Assignment 7 Paallel Resnance OBJECTIVE T investigate the paallel cnnectin f R,, and C. EQUIPMENT REQUIRED Qty Appaatus 1 Electicity & Electnics Cnstuct EEC470 1 Basic Electicity and Electnics Kit EEC471-1 channel scillscpe 1 Functin geneat 100 Hz 1 khz, 0 V pk pk sine (eg, Feedback FG601) PREREQUISITE ASSIGNMENTS Assignments 4 and 6 KNOWEDGE EVE See peequisite assignments. EEC471- -7-1
Assignment 7 Paallel Resnance Fig 7.3 EEC471- -7-
Assignment 7 Paallel Resnance EXPERIMENTA PROCEDURE The cicuit t be investigated is shwn in fig 7.1. Hweve, we will stat with the simple case whee R is mitted. Fig 7.1 Cnnect up the cicuit as shwn in the patching diagam f fig 7.3 cespnding t the cicuit diagam f fig 7.. Fig 7. Set the functin geneat t give an utput, V 1, f 6 V pk pk at 100 Hz.. Set the scillscpe t: Y1 channel (V1) t 0.1 V/cm Y channel (V) t 00 mv/cm Timebase t 1 ms/cm Vay the fequency f the geneat slwly fm 100 Hz t 1 khz, and ntice the vaiatin f the tw vltages shwn n the scillscpe. EEC471- -7-3
Assignment 7 Paallel Resnance Questins 1. Des the vltage utput f the geneat vay appeciably?. Des the vltage acss the capacit and induct vay? 3. What is the elatinship between V 1, V and I (fig 7.)? 4. Des I vay with fequency? 5. Is V a maximum minimum at the esnant fequency f the cicuit? 6. Is I a maximum a minimum at f? Set the geneat t the fequency that gives a minimum in I (maximum V ) and measue the fequency. 7. What is the esnant fequency? Calculate the cuent I at esnance fm 1 - V I = 10kΩ V Since the scillscpe is being used f measuements, it will be fund cnvenient t wk in peak-t-peak values f vltage and cuent thughut. The values f impedance calculated will be the same as if ms values wee used, pvided that the same kind f measue is used f bth vltage and cuent. Questins 8. What is the impedance f the paallel C cicuit at esnance? (Find this fm V ) I 9. Is the impedance high lw at esnance? 10. Hw des this cmpae with the seies esnant cicuit? EEC471- -7-4
Assignment 7 Paallel Resnance Set the geneat fequency t 150 Hz and the utput amplitude t give 8 V pk pk. If the dial accuacy f the geneat used is nt thught sufficient, a digital fequency mete may be used f geate accuacy. Measue the vltage acss the paallel C cicuit, V. Cpy the esults table as shwn in fig 7.4, epduced at the end f this assignment, and tabulate yu esults. Incease the geneat fequency t 00 Hz, and eset the utput amplitude t 8 V pk pk. Measue and ecd the esulting V. Repeat this pcedue f fequencies f, 50, 300, 350, 400, 450, 500, 550, 600, 700, 800, 900 and 1000 Hz. Ensue that the amplitude f V 1 emains cnstant f each fequency setting. Find the esnant fequency again and take a set f eadings at f. Calculate I and Z f each step, and ente yu esults in the apppiate spaces. On a sheet f single-cycle lgaithmic gaph pape, daw a cuves f Z against fequency, using the axes shwn in fig 7.5. EEC471- -7-5
Assignment 7 Paallel Resnance Z (Ω) 3k k 1k 0 0 00 300 400 600 1000 f Fig 7.5 Nw cnnect a 1 kω esist between pints A and B, in paallel with the esnant cicuit, as shwn in fig 7.6. Fig 7.6 Set the geneat fequency back t 150 Hz and the utput t give 8 V pk pk as befe, and measue and ecd the esultant V. Repeat the pcedue f the same fequency steps as befe, and daw the impedance cuve n the same piece f gaph pape as used f the pevius pat f the assignment. Ntice the diffeent shapes f the tw impedance cuves. EEC471- -7-6
