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Theoretical foundations of electrical engineering

Solution of the transient processes with the help of the Duhamel’s integral

Problem situation:
"The single impulse of the voltage operates at terminals of the circuit. It is necessary to find with the help of the Duhamel’s integral a transient current in one of branches of the given circuit, arising under the influence of the voltage pulse and calculated transient current".

Solution of the transient processes with the help of the Duhamel’s integral

Given
R=50 ohms;
L=100 mH=0,1 H.

Find
i2(t) — ?

Solution

We find the transient function of the branch conductivity with the inductor:
Solution of the transient processes with the help of the Duhamel’s integral

We find the current in the branch with the inductor when you turn on single voltage source:


We find the constant A from initial conditions. According to the first switching law – the current through the inductor cannot change abruptly. Therefore, if there is no current in the inductor before the power supply, so there will be no current and in the time point t=0.


We find characteristic impedance of the circuit p:


We substitute all known values in the formula for the current:
Solution of the transient processes with the help of the Duhamel’s integral

We write down the input function of the voltage in the form of a formula after breaking it in two parts:


Current in the inductor for the first time period:


Current in the inductor for the second time period:
Solution of the transient processes with the help of the Duhamel’s integral

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