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Chapter
18
Capacitive Circuits
Topics Covered in Chapter 18
18-1: Sine-Wave VC Lags iC by 90o
18-2: XC and R in Series
18-3: Impedance Z Triangle
18-4: RC Phase-Shifter Circuit
18-5: XC and R in Parallel
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered in Chapter 18
 18-6: RF and AF Coupling Capacitors
 18-7: Capacitive Voltage Dividers
 18-8: The General Case of Capacitive Current iC
McGraw-Hill
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
18-1: Sine-Wave VC Lags iC by 90o
 For any sine wave of applied voltage, the capacitor’s charge
and discharge current ic will lead vc by 90°.
Fig. 18-1: Capacitive current ic leads vc by 90°. (a) Circuit with sine wave VA across C. (b)
Waveshapes of ic 90° ahead of vc. (c) Phasor diagram of ic leading the horizontal reference vc by
a counterclockwise angle of 90°. (d) Phasor diagram with ic as the reference phasor to show vc
lagging ic by an angle of −90°.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-1: Sine-Wave VC Lags iC by 90o
 The value of ic is zero when VA is at its maximum value.
 At its high and low peaks, the voltage has a static value
before changing direction. When V is not changing and
C is not charging or discharging, the current is zero.
 ic is maximum when vc is zero because at this point the
voltage is changing most rapidly.
18-1: Sine-Wave VC Lags iC by 90o
 ic and vc are 90° out of phase because the maximum
value of one corresponds to the zero value of the other.
 The 90° phase angle results because ic depends on the
rate of change of vc. ic has the phase of dv/dt, not the
phase of v.
 The 90° phase between vc and ic is true in any sine
wave ac circuit. For any XC, its current and voltage are
90° out of phase.
18-1: Sine-Wave VC Lags iC by 90o
 The frequency of vc and ic are always the same.
 The leading phase angle only addresses the voltage
across the capacitor. The current is still the same in all
parts of a series circuit. In a parallel circuit, the voltage
across the generator and capacitor are always the
same, but both are 90° out of phase with ic.
18-2: XC and R in Series
 When a capacitor and a resistor are connected in
series, the current is limited by both XC and R.
 Each series component has its own series voltage
drop equal to IR for the resistance and IXC for the
capacitive reactance.
 For any circuit combining XC and R in series, the
following points are true:
1. The current is labeled I rather than IC, because I
flows through all the series components.
18-2: XC and R in Series
2. The voltage across XC, labeled VC, can be
considered an IXC voltage drop, just as we use VR
for an IR voltage drop.
3. The current I through XC must lead VC by 90°,
because this is the phase angle between the
voltage and current for a capacitor.
4. The current I through R and its IR voltage drop are
in phase. There is no reactance to sine-wave
alternating current in any resistance. Therefore, I
and IR have a phase angle of 0°.
18-2: XC and R in Series

Phase Comparisons

For a circuit combining series resistance and
reactance, the following points are true:
1. The voltage VC is 90° out of phase with I.
2. VR and I are in phase.
3. If I is used as the reference, VC is 90° out of phase
with VR.
 VC lags VR by 90° just as voltage VC lags the current I
by 90°.
18-2: XC and R in Series
 Combining VR and VC ; the Phasor Voltage Triangle
 When voltage wave VR is combined with voltage wave
VC the result is the voltage wave of the applied voltage
VT.
 Out-of-phase waveforms may be added quickly by
using their phasors. Add the tail of one phasor to the
arrowhead of another and use the angle to show their
relative phase.
VT =
VR2 + VC2
18-2: XC and R in Series
Fig. 18-3: Addition of two voltages 90° out of phase. (a) Phasors for VC and VR are 90° out of
phase. (b) Resultant of the two phasors is the hypotenuse of the right triangle for VT.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-2: XC and R in Series
Phasor Voltage Triangle for Series RC Circuits
VR
q
VC
VT
q
R
ZT
XC
Voltage Phasors
Impedance Phasor
VT 
ZT 
VC2  VR2
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
X 2C  R 2
18-2: XC and R in Series
Waveforms and Phasors for a Series RC Circuit
q  0
VR
I
I
q  - 90
I
VC
Note: Since current is constant in a series circuit, the current waveforms and current
phasors are shown in the reference positions.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-3: Impedance Z Triangle
 R and XC may be added
using a triangle model as
was shown with voltage.
