# 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".

**Given**

R=50 ohms;

L=100 mH=0,1 H.

**Find**

i

_{2}(t) — ?

## Solution

We find the transient function of the branch conductivity with the inductor:

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

We find characteristic impedance of the circuit p:

We substitute all known values in the formula for the current:

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:

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:

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: