# 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 p_{1}=-δ+jω_{0}, then p_{2}=-δ-jω_{0}.
In this case the solution of the desired voltage or current is found as follows:

x(t)=A·e^{-δt}·sin(ω_{0}t+φ)+x_{уст}

In this case, there is an **oscillatory transient process**. This characteristic describes damped sine wave at an angular frequency ω_{0} 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 ω_{0} depend only on the parameters of the circuit after the switching time. ω_{0} - angular frequency of free oscillations; δ - damping factor.

2. Roots of the characteristic equation are real negative and different. For example, p_{1}=-a and p_{2}=-b.

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

x(t)=A_{1}·e^{p1·t}+A_{2}·e^{p2·t}+x_{уст}

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)=A_{1}·e^{p·t}+A_{2}·t·e^{p·t}+x_{уст}

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