Finding the equivalent resistance, inductance, capacitance, and impedance: A new powerful pedagogical method

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1 Finding the equivlent resistnce, inductnce, cpcitnce, nd impednce: A new powerful pedgogicl method G. E. Chtzris, 1 E. G. Chtzris 2 nd M. M. Kntonidou 1 1 Electricl Engineering Deprtment, School of Pedgogicl nd Technologicl Eduction (ASPETE), Athens, Greece 2 Deprtment of Mthemtics, University of Ionnin, Greece E-mil: gextz@otenet.gr, ge.xtz@spete.gr Astrct This pper presents new method for the clcultion of the equivlent resistnce, inductnce, cpcitnce, nd impednce of liner electric circuit. It is sed on new pproch for finding the Thevenin equivlent circuit in comintion with mesh or nodl nlysis. It is powerful pedgogicl tool which leds to n esy, systemtic, time-sving nd solutely ccurte clcultion of the ove prmeters, regrdless of the circuit topology nd complexity. Keywords conductnce mtrix determinnt; inspection; mesh nlysis; nodl nlysis; resistnce mtrix determinnt; Thevenin equivlent circuit One of the principl nd fundmentl suject res tught to electricl nd electronic engineering students within their electric circuits course is the comintion of resistors. Within the sid frmewor, students re tught the series, prllel, delt ( ), nd wye (Y) comintions. Bsed on these, they my e le to clculte the equivlent resistnce of reltively complicted comintion of resistors, ming successive steps towrds the simplifiction of the comintion depending on the circuit topology. The ility to find the equivlent resistnce will prove necessry to the students for the clcultion of the Thevenin equivlent circuit nd the mtching lod t lter stge. This is ecuse TH (or mtching lod) is, y definition, the equivlent resistnce of circuit with ll its sources zeroed. The simplifiction process of circuit is etter suited to reltively simple comintion of resistors. If the topology of the circuit is complicted due to the existence of short-circuits, Y, nd/or -comintions, or if the comintion is nonplnr, finding the equivlent resistnce ecomes rther difficult nd t times even impossile ts for the students. This difficulty entils riss for the ccurcy of the clcultion nd it is time consuming. This is ecuse the students re expected to ct depending closely on the topology of the circuit, something which might include mny Y conversions; this might cuse the students nxiety nd me them prone to mistes in clcultions. Similr prolems ut with greter degree of difficulty re encountered y the students in the cse of comintions of inductors, mutul-coupled coils, cpcitors, nd impednces. This rticle presents new method for the clcultion of the ove prmeters, which is sed on new pproch for finding the Thevenin equivlent Interntionl Journl of Electricl Engineering Eduction 44/1

2 A new pedgogicl method 65 circuit in comintion with mesh or nodl nlysis. It is powerful pedgogicl tool which helps students cope with the difficulties mentioned ove, regrdless of the circuit topology nd complexity. Method description According to the method proposed y Chtzris et l. 1,2 for determining the Thevenin equivlent circuit of d.c. liner electric circuit (d.c. LEC), resistnce is connected to the terminls - of d.c. LEC, s shown in Fig. 1. Since d.c. LEC my e either plnr or nonplnr, the method to e used for clculting I nd V in ech of the two cses my e mesh or nodl nlysis for the former, nd nodl nlysis for the ltter. Cse (i) If mesh nlysis is used for plnr circuits, the resistnce mtrix determinnt D, the current I nd the voltge V re lwys of the form: D ( )= +l (1) m I = (2) +l m V = (3) +l where the constnts,, m. SC OC But, since I N = I nd V TH = V, (2) nd (3) give I N m VTH = limv = lim + = l m m = lim I = lim + = 0 0 l l TH VTH = N = = l I N m (4) (5) (6) dc LEC + V I Fig. 1 Ad.c. LEC. Interntionl Journl of Electricl Engineering Eduction 44/1

