Network Theorems - J. R. Lucas. Z(jω) = jω L

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1 Netwo Theoems - J.. Lucas The fudametal laws that gove electic cicuits ae the Ohm s Law ad the Kichoff s Laws. Ohm s Law Ohm s Law states that the voltage vt acoss a esisto is diectly ootioal to the cuet it flowig though it. vt it it o vt. it vt This geeal statemet of Ohm s Law ca be eteded to cove iductaces ad caacitos as well ude alteatig cuet coditios ad tasiet coditios. This is the ow as the Geealised Ohm s Law. This may be stated as it vt. it, whee d/dt diffeetial oeato vt is ow as the imedace fuctio of the cicuit, ad the above euatio is the diffeetial euatio goveig the behaviou of the cicuit. Fo a esisto, Fo a iducto L Fo a caacito, C I the aticula case of alteatig cuet, jω so that the euatio goveig cicuit behaviou may be witte as jω. I, Fo a esisto, Fo a iducto ad jω jω jω L Fo a caacito, jω jω C We caot aalyse electic cicuits usig Ohm s Law oly. We also eed the Kichoff s cuet law ad the Kichoff s voltage law. Kichoff s Cuet Law Kichoff s cuet law is based o the icile of cosevatio of chage. This euies that the algebaic sum of the chages withi a system caot chage. Thus the total ate of chage of chage must add u to zeo. ate of chage of chage is cuet. This gives us ou basic Kichoff s cuet law as the algebaic sum of the cuets meetig at a oit is zeo. i.e. at a ode, Σ I 0, whee I ae the cuets i the baches meetig at the ode This is also sometimes stated as the sum of the cuets eteig a ode is eual to the sum of the cuet leavig the ode. The theoem is alicable ot oly to a ode, but to a closed system. Netwo Theoems Pofesso J Lucas Novembe 00

2 i i i 3 i 4 i 5 0 i i i 4 i 3 i 5 i 4 i 5 i i i d i e i a lso fo the closed bouday, i 3 i c i a i b i c i d i e 0 Kichoff s oltage Law Kichoff s voltage law is based o the icile of cosevatio of eegy. This euies that the total wo doe i taig a uit ositive chage aoud a closed ath ad edig u at the oigial oit is zeo. This gives us ou basic Kichoff s law as the algebaic sum of the otetial diffeeces tae oud a closed loo is zeo. v d v e i.e. aoud a loo, Σ 0, v c whee ae the voltages acoss the baches i the loo. loo v a v b v c v d v e 0 This is also sometimes stated as the sum of the emfs tae v b aoud a closed loo is eual to the sum of the voltage dos aoud the loo. lthough all cicuits could besolved usig oly Ohm s Law ad Kichoff s laws, the calculatios would be tedious. aious etwo theoems have bee fomulated to simlify these calculatios. amle Fo the uoses of udestadig the icile of the Ohm s Law ad the Kichoff s Laws ad thei alicability, we will coside oly a esistive cicuit. Howeve it must be emembeed that the laws ae alicable to alteatig cuets as well. I 0 Ω 0 Ω I i b v a Ω I 70 Fo the cicuit show i the figue, let us use Ohm s Law ad Kichoff s Laws to solve fo the cuet I i the 60 Ω esisto. Usig Kichoff s cuet law I I I Usig Kichoff s voltage law 00 0 I 60 I I 0 8 I 6 I 70 0 I 60 I I 7 6 I 8 I which has the solutio I, I 0.5 ad the uow cuet I 0.5. Netwo Theoems Pofesso J Lucas Novembe 00

