1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits


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1 E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng resstve and reactve components and ther effect on the crcut mpedance are smlar whle the result can often be the opposte of the case for seres confguratons. Some comparable parallel combnatons wll therefore be examned.. Ideal Inductor and apactor n Parallel onsder the nductor and capactor connected n parallel as shown n Fg. below. The reactve elements are taken to be deal. In ths case the end nodes of both elements are connected together whch means that the voltage across the nductor s dentcal to the voltage across the capactor so that V V. If ths s taken as reference zero phase, then t can be seen that the current through the capactor leads the voltage so that t appears 90 0 ahead on the phasor dagram of Fg., whle the current through the nductor lags the voltage so that t appears 90 0 behnd. It can be seen therefore that n ths case the currents through the nductor and the capactor are n antphase or 80 0 out of phase wth each other. The relatve magntudes of the currents are dfferent and depend on the values of nductance and capactance of these elements at the frequency of exctaton. Ths stuaton s smlar to the seres combnaton except that n ths case t s the voltage rather than the current whch s the common component. I I I V V current leads voltage for V V I current lags voltage for Fg. An Inductor and apactor onnected n Parallel
2 I I V, V V, V I I t Fg. Phasor Dagram and Waveforms for Inductor and apactor n Parallel Waveforms are shown for snusodal exctaton of the crcut n Fg.. From ths t s clearly evdent that the phasors representng the currents through the nductor and the capactor are exactly 80 0 out of phase, showng excursons on opposte sdes of the abscssa axs. The dfference n the ampltudes depends on the relatve magntudes of the reactances as functons of frequency and hence also on the values of nductor and capactor used. The mpedance of the parallel combnaton can be found as for the case of the seres combnaton. As the elements are n parallel, the voltages across both elements are dentcal and the current through the parallel combnaton s the sum of the voltage drops across the ndvdual elements. Then the mpedance s gven as: Ζ where Then: or Ζ
3 Then: It can be seen that for the parallel combnaton of deal nductor and capactor the overall mpedance of the network s also purely reactve wth no resstance. It can be seen ths tme that a crtcal pont exsts when the demonnator s zero. Ths occurs when: or as before when: 0 The value of ths frequency s agan referred to as the resonant frequency, and depends entrely of the values of the components used. Ths tme the result mples that at ths frequency the mpedance of the parallel combnaton s nfnte for deal components. The magntude of the mpedance s gven as: 3
4 0 Fg. 3 The Magntude of the Impedance as a Functon of Frequency The magntude of the mpedance s shown as a functon of frequency n Fg. 3. It can be seen that the mpedance s very low at hgh and low frequences but tends towards nfnty at the resonant frequency 0. The stuaton at resonance can also be shown as n the phasor dagram and waveforms of Fg. 4. In essence, at the resonant frequency the effect of the nductve reactance counteracts that of the capactve reactance. Therefore the same voltage s present across both elements but the currents flowng through each element have equal magntude and opposte polarty at resonance. The net effect of ths s that zero current flows nto the parallel combnaton, gvng the resultant nfnte mpedance shown n Fg. 3. What happens s that energy s ntally drawn from the source feedng the crcut. Ths energy then oscllates between the nductor and capactor so that when current s flowng nto one element t s flowng out of the other. The precse magntudes of the ndvdual currents depend on the values of these components. 0 V, V 0 I V,V I 0 0 I I t Fg. 4 Phasors / Waveforms for Seres Inductor and apactor at esonance 4
5 .3 esstor, Inductor and apactor n Parallel Fg. 5 shows a resstor added n parallel wth the prevous nductor and capactor already connected n parallel. Agan, the same voltage s developed across all of the elements so that V V V. The same relatonshps hold between voltage and current n the nductor and the capactor so ther phase relatonshps are unaltered. The current flowng through the resstor s n phase wth the voltage across t so that ther phasors appear supermposed n Fg. 5. I I V V I V I I I current leads voltage for V V V current lags voltage for Fg. 5 A esstor, apactor and Inductor,, onnected n Parallel In ths case the mpedance has an added element n the resstor whch s present. The net current flowng through the crcut s the vector sum of the three ndvdual components of current so that the mpedance s gven as: Ζ where Then: Ζ Invertng as before 5
6 Snce the mpedance of the parallel combnaton of the nductor and capactor has already been evaluated above then the product over sum rule can be used to add the resstance n parallel wth ths whch gves: ( ) In order to obtan the mpedance n proper complex form ths expresson must be multpled n the numerator and denomnator by ts complex conugate. Ths gves: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ths s a much more complcated expresson than n the case of the seres combnaton. It s complex havng a real or resstve part and an magnary or reactve part. Agan, the reactve part can be domnated by the nductve reactance or the capactve reactance dependng on the values of these components and the frequency of operaton. The mpedance therefore has an assocated magntude and phase whch can be found n the tradtonal manner but havng a more complcated form of algebra: 6
7 ( ) ( ) ( ) φ Tan ( ) The values of both the magntude and phase depend on the values of all of the components as well as the frequency. onsder the case as before when: or 0 At ths frequency the mpedance becomes purely real and reduces to the value of the resstance alone. Ths s shown n the plot of the magntude of the mpedance of the combnaton shown n Fg. 6. In effect what has happened here s that the parallel combnaton of the nductor and capactor at resonance has produced nfnte mpedance between them, leavng only the resstance present at ths frequency. Fg The Magntude of the Impedance of the Seres ombnaton 7
8 The net resultant current flowng through the parallel combnaton s gven as: φ φ If the voltage phasor s taken as the reference zero phase vector, then the magntude and phase of ths and all of the currents nvolved, ncludng the resultant current can be shown as n Fg. 7. Note that the resultant net current has a magntude and phase whch depends on all three components n the combnaton and can lead or lag the current dependng on whether the net reactance s nductve or capactve. V, V, V I I I I I I V, V, V I t Fg. 7 Phasor Dagram and Waveforms for Seres ombnaton.4 Applcaton The parallel combnaton s often used as a load on the transstor n the ntermedate frequency amplfer of a rado recever. When ths s the case, the IF amplfer gan takes on a frequency response whch mrrors the frequency dependence of the mpedance of the parallel network. Ths provdes a frequencyselectve tuned gan stage whch amplfes only the wanted narrow band of frequences surroundng the ntermedate frequency whch s assocated wth the rado staton that the recever s tuned to. 8
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