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Theoretical foundations of electrical engineering

Transient processes of the second order, example of solution – Classical and operator methods

Given
R=5 ohms;
L=0,01 H;
C=2·10-4 F;
E=98 V;

Find
iC (t)—?(classical method)
iC (t)—?(operator method — transient components)
iC (t)—?(diagram)

Solution

Circuit arrangement:
Переходный процесс второго рода (второго порядка)

Classical method.

The calculation of the characteristic impedance of the circuit p.
We find the input impedance of the circuit and equate it to zero. By finding the input impedance it is assumed that all switching occurred, the electromotive force is replaced with the short circuit, and current sources are replaced with the break. Then any point of the scheme is transformed into two terminals and we find concerning them the circuit resistance:
Transient processes of the second order solution using Classical method
Transient processes of the second order solution using Classical method
The type of the transient process at complex-conjugate roots is defined by the formula:

We find final values of the required parameters. We construct an equivalent diagram after long time. The key worked, all transient processes are complete. At DC the inductor is equal to the short circuit, the capacitor is equal to the break:
Transient processes of the second order solution using Classical method
Final value of the unknown current:
ic уст=0 А Analysis of the condition before switching for finding the conserved quantities: the voltage across the capacitor and current through the inductor.
At DC the inductor is equal to the short circuit; the capacitor is equal to the break. Circuit arrangement before switching:
Transient processes of the second order solution using Classical method

Analysis of the initial conditions of the zero order. We consider the time right after the switching.
Transient processes of the second order solution using Classical method
It is possible to write down according to the laws of switching:
ILafter switching0=IL before switching=6,53 А;
UCafter switching0=UC before switching=65,33 V;

Тогда:
IR after switching0=UC after switching0/R=65,33/5=13,07 А;
IC after switching0=IL after switching0-IR after switching0=6,53-13,07=-6,53 А;
UL after switching0=E-RIL after switching0-UC after switching0=98-5·6,53-65,33 =0 V.
Analysis of initial conditions of the first order (derivatives from initial values). We consider the time right after the switching. Since the electromotive force is constant and equivalent diagram is constructed for derived values, so the source will be absent.
Transient processes of the second order solution using Classical method
Transient processes of the second order solution using Classical method
Definition of the independent constants from initial conditions. We write down the formula for the current through the capacitor for initial time. We write down also the formula for the current derivative through the capacitor in initial time. We solve the system of equations and find unknown A and φ. iC(0)=-6,53 А; iC'(0)=6530 А⁄s; iC уст=0 А; Transient processes of the second order solution using Classical method
From the second equation follows:

Operator method.

We construct the operator equivalent circuit:
Transient processes of the second order solution using the operator method
We make Laplace transformation for electromotive force sources:
E(t)=98;
E(p)=98/p;
We take the initial values for the current through the inductor and voltage across the capacitor from the previous calculation:
IL(0)=6,53 А;
UC(0)=65,33 V;
We find the potential of point a by the nodal pair method.
We mark the non-grounded node "a", then:
Y·Ua=I;
Ua=I/Y,where I — entering currents in a node, and Y — conductance of branches.
Transient processes of the second order solution using the operator method
Transient processes of the second order solution using the operator method
Transient processes of the second order solution using the operator method
Using the formula of the supplementary angle:

According to the task it is necessary to find only free current components, but because its final value is equal to zero, then the total current is received. The characteristic of the capacitive currents coincides with the received characteristic by a classical method that confirms the correctness of calculations::
Diagram of Transient processes of the second order
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