Resonance

Wednesday, 6 March 2019

4:10 PM

Machine generated alternative text: ELEC2134: Circuits and Signals School of Electrical Engineering and Telecommunications AC Resonance Series Resonance Parallel Resonance Series to Parallel Conversion Professor Eliathamby Ambikairajah
Machine generated alternative text: Resonance Resonanceis the tendency of a system to oscillate at maximum amplitude at a certain frequency known as the resonant frequency. The possibility of resonance always exists wherever there is periodic motion -movement that is repeated at regular intervals. The resonant system may be electrical, mechanical, hydraulic , acoustic or some other kind. Electrical resonanceoccurs in an electric circuitat a particularfrequencywhen the imaginary parts ofimpedances of circuit elements cancel each other. At resonance, the voltage and current at the circuit input terminals are in phase. Resonant circuits can generate higher voltages and currents than the input voltage or current. 2 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Series Resonance RLC circuits posses the distinctive ability to provide natural responses of the oscillatory type. The energy flows back and forth between the capacitance and the inductance. Consider a series RLC circuit The impedance seen by the source is 3 Prof E Ambikairajah, UNSW, Australia net reactance
Machine generated alternative text: Depending on the frequency of the applied source, we have three possibilities This behaviour, referred to as unity power - factor resonance , occurs at the special frequency that makes or . 4 Prof E Ambikairajah, UNSW, Australia rad/s is called the resonance frequency
Machine generated alternative text: For obvious reasons, is called the resonance frequency. Even though the individual impedances and are not zero, at the combination act as a short circuit. The component of in a series RLC circuit is 5 Prof E Ambikairajah, UNSW, Australia Magnitude of Impedance L X L R Z o C X C 1 Resonant frequency C L 1 0 LC reactance L 1/( C ) -
Machine generated alternative text: The current frequency response near resonance 6 Prof E Ambikairajah, UNSW, Australia BW-Bandwidth Half power points
Machine generated alternative text: At Resonance 7 Prof E Ambikairajah, UNSW, Australia Current at resonance At resonance Atseriesresonance equalamplitude) Resonant frequency V LO : Voltage across the inductor at resonance (amplitude ) V CO : Voltage across the capacitor at resonance (amplitude ) + - Ê ³ ð ± õ õ õ
Machine generated alternative text: At Resonance We define the quality factor of the series RLC circuit as We have At resonance , voltage across the inductor at resonance, At resonance , voltage across the capacitor at resonance, 8 Prof E Ambikairajah, UNSW, Australia Atseriesresonance equalimpedance) The peak voltage amplitude across the reactive elements (at resonance) can exceed the peak amplitude of the applied signal ( ).
Machine generated alternative text: Resonance Voltage Rise At resonance Clearly if , the peak voltage across the reactive elements will be greater than that of the applied voltage, a phenomenon known as resonance voltage rise . 9 Prof E Ambikairajah, UNSW, Australia and Note: Maximum current occurs in the circuit at resonance = 0
Machine generated alternative text: Resonance Voltage Rise The amount of magnification depends on the quality factor. For example if , , then It is interesting to note that the smaller the resistance in a series RLC circuit, the higher the value of and hence greater amount of voltage rise at resonance. In the limit , we would have and this would result in an infinite voltage rise. The quality factor is also a measure of frequency selectivity. Thus, we say that a circuit with a high has a high selectivity, whereas a low circuit has low selectivity. 10 Prof E Ambikairajah, UNSW, Australia Much greater than the applied voltage
Machine generated alternative text: Figure below shows the relative response versus for ã îëô ëðô and ïðð where we observe that highest provides the best frequency selectivity, i.e. the higher rejection of signal component outside the bandwidth ( ) which is the difference in 3 dB frequencies. 11 Prof E Ambikairajah, UNSW, Australia The quality factor (Q) is the ratio of its resonant frequency to its bandwidth. =