3 Sueositio Theoem The sueositio theoem tells us that if a etwo comises of moe tha oe souce, the esultig cuets ad voltages i the etwo ca be detemied by taig each souce ideedetly ad sueosig the esults. e t Liea Passive ilateal Netwo t Liea Passive ilateal Netwo t e t Liea Passive ilateal Netwo t e t e t If a ecitatio e t aloe gives a esose t, ad a ecitatio e t aloe gives a esose t, the, by sueositio theoem, if the ecitatio e t ad the ecitatio e t togethe would give a esose t t t The sueositio theoem ca eve be stated i a moe geeal mae, whee the sueositio occus with scalig. Thus a ecitatio of e t ad a ecitatio of e t occuig togethe would give a esose of t t. amle Let us solve the same oblem as ealie, but usig Sueositio theoem. 0 Ω 0 Ω Ω 70 Solutio 0 Ω 0 Ω 0 Ω 0 Ω Ω i 60 Ω i 70 fo cicuit, souce cuet 0 i // Netwo Theoems Pofesso J Lucas 3 Novembe 00

4 Similaly fo cicuit, souce cuet 0 i // uow cuet i i i which is the same aswe that we got fom Kichoff s Laws ad Ohm s Law. Thevei s Theoem o Helmholtz s Theoem The Thevei s theoem, basically gives the euivalet voltage souce coesodig to a active etwo. If a liea, active, bilateal etwo is cosideed acoss oe of its ots, the it ca be elaced by a euivalet voltage souce Thevei s voltage souce ad a euivalet seies imedace Thevei s imedace. Liea ctive ilateal Netwo i oc thevei thevei Sice the two sides ae idetical, they must be tue fo all coditios. Thus if we comae the voltage acoss the ot i each case ude oe cicuit coditios, ad measue the iut imedace of the etwo with the souces emoved voltage souces shot-cicuited ad cuet souces oe-cicuited, the thevei oc, ad thevei i amle Let us agai coside the same eamle to illustate Thevei s Theoem. 0 Ω 0 Ω Ω 70 Solutio Sice we wish to calculate the cuet i the 60 Ω esisto, let us fid the 00 Thevei s euivalet cicuit acoss the temials afte discoectig oe cicuitig the 60 Ω esisto. Ude oe cicuit coditios, cuet flowig is 00 70/ oc, Ω 0 Ω 60 Ω 70 Netwo Theoems Pofesso J Lucas 4 Novembe 00

5 Th 85 The iut imedace acoss with souces emoved 0//0 0 Ω. Th 0 Ω. Th 0 Ω Theefoe the Thevei s euivalet cicuit may be daw with bach i eitoduced as follows. Th Ω Fom the euivalet cicuit, the uow cuet i is detemied as 85 i which is the same esult that was obtaied fom the ealie two methods. Noto s Theoem Noto s Theoem is the dual of Thevei s theoem, ad states that ay liea, active, bilateal etwo, cosideed acoss oe of its ots, ca be elaced by a euivalet cuet souce Noto s cuet souce ad a euivalet shut admittace Noto s dmittace. Liea ctive ilateal Netwo i I sc oto I oto Sice the two sides ae idetical, they must be tue fo all coditios. Thus if we comae the cuet though the ot i each case ude shot cicuit coditios, ad measue the iut admittace of the etwo with the souces emoved voltage souces shot-cicuited ad cuet souces oe-cicuited, the I oto I sc, ad oto i amle 3 Let us agai coside the same eamle to illustate Noto s Theoem. 0 Ω 0 Ω Ω 70 Solutio Sice we wish to calculate the cuet i the 60 Ω esisto, let us fid the Noto s euivalet cicuit acoss the temials afte shot-cicuitig the 60 Ω esisto Ω 0 Ω 60 Ω I sc 70 Netwo Theoems Pofesso J Lucas 5 Novembe 00