3 66 G. E. Chtzris et l. Bsed on (6), (1) gives D ( )= + l = l + = ( + ) TH (7) Thus, the clcultion of the determinnt D leds to the finding of TH, tht is, of the equivlent resistnce eq = when ll circuit sources re zeroed. Bsed on the ove, this method my prove to e powerful pedgogicl tool in the following suunits of electric circuits: Cse (i). Finding the equivlent resistnce of comintion of resistors Let us suppose tht in rther complicted plnr circuit (Fig. 2) we need to determine the equivlent resistnce eq t terminls. Connecting resistnce t terminls (Fig. 2), we cn otin y inspection the resistnce mtrix determinnt D, which could result from the specific circuit if the circuit lso included sources. The clcultion of D leds to the finding of coefficients, ; thus the rtio / gives the equivlent resistnce eq t terminls. The clcultion of D cn e mde esily nd quicly y mens of pocet clcultor tht cn process determinnts including symolic vriles. In cse the student hs pocet clcultor tht cn process determinnts including lgeric vlues only, he cn wor eqully well sed on the following: Since the determinnt D is liner function of (eqns 1, 7), D( 0)= l D 0 D()= 1 + (8) l = ( ) l = D() 1 D( 0) The clcultion of D(0), D(1) gives the rtio / nd thus the equivlent resistnce eq t terminls is otined. Exmple 1 Find eq t terminls for the circuit in Fig. 3(). Ech resistor is 10 Ω (ten nd dpted from ef. 3, p. 69, where it is referred to s chllenging prolem ). Solution In Fig. 3(), mesh nlysis gives y inspection plnr plnr comintion comintion of resistors of resistors ( ) ( ) Fig. 2 Circuit for cse (i). Interntionl Journl of Electricl Engineering Eduction 44/1

4 A new pedgogicl method ( ) ( ) Fig. 3 Circuit for exmple 1. D = = l = l = = = = 12. 5Ω eq eq = Cse (i). Finding the equivlent inductnce of comintion of inductors Since the comintions of inductors re descried y similr mthemticl equtions s regrds the equivlent inductnce for the series, prllel, Y, nd comintions, the equivlent inductnce is otined in wy similr to the one descried for the equivlent resistnce ove; in this cse, n inductnce L is connected t terminls. the inductnce mtrix determinnt D will e DL ( )= L + l = l L+ L L = + ( ) eq (9) Specil ttention is needed in cse of comintions tht include mgneticlly coupled coils. This is ecuse the inductnce mtrix determinnt includes terms which represent the couplings etween the coils, nd which re defined in ccordnce with the dot convention. In this cse, the clcultion of the equivlent inductnce in the time domin is very difficult, nd t times impossile, for the student. This mes the proposed method solutely necessry nd effective. Interntionl Journl of Electricl Engineering Eduction 44/1

5 68 G. E. Chtzris et l. 0.5H 0.5H 1H 1H 1H 1H 0.5H 0.5H 1H 0.5H 0.5H L 1 1H 2 () () Fig. 4 Circuit for exmple 2. Exmple 2 Find L eq t terminls for the circuit in Fig. 4() (ten from ef. 4, sec. II, B). Solution In Fig. 4, mesh nlysis gives y inspection D = L = L + = l = 275. l 275. Leq = L = = Leq = H 3 1 L = Cse (ic). Finding the equivlent cpcitnce of comintion of cpcitors Since the comintions of the cpcitors re descried y similr mthemticl equtions (lthough in swpping mode) s regrds the equivlent cpcitnce for the series, prllel, Y, nd comintions, the equivlent cpcitnce is otined in wy similr to the one descried for the equivlent resistnce ove; in this cse, cpcitnce C is connected t terminls. The cpcitnce mtrix determinnt D consists of terms resulting from the reciprocl vlues of the cpcitnces of the comintion, nd will e D( 1 C)= C+ = l + (10) C = C + l Ceq Notice tht if the student hs pocet clcultor tht cn process determinnts including lgeric vlues only, he cn wor eqully well ting into considertion tht, since the determinnt D is liner function of 1/C (eqn 10), eqn (8) ove ecomes D( )= l D D()= + l = ( ) 1 l = D() 1 D( ) (11) Interntionl Journl of Electricl Engineering Eduction 44/1

6 A new pedgogicl method 69 The clcultion of D( ), D(1) gives the rtio / nd thus the equivlent cpcitnce C eq t terminls is otined. Exmple 3 Find C eq t terminls for the circuit in Fig. 5(). Ech cpcitor is 1µF (ten from ef. 5, p.148). Solution In Fig. 5(), mesh nlysis gives y inspection D = C = = C l = Ceq = C = = Ceq = 04. mf l 40 Cse (id). Finding the equivlent impednce of circuit including pssive elements, L, C, M In this cse we dopt exctly the sme nlysis s in cse (i); the only difference is tht we lwys wor in the frequency domin nd tht the impednce mtrix determinnt D consists of complex numers (constnt nd/or vrile). Thus D ( )= + l = l + Z = + ( ) eq (12) where the constnts, re complex numers. Exmple 4 Find Z eq t terminls for the circuit in Fig. 6() (ten from ref. 4, sec. II, C) () C () 1 Fig. 5 Circuit for exmple 3. Interntionl Journl of Electricl Engineering Eduction 44/1

