Page 299 - Handbook of Electrical Engineering

FAULT CALCULATIONS AND STABILITY STUDIES 285

The term in brackets is again called the ‘doubling factor’ but it is now less than 2.0 when
t = π/ω. Table H.1b shows the doubling factor for different ratios of X to R.
√
Note: The doubling factor is sometimes combined with 2when V is given as the root-mean-square
value. In which case the doubling factor has a maximum value of 2.8284 and a minimum value
of 1.4142.

11.6.1.2 Resistance larger than inductive reactance

This case represents the least onerous duty for the switchgear. The angle φ becomes small as the
resistance increases. The worst-case switching angle θ approaches zero. The conditions that produce
a minimum or a maximum can be found by differentiating i in equation (11.5) with respect to the
time t and equating the result to zero. This yields the following conditions,

−Rt
+Re L −ω cos (ωt + θ − φ)
= (11.7)
L sin (θ − φ)

−Rt
When R>> L, e L approaches zero for t in the range of one or two periods.
The angle φ approaches zero.
Transposing equation (11.7) for the cosine term gives,

R
cos(ωt + θ) =− sin θ
ωL
−Rt
Where is the small value of e L , which approaches zero.
The right-hand side approaches zero as becomes very small. Therefore the left-hand side
becomes,
cos(ωt + θ) = 0

Now since θ also approaches zero cos ωt equals zero for the first time when ωt = π/2.
If the above conditions are substituted into (11.5) the current becomes,

V ˆ V ˆ
i = sin (ωt + (0 − 0)) = sin ωt
Z R
Which is in phase with V as can be expected. Note, the switching angle θ need not be zero
when the inductance is negligible, see Figure 11.9.

11.6.1.3 The doubling factor

The conditions given by equation (11.7) apply to all combinations of resistance and inductance, and
the switching angle θ. Equation (11.7) can be used with little error for cases where the resistance