6 the shot cicuit cuet I sc is give by I sc 00/0 70/0 8.5 I oto 8.5 Noto s admittace /0 /0 0. S S 60 Ω o S Noto s euivalet cicuit is ad the cuet i the uow esisto is which is the same esult as befoe. eciocality Theoem The eciocality theoem tells us that i a liea assive bilateal etwo a ecitatio ad the coesodig esose may be itechaged. I a two ot etwo, if a ecitatio et at ot oduces a cetai esose t at a ot, the if the same ecitatio et is alied istead to ot, the the same esose t would occu at the othe ot. t t Liea Liea et Passive Passive Pot Pot Pot Pot et ilateal ilateal Netwo Netwo amle 4 0 Ω 0 Ω Ω Coside the ealie eamle, but with oly oe souce. Detemie the cuet i the 60 Ω bach. Now elace the 60 Ω esisto with the souce i seies with it ad afte shotcicuit the souce at the oigial locatio, fid the cuet flowig at the oigial souce locatio. Show that it veifies the eciocality theoem. Solutio 0 Ω 0 Ω 0 Ω 0 Ω Ω I I Ω Fo the oigial cicuit, cuet I // Netwo Theoems Pofesso J Lucas 6 Novembe 00

7 similaly fo the ew cicuit, cuet I // It is see that the idetical cuet has aeaed veifyig the eciocality theoem. The advatage of the theoem is whe a cicuit has aleady bee aalysed fo oe solutio, it may be ossible to fid a coesodig solutio without futhe wo. Comesatio Theoem I may cicuits, afte the cicuit is aalysed, it is ealised that oly a small chage eed to be made to a comoet to get a desied esult. I such a case we would omally have to ecalculate. The comesatio theoem allows us to comesate oely fo such chages without sacificig accuacy. I ay liea bilateal active etwo, if ay bach cayig a cuet I has its imedace chaged by a amout, the esultig chages that occu i the othe baches ae the same as those which would have bee caused by the ijectio of a voltage souce of - I. i the modified bach. Liea ctive ilateal Netwo I I Ζ Liea ctive ilateal Netwo I I I. Ζ Liea ctive ilateal Netwo I Liea Passive ilateal Netwo I I. Ζ Ζ Coside the voltage do acoss the modified bach. I I. I. I. I fom the oigial etwo,. I. I. I Sice the value I is aleady ow fom the ealie aalysis, ad the chage euied i the imedace,, is also ow, I. is a ow fied value of voltage ad may thus be eeseted by a souce of emf I.. Usig sueositio theoem, we ca easily see that the oigial souces i the active etwo give ise to the oigial cuet I, while the chage coesodig to the emf I. must oduce the emaiig chages i the etwo. amle 5 0 Ω 0 Ω Ω I 70 Netwo Theoems Pofesso J Lucas 7 Novembe 00

8 Fom eamle 4, we saw that the cuet i the 60 Ω esisto is 0.5. Let us say that we wat to chage the esisto by a uatity such that the cuet i the 60 Ω esisto is The the cicuit fo chages ca be witte as I Ω 0 Ω I 0.5 I 0.5 I 60 Ω 60 0 // 0 I. 0.5 i.e i.e. -7/ Ω Theefoe the euied value of Ω This could have bee calculated usig Kichoff s ad Ohm s laws but would have bee moe comlicated. We ca also chec this aswe with Thevei s theoem as follows. Fom amle, we had the Thevei s cicuit as show, with the 60 Ω elaced by 3.67 Ω. Th 0 Ω The cuet fo this value ca be uicly obtaied 85 i as i Th 85 So you ca also see that by owig Thevei s euivalet cicuit fo a give etwo, we ca obtai solutios fo may coditios with little additioal calculatios. The same is tue with Noto s theoem. Maimum Powe Tasfe Theoem s you ae obably awae, a omal ca battey is ated at ad geeally has a oe cicuit voltage of aoud 3.5. Similaly, if we tae 7 e-toch batteies, they too will have a temial voltage of Howeve, you would also be awae, that if you ca battey is dead, you caot go to the eaest sho, buy 7 e-toch batteies ad stat you ca. Why is that? ecause the e-toch batteies, although havig the same oe cicuit voltage does ot have the ecessay owe o cuet caacity ad hece the euied cuet could ot be give. O if stated i diffeet tems, it has too high a iteal esistace so that the voltage would do without givig the ecessay cuet. This meas that a give battey o ay othe eegy suly, such as the mais ca oly give a limited amout of owe to a load. The maimum owe tasfe theoem defies this owe, ad tells us the coditio at which this occus. Fo eamle, if we coside the above battey, maimum voltage would be give whe the cuet is zeo, ad maimum cuet would be give whe the load is shot-cicuit load voltage is zeo. Ude both these coditios, thee is o owe deliveed to the load. Thus obviously i betwee these two etemes must be the oit at which maimum owe is deliveed Ω Netwo Theoems Pofesso J Lucas 8 Novembe 00