7 70 G. E. Chtzris et l. j2ω j2ω 1Ω j2ω j5ω j2.5ω j5ω 0.5Ω j5ω 1Ω j2.5ω j0. 4Ω 1Ω 2Ω 1Ω j2ω j5ω j2.5ω ( ) ( ) Fig. 6 Circuit for exmple 4. 1 j5ω 0.5Ω j5ω 1Ω j2.5ω j0. 4Ω 2 1Ω 3 2Ω Solution In Fig. 6(), mesh nlysis gives y inspection j j3 0 D = 05. j j j j j0. 4 j = ( j43 6) + ( 87 + j90 4) = l = 87 + j90. 4 Z Cse (ii) If nodl nlysis is used specificlly for nonplnr circuits, the conductnce mtrix determinnt D, the voltge V AB nd the current I AB re lwys of the form: D( 1 )= +l (13) m VAB = (14) + l m IAB = (15) + l where the constnts,, m. But, since V TH = V OC AB nd I N = I AB, SC (14) nd (15) give V TH l j = Z = = Z = j Ω j43. 6 eq eq m m = limvab = lim + = l l (16) I N m = lim IAB = lim + = 1 1 l m (17) Interntionl Journl of Electricl Engineering Eduction 44/1

8 A new pedgogicl method 71 TH VTH = N = = I l Bsed on (18), (13) gives N D( 1 )= + l = l = + ( ) TH (18) (19) Thus, the clcultion of the determinnt D leds to the finding of TH, tht is, of the equivlent resistnce eq = AB when ll circuit sources re zeroed. Bsed on the ove, the proposed method my prove to e s effective nd suitle for cse (ii), s for cse (i). Notice tht if for the clcultion of the determinnt D in cses (i), (i), nd (id) ove the student hs pocet clcultor tht cn process determinnts including lgeric vlues only, the clcultion of rtio / for finding eq, L eq, nd Z eq cn result from eqn (11), ecuse the determinnt D is liner function of 1/, 1/L nd 1/Z respectively. In these cses, the determinnt D consists of terms resulting from the reciprocl vlues of the resistnces, inductnces, nd cpcitnces of the comintions. The clcultion, however, of rtio / for finding C eq (cse(ic)) results from eqn (8), ecuse the determinnt D is liner function of C. As follow-up to the ove considertions, we will provide single exmple for the clcultion of the equivlent resistnce. Given tht the proposed method holds true for the clcultion of ll the prmeters of cse (i), this exmple is ten to e n illustrtion for them, s well. Exmple 5 Find eq t terminls for the nonplnr circuit in Fig. 7(). Ech resistor is 1Ω (ten from ef. 5, p. 146). Solution In Fig. 7, nodl nlysis gives y inspection () 1 2 () Fig. 7 Circuit for exmple 5. Interntionl Journl of Electricl Engineering Eduction 44/1

9 72 G. E. Chtzris et l D = = = = = 2Ω l 64 eq AB eq = = l = 64 Evidence of the method s effectiveness Evidence of the effectiveness of the proposed method on the student lerning outcomes cme from dt gthered during n interpretive oservtion process, followed y student nd techer survey. The process ws designed to encompss 5 cycles of clssroom oservtions, corresponding to the 5 exmples exmined within cses i (i i) nd ii ove. The clssroom oservtions were conducted during the 4th nd the 9th wee of 14-wee semester (fll 2003), with the im of encourging comprison etween the proposed method nd the method trditionlly descried in textoos on electric circuits. The smple ws drwn from ASPETE, School of Pedgogicl & Technologicl Eduction locted in Athens, nd consisted of 3 groups of electricl nd 3 groups of electronic engineering students with men ge of 21 yers. Ech group ws limited to mximum of 20, nd there were no chnges etween cycles s to the groups internl rrngements. During ech 45-minute oservtion period, the reserchers too notes focusing on the impct of the proposed method on two vriles: the time spent y the students when sed to del with the specific exmples (1 5), nd the ccurcy of their nswers. At the end of the 5th wee the reserchers dministered student nd techer survey to cross-chec the outcomes. Figs 8 nd 9 elow show the tremendous effect of the proposed method on the time vrile. The correltion results of these two figures demonstrte the drmtic decrese in the time needed y the students to rrive t solution when pplying the proposed method n oservtion finding vlid for ll 5 exmples. Interestingly, this finding ws reinforced y findings from the student survey. Wht is worth noticing is tht the mjority of students too less thn 15 minutes to respond to ech of the prolem-solving situtions nd, with the exception of exmple 4, only 25 minutes to rrive t solution (Fig. 9). By contrst, the proportion of students who mnged to do tht woring with the trditionl method in the sme prolem-solving situtions ws considerly lower (Fig. 8). This sttisticlly significnt difference is prticulrly true for exmple 4. Interntionl Journl of Electricl Engineering Eduction 44/1

