Lesson 15 – Capacitors Transient Analysis

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

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.

Capacitor Current and Voltage Capacitor v-i relationship

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.

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

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

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

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

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

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

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.

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

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

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

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.

More complex circuits Thèvenin's Equivalent of charging circuit:

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

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).

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.

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.