 Adding phasors XC and R
results in their total
opposition in ohms, called
impedance, using symbol
ZT.
Fig. 18-4: Addition of R and XC 90° out of phase in a
series RC circuit to find the total impedance ZT.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-3: Impedance Z Triangle
 Z takes into account the 90° phase relationship between
R and XC.
ZT =
R2 + XC2
Phase Angle with Series XC and R
 The angle between the applied voltage VT and the series
current I is the phase angle of the circuit.
 The phase angle may be calculated from the impedance
triangle of a series RC circuit by the formula
tan ΘZ = −
XC
R
18-3: Impedance Z Triangle
The Impedance of a Series RC Circuit
I=2A
R
R = 30 W
VT = 100
XC
XC = 40 W
Z=
R2 + XC2 =
I=
VT
Z
=
Z
302 + 402 = 50 W
100
50
=2A
The impedance is the total opposition to current flow. It’s the phasor sum of resistance and
reactance in a series circuit
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-3: Impedance Z Triangle
The Tangent Function
adjacent
opposite
Tanq  adjacent
opposite
-1
q  Tan adjacent
opposite
Tanq 
adjacent
q  Tan
-1
Θ
opposite
negative
angle
positive
angle
opposite
adjacent
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Θ
adjacent
opposite
18-3: Impedance Z Triangle
The Phase Angle of a Series RC Circuit
I=2A
q
30 W
40 W
50 W
R = 30 W
VT = 100
−53°
I
XC = 40 W
Θ=
Tan−1
-
XC
=
Tan−1 -
R
VC
VT
VT lags I by 53°
40
30
= −53°
18-3: Impedance Z Triangle
Source Voltage and Current Phasors
XC < R
Θ < 0
I
VT
VT
I
XC = R
Θ = −45
I
VT
Note: The source voltage lags the current by an
amount proportional to the ratio of capacitive
reactance to resistance.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
XC > R
Θ < − 45
VT
I
18-4: RC Phase-Shifter Circuit
 The RC phase-shift circuit is used to provide a voltage
of variable phase to set the conduction time of
semiconductors in power control circuits.
 Output can be taken across R or C depending on
desired phase shift with respect to VIN.
 VR leads VT by an amount depending on the values of
XC and R.
 VC lags VT by an amount depending on the values of XC
and R.
18-4: RC Phase-Shifter Circuit
Fig. 18-5: An RC phase-shifter circuit. (a) Schematic diagram. (b) Phasor triangle with IR, or VR,
as the horizontal reference. VR leads VT by 46.7° with R set at 50 kΩ. (c) Phasors shown with VT
as the horizontal reference.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-5: XC and R in Parallel
 The sine-wave ac charge and discharge currents for a
capacitor lead the capacitor voltage by 90°.
 The sine-wave ac voltage across a resistor is always in
phase with its current.
 The total sine-wave ac current for a parallel RC circuit
always leads the applied voltage by an angle between
0° and 90°.
18-5: XC and R in Parallel
 Phasor Current Triangle
 The resistive branch current IR
is used as the reference
phasor since VA and IR are in
phase.
 The capacitive branch current
IC is drawn upward at an angle
of +90° since IC leads VA and
thus IR by 90°.
Fig. 18-7: Phasor triangle of capacitive and resistive branch currents 90° out of phase in a
parallel circuit to find the resultant IT.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-5: XC and R in Parallel
 Phasor Current Triangle (Continued)
 The sum of the IR and IC phasors is indicated by the
phasor for IT, which connects the tail of the IR phasor to
the tip of the IC phasor.
