Analysis and Modeling of Magnetic Coupling

Transcription

1 Analyss and Modelng of Magnetc Couplng Bryce Hesterman Adanced Energy Industres Tuesday, Aprl 7 Dscoery earnng Center Unersty Of Colorado, Boulder, Colorado Dener Chapter, IEEE Power Electroncs Socety

2 SIDE # Analyss and Modelng of Magnetc Couplng Dener Chapter of IEEE PES Dscoery earnng Center Unersty of Colorado Aprl, 7 Bryce Hesterman Member of Techncal Staff I Adanced Energy Industres, Inc. Fort Collns, Colorado

3 Presentaton Outlne SIDE # Introducton Modelng magnetc couplng wth electrc crcut equatons Measurng electrc crcut model parameters Equalent crcuts for transformers and coupled nductors Magnetc crcut modelng oerew Tps for creatng magnetc crcut models Derng electrc model parameters from magnetc model parameters Matrx theory requrements for couplng stablty Examples

4 Motaton For Ths Presentaton SIDE # 3 Magnetc couplng often seems to be mysterous and hard to quantfy I had the good fortune of hang a mentor, Dr. James H. Spreen, who taught me how to analyze magnetc couplng Goal: help make magnetc couplng less mysterous by showng how to model t, measure t and use t n crcut analyss and smulaton

5 What Is Magnetc Couplng? SIDE # 4 Two wndngs are coupled when some of the magnetc flux produced by currents flowng n ether of the wndngs passes through both wndngs Only part of the flux produced by a current n one wndng reaches other wndngs

6 Magnetc Couplng Modelng Optons SIDE # 5 Electrc crcut: nductances and couplngs near model Model parameters can be determned from crcut measurements Parameters can be measured wth farly hgh accuracy f approprate measurement procedures are followed No nformaton on flux paths or flux leels

7 Magnetc Couplng Modelng Optons SIDE # 6 Magnetc crcut: reluctance crcut and electrcal-to-magnetc nterfaces for each wndng Explctly shows flux paths as magnetc crcut elements Flux paths and reluctances are only approxmatons Works wth lnear or nonlnear reluctance models Electrcal crcut parameters can be calculated from magnetc crcut parameters, but not ce-ersa

8 V-I Equatons For an Isolated Inductor SIDE # 7 (Current flows nto poste termnal) Tme doman d dt Frequency doman jω

9 Tme-doman Equatons For Two wndngs SIDE # 8 Self nductance of wndng Self nductance of wndng Mutual nductance d d + dt dt d d + dt dt

10 SIDE # 9 Tme-doman Equatons For N Wndngs dt d dt d dt d dt d N N dt d dt d dt d dt d N N dt d dt d dt d dt d N NN N N N N [] [ ] [] dt d N NN N N N N N dt d

11 SIDE # Frequency-doman Equatons For Two Wndngs Self nductance of wndng Self nductance of wndng Mutual nductance + jω jω + jω jω

12 SIDE # Frequency-doman Equatons For N Wndngs jω + jω + jω jω N N jω + jω + jω jω N N N jωn + jωn + jωn jω NN N N jω N N N N N N N [ ] jω[ ][ ]

13 SIDE # Inductance Matrx Symmetry The nductance matrx s symmetrc due to the recprocty theorem (Ths s requred for conseraton of energy) qr rq. O. Chua, near and Nonlnear Crcuts. New York: McGraw-Hll, 987, pp Ths prncple can be dered from Maxwell s equatons: C. G. Montgomery, R. H. Dcke, and E. M. Purcell, Ed., Prncples of Mcrowae Crcuts. New York: Doer Publcatons, 965 It can also be dered from a stored energy argument: R. R. awrence, Prncples of Alternatng Currents. New York: McGraw- Hll, 935, pp

14 Couplng Coeffcent For Two Wndngs SIDE # 3 M k k

15 Couplng Coeffcents For N Wndngs SIDE # 4 k qr qr qq rr k qr k rq K k k 3 k k 3 k k

16 Couplng Measurement Technques: Seres-adng Seres-opposng Method SIDE # 5 Couplng measurements are made for each par of wndngs Measure the nductance of each par of wndngs connected n the seresadng manner Measure the nductance of each par of wndngs connected n the seresopposng manner ad opp 4 k ad 4 opp seres-adng, ad seres-opposng, opp

