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Chapter 10RC Circuits

Sinusoidal Response of RC

Circuits• When a circuit is purely resistive, the phase angle between applied

voltage and total current is zero

• When a circuit is purely capacitive, the phase angle between applied

voltage and total current is 90° (Current Leads)

• When there is a combination of both resistance and capacitance in a

circuit, the phase angle between resistance (R) and capacitive

reactance (XC) is 90 ° and the phase angle for total impedance (Z) is

somewhere between 0° and 90°

• When there is a combination of both resistance and capacitance in a

circuit, the phase angle between total current (IT) and the capacitor

voltage (VC) is 90 ° and the phase angle between the applied voltage

(VS) and total current (IT) is somewhere between 0° and 90°, depending on relative values of resistance and capacitance

Thomas L. Floyd

Electronics Fundamentals, 6e

Electric Circuit Fundamentals, 6e

Copyright ©2004 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458

All rights reserved.

Purely Resistive Purely Capacitive A Phasor Quantity of Both

R00

-900

XC Z

Impedance in Phase with Voltage

Capacitive Impedance out of Phase with Voltage

by -90 Degrees

Total Impedance out of Phase with Voltage

by Phasor Sum

Impedance Phase Angles of Series R, C and RC Circuits(Impedance is what the source “Sees” as total Opposition to Current)

R

XC

Voltage and Current Phasor Diagram for the Waveforms

Thomas L. Floyd






It is the used as the Reference Wave

It,Vr

Vc

Vs

Current, Resistance and Voltage Phase Angles of

Series RC Circuits

Reference

Reference

Impedance and Phase Angle of

Series RC Circuits• In the series RC circuit, the total impedance is the phasor sum of R and XC

• Impedance magnitude: Z = √R2 + X2C

• Phase angle: θ = tan-1(XC/R)

Impedance of a series RC circuit.

Thomas L. Floyd






FIGURE 10-6

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The source voltage lags the current by 64.8 Degrees

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Analysis of Source Voltage, Total

Impedance and Total Current in

Series RC Circuits

• The application of Ohm’s law to an entire series RC circuit involves

the use of the quantities Z, Vs, and Itot as:

Itot = Vs/Z

Z = Vs/Itot

Vs = ItotZ

Thomas L. Floyd






Determine the Source Voltage and Phase Angle

.2mA

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FIGURE 10-10 P 451

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Determine the Current

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18V

VR and VC cannot be added directly

VS is the phasor sum of the two

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Thomas L. Floyd






Frequency Up –

Circuit becomes more resistive

•XC Down

•VC Down

•Z Down

•I Up

•VR Up

Variation of Impedance with Frequency

Frequency Down –

Circuit becomes more capacitive

•XC Up

•VC Up

•Z Up

•I Down

•VR Down

Variation of Impedance and

Phase Angle with Frequency• For a series RC circuit; as

frequency increases:

– XC decreases

– Z decreases

– θ decreases

– R remains constant

0° = Purely Resistive

An illustration of how Z and XC

change with frequency

Thomas L. Floyd






Thomas L. Floyd






Determine the Impedance and Phase Angle for the following Frequencies

10 kHz

20 kHZ

30 kHz

Power in RC Circuits

• When there is both resistance and capacitance, some of the energy is alternately stored and returned

(shifted back and forth) by the capacitance (Reactive

Power) and some is actually dissipated by the resistance (True Power)

• Apparent power consists of both components :

– It is the phasor sum of reactive and true power

– It is the total power that the source must provide

Thomas L. Floyd






NOTE: It is important to understand that, in a

reactive circuit, more power is often required from

the power supply than is dissipated by the load.

When selecting a source (Power Supply), this must

be taken into consideration or the source can

become overloaded/overheated.

