1. Lesson 12: Capacitors Transient Analysis

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

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 Voltage and Current Relationship
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 the switch is closed to 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 reduces the voltage across the resistor, therefore current in the circuit drops.

9. Capacitor Charging Equations
Voltages and currents in a charging circuit change exponentially over time.
Voltages Increases Exponentially Over Time
Current Decreases Exponentially Over Time

10. Steady State Condition (Fully Charged)
When a circuit is at steady state:
The voltage and current reach their final values and stop changing.
A capacitor, at steady state, will have voltage across it, but no current will flows through the capacitor.
Capacitor ‘looks’ (electrically) like an open circuit.

11. The Time Constant
The rate at which a capacitor charges and discharges depends on resistance (R) and capacitance (C).
This discharge rate is called the TIME CONSTANT (τ):
As the voltage increases, there will be transients, which 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.
c) V

13. Capacitor Discharging
When a capacitor is initially fully charged:
It acts like an open circuit.
When the switch is moved to the discharge position, the current instantaneously drops to

14. Capacitor Discharging
As charge dissipates from the capacitor, the voltage reduces across the capacitor.
This makes the voltage across the resistor go down, so current in the circuit reduces until the point that the capacitor is fully discharged (V=0 and I=0).

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:
Now you can write the charging equation using the Thèvenin Voltage.

19. More Complex Circuits
The discharge portion of the circuit operates the same as 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 circuit below will not have the same charging equation due to the voltage divider in the system.
After the switch is shut: (CHARGING)
a. Transform the circuit into the Thèvenin equivalent as seen by the capacitor.
b. Determine the charging constant.
c. Write the charging equation for vc(t).
1.82kΩ
40V

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. 2
20kΩ

22. QUESTIONS?