ISU EE C.Y. Lee

Chapter 10RC Circuits

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ObjectivesDescribe the relationship between current and voltage in an RC circuitDetermine impedance and phase angle in a seriesRC circuitAnalyze a series RC circuitDetermine the impedance and phase angle in a parallel RC circuitAnalyze a parallel RC circuit Analyze series-parallel RC circuitsDetermine power in RC circuits

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Sinusoidal Response of RC Circuits

The capacitor voltage lags the source voltage Capacitance causes a phase shift between voltageand current that depends on the relative values of the resistance and the capacitive reactance

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Impedance and Phase Angle of Series RC Circuits

The phase angle is the phase difference between the total current and the source voltageThe impedance of a series RC circuit is determined by both the resistance (R) and the capacitive reactance (XC)

(Z = RjXC)(Z = jXC)(Z = R)

(ZC= 1/jC= jXC) (Z= R+ZC= RjXC)

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Impedance and Phase Angle of Series RC Circuits

In the series RC circuit, the total impedance is the phasor sum of R and jXCImpedance magnitude: Z = R2 + X2CPhase angle: = tan-1(XC/R)

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Impedance and Phase Angle of Series RC Circuits

Z = (47)2 + (100)2 = 110 = tan-1(100/47) = tan-1(2.13) = 64.8

Example: Determine the impedance and the phase angle

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Analysis of Series RC Circuits

The application of Ohms law to series RC circuits involves the use of the quantities Z, V, and I as:

IZV =

ZVI =

IVZ =

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Analysis of Series RC CircuitsExample: If the current is 0.2 mA, determine the source

voltage and the phase angle

XC = 1/2(1103)(0.0110-6) = 15.9 kZ = (10103)2 + (15.9103)2 = 18.8 kVS = IZ = (0.2mA)(18.8k) = 3.76 V = tan-1(15.9k/10k) = 57.8

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Relationships of I and V in a Series RC Circuit

In a series circuit, the current is the same through both the resistor and the capacitorThe resistor voltage is in phase with the current, and the capacitor voltage lags the current by 90

I

VR

VC

VS

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KVL in a Series RC Circuit

From KVL, the sum of the voltage drops must equal the applied voltage (VS)

Since VR and VC are 90out of phase with each other, they must be added as phasor quantities

VS = V2R + V2C = tan-1(VC/VR)

VR

VC

VS

I

VR= IR

VS= IZ = I(RjXC)VC= I(jXC)

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KVL in a Series RC CircuitExample: Determine the source voltage and the phase angle

VS = (10)2 + (15)2 = 18 V = tan-1(15/10) = tan-1(1.5) = 56.3

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Variation of Impedance and Phase Angle with Frequency

For a series RC circuit; as frequency increases: R remains constant XC decreases Z decreases decreases

f Z

R

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Variation of Impedance and Phase Angle with Frequency

Example: Determine the impedance and phase angle for each of the following values of frequency: (a) 10 kHz (b) 30 kHz

(a)XC = 1/2(10103)(0.0110-6) = 1.59 kZ = (1.0103)2 + (1.59103)2 = 1.88 k = tan-1(1.59k/1.0k) = 57.8

(b)XC = 1/2(30103)(0.0110-6) = 531 kZ = (1.0103)2 + (531)2 = 1.13 k = tan-1(531/1.0k) = 28.0

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Impedance and Phase Angle of Parallel RC Circuits

Total impedance in parallel RC circuit:Z = (RXC) / ( R2 +X2C)

Phase angle between the applied V and the total I: = tan-1(R/XC)

( )

jRXRXZ

jXRRjX

Z

jXRZ

C

C

C

C

C

+=

+

=

+=

1

111

+

==jRX

RXIIZVC

C

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Conductance, Susceptance and Admittance

Conductance is the reciprocal of resistance:G = 1/R

Capacitive susceptance is the reciprocal of capacitive reactance:

BC = 1/XCAdmittance is the reciprocal of impedance:

Y = 1/Z

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Ohms Law

Application of Ohms Law to parallel RC circuits using impedance can be rewritten for admittance (Y=1/Z):

YIV =

VYI =

VIY =

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Relationships of the I and V in a Parallel RC Circuit

