# Transient processes of the first order, solution example

**Given**

E

_{0}=100 V;

a·τ=2;

L=2 mH=0,02 H;

C=10 μF=10

^{-5}F;

R

_{1}=1 ohm;

R

_{2}=4 ohms;

R

_{3}=1 ohm.

Find

u

_{C}(t)-?(numerically and diagram);

## Solution

Find characteristic resistance of the circuit p, for this purpose we equate to zero the input resistance:

Circuit time constant:

Find steady voltage across the capacitor. At constant value of electromotive force the capacitor is equal to the break – therefore, current does not flow in the circuit: u

Find the voltage value across the capacitor before switching. The circuit is open, there are no energy sources in it , therefore, all voltage drops on elements are equal to zero: u

General solution for voltage across the capacitor can be written in the form:

Find constant of integration A from the initial conditions:

All unknown are defined, it is possible to write the answer:

Find the limit of the diagram construction:

The dotted line showed electromotive force value in the circuit.

Circuit time constant:

Find steady voltage across the capacitor. At constant value of electromotive force the capacitor is equal to the break – therefore, current does not flow in the circuit: u

_{C уст}=E;Find the voltage value across the capacitor before switching. The circuit is open, there are no energy sources in it , therefore, all voltage drops on elements are equal to zero: u

_{C}(0_{-})=u_{C(0)=0;}General solution for voltage across the capacitor can be written in the form:

Find constant of integration A from the initial conditions:

All unknown are defined, it is possible to write the answer:

Find the limit of the diagram construction:

_{max}=4τ=4·2·10-5=8·10-5 с=80 мкс;The dotted line showed electromotive force value in the circuit.