capacitive circuits topics covered in chapter 18 18-1: sine-wave v c lags i c by 90 o 18-2: x c and...

Capacitive CircuitsCapacitive Circuits

Topics Covered in Chapter 1818-1: 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

ChapterChapter1818

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 18Topics 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 18-1: Sine-Wave VVCC Lags Lags iiCC by 90by 90oo

For any sine wave of applied voltage, the capacitor’s charge and discharge current ic will lead vc by 90°.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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°.


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.


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.


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: 18-2: XXCC and and RR in Series 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.


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°.


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°.


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.

VR2 + VC

2VT =



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.


VR

VC

VT

Voltage Phasors

R

XC

ZT

Impedance Phasor


Phasor Voltage Triangle for Series RC Circuits

Amy Hill

Auth: Did you create these figures yourself, or were they taken from a previous edition text or other source? Please clarify for permissions.


Waveforms and Phasors for a Series RC Circuit

Note: Since current is constant in a series circuit, the current waveforms and current phasors are shown in the reference positions.

VR

I

I

I

VC


Amy Hill

Author: Again, please advise source of figure for permissions. Thanks.Also, is this the best place for this slide? Please advise.

18-3: Impedance 18-3: Impedance ZZ Triangle 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.


Z takes into account the 90° phase relationship between R and XC.

R2 + XC2ZT =

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


The Impedance of a Series RC Circuit

The impedance is the total opposition to current flow. It’s the phasor sum of resistance and reactance in a series circuit

I = 2 A

VT = 100R = 30

XC = 40

= 2 AZ

VTI =

50

100=

R

XC Z

= 50 W302 + 402R2 + XC2 =Z =



The Tangent Function

adjacent

oppositeTan

adjacent

oppositeTan 1

Θ

opposite

adjacent

negativeangle

adjacent

oppositeTan

adjacent

oppositeTan 1 Θ

opposite

adjacent

positiveangle



The Phase Angle of a Series RC Circuit

I = 2 A

VT = 100R = 30

XC = 40

30

40 50

= −53°30

40 = Tan−1

R

XCΘ = Tan−1

VT lags I by 53°

I

VC VT

−53°


Source Voltage and Current Phasors

Note: The source voltage lags the current by an amount proportional to the ratio of capacitive reactance to resistance.

IVT

Θ < 0I

VT

XC < R

Θ = −45I

VT

XC = R

Θ < − 45I

VT

XC > R


18-4: 18-4: RCRC Phase-Shifter Circuit 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: 18-4: RCRC Phase-Shifter Circuit 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.

18-5: 18-5: XXCC and and RR in Parallel 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°.


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.


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.

The IT phasor is the hypotenuse of the right triangle.

The phase angle between IT and IR represent the phase angle of the circuit.

IR2 + IC

2IT =


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


Impedance in a Parallel RC Circuit

VA = 120 R = 30 XC = 40

IT = 5 A

4 A

3 A 5 A

= 24 ΩZEQ = IT

VA

5

120



Phase Angle in Parallel Circuits


Fig. 18-7

Tan ΘI = IC

IR

Use the Tangent form to find Θ from the current triangle.

Tan ΘI = 10/10 = 1 ΘI = Tan 1 ΘI = 45°


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.


Series circuit impedance (ZT) in Ohms, Ω

Voltage lags current. Becomes more resistive

with increasing f. Becomes more capacitive

with decreasing f.

Parallel circuit impedance (ZEQ) in Ohms, Ω

Voltage lags current. Becomes more resistive

with decreasing f. Becomes more capacitive

with increasing f.

Parallel Combinations of XC and RResistance (R) in Ohms, Ω

Voltage in phase with current.Capacitive Reactance (XC) in Ohms, Ω

Voltage lags current by 90°.


Summary of Formulas

Series RC Parallel RC

XC = 1

2 π f CXC =

1

2 π f C

VT = VR2 + VC

2 IT = IR2 + IC

2

R2 + XC2ZT = ZEQ =

VA

IT

tan Θ = −XC

Rtan Θ =

IC

IR

18-6: RF and AF 18-6: RF and AF Coupling CapacitorsCoupling 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 one-tenth or less of the series R.

18-7: Capacitive Voltage Dividers18-7: 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.



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.


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 18-8: The General Case of Capacitive Current Capacitive Current iiCC

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 18-8: The General Case of Capacitive Current Capacitive Current iiCC

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).

capacitive circuits topics covered in chapter 18 18-1: sine-wave v c lags i c by 90 o 18-2: x c and...

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