Assignment 7 Paallel Resnance Questins 11. Which cicuit has the highe Q? 1. What is the esnant fequency f the cicuit with R = 1 kω inseted? 13. Des this diffe fm when R was excluded? Detemine, fm yu cuves, the bandwidths f the tw cicuits. Using the expessin elating Q with bandwidths and f, detemine the Q values f the tw cicuits. et us analyse the cicuit futhe: Fig 7.7 Refeing t fig 7.7, at esnance the inductive eactance is numeically equal t the capacitive eactance, thus I C and I will be equal and ppsite. They will cancel each the ut, as shwn in the phas diagam f fig 7.8. Fig 7.8 The esultant cuent at esnance is theefe nly that equied t supply the esistance, R. EEC471- -7-7
Assignment 7 Paallel Resnance Questin 14. What was the measued value f this cuent? Refeing t fig 7.7 at esnance I C = I thus I R = I at f. F a paallel cicuit we define the Magnificatin Fact Q as the ati f eactive cuent t esistive cuent, at esnance: ie, Q = I I I = C I Nw I = V V and I = R X I V R =. I X V = R X Q = R ω Similaly: Q = ω CR Cmpae these expessins with thse fund f Q in the seies esnant cicuit. Questin 15. What elatinship have they? Remembe, in the seies case the R is the seies esistance in the cicuit, in the paallel case the R is the paallel esistance. F claity we may dente these by R s and R p f seies and paallel esistance espectively. The equatins then becme: Seies cicuit: Q = ω 1 = R s ω CR s Paallel cicuit: Q = R p = ω CR ω p EEC471- -7-8
Assignment 7 Paallel Resnance Using the expessins just fund, calculate the Q value f the cmpnent values used. Questins 16. Hw des this calculated value cmpae with the Q value fund fm the bandwidth? 17. What was the impedance f the RC paallel cicuit at esnance? The fmula f the impedance f a paallel RC cicuit is given by: 1 Z = 1 1 + ( fm Assignment 4) R X p But at esnance the eactive tems cancel each the ut, and the ttal esultant eactance is ze. at f 1 Z = 1 1 = R p R p Z = R p Questins 18. D yu expeimental findings agee with this? In the case when thee is n paallel esistance cnnected acss the capacit and induct then Rp = (infinity). 19. What wuld the theetical value f the impedance at esnance be? 0. Is it this in pactice? 1. What wuld the theetical value f Q be when R p =. Is it this in pactice? Thee must theefe be sme esistance pesent t limit Q and Z. 3. What the paallel esistances impedances ae thee pesent acss the C cicuit that might limit Q and Z? EEC471- -7-9
Assignment 7 Paallel Resnance An equivalent cicuit may be dawn shwing all the paallel esistances. This is shwn in fig 7.9. Fig 7.9 All these paallel paths cause a eductin in Q and in the impedance at esnance. Questin 4.. What the cmpnent has esistance in the cicuit? An equivalent cicuit including the intenal esistance f the induct may be epesented as in fig 7.10. Fig 7.10 epesents the intenal esistance f the induct. et us investigate the effect f the intenal esistance f the induct n the Q and impedance by exaggeating the value f by adding seies esistance. EEC471- -7-10
Assignment 7 Paallel Resnance Remve the 1 kω esist and cnnect a 100 Ω esist between pints C and D as shwn in fig 7.11. Fig 7.11 As peviusly, take eadings f vltage f the same fequencies between 150 Hz and 1 khz, and ecd them in anthe cpy f the esults table as in fig 7.4. Daw the impedance cuve n the same sheet f gaph pape as befe. Questins 5. Hw des the Q f the cicuit just tested (fig 7.11) cmpae with thse f the the cicuits? 6. What wuld yu say must be dne if a high Q cicuit is t be achieved? Find the esnant fequency f the cicuit. 7. Is this the same as f the the tw cicuits? Thus, nt nly des an appeciable intenal esistance in the induct dastically educe the Q f a paallel tuned cicuit, but als it changes the esnant fequency f the cicuit as well. EEC471- -7-11
