1. Lesson 15 – Capacitors Transient Analysis

2. Learning Objectives
Calculate capacitor voltage and current as a function of time.
Explain Capacitor DC characteristics.

3. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE
The placement of charge on the plates of a capacitor does not occur instantaneously.
Instead, it occurs over a period of time determined by the components of the network. This period of time is called the Transient Phase.

4. Capacitor Current and Voltage
Capacitor v-i relationship

5. Capacitor Current and Voltage
The charge on a capacitor is given by:
Current (iC) is the rate of flow of charge:
Current through a capacitor is equal to C times the rate of change of voltage across it.

6. Circuit Analysis (for Physics Majors)
Using KVL:
Substituting in using ohm’s law and the capacitor current relationship:
Using Calculus:

7. Capacitor charging Capacitor is initially fully discharged
acts like a short circuit
When switch is closed (position 1), the current instantaneously jumps to:

8. Capacitor charging
As charge is stored in the capacitor, the voltage across the capacitor starts to rise.
This makes the voltage drop across the resistor drop, so current in the circuit drops

9. Capacitor Charging Equations
Voltages and currents in a charging circuit change exponentially over time

10. Steady State Condition (Fully Charged)
Circuit is at steady state
When voltage and current reach their final values and stop changing
Capacitor has voltage across it, but no current flows through the circuit
Capacitor looks like an open circuit

11. The Time Constant
Rate at which a capacitor charges and discharges depends on R and C, which is called the TIME CONSTANT:
Transients can be considered to last for five time constants

12. Example Problem 1
The capacitor in the circuit below is initially uncharged.
After the switch is shut:
a. determine how long it will take for the capacitor to reach a steady-state condition (>99% of final voltage). b. Write the equation for vc(t).
c. Sketch the transient.

13. Capacitor Discharging
Capacitor is initially fully charged
acts like a open circuit
When switch is moved to discharge, the current instantaneously jumps to -E/R

14. Capacitor Discharging
As charge flows out of the capacitor, the voltage across the capacitor drops.
This makes the voltage drop across the resistor drop, so current in the circuit drops until the capacitor is fully discharged

15. Capacitor Discharging Equations
Voltages and currents in a discharging circuit also change exponentially over time

16. More complex circuits
If the circuit does not look like the simple charge-discharge circuit, then you will need to use Thèvenin's Equivalent to make it into the simple circuit.
The circuit below does not have the same charging equation as the previous circuits, since the voltage drop across the capacitor is controlled by the voltage divider circuit.

17. More complex circuits
Thèvenin's Equivalent of charging circuit:

18. More complex circuits
Now you can calculate the charging time constant using the Thèvenin Equivalent resistance.
You write the charging equation using Thèvenin Voltage.

19. More complex circuits
The discharge portion of the circuit operates the same as we previously analyzed.
The steady-state (fully charged) voltage across the capacitor can be determined by the VDR (this is the Thèvenin voltage found earlier).

20. Example Problem 2
The capacitor in the circuit below is initially at steady state with the switch open and capacitor fully discharged.
After the switch is shut: (CHARGING)
a. determine how long it will take for the capacitor to fully charge (>99% of final voltage).
b. Write the equation for vc(t). Sketch the transient.

21. Example Problem 2b
The capacitor is now fully charged and at steady-state condition. The switch is opened to start the discharge cycle.
After the switch is open:(DISCHARGING)
a. determine how long it will take for the capacitor to fully discharge .
b. Identify the direction of current flow.
c. Write the equation for vc(t). Sketch the transient.