of the squares doesn't equal the square of the sum?

Near the end of this link, a kindly professor speaks to power associated with sequence components. Each sequence gets its power calculated independently and they are summed.

http://www.elect.mrt.ac.lk/EE201_3phase_sym_comp.pdf

I hope to learn along with you on this one.

old jim
After litter more pondering today, I think I found the answer.
Here is what I got.
Resolving a set of 3 phase unbalanced currents into sequence components is purely a mathematical thing.
You have right to say
Ia = Ia0 + Ia1 + Ia2;
Ib = Ib0 + Ib1 + Ib2;
Ic = Ic0 + Ic1 + Ic2;
where Ia1, Ib1 and Ic1 form +ve sequnce balanced set (i.e. phase sequence of abc) whereas Ia2, Ib2 and Ic2 form -ve sequence balanced set (i.e phase sequence of acb)
Ia0, Ib0 and Ic0 form zero sequence. (all thre currents are in phase)
The problem is just finding correct magnitude for |Ia0|, |Ia1| and |Ia2|. But mathematics has the answer.

Now when unbalanced current Ia, Ib and Ic is flowing in motor, you can substitute Ia, Ib and Ic with their respective sums of sequence components.
The actual loss in winding is Ia^2*r = (Ia0 + Ia1 + Ia2)^2*r
So, its not equal to Ia0^2*r + Ia1^2*r + Ia2^2*r.
Simple and easy.
The magnetic field produced is Ia*K = (Ia0+Ia1+Ia2)*K = Ia0*K + Ia1*K + Ia2*K
where K is some constant.

So it is evident that, if the feature being calculated is proportional to I^2 then we can't calculate it separately with different sequence components and add(The former case i.e I^2*r case)

But if its is proportional to just I, then we can calculate the quantity separately for 3 sequence components and then Add. (the second case, i.e. Magnetic field case)

So, we can say that The magnetic fields, The torque produced, The voltage induced in rotor, the current induced in rotor, all can be calculated separately and then added.
However the power induced (lost) in rotor (or stator) can't be separately calculated and added.

So, yes jim, it basically boils down to "sum of the squares doesn't equal the square of the sum".

But texts sometimes seem to miss this point. Its true that negative sequence currents in stator windings create large induced currents in the rotor.
But it would be wrong to say this large induced current will create large loss in rotor.
Because, before finding the losses due to induced current by -ve sequence currents in stator, we need to find induced current by +ve and 0 sequence currents in stator and add all three induced currents. I_total = I_induced_-ve + I_induced_+ve + I_induced_0
The loss will then be I_total^2*r.

So, although I_induced_-ve might be large, it might be cancelled or reduced by I_induced_+ve and/or I_induced_0

But I don't see that being mentioned in texts. See for eg. in this document. Page 13
The current induced at the rotor center is of double frequency. The induced current could cause rapid
heating in the rotor, which may result in insulation failure and/or mechanical problems.
Although in the case analyzed in the document, I_induced_+ve and I_induced_0 is actually 0 and I_total is indeed I_induced_-ve alone; its generally wrong to jump to rotor heating or other power losses conclusion based on induced current by only one sequence components.

Even on other texts, where I_induced_+ve isn't 0 (induction motors) I see similar conclusions being made. And that was the reason for starting this thread in the first place.

I_induced_0 is always 0 in an electric machine because zero sequence current don't produce any net magnetic fields.
Correcting my OP
3. Zero sequence
The magnetic field produced by 3 coils carrying Zero sequence currents cancel each other and produce no net magnetic field.

Thanks for showing interest.

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jim hardy
Gold Member
2019 Award
Dearly Missed
i have thought aboout this... am away from home so no books handy

but you're right math has the answer.

i finally realized your currents are complex
so when you square them you CAN get a negative result, unlike squaring a real number.

Square (1 +j 3) and see what you get
1 +j6 - 9 ?

so powers might very well sum to zero ?

1. Positive sequence currents.
These produce rotating magnetic fields in usual direction and produces +ve torque and power
2. Negative sequence currents
There produce rotating magnetic fields in opposite direction and produce -ve torque
BEWARE that the rotor has only single direction of rotation for these two current rotations!

Then, the reversed current rotation produces a braking torque, but not of the same strength as the normal moment! In a squirrel cage motor, the braking torque would be much smaller because of the high slip, which induces high frequency in the rotor, whose inductance limits the induced current. Similar to the small starting couple but worse.