Assignment 7 Paallel Resnance The easn is shwn belw. The cuents in each banch ae shwn in fig 7.1. Fig 7.1 Nw, the inductive banch is n lnge a pue inductance, but an inductance and esistance in seies. Thus the cuent I will n lnge be equal and ppsite t I C. Thee will be a phase shift due t the pesence f. (See Assignment 0). The cuent thugh the induct will lag the applied vltage by less than 90. This is shwn in fig 7.13. Fig 7.13 EEC471- -7-1
Assignment 7 Paallel Resnance The induct cuent, I, may be eslved int tw cmpnents. I sin θ, equal and ppsite t I C, and I cs θ, the cuent taken fm the supply at esnance. Thus: I C = V = X C V 1 = V ω C at esnance ω C and I = + V ( ω ) = V + ω at esnance V ωc = V + ω sinθ (1) et us cnside the induct, as in fig 7.14. Fig 7.14 Nw fm the phas diagam sin θ = ω + ω EEC471- -7-13
Assignment 7 Paallel Resnance theefe, substituting this in equatin (1) V wc = V + ω x ω + ω C = + ω + ω = C ω = C 1 ω = C 1 ω = C f = 1 π 1 C Als we knw Z = V at esnance I At esnance, the cuent I 0 = I cs θ (fig 7.13) and fm fig 7.14: csθ = + ω I csθ = V + ω x + ω = V + ω EEC471- -7-14
Assignment 7 Paallel Resnance Z = V V + ω = + ω Substituting f ω Z = + 1 C = + C Z = C ------------------- This is the value f the impedance at esnance and since the cuent and the vltage ae in phase at esnance the impedance appeas as a pue esistance. It is ften called the Dynamic Resistance f the cicuit. EEC471- -7-15
Assignment 7 Paallel Resnance PRACTICA CONSIDERATIONS AND APPICATIONS The pints cncening lsses in a esnant cicuit which wee enumeated in the applicatins ntes f Assignment 6 apply equally f paallel tuned cicuits as f seies cicuits. Hweve the equivalent cicuit becmes as in fig 7.15. Fig 7.15 Again, these ae nmally epesented by a single paallel lss esistance, R p. The diffeence in esnant fequency caused by the taking int accunt the seies esistance f the cil is nmally vey small. This is because: ω = Q = ω Q 1 But ω = C 1 ω = C ω Q EEC471- -7-16
Assignment 7 Paallel Resnance 1 1 ω ( 1 ) = Q C Nmally Q is in the egin 10 t 00. With a typical value f 1 Q = 50 the tem makes the equatin Q ω 1 C inaccuate by nly ne pat in 500, s f 1 π C with an accuacy in this case f ne pat in 5000. This is the same equatin as used in the seies esnance case, and in pactice pvides a vey clse appximatin t the tue f. The impedance f a paallel tuned cicuit is at its highest at esnance. Thus the cicuit is smetimes efeed t as a 'ejecte cicuit'. The paallel esnant cicuit is pehaps me ften used than the seies cicuit, and it is cmmnly used t pvide a fequency dependent lad acss which a high vltage will be pesent at esnance, but a lw vltage at fequencies away fm esnance. This is shwn in fig 7.16. Fig 7.16 The paallel tuned cicuit is used extensively in adi eceives and tansmittes, and many the types f equipment. EEC471- -7-17
Assignment 7 Typical Results and Answes RESUTS FOR ASSIGNMENT 7 V1 V i = 3 10 10 6 1.1 10 10 = 3 i = 0.49 ma pk pk fequency (Hz) V1 (Vp p) V (Vp p) V1 V (Vp p) I (ma p p) Z (Ω) 150 8 0.115 7.885 0.789 145 00 8 0.10 7.790 0.779 69 50 8 0.560 7.440 0.744 759 300 8 1.160 6.840 0.684 170 350 8 0.570 7.430 0.743 760 400 8 0.360 7.640 0.764 474 450 8 0.50 7.750 0.775 31 500 8 0.190 7.810 0.781 44 550 8 0.160 7.840 0.784 05 600 8 0.140 7.860 0.786 179 700 8 0.100 7.900 0.790 17 800 8 0.084 7.916 0.79 106 900 8 0.074 7.96 0.793 94 1000 8 0.064 7.936 0.794 81 f0 = 75 8 1.80 6.70 0.67 1904 Fig 7.4 Paallel C cicuit EEC471- -7-18
