# Discharging capacitor - transient state

Electrical circuit is built with:
• voltage source E
• resistor R
• switches X1, X2
• capacitor C
Capacitor was being charged with to its full capacity and it is in stable state. Capacitor initial voltage is equal to voltage source UC0=E.

Switch X2 is being closed and switch X1 is being opened, both switches change their state in the same time. As a result capacitor's discharging current i(t) starts to flow in circuit. Immediately after closing the switch X2 the transient state occurs in electrical circuit, therefore, commutation principles have to taken under consideration. Kirchhoff's voltage law equation will be written for circuit. General formula for capacitor's current will be taken under consideration.

Kirchhoff's voltage law equation will be rewritten to create differential equation.

Differential equation with separated variables which has to be solved. To solve this differential equation substitution method will be applied.

## Capacitor's discharging voltage

After solving differential equation a relation for capacitor's voltage uC(t) is obtained. Formula is related to capacitor discharging time's transient state. Capacitor discharging circuit's time constant appears in formula → T=R·C.

## Capacitor's discharging current

To obtain formula for capacitor's current iC(t) a derivative of capacitor's voltage has to be computed → iC(t)=C·duC(t)/dt. Formula is related to capacitor charging time's transient state. Capacitor discharging circuit's time constant appears in formula → T=R·C.

## Capacitor's discharging voltage - time characteristic

Capacitor's discharging voltage uC(t) received from simulation for following elements values:
• voltage source E=12V
• resistor R=1kΩ
• capacitor C=1μF
Characteristic could be divided into two stages. The first stage represents transient state of capacitor's voltage uC(t) during charging until capacitor's voltage is equal to power supply voltage. The second stage represents transient state of capacitor's voltage uC(t) during discharging the capacitor. Both stages last the same period of time. During discharging process capacitor releases energy stored between its electrodes. Capacitor has ability to store energy in electric field. Both stages last the same period of time.

## Capacitor's discharging current - time characteristic

Capacitor's charging current iC(t) received from simulation for following elements values:
• voltage source E=12V
• resistor R=1kΩ
• capacitor C=1μF
Characteristic could be divided into two stages. The first stage represents transient state of capacitor's current iC(t) during charging until capacitor's voltage is equal to power supply voltage. The second stage represents transient state of capacitor's current iC(t) during discharging the capacitor. Reader should note that capacitor's current flows in opposite direction during discharging than it did during charging. During discharging process capacitor releases energy stored between its electrodes. Capacitor has ability to store energy in electric field. Both stages last the same period of time.