1	Chapter 11 Capacitive Charging, Discharging, and Simple Waveshaping Circuits
2	Introduction Circuit Capacitor charging and discharging Transient voltages and currents result when circuit is switched
3	Introduction
4	Capacitor Charging Charging a capacitor that is discharged When switch is closed, the current instantaneously jumps to E/R Exponentially decays to zero When switching, the capacitor looks like a short circuit
5	Capacitor Charging Voltage begins at zero and exponentially increases to E volts Capacitor voltage cannot change instantaneously
6	Capacitor Charging Capacitor voltage has shape shown:
7	Steady State Conditions 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
8	Capacitor Discharging Assume capacitor has E volts across it when it begins to discharge Current will instantly jump to –E/R Both voltage and current will decay exponentially to zero
9	Capacitor Discharging Here are the decay waveforms:
10	Capacitor Charging Equations Voltages and currents in a charging circuit do not change instantaneously These changes over time are exponential changes
11	Capacitor Charging Equations Equation for voltage across the capacitor as a function of time is
12	Capacitor Charging Equations Voltage across resistor is found from KVL: E - v_C The current in the circuit is
13	Capacitor Charging Equations Values may be determined from these equations Waveforms are shown to right
14	The Time Constant Rate at which a capacitor charges depends on product of R and C Product known as time constant t = RC t (Greek letter tau) has units of seconds
15	Duration of a Transient Length of time that a transient lasts depends on exponential function e^-^t^/^t As t increases Function decreases When the t reaches infinity, the function decays to zero
16	Duration of a Transient For all practical purposes, transients can be considered to last for only five time constants
17	Capacitor with an Initial Voltage Voltage denoted as V₀ Capacitor has a voltage on it Voltage and current in a circuit will be affected by initial voltage
18	Capacitor with an Initial Voltage
19	Capacitor Discharging Equations If a capacitor is charged to voltage V₀ and then discharged, the equations become
20	Capacitor Discharge Equations Current is negative because it flows opposite to reference direction Discharge transients last five time constants All voltages and currents are at zero when capacitor has fully discharged
21	Capacitor Discharge Equations Curves shown represent voltage and current during discharge
22	More Complex Circuits You may have to use Thévenin’s theorem (those with multiple resistors) Remove capacitor as the load and determine Thévenin equivalent circuit
23	More Complex Circuits Use R_Th to determine t t = R_Th∙C Use E_Th as the equivalent source voltage
24	An RC Timing Application RC circuits Used to create delays for alarm, motor control, and timing applications Alarm unit shown contains a threshold detector When input to this detector exceeds a preset value, the alarm is turned on
25	An RC Timing Application
26	Pulse Response of RC Circuits Pulse Voltage or current that changes from one level to another and back again Periodic waveform Pulse train is a repetitive stream of pulses
27	Pulse Response of RC Circuits Square wave Waveform’s time high equals its time low Length of each cycle of a pulse train is its period
28	Pulse Response of RC Circuits Number of pulses per second is its pulse repetition frequency Width of pulse compared to its period is its duty cycle Usually given as a percentage
29	Pulse Response of RC Circuits Pulses have a rise and fall time Because they do not rise and fall instantaneously Rise and fall times are measured between the 10% and 90% points
30	The Effect of Pulse Width Width of pulse relative to a circuit’s time constant Determines how it is affected by an RC circuit If pulse width >> 5t Capacitor charges and discharges fully With the output taken across the resistor, this is a differentiator circuit
31	The Effect of Pulse Width If pulse width = 5t Capacitor fully charges and discharges during each pulse If the pulse width << 5t Capacitor cannot fully charge and discharge This is an integrator circuit
32	Simple Waveshaping Circuits Circuit (a) provides approximate integration if 5t >>T Circuit (b) provides approximate differentiation if T >> 5t
33	Simple Waveshaping Circuits
34	Capacitive Loading Capacitance Occurs when conductors are separated by insulating material Leads to stray capacitance In high-speed circuits this can cause problems