Assignment 7 Typical Results and Answes fequency (Hz) V1 (Vp p) V (Vp p) V1 V (Vp p) I (ma p p) Z (Ω) 150 8 0.10 7.898 0.790 19 00 8 0.180 7.80 0.78 31 50 8 0.335 7.665 0.767 435 300 8 0.550 7.450 0.745 733 350 8 0.460 7.540 0.754 613 400 8 0.35 7.675 0.768 4 450 8 0.35 7.765 0.777 301 500 8 0.190 7.810 0.781 44 550 8 0.155 7.845 0.785 199 600 8 0.13 7.868 0.787 167 700 8 0.10 7.898 0.790 19 800 8 0.084 7.916 0.79 106 900 8 0.074 7.96 0.793 94 1000 8 0.064 7.936 0.794 81 f0 = 310 8 0.560 7.440 0.744 757 Fig 7.4 Paallel RC cicuit EEC471- -7-19
Assignment 7 Typical Results and Answes F a paallel cicuit the bandwidth is defined as the diffeence between tw fequencies f 1 and f at which the impedance has fallen t 0.707 x impedance at f 0. Fm the gaphical esults: withut R f 0 =75 Hz with R = 1 kω f 0 = 310 Hz Z 0 = 1904 Ω Z 0 = 757 Ω 0.707 x Z 0 = 1346 Ω 0.707 x Z 0 = 535 Ω f 1 = 65 Hz f = 305 Hz f 1 = 70 Hz f = 360 Hz BW = 40 Hz BW = 90 Hz f Fm the fmula Q = BW withut R Q = 75 with R = 1 kω Q = 310 40 90 Q = 6.8 Q = 3.4 EEC471- -7-0
Assignment 7 Typical Results and Answes fequency (Hz) V1 (Vp p) V (Vp p) V1 V (Vp p) I (ma p p) Z (Ω) 150 8 0.100 7.900 0.790 17 00 8 0.165 7.835 0.784 10 50 8 0.60 7.740 0.774 336 300 8 0.335 7.665 0.777 431 350 8 0.300 7.700 0.770 390 400 8 0.50 7.750 0.775 33 450 8 0.00 7.800 0.780 56 500 8 0.170 7.830 0.783 17 550 8 0.140 7.860 0.786 178 600 8 0.10 7.880 0.788 15 700 8 0.096 7.904 0.790 1 800 8 0.080 7.90 0.79 101 900 8 0.070 7.930 0.793 88 1000 8 0.060 7.940 0.794 76 f0 = 315 8 0.340 7.660 0.766 444 Fig 7.4 Paallel C cicuit with inceased induct esistance EEC471- -7-1
Assignment 7 Typical Results and Answes Results gaph f Fig 7.4 (thee paallel cicuits) EEC471- -7-
Assignment 7 Typical Results and Answes ANSWERS TO ASSIGNMENT 7 1. N. Yes 3. i = V 1 - V 10kΩ A 4. Yes, because V 1 and the esistance ae cnstant while V vaies with fequency. 5. V is a maximum at f 0. 6. i is a minimum at f 0. 7. f 0 = 77 Hz 8. F the paallel cicuit: Z = V i 1.1 0.49 10 = 3 Z = 45 Ω 9. Z is high at f0 10. In a seies RC cicuit Z is lw at f 0. 11. The cicuit withut the esist in paallel has the highe Q. 1. f 0 = 310 Hz 13. Yes 14. 0.74 ma 15. The expessins f Q in tems f w 0, R, and C f a paallel cicuit ae the invese f thse f a seies cicuit. 16. Fm the given value f the cmpnents: R Q = ω Q = ω 0 CR p EEC471- -7-3
Assignment 7 Typical Results and Answes 3 10 π 310 100 10 = 3 =π x 310 x. x 10 6 x 10 3 Q = 5.14 Q = 4.8 The diffeence in value is due t the actual esnant fequency, 310 Hz, used in the calculatin nt being the same as that which may be calculated using the given cmpnent values. In cmpaisn, fm the bandwidth Q = 3.4 17. 757 Ω 18. At f 0 : Fm the expeimental esults R p =Z = 757 Ω 19. Infinity 0. N 1. Infinity. N Fm the given cmpnent value R p = 1 kω 3. The fllwing impedances ae effectively acss the paallel C cicuit: Input impedance f scillscpe Y amplifies. Output impedance f functin geneat. 10 kω cuent mniting esist. 4. The induct has esistance. 5. The Q f the cicuit with exta esistance in seies with the induct is lwe than thse f the the cicuits. 6. T achieve a high Q cicuit, the value f esistance must be as lw as pssible. EEC471- -7-4
Assignment 7 Typical Results and Answes 7. f 0 = 315 Hz f the cicuit with inceased induct esistance. In cmpaisn: f 0 = 75 Hz f the basic C paallel cicuit. f 0 = 310 Hz f the cicuit with a 1 kω esist in paallel with the C cicuit. EEC471- -7-5
Assignment 7 Typical Results and Answes fequency (Hz) V1 (Vp p) V (Vp p) V1 V (Vp p) I (ma p p) Z (Ω) 150 00 50 300 350 400 450 500 550 600 700 800 900 1000 Fig 7.4 EEC471- -7-6