9 The Maimum Powe Tasfe theoem states that fo maimum active owe to be deliveed to the load, load imedace must coesod to the cojugate of the souce imedace o i the case of diect uatities, be eual to the souce imedace. Let us aalyse this, by fist statig with the basic case of a esistive load beig sulied fom a souce with oly a iteal esistace this is the same as fo d.c. esistive Load sulied fom a souce with oly a iteal esistace Coside a souce with a iteal emf of ad a iteal esistace of ad a load of esistace. cuet I I Load Powe P I. The souce esistace is deedat uely o the souce ad is a costat, as is the souce emf. Thus oly the load esistace is a vaiable. To obtai maimum owe tasfe to the load, let us diffeetiate with esect to. dp d [ ] 0 4 [Note: I said maimum, athe tha maimum o miimum, because fom hysical cosideatios we ow that thee must a maimum owe i the age. So we eed ot loo at the secod deivative to see whethe it is maimum o miimum]. 0 o 0 i.e. fo maimum owe tasfe. value of maimum owe P ma fo maimum load voltage at maimum owe.. It is to be oted that whe maimum owe is beig tasfeed, oly half the alied voltage is available to the load, ad the othe half dos acoss the souce. lso, ude these coditios, half the owe sulied is wasted as dissiatio i the souce. Thus the useful maimum owe will be less tha the theoetical maimum owe deived ad will deed o the voltage euied to be maitaied at the load. I P 4 Load sulied fom a souce with a iteal imedace Coside a souce with a iteal emf of ad a iteal imedace of z j ad a load of imedace j. j j Netwo Theoems Pofesso J Lucas 9 Novembe 00

10 Netwo Theoems Pofesso J Lucas 0 Novembe 00 cuet I j j j magitude of I Load Powe P I. Sice thee ae two vaiables ad, fo maimum owe 0 0 P ad P i.e. [ ] [ ] [ ] [ ] 0 ad [ ] [ ] The secod euatio gives 0 o - Substitutig this i the fist euatio gives. 0 Sice caot be egative, 0. 0 i.e. j j z * Theefoe fo maimum owe tasfe, the load imedace must be eual to the cojugate of the souce imedace. Load of fied owe facto sulied fom a souce with a iteal imedace Coside a souce with a iteal emf of ad a iteal imedace of z j ad a load of imedace j which has a give owe facto. [This situatio is ot ucommo, as fo eamle if the load was a iductio moto load, the owe facto would have a fied value such as 0.8 lag] cuet I j j j ad ae o loge ideedet but have the elatioshi owe facto f o f. magitude of I. Load Powe P I.. j j I

11 dp Sice thee is oly oe vaiable, fo maimum owe 0 d i.e. [. ] [. ] [.. ] i.e i.e i.e.. 0 i.e.. i.e. i.e. z So eve whe the owe facto of the load is diffeet fom that of the souce, a coditio that eeds to be satisfied is that the magitude of the load imedace must be eual to the magitude of the souce imedace. Note: If limits ae laced o the voltage, the maimum owe will ot always occu ude the above coditio, but at the limit of the voltage closest to the desied solutio. Millma s Theoem 0 N S efeece 3 3 Coside a umbe of admittaces,, 3.., ae coected togethe at a commo oit S. If the voltages of the fee eds of the admittaces with esect to a commo efeece N ae ow to be N, N, 3N. N. N, N, the Millma s theoem gives the voltage of the commo oit S with esect to the efeece N as follows. lyig Kichoff s Cuet Law at ode S I i.e. 0, I N SN 0, N SN i.e. N SN so that SN N etesio of the Millma theoem is the euivalet geeato theoem. I L I L L e L e Netwo Theoems Pofesso J Lucas Novembe 00

