# Why is there a Sub-transient, transient periods during a short-circuit event?

I am currently studying short-circuit analysis and it was mentioned that the short-circuit current can be divided into three periods, the sub-transient, transient and the steady state. However there are no explanations why the two first periods happen in the first place. The only explanation mentioned was:

Before the fault, only AC voltages and currents are present, but immediately after the fault, both AC and DC currents are present. And this DC component will quickly decay over time.

I understand the explanation nice and clear however, I have no clue where the DC component came from that causes the "spike" of current. So as the title says my question is Why is there a Sub-transient, transient periods during a short-circuit event?

You have two somewhat conflated ideas here. One is dc offset, and the other is generator decrement. I'll address dc offset first.

Considering a circuit like this below, where $$\v(t)=V_Msin(\omega t+\theta)\$$.

When the switch is closed, by KVL we have $$\v(t)=Ri+L\frac{di}{dt}\$$ which is a first order d.e. with solution:

where $$\\tau=\frac{L}{R}\$$ and $$\|Z|=\sqrt{R^2+(\omega L)^2}\$$.

The first term is the ac component of the resulting current, the second term is the dc component. Power systems are dominantly inductive, so if the switch is closed near a voltage minimum the current through the inductor should be near a maximum (if it where flowing). Closing at or near this time gives the largest dc offset.

If the switch happened to be closed at or near voltage maximum (where the current through the inductor should be at a current minimum if it where flowing) then little or no dc offset occurs.

The time constant is $$\\tau=\frac{L}{R}\$$ seconds and $$\\tau=\frac{\frac{X}{R}}{2\pi}\$$ in cycles. After one time constant the dc component has decayed to $$\\approx\$$37% of initial value. After 5 time constants it is essentially gone.

It is interesting to note that in real practice engineers do not simply use the $$\X\$$ and $$\R\$$ from the Thévenin equivalent, they use short-circuit programs that calculate these values from an "X-only" solution and an "R-only" solution. It turns out (refer to IEEE C37.010-1999, “Application Guide for AC High-Voltage Circuit Breakers Rated on a Symmetrical Current Basis") that calculating the $$\\frac{X}{R}\$$ ratio with this approach produces results closer to actual. The "X-only" network ignores all resistance and the "R-only" network ignores all inductance.

Generator Decrement - there is also the phenomenon of generator decrement to consider. The figure below is from page 258 of "Symmetrical Components for Power Systems Engineering", by J.L. Blackburn. The bottom 3 traces show the phase currents from the synchronous generator. You have the dc offset as previously described, but you also have a decreasing amplitude of ac as time progresses.

The subtransient component of fault current is dictated by the subtransient reactance, $$\{{X^"}_d}\$$. The air-gap flux at the moment of fault application cannot change immediately due to features of the machine (damper windings etc. - see p. 262 of above referenced text). This time frame is relatively short lived though, on the order of 0.01 to 0.05 seconds.

The transient component of fault current is dictated by the transient reactance, $$\X_d^{'}\$$. The fault current flowing tends to demagnetize the field and decrease flux linkage between the stator and field winding. This time frame is longer duration than the subtransient and lasts on the order of 0.35 to 3.3 seconds.

The steady-state component of fault current is dictated by the synchronous reactance, $$\X_d\$$. This period begins after the dynamics of the transient period are largely over. It is possible for a modern machine to have an $$\X_d\$$ larger than 1.0 pu so that fault current from the machine during this period is actually lower than full load current from the machine.

The specifics for calculating the decrement depends on many things and is well beyond answering in a post (things like field-forcing, magnetic circuits etc.). The above referenced text covers this well as does the classic "Power System Stability and Control" by Prahba Kundur.

• Good point for the example ... Commented Mar 20, 2022 at 20:45

I assume you are talking about short circuits in AC distribution (like symmetric components techniques and so on). The DC component (the sub transient) come from the stored energy in the system, both load and wires (wires are not negligible inductors in power distribution). It also varies depending on the angular position of the phase during the short.

The duration of the "phases" is linked to the reactances involved.
There are in fact three types of reactances:
sub transient X"d ("much" shorter than the X'd), transient X'd, and Xd.

Short-circuit current can be written as this : link

Sorry, only in french ... use "Google translate".

From this

I found also this

And this : Short-circuits in power systems a practical guide to iec-60909, Kasikci. §9-2-1