Power Triangle for RC Circuits

Power Factor

• Power Factor is the efficiency of the circuit:

• The power factor can vary from 0 for a purely reactive circuit to 1 for a purely resistive circuit

• It is the ratio of the dissipated power to the total power required of the power supply:

PF = P true/Pa (Power Used/Total Power Required)

PF = R/Z

PF = cos θ

• A high Power Factor is more efficient than a low Power Factor

Thomas L. Floyd






When you add a reactive load, the power factor goes

down causing apparent power and over all current to

go up. This can harm the power supply if it is not rated for

that much current

Source Rated at 5A Max

800.4W

Resistor

Dissipating

480 Watts of

Heat

Source must

supply equivalent

of 800.4 Watts

Thomas L. Floyd






(I2)(R) => (4.25mA2)(1K Ohm) = 18mW

(I2)(Z) => (4.25mA2)(3.53K Ohm) = 63.8mVA

R/Z => 1K Ohm/3.53K Ohm = .283

(I2)(XC) => (4.25mA2)(3.39K Ohm) = 61.2mVAR

18mW

61.2m VAR 63.8m VA

73°

Low and High Pass RC Circuits

• Frequency-selective circuits permit signals of certain frequencies to pass from

the input to the output, while blocking all others

• A low-pass circuit is realized by taking the output across the capacitor, just as

in a lag network.

– Low frequencies are passed on to the next circuit (accepted) and high

frequencies are not (rejected).

• A high-pass circuit is implemented by taking the output across the resistor, as

in a lead network

– High frequencies are passed on to the next circuit (accepted) and low

frequencies are not (rejected).

Low Pass Circuit (RC Lag Network)

• The RC lag network is a phase shift circuit in which the output voltage lags the

input voltage

When the output is taken across the capacitor, the output

voltage lags the input voltage by the phase lag of the circuit.

It is also is lower in amplitude dependent upon the frequency

(VC)Scope

Ch A

Scope

Ch B

Illustration of how the frequency affects the phase lag and the output voltage in an RC

Low Pass lag network with the amplitude of Vin

held constant.

Thomas L. Floyd






Thomas L. Floyd






In a low-pass filter, the signal output across the capacitor

decreases as the source frequency increases

Example of low-pass filtering action. As frequency increases, Vout

decreases

Frequency response curve for the low-pass RC circuit in Figure 10-50

Thomas L. Floyd






In a low-pass filter, the signal output across the capacitor

decreases as the source frequency increases

High Pass Filter (RC Lead Network)

• The RC lead network is a phase shift circuit in which the output voltage leads

the input voltage

When the output is taken across the resistor, the output

leads and source voltage

(VR)

Scope

Ch A

Scope

Ch B

Illustration of how the frequency affects the phase lead and the output voltage in an RC

lead network with the amplitude of Vin held constant.

Thomas L. Floyd






Example of high-pass filtering action. As frequency increases, Vout

increases

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In a high-pass filter, the signal output across the capacitor

increases as the source frequency decreases

Frequency response curve for a high-pass RC circuit

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Frequency Selectivity of RC

Circuits• The frequency at which the

capacitive reactance equals the

resistance in a low-pass or high-

pass RC circuit is called the cutoff

frequency:

fc = 1/(2ππππRC)• All frequencies below (high pass) or

all frequencies above (low pass) are

reduced to 70.7% of the maximum

value and are considered rejected

by the circuit

Normalized general response curve of a low-pass RC circuit showing the cutoff

frequency and the bandwidth

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Accept

Reject

½ Power

-3db

Troubleshooting: Effect of an open resistor.

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Troubleshooting: Effect of an open capacitor

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Note: An open Capacitor has lost it’s ability to hold a charge.

It would not have little or no capacitance. As a result there would be little

or no current in the circuit and little or no voltage developed across the resistor.

Troubleshooting: Effect of a shorted capacitor

Thomas L. Floyd



Note: A leaky Capacitor is somewhere between normal and shorted and

it would not have little or no capacitance. As a result, circuit current would be high

and there would be a higher developed voltage across the resistor.

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CH10 Mine No Paralleleon.sdsu.edu/~johnston/EE204_PDF_Slides/Chapters 8-10/CH10.pdf · Analysis of Source Voltage, Total Impedance and Total Current in Series RC Circuits • The