Chapter 12

# Charging a capacitor with a voltage source

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

For , the circuit is at rest and is for and for .

Charging a capacitor with a voltage source

## Question

We are interested in the circuit just after application of voltage ; determine , , and .

### Solution

It is known that the voltage and the charge of a capacitor are continuous functions. Therefore :

The Kirchhoff's Voltage Law (CVL) and Kirchhoff's Current Law (KCL) then gives :

## Question

We are interested in the permanent regime ; determine .

### Solution

The permanent regime, , then :

## Question

Determine expression of .

### Solution

The KCL and KVL allow you to write :

We can deduce the differential equation checked by  :

The voltage across the capacitor is then given by taking into account the initial conditions :

With :

And :

## Question

Make an energy balance.

### Solution

The energy balance is :

It merely reflects that the energy supplied by the generator ( ) during the charging of the capacitor is found only partially stored by the capacitor ; a part is in fact dissipated by Joule effect in the two resistors.

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Study of a RLC circuit - Impedances