12 This theoem states that a system of voltage souces oeatig i aallel may be elaced by a sigle voltage souce i seies with a euivalet imedace give as follows this is effectively the Thevei s theoem alied to a umbe of geeatos i aallel. e, e amle 6 The figue show also used i ealie 0 Ω 0 Ω eamles ca be cosideed euivalet to I two souces of 00 ad 70, with 00 iteal esistaces 0 Ω each, feedig a 60 Ω load of 60 Ω. Usig Millma s theoem o euivalet geeato theoem fid the cuet I. Solutio Fom uivalet geeato theoem, it has bee show that the euivalet geeato has e e Ω 85 Hece cuet I same aswe was obtaied with Thevei s Th m agai same aswe was obtaied with Thevei s Th m 70 ose s Theoem Nodal-Mesh Tasfomatio Theoem I I I I I S I N I I I N I ose s theoem tells us how we could fid the mesh euivalet of a etwo whee all the baches ae coected to a sigle ode. [I the mesh euivalet, all odes ae coected to each othe ad ot to a commo ode as i the odal etwo]. Whe the euivalet is obtaied the eteal coditios ae ot affected as see fom the eteal cuets i the above diagams. Netwo Theoems Pofesso J Lucas Novembe 00

13 Fo the odal etwo, fom Millma s theoem SN N, so that I N SN. N N I N - N N - N Fo a defiite summatio, whethe the vaiable is used o the vaiable is used maes o diffeece. Thus I N - N Fo the mesh etwo, fom Kichoff s cuet law, the cuet at ay ode is I Comaig euatios, it follows that a solutio to euatio is which is the statemet of ose s theoem The covese of this theoem is i geeal ot ossible as thee ae geeally moe baches i the mesh etwo tha i the odal etwo. Howeve, i the case of the 3 ode case, thee ae eual baches i both the odal etwo also ow as sta ad the mesh etwo also ow as delta. Sta-Delta Tasfomatio S C C CS S C C sta coected etwo of thee admittaces o coductaces S, S, ad CS coected togethe at a commo ode S ca be tasfomed ito a delta coected etwo of thee admittaces, C, ad C usig the followig tasfomatios. This has the same fom as the geeal eessio deived ealie. S. S S S CS, C S. CS S S CS, C CS. S S S CS Netwo Theoems Pofesso J Lucas 3 Novembe 00

14 Note: ou ca obseve that i each of the above eessios if we eed to fid a aticula delta admittace elemet value, we have to multily the two values of admittace at the odes o eithe side i the oigial sta-etwo ad divide by the sum of the thee admittaces. I the secial case of thee odes, evese tasfomatio is also ossible. Delta-Sta Tasfomatio C S C C C CS S delta coected etwo of thee imedaces o esistaces, C, ad C ca be tasfomed ito a sta coected etwo of thee imedaces S, S, ad CS coected togethe at a commo ode S usig the followig tasfomatios. [ou will otice that I have used imedace hee athe tha admittace because the the fom of the solutio emais simila ad easy to emembe.]. C. C C. C S, S, CS C C C C C C Note: ou ca obseve that i each of the above eessios if we eed to fid a aticula delta elemet value, we have to multily the two imedace values o eithe side of ode i the oigial sta-etwo ad divide by the sum of the thee imedaces. Poof: Imedace betwee ad C with zeo cuet i ca be comaed i the two etwos as follows. C // C CS S i.e. similaly C C C C C C C C CS S C C ad S CS C C elimiatio of vaiables fom the above euatios gives the desied esults. S S Netwo Theoems Pofesso J Lucas 4 Novembe 00