Chapter 11 Capacitive Charging, Discharging, and Waveshaping Circuits

Introduction Circuit for studying capacitor charging and discharging. Transient voltages and currents result when the circuit is switched.

Capacitor Charging For an uncharged capacitor, at the instant the switch is closed, the current jumps to E/R, then decays to zero. At the instant of switching, the circuit looks like a short circuit. The voltage across the capacitor begins at zero and gradually climbs to E volts. The capacitor voltage cannot change instantaneously.

Capacitor Charging Because the voltage cannot change instantaneously, the graphs have the shapes shown:

Steady State Conditions When the voltage and current reach their final values and stop changing, the circuit is at steady state. The capacitor has voltage across it, but no current flows through the circuit. The capacitor looks like an open circuit to steady state dc.

Capacitor Discharging Assume the capacitor has E volts across when it begins to discharge. The current will jump to -E/R. Both voltage and current will decay to zero.

Capacitor Discharging Here are the decay waveforms:

Capacitor Charging Equations The voltages and currents in a charging circuit do not change instantaneously. These changes over time are exponential changes. The equation for voltage across the capacitor over time is

Capacitor Charging Equations The voltage across the resistor is found from KVL: E - v C. The current in the circuit is

Capacitor Charging Equations Values at any time may be determined from these equations. The waveforms are shown:

The Time Constant The rate at which a capacitor charges depends on the product of R and C. This product is known as the time constant.  = RC  has units of seconds.

Duration of a Transient The length of time that a transient lasts depends on the exponential function e -t/ . As t increases, the function decreases, and when the t reaches infinity, the function decays to zero. For all practical purposes, transients can be considered to last for only five time constants.

Capacitor with an Initial Voltage If the capacitor already has a voltage on it, this voltage is denoted as V 0. The voltage and current in a circuit will be affected by the initial voltage

Capacitor Discharging Equations If a capacitor is charged to voltage V 0 and then discharged, the equations become

Capacitor Discharge Equations Note that the current is negative because it flows opposite to the reference direction. As for the charging phase, discharge transients last five time constants. All voltages and currents are at zero when the capacitor has fully discharged.

Capacitor Discharge Equations The curves shown represent voltage and current during discharge:

More Complex Circuits For complex circuits (those with multiple resistors), you may have to use Thévenin’s theorem. Remove the capacitor and determine the Thévenin equivalent of the circuit. Use R Th to determine . Use E Th as the equivalent source voltage.

An RC Timing Application RC circuits are used to create delays for alarm, motor control, and timing applications. The alarm unit shown contains a threshold detector, and when the input to this detector exceeds a preset value, the alarm is turned on.

Simple Waveshaping Circuits Circuit (a) provides approximate integration if 5  >>T. Circuit (b) provides approximate differentiation if 5  >>T.