# resonant circuits

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Resonant Circuits. SEE 1023 Circuit Theory. Frequency Response. L. R. I. + V L -. + V R -. + V C -. V s. C. w (varied). Series RLC Circuit. When w varies, the impedance of the circuit will vary. Then, the current and the real power will also vary. - PowerPoint PPT PresentationTRANSCRIPT

Resonant CircuitsSEE 1023 Circuit TheoryFrequency Response

Series RLC CircuitWhen w varies, the impedance of the circuit will vary.Then, the current and the real power will also vary.We would like to study the frequency response of these quantities.

Series RLC Circuit

Series RLC CircuitExcitation(Input)Response(Output)Series RLC CircuitConstant input voltage: VsVariable Source angular frequency: wMain response: currentOther responses: Power, Impedance, reactance, etc.

Series RLC Circuit in PSpiceIt is too hard to study the frequency response of these quantities manually.It is too easy to study the frequency response of these quantities PSpicely.0123

Series RLC Circuit in PSpiceSeries resonant CircuitVs 1 0 AC 10VR1 1 2 10L1 2 3 100mHC1 3 0 10uF.AC LIN 1001 100Hz 220Hz.Probe.end

In the Probe windows Trace ExpressionM(V(1)/I(R1))Magnitude of ZResponseP(V(1)/I(R1))Phase of ZR(V(1)/I(R1))Real part of ZIMG(V(1)/I(R1))Imaginary part of Z

In the Probe windows Trace ExpressionM(I(R1))Magnitude of IResponseP(I(R1))Phase of IR(I(R1))Real part of IIMG(I(R1))Imaginary part of I

In the Probe windows Trace ExpressionV(1,2)Magnitude of VRResponseV(2,3)Magnitude of VLV(3)Magnitude of VCI(R1)*I(R1)*10Real power, P

Run Pspice FileFrequency Response of The Current

(Variation of the current with frequency)Frequency Response of The CurrentAt Resonance, the current is maximumAt Resonance, the current is maximum

Basic QuestionsWhat is the minimum value of Z?What is the maximum value of I?What is the maximum value of P?Z= R

Basic QuestionsThe magnitude of I?When the power P = Po/2, what isThe magnitude of Z?The magnitude of X?The angular frequency?w1 lower half power frequencyw2 higher half power frequencyat w1at w2

Resonant ConditionBy definition the resonant angular frequency, wo, for the RLC series circuit occurs at the peak of the current response. Under this condition:

The real power is maximum The magnitude of impedance is minimum The circuit is purely resistive The imaginary part of the impedance is zero The pf = 1 The current is in phase with the voltage source

Lower half-power angular frequency, w1, conditionBy definition lower half-power angular frequency, w1, occurs when the power is Po/2 and the angular frequency is below the resonant angular frequency.

The real power is Po/2 The current is Io /2 The magnitude of impedance is 2R X = -R The circuit is predominantly capacitive The pf = cos(45) leading

By definition lower half-power angular frequency, w2, occurs when the power is Po/2 and the angular frequency is above the resonant angular frequency.

The real power is Po/2 The current is Io /2 The magnitude of impedance is 2R X = +R The circuit is predominantly inductive The pf = cos(45) lagging

Lower half-power angular frequency, w2, condition

The Voltage Phasor Diagram at woFor R: I is in phase with VRFor L:I lags VL by 90For C:I leads VC by 90For series circuit, use I as the reference.VR = VSIVLVCat woThe circuit is purely resistive.

The Voltage Phasor Diagram at w1 For R: I is in phase with VRFor L:I lags VL by 90For C:I leads VC by 90For series circuit, use I as a reference.VSIVLVL+VCat w1VRVCThe circuit is predominantly capacitive.

The Voltage Phasor Diagram at w2For R: I is in phase with VRFor L:I lags VL by 90For C:I leads VC by 90For series circuit, use I as the reference.VSIVLVL+VCat w2VRVCThe circuit is predominantly inductive.

Learning Sheet 3Five Resonant Parameters:1. Resonant Angular frequency, 2. Lower cut-off angular frequency, 4. Bandwidth of the resonant circuit, 3. Upper cut-off angular frequency, 5. Quality factor of the resonant circuit,

Learning Sheet 3Five Resonant Parameters:1. Resonant Angular frequency, 2. Lower cut-off angular frequency, 4. Bandwidth of the resonant circuit, 3. Upper cut-off angular frequency, 5. Quality factor of the resonant circuit, Note: Lower cut-off angular frequency is also popularly known aslower half-power angular frequency. The same is true for the upper.

Learning Sheet 3We know that, Lower cut-off angular frequency, Upper cut-off angular frequency, Are the half-power frequencies symmetrical around wo? Generally No. The resonant frequency is the geometric mean of the half-power frequencies. But, If Q 10, the half-power frequencies can be approximately considered as symmetrical around wo . Then

Example: Series RLC Resonant CircuitVs = 10 Vrms, R = 10 W, L = 100 mH, C = 10 mF

(xii) The current at w2 in polar form(xiii) The real power P at w2(xiv) The expression for i(t) at w2(xv) The expressions for vL(t), vC(t) and vL(t)+vC(t) at w2(xi) The impedance of the circuit at w2 in polar form

(xvi) Draw the voltage phasor diagram at wo(xvii) Draw the voltage phasor diagram at w1(xviii) Draw the voltage phasor diagram at w2(ixx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at wo(xx) Draw the waveforms of vC(t), vL(t) and vC(t)+vL(t) at w1(xxi) Draw the waveforms of vL(t), vC(t) and vL(t)+vC(t) at w2

(xxii) The resonant frequency, fo(xxiii) The lower cut-off frequency, f1(xxiv) The upper cut-off frequency, f2(xxv) The bandwidth, BW in Hertz(xxvi) The Quality factor, Q

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