Machine generated alternative text: The plot of versus for a given has the familiar bell shaped profile of the band-pass function. 12 Prof E Ambikairajah, UNSW, Australia If and ,then ¿²¼ ãÏ
Machine generated alternative text: At resonance, the average power dissipated in the resistance is also maximised, The frequencies at which is down to half its maximum value are the half-power frequencies and . Half-power 13 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: We can show that, If , then the half-power frequencies are given by the approximate expressions: 14 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Example 5.1 Let , , , . Calculate the phasor voltages across the elements (if is the resonant frequency) for (a) (b) (c) 15 Prof E Ambikairajah, UNSW, Australia We have
Machine generated alternative text: 16 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Example 5.2 Aseriesresonantcircuitisshownbelow.Calculatetheresonantfrequency, thebandwidthandthehalf-powerfrequencies.Assumingthatthefrequency ofthesourceisthesameastheresonantfrequency,findthephasorvoltages acrosstheelementsanddrawaphasordiagram. 17 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Thetotalimpedanceofthecircuit Phasorcurrent Voltageacrosstheelements 18 Prof E Ambikairajah, UNSW, Australia Noticethatthevoltageacrossthe inductanceandcapacitancearemuch largerthanthesourcevoltagein magnitude;Nevertheless, voltagelawissatisfiedbecause and areoutofphaseandcancel .
Machine generated alternative text: Parallel Resonance Another type of circuit known as a parallel resonant circuit is shown below: The admittance seen by the source is the sum of the three admittances. 19 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: We expect a resonance behaviourwhen the susceptance and cancel each other out and this occurs when . Hence, we have The resonance frequency is exactly the same as the expression for the resonant frequency of the series circuit. We observe that at parallel resonant frequency (resistive) and 20 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: At resonance (magnitude) (magnitude) (magnitude) At parallel resonance, By definition quality factor at parallel resonance 21 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: If , , , then and The plot versus is shown below. 22 Prof E Ambikairajah, UNSW, Australia Taking magnitude,
Machine generated alternative text: Eventhoughtheindividualadmittances and arenonzero, theirsumvanishesatresonance,makingtheparallel combinationof and effectivelybehaveasanopencircuit. Theaveragepowerdissipatedintheresistance Thefrequenciesatwhich isdowntohalfitsmaximumvalue arethehalf-powerfrequencies and andthehalf-power ,suchthat 23 Prof E Ambikairajah, UNSW, Australia Combining these equations, we get
Machine generated alternative text: Example 5.3 Findthe and valuesforaparallelresonantcircuitthathas , , .If ,drawthephasordiagram showingthecurrentsthrougheachoftheelementsinthecircuitat resonance. 24 Prof E Ambikairajah, UNSW, Australia Notice that, the current through the inductance and capacitance are larger in magnitude than the applied source current. However, since and are out of phase, they cancel.
Machine generated alternative text: Practical Resonant Circuit The parallelRLC configuration is known as tuned circuit, such as tuned oscillators and tuned amplifiers in radio communications. Of the three components making up an RLC circuit, the inductor is the least ideal because of the winding resistance ( ). We wish to investigate the impact of upon the resonance frequency. For an ideal inductor For non ideal inductor, we have By realisingthe denominator of the last term, we obtain 25 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: We are interested in the frequency that makes purely real Imaginary part = 0 In the limit (ideal case) The quality factor of the coils is Clearly, the higher the value of the is close to 26 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: When a radio or TV receivers is tuned to a particular station or a channel, it is set to operate at the resonant frequency of that station. A parallel circuit has high impedance at its resonant frequency. Therefore it attenuates signals at all frequencies except the resonant frequency. Varying the capacitance of the tuned circuit, will change the resonant frequency. Generally inductance is kept constant and the capacitor value is changed as we select different stations or channels. 27 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Series to Parallel Conversion It is possible to select and so that, at a given frequency , the admittance of the parallel structure equals that of the series structure. 28 Prof E Ambikairajah, UNSW, Australia Series to parallel conversion of an RL circuit