IT =
IR2 + IC2
 The IT phasor is the hypotenuse of the right triangle.
 The phase angle between IT and IR represent the phase
angle of the circuit.
18-5: XC and R in Parallel
 Impedance of XC and R in Parallel
 To calculate the total or equivalent impedance of XC and
R in parallel, calculate total line current IT and divide into
applied voltage VA:
ZEQ =
VA
IT
18-5: XC and R in Parallel
Impedance in a Parallel RC Circuit
IT = 5 A
3A
VA = 120
R = 30 W
5A
XC = 40 W
4A
ZEQ =
VA
IT

120
5
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
= 24 Ω
18-5: XC and R in Parallel
 Phase Angle in Parallel
Circuits
Use the Tangent form to find Θ
from the current triangle.
Tan ΘI =
IC
IR
Tan ΘI = 10/10 = 1
ΘI = Tan 1
ΘI = 45°
Fig. 18-7
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-5: XC and R in Parallel
 Parallel Combinations of XC and R
 The series voltage drops VR and VC have individual values
that are 90° out of phase.
 They are added by phasors to equal the applied voltage
VT.
 The negative phase angle −ΘZ is between VT and the
common series current I.
 The parallel branch currents IR and IC have individual values
that are 90° out of phase.
 They are added by phasors to equal IT, the main-line
current.
 The positive phase angle ΘI is between the line current IT
and the common parallel voltage VA.
18-5: XC and R in Parallel
Parallel Combinations of XC and R
Resistance (R) in Ohms, Ω
Voltage in phase with current.
Capacitive Reactance (XC) in Ohms, Ω
Voltage lags current by 90°.
 Series circuit impedance
(ZT) in Ohms, Ω
 Parallel circuit impedance
(ZEQ) in Ohms, Ω
 Voltage lags current.
 Voltage lags current.
 Becomes more resistive
 Becomes more resistive
with increasing f.
 Becomes more capacitive
with decreasing f.
with decreasing f.
 Becomes more capacitive
with increasing f.
18-5: XC and R in Parallel
Summary of Formulas
Series RC
Parallel RC
1
1
XC =
VT =
ZT =
2πfC
VR2 + VC2
R2 + XC2
tan Θ = −
XC
R
XC =
IT =
2πfC
IR2 + IC2
ZEQ =
tan Θ =
VA
IT
IC
IR
18-6: RF and AF
Coupling Capacitors
 CC is used in the application of a
coupling capacitor.
 The CC’s low reactance allows
developing practically all the ac
signal voltage across R.
 Very little of the ac voltage is
across CC.
 The dividing line for CC to be a
coupling capacitor at a specific
frequency can be taken as XC onetenth or less of the series R.
18-7: Capacitive Voltage Dividers
 When capacitors are connected in series across a
voltage source, the series capacitors serve as a
voltage divider.
 Each capacitor has part of the applied voltage.
 The sum of all the series voltage drops equals the
source voltage.
 The amount of voltage across each capacitor is
inversely proportional to its capacitance.
18-7: Capacitive Voltage Dividers
Fig. 18-9: Series capacitors divide VT inversely proportional to each C. The smaller C has more V.
(a) An ac divider with more XC for the smaller C.
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18-7: Capacitive Voltage Dividers
Fig. 18-9: Series capacitors divide VT inversely proportional to each C. The smaller C has more V.
(b) A dc divider.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
18-8: The General Case of
Capacitive Current iC
 The capacitive charge and discharge current ic is
always equal to C(dv/dt).
 A sine wave of voltage variations for vc produces a
cosine wave of current i.
 Note that vc and ic have the same waveform, but they
are 90° out of phase.
18-8: The General Case of
Capacitive Current iC
 XC is generally used for calculations in sine-wave
circuits.
 Since XC is 1/(2πfC), the factors that determine the
amount of charge and discharge current are included
in f and C.
 With a nonsinusoidal waveform for voltage vc, the
concept of reactance cannot be used. (Reactance XC
applies only to sine waves).