Current Interruption Transients Calculation, 2nd Edition

Appendix EAsymmetrical Current‐Calculating Areas Under Curves

Maximum asymmetrical current is given by Eq. (4.23) which, to account for polarity, can be written as:

where I is the circuit breaker short‐circuit current rating and τ is the system DC time constant. The area under any loop of current can be obtained by integration of i_max between the two applicable current zero crossing times t₁ and t₂

(E.1)

Taking the case of 63 kA, 50 Hz and τ = 45 ms and the major loop shown in Figure 4.37, t₁ and t₂ are 23 ms and 36.7 ms, respectively. Using the negative version of Eq. (E.1), the reader can readily determine that the loop area is 1.105 kAs.

Having plotted the asymmetrical current as shown for the example in Figure 4.35, the area under any particular loop of current can be estimated using the so‐called Trapezoidal Rule. With reference to Figure E.1, any area under a curve can be split up into a series of trapezoids of width t_s and the area A_TR under the curve is the combined areas of the individual trapezoids:

(E.2)

A second and more subtle approach is the use of Simpson's Rule. Simpson proposed the area under a curve could be approximated by means of the parabolic equation as ...