Transformer Energization

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No-load energization of a saturable 230/115 kV transformer — magnetizing inrush and its decay.

Per-phase equivalent of the energization (one of three identical phases shown). The source snaps on at $t=0$ behind 0.5 Ω $+$ 14 mH and drives the HV winding through 0.005 $+j0.1$ pu of leakage. Everything interesting happens in the shunt magnetizing branch: below the knee ${\psi }_{\mathrm{knee}}=1.2$ pu it is a linear 1 %-of-rated inductance, and above it the core is gone and the branch collapses to the air-core inductance $L_{\mathrm{air}}=0.1$ pu — a thousand times stiffer. The LV side sits on 1 MΩ per phase, so the unit is energized at no load.

A 230 kV source is switched onto an unloaded 100 MVA 230/115 kV transformer at $t=0$, with no ramp, at the instant phase A's voltage passes through zero. The LV winding is terminated on 1 MΩ per phase, which is open-circuit for practical purposes, so every amp you see is going into magnetizing the core. Both neutrals are solidly grounded. The plot follows the three magnetizing currents Imag_a, Imag_b, Imag_c and the three core flux linkages Psi_a, Psi_b, Psi_c.

For reference, rated HV current for this unit is 251 A RMS, or 355 A peak, and every multiple quoted below is against that peak.

The first three cycles

Core flux is the integral of applied voltage, so the instant of closing sets how much DC offset the flux carries. Closing at a voltage zero is the worst case: the flux has nowhere to start but zero and has to swing toward twice its normal peak.

Phase A crests at 1745 A — 4.9 times the 355 A rated peak — 8.4 ms after closing, then repeats every cycle. Note the shape: current is essentially zero for most of the cycle and appears as a one-sided pulse lasting a few milliseconds, because the branch only conducts while the flux is over the knee. Phases B and C carry the opposite-sign offset and peak near 680 A and 643 A.

That waveform only makes sense once you look at the flux underneath it.

Top: the three flux linkages over the same first three cycles, with the $\pm 1.2$ pu knee marked. Phase A rides an offset of about $+0.83$ pu, so its crest reaches 1.69 pu and spends a few milliseconds per cycle above the knee — that is the window in which the current pulses above happen. Bottom: the per-cycle crest, trough and midpoint of ${\psi }_a$ over the full 8 s. The offset bleeds away quickly at first and then stalls: it stops falling the moment the crest drops onto the knee, because that is the moment the core stops saturating.

Phase A's flux crests at 1.690 pu. Above the knee the magnetizing branch is no longer a core — it is a coil in air, and the sample's l_air_pu of 0.1 pu makes that branch a thousand times stiffer than the 100 pu unsaturated $L_m$:

The two-slope (air-core) magnetizing characteristic this sample uses. Below ${\psi }_{\mathrm{knee}}$ the branch is the unsaturated $L_m$ — 1 % of rated, which is 3.55 A at 1.0 pu flux and only 4.3 A at the knee, so flat against this axis it looks like nothing at all. Above the knee the slope is $1/L_{\mathrm{air}}$: every extra 0.1 pu of flux costs another 355 A. The marked point is the run's own crest, 1.69 pu and 1745 A, which is where the two curves meet.

Read the crest off that curve and the current follows arithmetically: 1.690 pu is 0.490 pu past the knee, which at 3550 A per pu is 1740 A, on top of the 4 A the core was already drawing at the knee. The simulation reports 1745 A. It is also a converged number, not a step-size artifact — refining from the sample's 50 µs to 12.5 µs moves the peak from 1745.0 A to 1744.8 A.

The decay

Per-cycle peak of each magnetizing current over the whole 8 s window. Phase A halves in 0.38 s, is under a tenth of its first peak by 2.0 s, and finishes the run at 8 A. The decay is visibly faster than exponential early on and slower than exponential late, because the mechanism switches off: the offset only bleeds while the crest is over the knee.

Nothing here is a fault. It is a perfectly healthy transformer drawing nearly five times rated current on its first cycle and staying above rated for a full second, which is why energization inrush drives transformer differential-relay settings and inrush-restraint logic.

The shape of the decay is worth a second look, because it is not a single exponential. What removes the DC offset is $I\cdot R$ in the source resistance and the winding resistance, and that only bites while there is a large $I$ — that is, while the crest is over the knee and the branch is drawing hundreds of amps. As the offset shrinks the crest sinks toward the knee, the pulses get smaller, and the mechanism damping the offset weakens along with them. It self-arrests: by 3 s the crest is down to 1.22 pu and the offset has all but stopped at 0.23 pu (eight seconds in they are 1.20 and 0.20). What is left to remove that residue is the unsaturated $L_m/R$ of the loop — roughly 140 H over 1.8 Ω, a time constant near 80 s — so on an 8 s window it looks frozen. That is physical, not a numerical artifact.

What saturation is actually contributing

The same energization with the transformer's saturation parameter on and off, on a log axis. Without saturation the flux still doubles — it reaches 1.998 pu — but a linear core answers a doubled flux with a doubled current, so the peak is 7.1 A instead of 1745 A: a factor of 250, from one nonlinearity. The two traces converge at the right-hand edge because by then the saturated run has fallen back onto the linear magnetizing branch.

What to try

Close at the voltage crest instead. Set the source's phase to 90° and phase A's inrush disappears completely — 3.6 A, indistinguishable from normal magnetizing current, because the flux starts at its own crest and never needs an offset. Phases B and C then take the hit instead, at 1446 A and 1466 A. You cannot null all three at once with a single simultaneous close; that is why controlled-switching schemes stagger the poles.

Ramp the source. A ramp_time of 40 ms brings the voltage up over a couple of cycles instead of snapping it on, and the flux tracks it without ever building an offset: the peak falls to 3.7 A on every phase. This is the standard way to suppress inrush in a study when it is not what you are looking at.

Try the exponential curve. Switch sat_definition to Exponential and the same energization peaks at only 347 A while the flux climbs to 1.94 pu. That is not a bug — it is the honest behaviour of a curve fitted between 1.0 and 1.2 pu. The exponential has no air-core asymptote, so extrapolated out to 1.9 pu it still reports an inductance far above air core and understates the inrush by a factor of five. Use it for steady-state over-excitation; use the two-slope curve for energization.

Change l_air_pu. It is the single number that sets the peak, and the relationship is close to inverse: 0.2 pu gives 1083 A, 0.1 pu gives 1745 A, 0.08 pu gives 1989 A. Air-core inductance for a power transformer typically lands between one and three times the short-circuit reactance, so for this unit's 0.1 pu leakage the plausible range is roughly 0.1 – 0.3 pu.

Stiffen the source. Dropping the source to 0.05 Ω + 1.4 mH raises the first peak slightly (1856 A) and, more interestingly, slows the decay: the run is still at 22 A after 8 s instead of 8 A, because source resistance is the main thing draining the offset.

The run is 8 s at a 50 µs step — 480 cycles, which is what it takes for the inrush to fall back to the transformer's ordinary magnetizing current.