The applied voltage, VS, appears across both the resistive and the capacitive branchesTotal current, Itot, divides at the junction into the two branch current, IR and IC

Vs, VR, VC

IR= V/R

Itot= V/Z = V((XC+jR)/RXC)IC= V/(jXC)

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KCL in a Parallel RC Circuit

From KCL, Total current (IS) is the phasor sum of the two branch currents

Since IR and IC are 90out of phase with each other, they must be added as phasor quantities

Itot = I2R + I2C = tan-1(IC/IR)

IR= V/R

Itot= V/Z = V((XC+jR)/RXC)IC= V/(jXC)

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KCL in a Parallel RC CircuitExample: Determine the value of each current, and describe

the phase relationship of each with the source voltage

IR = 12/220 = 54.5 mAIC = 12/150 = 80 mAItot = (54.5)2 + (80)2 = 96.8 mA = tan-1(80/54.5) = 55.7

Vs

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Series-Parallel RC CircuitsAn approach to analyzing circuits with combinations of both series and parallel R and C elements is to: Calculate the magnitudes of capacitive reactances (XC) Find the impedance (Z) of the series portion and the

impedance of the parallel portion and combine them to get the total impedance

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(Z)(Z')(Z"):

Z= Z' jZ"

( )mi jXRRZ ++=

111

( ) 22222

mie

meieie

XRRXRRRRRZ

++++

=

( ) 222

mie

me

XRRXRj++

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c c

Z-plot

Z:

Zs:

fC ():Z"

( ) ( )22 ZZZ +=

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Z-plot

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RC Lag NetworkThe RC lag network is a phase shift circuit in which the output voltage lags the input voltage

=

RXC1tan90o

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RC Lead NetworkThe RC lead network is a phase shift circuit in which the output voltage leads the input voltage

=

RXC1tan

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Frequency Selectivity of RC CircuitsA low-pass circuit is realized by taking the output across the capacitor, just as in a lag network

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Frequency Selectivity of RC CircuitsThe frequency response of the low-pass RC circuit is shown below, where the measured values are plotted on a graph of Vout versus f.

10 V

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Frequency Selectivity of RC CircuitsA high-pass circuit is implemented by taking the output across the resistor, as in a lead network

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Frequency Selectivity of RC CircuitsThe frequency response of the high-pass RC circuit is shown below, where the measured values are plotted on a graph of Vout versus f.

10 V

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Frequency Selectivity of RC CircuitsThe frequency at which the capacitive reactance equals the resistance in a low-pass or high-pass RC circuit is called the cutoff frequency:

RCfC 2

1=

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Coupling an AC Signal into a DC Bias Network

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Summary

When a sinusoidal voltage is applied to an RC circuit, the current and all the voltage drops are also sine wavesTotal current in an RC circuit always leads the source voltageThe resistor voltage is always in phase with the currentIn an ideal capacitor, the voltage always lags the current by 90

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Summary

In an RC circuit, the impedance is determined by both the resistance and the capacitive reactance combinedThe circuit phase angle is the angle between the total current and the source voltageIn a lag network, the output voltage lags the input voltage in phaseIn a lead network, the output voltage leads the input voltageA filter passes certain frequencies and rejects others

Chapter 10ObjectivesSinusoidal Response of RC CircuitsImpedance and Phase Angle of Series RC CircuitsImpedance and Phase Angle of Series RC CircuitsAnalysis of Series RC CircuitsAnalysis of Series RC CircuitsRelationships of I and V in a Series RC CircuitKVL in a Series RC CircuitKVL in a Series RC CircuitVariation of Impedance and Phase Angle with FrequencyVariation of Impedance and Phase Angle with FrequencyImpedance and Phase Angle of Parallel RC CircuitsConductance, Susceptance and AdmittanceOhms LawRelationships of the I and V in a Parallel RC CircuitKCL in a Parallel RC CircuitKCL in a Parallel RC CircuitSeries-Parallel RC CircuitsRC Lag NetworkRC Lead NetworkFrequency Selectivity of RC CircuitsFrequency Selectivity of RC CircuitsFrequency Selectivity of RC CircuitsFrequency Selectivity of RC CircuitsFrequency Selectivity of RC CircuitsCoupling an AC Signal into a DC Bias NetworkSummarySummary