Chapter
22
RC and L/R Time Constants
Topics Covered in Chapter 22
22-1: Response of Resistance Alone
22-2: L/R Time Constant
22-3: High Voltage Produced by Opening an RL Circuit
22-4: RC Time Constant
22-5: RC Charge and Discharge Curves
22-6: High Current Produced by Short-Circuiting RC Circuit
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered in Chapter 22
 22-7: RC Waveshapes
 22-8: Long and Short Time Constants
 22-9: Charge and Discharge with Short RC Time
Constant
 22-10: Long Time Constant for RC Coupling Circuit
 22-11: Advanced Time Constant Analysis
 22-12: Comparison of Reactance and Time Constant
McGraw-Hill
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
22-1: Response of Resistance Alone
 Resistance has only opposition to current.
 There is no reaction to a change.
 R has no concentrated magnetic field to oppose a
change in I, like inductance, and no electric field to
store charge that opposes a change in V, like
capacitance.
22-1: Response of Resistance Alone
 When the switch S is closed in Fig. 22-1 (a), the battery supplies 10 V
across the 10-Ω R and the resultant I is 1 A.
 The graph in Fig. 22-1 (b) shows that I changes from 0 to 1 A instantly
when the switch is closed.
Fig. 22-1:
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22-2: L/R Time Constant
 The action of an RL circuit during the time current




builds up to a specific value is its transient response.
Transient response is a temporary condition that
exists only until the steady-state current is reached.
The transient response is measured in terms of the
ratio L/R, which is the time constant T of an inductive
circuit.
T = L/R
The time constant is a measure of how long it takes
the current to change by 63.2%.
22-2: L/R Time Constant
 When S is closed, the current changes as I increases from zero.
 Eventually, I will reach the steady value of 1 A, equal to the battery
voltage of 10 V divided by the circuit resistance of 10 Ω.
Fig. 22-2:
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22-3: High Voltage Produced by
Opening an RL Circuit
 When an inductive circuit is opened, the time constant
for current decay becomes very short because L/R
becomes smaller with the high resistance of the open.
 Then the current drops toward zero much faster than
the rise of current when the switch is closed.
 The result is a high value of self-induced voltage
across a coil whenever an RL circuit is opened.
 This high voltage can be much greater than the
applied voltage.
22-3: High Voltage Produced by
Opening an RL Circuit
 In Fig. 22-3, the neon bulb requires 90 V for ionization, at which time it
glows.
 The source is only 8 V, but when the switch is opened, the self-induced
voltage is high enough to light the bulb for an instant.
The sharp voltage pulse or spike is more than 90 V just after the switch is
opened.
Fig. 22-3:
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22-4: RC Time Constant
 The transient response of capacitive circuits is
measured in terms of the product R x C.
 To calculate the time constant,
T=RxC
where R is in ohms, C is in farads, and T is in seconds.
22-4: RC Time Constant
T = 3 x 106 x 1 x 10−6
=3s
Fig. 22-4:
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22-5: RC Charge and
Discharge Curves
 In Fig. 22-4, the rise is shown in
the RC charge curve because the
charging is fastest at the start and
then tapers off as C takes on
additional charge at a slower rate.
 As C charges, its potential
difference increases.
 Then the difference in voltage
between VT and v C is reduced.
 Less potential difference
reduces the current that puts the
charge in C.
 The more C charges, the more
slowly it takes on additional
charge.
Fig. 22-4
22-5: RC Charge and
Discharge Curves
 On discharge, C loses its
8
vC in Volts
charge at a declining rate.
 At the start of discharge, vC has
its highest value and can produce
maximum discharge current.
 As the discharge continues, vC
goes down and there is less
discharge current.
 The more C discharges, the
more slowly it loses the
remainder of its charge.
6
4
2
0
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0
1
2
3
T in ms
4
5
22-6: High Current Produced by
Short-Circuiting RC Circuit
 A capacitor can be charged slowly by a small charging
current through a high resistance and then be
discharged quickly through a low resistance to obtain
a momentary surge, or pulse of discharge current.
 This idea corresponds to the pulse of high voltage
obtained by opening an inductive circuit.
22-6: High Current Produced by
Short-Circuiting RC Circuit
 The circuit of Fig. 22-5 illustrates the application of a battery-capacitor (BC)
unit to fire a flashbulb for a camera.
 The flashbulb needs 5 A to ignite, but this is too much load current for the
small 15-V battery.
 Instead of using the bulb as a load for the battery, the 100-μF capacitor is
charged.
 The capacitor is then discharged through the bulb in Fig. 22-5 (b).
With large capacitors,
this can be dangerous!
Fig. 22-5:
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22-7: RC Waveshapes
 Voltage and current waveshapes in RC circuits can
show when a capacitor is allowed to charge through a
resistance for RC time and then discharge through the
same resistance for the same amount of time.
 Waveshapes show some useful details about the
voltage and current for charging and discharging.
22-7: RC Waveshapes
Fig. 22-6:
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22-8: Long and Short Time
Constants
 Useful waveshapes can be obtained by using RC




circuits with the required time constant.
In practical applications, RC circuits are used more
than RL circuits because almost any value of an RC
constant can be obtained easily.
Whether an RC time constant is long or short depends
on the pulse width of the applied voltage.
A long time constant can be arbitrarily defined as at
least five times longer than the pulse width, in time.
A short time constant is defined as no more than onefifth the pulse width, in time.
22-8: Long and Short Time
Constants
R
VA
C
vOUT
vOUT
Integrators and Differentiators
C
VA
R
vOUT
vOUT
Integrators use a relatively long time constant.
Differentiators use a relatively short time constant.
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22-9: Charge and Discharge with
Short RC Time Constant
 Fig. 22-7 illustrates the charge and
discharge of an RC circuit with a short time
constant.
 Note that the waveshape of VR in (d) has
sharp voltage peaks for the leading and
trailing edges of the square-wave applied
voltage.
Fig. 22-7
22-10: Long Time Constant for RC
Coupling Circuit
 Fig. 22-8 illustrates the charge and
discharge of an RC circuit with a long
time constant.
 Note that the waveshape of VR in (d)
has the same waveform as the applied
voltage.
Fig. 22-8:
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22-11: Advanced Time
Constant Analysis
 Transient voltage and current values can be
determined for any amount of time with a universal
time-constant chart.
 A universal time-constant chart is a graph of curves
obtained by plotting time in RC or L/R time constants
versus percent of full voltage or current.
 An example of a universal time-constant chart for RC
and RL circuits is shown in Fig. 22-9 (next slide).
22-11: Advanced Time
Constant Analysis
.
Fig. 22-9:
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22-12: Comparison of Reactance
and Time Constant
Table 22-2
Comparison of Reactance XC and
RC Time Constant
Sine-Wave Voltage
Nonsinusoidal Voltage
Examples are 60-Hz power line, af signal Examples are dc circuit turned on and
voltage, rf signal voltage
off, square waves, rectangular pulses
Reactance XC = 1/(2πfC)
Time constant T = RC
Larger C results in smaller reactance XC
Larger C results in longer time constant
Higher frequency results in smaller XC
Shorter pulse width corresponds to
longer time constant
IC = VC/XC
iC = C(dv/dt)
XC makes IC and VC 90° out of phase
Waveshape changes between iC and vC