Chapter 11

Capacitive Charging, Discharging, and Simple Waveshaping Circuits

Introduction

Circuit

Capacitor charging and discharging

Transient voltages and currents result when circuit is switched

Introduction

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

Capacitor Charging

Voltage begins at zero and exponentially increases to E volts

Capacitor voltage cannot change instantaneously

Capacitor Charging

Capacitor voltage has shape shown:

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

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

Capacitor Discharging

Here are the decay waveforms:

Capacitor Charging Equations

Voltages and currents in a charging circuit do not change instantaneously

These changes over time are exponential changes

Capacitor Charging Equations

Equation for voltage across the capacitor as a function of time is

Capacitor Charging Equations

Voltage across resistor is found from KVL: E - v_C

The current in the circuit is

Capacitor Charging Equations

Values may be determined from these equations

Waveforms are shown to right

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

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

Duration of a Transient

For all practical purposes, transients can be considered to last for only five time constants

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

Capacitor with an Initial Voltage

Capacitor Discharging Equations

If a capacitor is charged to voltage V₀ and then discharged, the equations become

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

Capacitor Discharge Equations

Curves shown represent voltage and current during discharge

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

More Complex Circuits

Use R_Th to determine t

t = R_Th∙C

Use E_Th as the equivalent source voltage

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

An RC Timing Application

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

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

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

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

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

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

Simple Waveshaping Circuits

Circuit (a) provides approximate integration if 5t >>T

Circuit (b) provides approximate differentiation if T >> 5t

Simple Waveshaping Circuits

Capacitive Loading

Capacitance

Occurs when conductors are separated by insulating material

Leads to stray capacitance

In high-speed circuits this can cause problems


	Capacitive Charging, Discharging, and Simple Waveshaping Circuits


	Circuit
		Capacitor charging and discharging
		Transient voltages and currents result when circuit is switched


	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


	Voltage begins at zero and exponentially increases to E volts
	Capacitor voltage cannot change instantaneously


	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


	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


	Voltages and currents in a charging circuit do not change instantaneously
	These changes over time are exponential changes


	Equation for voltage across the capacitor as a function of time is


	Voltage across resistor is found from KVL: E - v_C

	The current in the circuit is


	Values may be determined from these equations
	Waveforms are shown to right


	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


	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


	For all practical purposes, transients can be considered to last for only five time constants


	Voltage denoted as V₀
		Capacitor has a voltage on it
	Voltage and current in a circuit will be affected by initial voltage


	If a capacitor is charged to voltage V₀ and then discharged, the equations become


	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


	You may have to use Thévenin’s theorem (those with multiple resistors)
	Remove capacitor as the load and determine Thévenin equivalent circuit


	Use R_Th to determine t
	t = R_Th∙C
	Use E_Th as the equivalent source voltage


	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


	Pulse
		Voltage or current that changes from one level to another and back again
	Periodic waveform
		Pulse train is a repetitive stream of pulses


	Square wave
		Waveform’s time high equals its time low
	Length of each cycle of a pulse train is its period


	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


	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


	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


	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


	Circuit (a) provides approximate integration if 5t >>T
	Circuit (b) provides approximate differentiation if T >> 5t


	Capacitance
		Occurs when conductors are separated by insulating material
		Leads to stray capacitance
		In high-speed circuits this can cause problems