Current Interruption Transients Calculation, 2nd Edition

Appendix ADifferential Equations

Current interruption transient oscillations in AC power systems can be treated by considering four basic RLC circuits as shown in Figure 2.1. The oscillations or other transient events that occur in these circuits due to sudden changes are described by second‐order differential equations. The equations have a common format and, rather than resorting to Laplace transforms in each case, it is possible to use a purely mathematical approach in general and then apply the general solutions to each transient case.

Two types of differential equations are applicable to the circuits shown in Figure 2.1. The first type is a second‐order linear homogeneous equation with constant coefficients which takes the form:

(A.1)

The solution of this equation must include two arbitrary constants and, therefore, in general is:

(A.2)

where k₁ and k₂ are the arbitrary constants dependent on the boundary conditions for the circuit under consideration.

The exponential function x(t) = e^rt is a solution for Eq. (A.1) and substitution gives:

(A.3)

Equation (A.3) is the so‐called auxiliary or characteristic equation and has two roots, r₁ and r₂:

(A.4)

(A.5)

and Eq. (A.2) becomes ...