10 A new pedgogicl method 73 Trditionl Method No. of Students t<15min 15min<t<25min t>25min Exmples Fig. 8 Time needed with the trditionl method. Proposed Method No. of Students t<15min 15min<t<25min t>25min Exmples Fig. 9 Time needed with the proposed method. Note Specificlly for exmples 1, 3 nd 5, the time needed y students woring with the trditionl method would e much greter if the Y nd comintions were not lnced something tht does not influence the proposed method. Figs 10 nd 11 show the effect of the proposed method on the degree of ccurcy of the students responses to the specific prolem situtions (exmples 1 5). In this cse, the correltion outcomes of the two figures show drmtic increse in the numer of correct nswers given y students employing the proposed method Interntionl Journl of Electricl Engineering Eduction 44/1

11 74 G. E. Chtzris et l. Trditionl Method No. of Students correct nswer incorrect nswer Exmples Fig. 10 Accurcy of nswers with the trditionl method. Proposed Method No. of Students correct nswer incorrect nswer Exmples Fig. 11 Accurcy of nswers with the proposed method. (Fig. 11), vs those woring with the trditionl one (Fig. 10). There is sttisticlly significnt difference etween the two, which holds true for ll 5 exmples exmined ove, nd most specificlly for exmples 2 nd 4. The sid difference is reinforced y findings from the techer survey. Note Specificlly for exmples 1, 3 nd 5, the numer of correct nswers given y students woring with the trditionl method would e much smller if the Y Interntionl Journl of Electricl Engineering Eduction 44/1

12 A new pedgogicl method 75 nd comintions were not lnced something tht does not influence the proposed method. To summrise our findings, when compred to the method trditionlly used to dte, the proposed method ppers to e time-sving nd y fr more ccurte. Conclusions The clcultion of the equivlent resistnce, inductnce, cpcitnce, nd impednce of liner electric circuit is topic tht should receive consistent ttention from engineering nd eductionl reserchers. To ddress the issue, this pper presents new pedgogicl method tht is sed on new pproch for finding the Thevenin equivlent circuit in comintion with mesh or nodl nlysis. Tringultion of dt gthered from clssroom oservtions, student survey nd techer survey demonstrte the method s suitility nd provide cross-vlidtion of the strong reltionship etween the student lerning outcomes nd its ppliction. Wht these outcomes suggest is tht the proposed method, compred to the method trditionlly descried nd employed to dte, leds to n esy, systemtic, time-sving nd solutely ccurte clcultion of the ove prmeters, regrdless of the circuit topology nd complexity. eferences 1 G. E. Chtzris, P. G. Cottis, M. D. Tortoreli, P. B. Mltests, N. J. Kolliopoulos nd S. N. Liviertos, Powerful pedgogicl pproches for finding Thevenin nd Norton equivlent circuits for liner electric circuits, Intl. J. Elect. Enging Educ., 42(4) (2005), G. E. Chtzris, A. D. Pngopoulos nd P. G. Cottis, Determining Thevenin nd Norton equivlent circuits: new powerful pedgogicl pproches, presented t 9 th Intl Conf. on Optimistion of Electricl nd Electronic Equipment OPTIM 04, Brsov, omni, My Ch. K. Alexnder nd M. N. O. Sdiu, Fundmentls of Electric Circuits (McGrw-Hill, New Yor, 2000). 4 G. E. Chtzris, M. D. Tortoreli nd A. D. Tziols, Thevenin nd Norton s Theorems: powerful pedgogicl tools for treting specil cses of electric circuits, Intl J. Elect. Enging Educ., 40(4) (2003), G. E. Chtzris, Electric Circuits, vol. 1 (Tziols, Thesslonii, 2002). Interntionl Journl of Electricl Engineering Eduction 44/1