harmonic content calculation due to in rush current 2

[EddyPach]

Somebody knows how to calculate the harmonic content of the inrush current during the initial energization of a high power trnasformer, from the high voltage side?

 

[electricuwe]

The harmonic content can be calculated using FFT (Fast Fourier Transform).

The inrush current of a large transfomer will take up to a few secondy to decay. Before applying an FFT to such an function you have to choose a window small enough so that the signal doesn't change much within this window (e.g. 10 periods of line frequency)

Apply the FFT with such a window at different times.
The harmonic content of the signal will change with time.

To my mind calculation of harmonic content for such a signal is not very useful. What is the purpose of calculating the harmonic content?

 

[electricpete]

You'll see the answer at page 15 (225) of the following document:

http://www.geindustrial.com/industrialsystems/pm/notes/artsci/art11.pdf

I calculated it using Matlab and got similar results. All I did was apply voltage to a saturable inductor which clamped at 120% flux.

 

[electricpete]

I will try to provide an example calculation this weekend. Assuming the transformer is unloaded, the basic problem is the same as determing the current of a saturable inductor which has an ac voltage applied suddenly.

In the linear range we can write:
N*Phi=L*I
v = N*dPhi/dt = L*dI/dt

A curve of N*Phi vs I would have a slope of L for current in the linear range, and approximate a slope of perhaps 0.1*L in the saturated range.

We can compute the slope dNPhi/di = L for I<Isat or = 0.1*L for I > Isat.

v=d(N*Phi)/dt = dNPhi/di * di/dt where dNphi/di as a function of I is known as described above.

di/dt = v/(dNPhi/di) = Vmaxsin(2pift)/(dNPhi/di)
i(0)=0.

Knowing the initial value of i and the derivative di/dt as a function of i,t, we can easily calculate the waveform numerically using finite time step i[k]=i[k-1] + di/dt((i[k-1],t))*deltaT.

The result will look like a sinusoid with dc offset, but somewhere near the peak the current jumps tremendously high.

In reality the dc would decay due to series resistance (not modeled above), but this does not affect the FFT.

Since the waveform does not have quarter wave symmetry, I would expect the even harmonics to play a big role.

electricuwe - one of the reasons that transformer inrush harmonic content is of interest is for protective relaying. A differential relay with harmonic restraint won't trip on differential current if harmonics are present (assumed inrush) and will trip on differential current if harmonics are not present (assumed internal fault).

 

[electricpete]

I posted an example solution. It is in the form of a Matlab notebook file which is also a microsoft word file.

There is a lot of explanation and graphs. It should be mildly interesting even if you're not into Matlab, IMHO.

http://geocities.com/pschimpf/temporary_stuff/transformer_inrush_simulation.htm

Here are my results:
2nd harmonic as percent of first:
0.4887

3nd harmonic as percent of first:
0.3260

4th harmonic as percent of first:
0.1797

Compare this to GE's result:
2nd harmonic - 63%
3rd harmonic 27%
4th harmonic 5%

Note there are many variables with regard to saturation characteristics. Also the phase at time of close in will affect the exact result. I don't believe that the failure to include resistance in the model makes much difference.

I started thinking a little bit about the effect of adding load to the transformer. I was first tempted to take the load impedance and combine with exciting impedance as an equivalent circuit. Even if there were not saturation to consider, that approach would still be totally wrong since the impedances of these two branches have a ratio which depends on frequency. The transient involves sudden application of voltage... and this cannot be analysed in the same manner as single-frequency sinusoidal steady state.

It it interesting to note that if we ignore the primary leakage reactance, we will apply the same voltage to the magnetizing branch and to the load/secondary leakage reactance. The resulting current will be close to the addition/superposition of the currents expected from these two branches. i.e. normal inrush as shown in attached, plus decaying dc-offset load inrush current.