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

√

2whenVis 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 angleφbecomes small as the
resistance increases. The worst-case switching angleθapproaches 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,

+Re

−Rt

L

L

=

−ωcos(ωt+θ−φ)

sin(θ−φ)

( 11. 7 )

WhenR>>L,e

−Rt

L approaches zero fortin the range of one or two periods.

The angleφapproaches zero.

Transposing equation (11.7) for the cosine term gives,

cos(ωt+θ)=−

R

ωL

sinθ

Whereis the small value of e

−Rt

L , which approaches zero.

The right-hand side approaches zero asbecomes very small. Therefore the left-hand side
becomes,
cos(ωt+θ)= 0

Now sinceθalso approaches zero cosωtequals zero for the first time whenωt=π/2.

If the above conditions are substituted into (11.5) the current becomes,

i=

Vˆ

Z

sin(ωt+( 0 − 0 ))=

Vˆ

R

sinωt

Which is in phase withV as can be expected. Note, the switching angleθneed 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 angleθ. Equation (11.7) can be used with little error for cases where the resistance