Page 502 - Handbook of Electrical Engineering

492 HANDBOOK OF ELECTRICAL ENGINEERING

In a multi-machine network the generators and motors should be considered in relation to the
‘source impedance’ to which they are connected. This impedance will also be dependent upon the
location and type of disturbance e.g. near to a generator, remote from a generator, three-phase fault,
line-to-ground fault, change in the state of the load such as starting a large motor direct-on-line.
The following discussion applies to a synchronous machine that has one field and two damper
windings.
There are various methods of solving the equations for a three-phase short circuit on the
basic that the set of equations are linear and where the use of Laplace transforms, or the Heaviside
calculus, is appropriate. See References 3, 5, 6 and 8 for examples. These methods are complicated
and appropriate assumptions concerning the relative magnitudes of resistances, inductances and time
constants need to be made in order to obtain practical solution. The relative magnitudes of the
parameters are derived from typical machinery data. Adkins in Reference 3 gives a solution of the
following form,

√ 1 1 1 −t 1 1 −t

i a = 2 V o/c + − e T d + − e T d cos(ωt + θ)

X d X d X d X d X d

X + X q −t (X − X ) −t
q
d
d
− e Ta cos θ − e Ta cos(2ωt + θ)

2X X 2X X

d q d q
√
= 2 V o/c (A + B + C + D + E)
Where A, B and C are the fundamental frequency synchronous, transient and sub-transient
AC components,
E is due to the sub-transient saliency and contributes a small double frequency component,
usually small enough to be neglected.
D is the DC offset caused by the switching angle θ and the values of the sub-transient
reactances.
θ is the angle of the open-circuit sinusoidal terminal voltage when the short circuit is applied.
All the reactances and time constants are the same as those defined in sub-section 20.2.1g)
and h)
In a situation where the disturbance is remote from the machine the short circuit time constants

and the derived reactances X d , X , X , X q ,(X ), X and X 2 are all functions of the external reactance

d d q q
X e since it should be added to X a . Likewise R e should be added to R a . R a does not appear in the
time constants except for T a .
An example of the decrement in the short-circuit current for a synchronous generator is given
in sub-section 7.2.10 where its relevance to switchgear is described.
The worst-case situation for calculating the fault current in phase A is when the switching
angle θ is zero, the DC offset is then at its maximum value.
The above expression is adequate for data that are typically available for the industry. The
armature resistance R a is only present in the time constant T a . (Krause offers a more complete
solution in which the omission of R a is minimised. The effect is then to modify the time constant T a