Machine generated alternative text: Rationalising both sides and equating the real and imaginary parts yields Note: 29 Prof E Ambikairajah, UNSW, Australia Equating the real parts If then R P = Parallel resistance Equating the imaginary parts If is very small
Machine generated alternative text: TheparallelRLCconfigurationfindsapplicationinaclassof circuitsknownastuned circuits,tunedamplifiersandtuned oscillators. 30 Prof E Ambikairajah, UNSW, Australia Smallcoilresistance provideslarge Small implieslarge . A B A B A B ¤¤ Small implies
Machine generated alternative text: Example 5.4 AparallelRLCcircuithas , , .Find resonantfrequency ,bandwidth ,andthequalityfactor for(a) and(b) . (a)With i.e. 31 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: (b)With i.e. rad/sec Note :Thequalityfactorofthecircuithaschangedfrom100(when or )to33.33(when or ) 32 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Summary Series ResonanceParallel Resonance Equation Circuit current Max current: Max voltage: Bandwidth Qualityfactor = = Resonantrise Voltage( or ) Current( or ) Phaseangle Power factor 33 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: Example 5.5 Aradioreceiverwithaparallel circuithasaninductance .It istunedtoaradiostationtransmittingat810kHzfrequency. (a)Whatisthevalueofthecapacitorofthecircuitatthisresonant frequency? resonantfrequency (b) What is the value of resistance if ? 34 Prof E Ambikairajah, UNSW, Australia
Machine generated alternative text: 35 Prof E Ambikairajah, UNSW, Australia (c)Ifanearbyradiostationtransmits740kHzandbothsignalsarepicked upbytheantennahavingthesamecurrentamplitude ,whatisthe ratioofthevoltageat810kHztothevoltageat740kHz. That is, the voltage developed across the parallel circuit when it is tuned at =810 is 14.9 times larger than the voltage developed at =740 .
Machine generated alternative text: 36 Prof E Ambikairajah, UNSW, Australia Exercise 5.1 Q1: Determine the and values for a series resonant circuit that has , , and . Find the bandwidth and approximate half-power frequencies of the circuit. Answer: , , , , and Q2:(a)Usinga inductance,specifysuitablevaluesfor and toimplementaseriesRLCcircuitthatresonatesat1MHz withaqualityfactorof100. Answer:(a) and , (b)Ifthecircuitisdrivenwithasourcehavingapeakamplitudeof 1Volt,findthepoweritdissipatesatresonance,aswellasthe amplitudesofthevoltagesacrossitsreactiveelements. Answer: ,
Machine generated alternative text: 37 Prof E Ambikairajah, UNSW, Australia Exercise 5.1 Q3:Forthecircuitshownbelow and Findtheresonancefrequency ,bandwidth ,uppercut-off frequency andthelowercut-offfrequency . Ans: rad/sec rad/sec, Q4:Findthe and valuesforaseriesresonantcircuitthathas =470 ,aresonantfrequencyof5MHz,andabandwidthof200 kHz. Answer:(a) andL , Ýãï kÚ î ã ïð ï ã ï ã ï ³Ø V s Q5:Aparallelresonantcircuithas and . FindLandC. Answer:L ,C=795.8pF
Machine generated alternative text: 38 Prof E Ambikairajah, UNSW, Australia Exercise 5.1 Q6:ThecircuitinFig.4.1canbeconvertedtoFig.4.2.UsingFig 4.2, (i)showthat where (ii)ShowthattheBandwidth(BW) + - ×ã× ³ ð ± ÊãÊ ³ Fig.4.1 Fig.4.2 + - ×ã× ³ ð ± ÊãÊ ³
Machine generated alternative text: 39 Prof E Ambikairajah, UNSW, Australia Exercise 5.1 Q7:Showthatprovided theimpedance seenby thesourceoffigurebelowbecomespurelyresistivefor andthat at ( - resonantfrequency) Ý î ï õ ó Q8:Aseriesresonantcircuitisshownbelow.Thehalf-power frequencies( and )canbeobtainedbysettingthe impedance|Z|equalto .Showthat: = + = + . + -
Machine generated alternative text: [1] Alexander, C. K., & Sadiku th edition, McGraw Hill. th edition, Wiley & sons. School of Electrical engineering and Telecommunications, UNSW, Australia. [4] Hambley Publishing. References

 

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