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Objectives• Describe the relationship between currentand voltage in an RC circuit• Determine impedance and phase angle in aseries RC circuit• Analyze a series RC circuit• Determine the impedance and phase anglein a parallel RC circuit
3.
Objectives• Analyze a parallel RC circuit• Analyze series-parallel RC circuits• Determine power in RC circuits
4.
Sinusoidal Response of RCCircuits• When a circuit is purely resistive, the phase anglebetween applied voltage and total current is zero• When a circuit is purely capacitive, the phaseangle between applied voltage and total current is90°• When there is a combination of both resistanceand capacitance in a circuit, the phase anglebetween the applied voltage and total current issomewhere between 0° and 90°, depending onrelative values of resistance and capacitance
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Impedance and Phase Angle ofSeries RC Circuits• In the series RC circuit, the total impedance is thephasor sum of R and jXC• Impedance magnitude: Z = √R2+ X2C• Phase angle: θ = tan-1(XC/R)
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Analysis of Series RC Circuits• The application of Ohm’s law to series RC circuitsinvolves the use of the quantities Z, V, and I as:V = IZI = V/ZZ = V/I
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Relationships of I and V in aSeries RC Circuit• In a series circuit, the current is the same throughboth the resistor and the capacitor• The resistor voltage is in phase with the current,and the capacitor voltage lags the current by 90°
8.
KVL in a Series RC Circuit• From KVL, the sum of the voltage drops mustequal the applied voltage (VS)• Since VR and VC are 90° out of phase with eachother, they must be added as phasor quantities• Magnitude of source voltage:VS = √V2R + V2C• Phase angle between resistor and source voltages:θ = tan-1(VC/VR)
9.
Variation of Impedance andPhase Angle with Frequency• For a series RCcircuit; as frequencyincreases:– XC decreases– Z decreasesθ decreases– R remains constant
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Impedance and Phase Angle ofParallel RC Circuits• Total impedance :Z = (RXC) / (√R2+X2C)• Phase angle:θ = tan-1(R/XC)
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Conductance, Susceptance andAdmittance• Conductance is the reciprocal of resistance:G = 1/R• Capacitive susceptance is the reciprocal ofcapacitive reactance:BC = 1/XC• Admittance is the reciprocal of impedance:Y = 1/Z
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Ohm’s Law• Application of Ohm’s Law to parallel RCcircuits using impedance can be rewrittenfor admittance (Y=1/Z):V = I/YI = VYY = I /V
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Relationships of the Currents andVoltages in a Parallel RC Circuit• The applied voltage, VS, appears across both theresistive and the capacitive branches• Total current Itot, divides at the junction into thetwo branch current, IR and IC
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Kirchhoff’s Current Law• Current through the resistor is in phase with thevoltage• Current through the capacitor leads the voltage,and thus the resistive current by 90°• Total current is the phasor sum of the two branchcurrents• Magnitude of total current is:Itot = √I2R + I2C• Phase angle: θ = tan-1(IC/IR)
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Conversion from Parallel toSeries Form• For every parallel RCcircuit there is anequivalent series RCcircuit for any givenfrequency• Equivalent resistanceand capacitive reactanceare indicated on theimpedance triangle
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Series-Parallel RC Circuits• An approach to analyzing circuits withcombinations of both series and parallel Rand C elements is to:– Calculate the magnitudes of capacitivereactances (XC)– Find the impedance of the series portion and theimpedance of the parallel portion and combinethem to get the total impedance
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Power in RC Circuits• When there is both resistance andcapacitance, some of the energy isalternately stored and returned by thecapacitance and some is dissipated by theresistance• The amount of energy converted to heat isdetermined by the relative values of theresistance and the capacitive reactance
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Power Triangle for RC Circuits• The Power can be written as:Ptrue = VsItotalcosθwhere: θ = 0° for a purely resistive circuitsince cos(0°) = 1, Ptrue = VsItotalθ = 90° for a purely capacitive circuitsince cos(90°) = 0, Ptrue = zero
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Power Factor• The term cos θ, in the previous slide, is called thepower factor:PF = cos θ• The power factor can vary from 0 for a purelyreactive circuit to 1 for a purely resistive circuit• In an RC circuit, the power factor is referred to asa leading power factor because the current leadsthe voltage
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Significance of Apparent Power• Apparent power is the power that appears to betransferred between the source and the load• Apparent power consists of two components; atrue power component, that does the work, and areactive power component, that is simply powershuttled back and forth between source and load• Apparent power is expressed in volt-amperes(VA)
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RC Lag Network• The RC lag network is a phase shift circuit inwhich the output voltage lags the input voltage
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RC Lead Network• The RC lead network is a phase shift circuit inwhich the output voltage leads the input voltage
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Frequency Selectivity of RCCircuits• Frequency-selective circuits permit signals ofcertain frequencies to pass from the input to theoutput, while blocking all others• A low-pass circuit is realized by taking the outputacross the capacitor, just as in a lag network• A high-pass circuit is implemented by taking theoutput across the resistor, as in a lead network
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Frequency Selectivity of RCCircuits• The frequency atwhich the capacitivereactance equals theresistance in a low-pass or high-pass RCcircuit is called thecutoff frequency:fc = 1/(2πRC)
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Summary• When a sinusoidal voltage is applied to an RCcircuit, the current and all the voltage drops arealso sine waves• Total current in an RC circuit always leads thesource voltage• The resistor voltage is always in phase with thecurrent• The capacitor voltage always lags the current by90°
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Summary• In an RC circuit, the impedance is determined byboth the resistance and the capacitive reactancecombined• Impedance is expressed in units of ohms• The circuit phase angle is the angle between thetotal current and the source voltage• The impedance of a series RC circuit variesinversely with frequency
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Summary• The phase angle (θ) of a series RC circuit variesinversely with frequency• For each parallel RC circuit, there is an equivalentseries circuit for any given frequency• The impedance of a circuit can be determined bymeasuring the applied voltage and the total currentand then applying Ohm’s law
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Summary• In an RC circuit, part of the power is resistive andpart is reactive• The phasor combination of resistive power andreactive power is called apparent power• Apparent power is expressed in volt-amperes(VA)• The power factor indicates how much of theapparent power is true power
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Summary• A power factor of 1 indicates a purely resistivecircuit, and a power factor of 0 indicates a purelyreactive circuit• In a lag network, the output voltage lags the inputvoltage in phase• In a lead network, the output voltage leads theinput voltage• A filter passes certain frequencies and rejectsothers
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