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Theoretical foundations of electrical engineering

# Problems solving in electrical engineering - General form of the transient process of the second order (aperiodic, vibrational, critical)

The nature of the transient process at two complex conjugate roots (the transient process of the second order) depends on the roots of the characteristic equation. There are three possible cases:

1. Complex roots are pairwise conjugated. For example, if p1=-δ+jω0, then p2=-δ-jω0. In this case the solution of the desired voltage or current is found as follows:

x(t)=A·e-δt·sin(ω0t+φ)+xуст

In this case, there is an oscillatory transient process. This characteristic describes damped sine wave at an angular frequency ω0 and the initial phase φ. The envelope of the oscillations is described by a curve A·e-δt. The higher δ is, the more likely the oscillation process will end.

It should be noted that the integration constants A and φ are determined by the parameters of the circuit and the initial conditions of the transient process, and δ and ω0 depend only on the parameters of the circuit after the switching time. ω0 - angular frequency of free oscillations; δ - damping factor.

2. Roots of the characteristic equation are real negative and different. For example, p1=-a and p2=-b.

Then there is the so-called aperiodic transient process. Its solution is found in the form of:

x(t)=A1·ep1·t+A2·ep2·t+xуст

This transient process is characterized that in it there are no vibrations.

3. The roots of the characteristic equation are negative and equal.

In this case, there is a so-called critical transient process, which is an intermediate case between options 1 and 2. Its solution is found in the form of:

x(t)=A1·ep·t+A2·t·ep·t+xуст

Diagrams showing the essence of the different types of transient processes of the second order are presented below:

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