An explanation of the origin of a generator’s subtransient reactance

The term “subtransient reactance” is denoted by the symbol X’’d and is used to calculate available short-circuit generator fault currents. But where does it come from? What’s its origin?

A generator’s short-circuit current is calculated from its subtransient reactance. The subtransient reactance is an impedance value that entirely neglects the resistance component.

This is justifiable based on the physical characteristics and construction of electric generators. The subtransient reactance is a transient effect that’s directly related to the electromagnetic relationships between the various physical components of the generator. The resistance of the windings of a synchronous generator are generally negligible compared to their reactance, but they do play a role in the decay rates of the transient currents in the form of L/R time constants.

THE TEXTBOOK EXPLANATION

During a short-circuit, the steady-state reactance is temporarily reduced due to the interaction of the magnetic flux between the damper windings and the armature windings. Just as a tuned-mass damper in a skyscraper stabilizes it from the effects of oscillations created by earthquakes, damper bars with damper windings are used to counteract the rotor’s tendency to stray from its synchronous speed (typically 60 Hz) when reacting to transient disturbances, such as load changes and short-circuits. The damper windings stabilize the rotor by inducing an electromagnetic torque that resists undesired rotor motion.

Though a damper bar benefits the machine by increasing its efficiency and stability, it also introduces a mutual inductance between it and the various other windings. The complex magnetic flux interactions created by the small resistances and inductances (self and mutual) between the various elements (damper windings, field windings, and rotor body) during a transient condition ultimately act to dramatically and suddenly reduce the machine’s reactance.

This reactance reduction is temporary, but its sudden drop allows much higher instantaneous short-circuit currents to form during the first few cycles of a fault. Even in machines that don’t have damper windings, during a transient condition time-changing magnetic fluxes induce currents directly in the rotor body that produce damping effects as if the machine did have damper windings.

Because of these complex physical and electromagnetic relationships, the subtransient reactance is typically determined by testing. White papers produced by generator manufacturers are usually a good source of information for learning about generator characteristics. They go into detail and provide specific subtransient reactance values for the generators they sell.

At the initiation of a fault, a generator’s current initially spikes to a very high value but then decays to steady-state after a few seconds. There are three main regions of time that mark-off the fault-current procession. The first is that which is governed by the sub-transient reactance, which lasts just a few cycles. The second is that which is governed by the transient reactance, which lasts several seconds. Finally, there is the steady-state region whose fault current is determined by the generator’s synchronous reactance. The current produced due to sub-transient reactance is relevant to choosing a circuit breaker’s instantaneous trip setting. The figure below illustrates the different components and the relative timeframes in which they dominate.

Click it for a larger (and readable) version.

subtransient reactance plot

There’s a good engineering book available that explains this entire concept in detail and develops the physical model and formula used to plot the graph shown here. It’s called Electric Machinery, 5th Ed., by Fitzgerald, Kingsley, and Umans (New York: McGraw Hill, 1980).  p.227. You can pick up a used copy on Amazon.

If you don’t want to buy the engineering book, you can find a pretty good explanation online published in a Cummins white paper. It also contains example calculations. It’s written by Timothy A. Loehlein and called “Calculating generator reactances.” Here’s the link:

http://cumminspower.eu/www/literature/technicalpapers/PT-6008-GeneratorReactances-en.pd

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