17 Seres-adng Seres-opposng Couplng Formula Deraton SIDE # 6 Start wth the fundamental VI equatons + jω jω + jω jω Wrte down what s known for the seres-adng confguraton ad + k ad jω ad seres-adng, ad

18 SIDE # 7 Substtute assumptons for seres adng nto V-I equatons + jω jω + jω jω ad + Recall Substtute ( jω + jω ) + ( jω j ) ad + ω ad jω + + Smplfy ( ) Recall ad jω ad Therefore ad + +

19 SIDE # 8 Wrte down what s known for the seres-opposng confguraton opp opp jω opp jω jω jω jω opp seres-opposng, opp Substtute assumptons for seres-opposng nto V-I equatons Recall Substtute opp ( jω jω ) ( jω j ) ω

20 opp jω + Smplfy ( ) SIDE # 9 Recall opp jω opp Therefore opp + ad + + Recall Subtract ad opp 4 Sole Recall Therefore k k ad opp ad 4 4 opp

21 Couplng Measurement Technques: Voltage Rato Method SIDE # Couplng measurements are made for each par of wndngs Measure the oltage at each wndng when a oltage s appled to the other wndng ocqr oltage measured at wndng q when wndng r s dren drq oltage measured at wndng r when wndng r s dren and the oltage at wndng q s beng measured k oc d oc d Measurements can be corrupted by capactance and loadng Partcularly suted for measurements between wndngs on dfferent legs of three-phase ungapped transformers

22 Voltage Rato Method Couplng Formula Deraton SIDE # Start wth the fundamental VI equatons + jω jω + jω jω Fnd and when wndng s dren and wndng s unloaded d jω oc jω jω d Substtute, smplfy and rearrange oc jω jω d oc d

23 SIDE # Fnd and when wndng s dren and wndng s unloaded d jω oc jω jω d Substtute, smplfy and rearrange Recall Multply oc oc d d jω oc oc d jω d k oc d d oc oc d

24 SIDE # 3 Take the square root d oc oc d k Therefore k oc d oc d

25 Couplng Measurement Technques: Self and eakage Inductance Method SIDE # 4 Couplng measurements are made for each par of wndngs Measure the self nductance of each wndng ( and ) The nductance measured at a wndng when another wndng s shorted s called the leakage nductance leakqr nductance measured at wndng q when wndng r s shorted k leak k leak If perfect measurements are made, k k

26 Self and eakage Inductance Method Couplng Formula Deraton SIDE # 5 Start wth the fundamental VI equatons + jω jω + jω jω Fnd the leakage nductance at wndng when wndng s shorted, s Wrte down what s known for that condton jω leak Smplfy + jω jω +

27 SIDE # 6 Sole for Combne equatons for leak + jω jω jω Smplfy leak + Substtute alue of leak

28 SIDE # 7 Dde by leak Rearrange s Take square root leak Therefore k leak Smlar formula leak when wndng s k shorted

29 Self Inductance Measurement Tps SIDE # 8 The ratos of the self nductances of wndngs on the same core wll be nearly equal to the square of the turns ratos The ratos of the nductances are more mportant than the exact alues Measure at a frequency where the Q s hgh (It doesn t hae to be at the operatng frequency) Aod measurng nductance close to self-resonant frequences Take all measurements at the same frequency

30 Self Inductance Measurement Tps SIDE # 9 For ungapped cores: The ratos of the nductances are more mportant than the exact alues Inductance measurements may ary wth ampltude of the test sgnal Best results are obtaned f the core exctaton s equal for all measurements If possble, adjust the measurement dre oltage to be proportonal to the turns n order to keep the flux densty constant Inductance measurements can be made at flux leels near normal operatng leels usng a power amplfer and a network analyzer

31 eakage Inductance Measurement Tps SIDE # 3 Aod measurng nductance close to self-resonant frequences Wndng resstances decrease the measured couplng coeffcents when usng the self and leakage nductance method Measure at a frequency where the Q s hgh (It doesn t hae to be at the operatng frequency) For each par of wndngs, shortng the wndng wth the hghest Q ges the best results.

32 Couplng Coeffcent Value Dscrepances SIDE # 3 Use the same alue for k and k when performng crcut calculatons If k k, the one wth the largest alue s usually the most accurate

33 Couplng Coeffcent Sgnfcant Dgts SIDE # 3 Inductances can typcally be measured to at least two sgnfcant fgures Use enough sgnfcant dgts for couplng coeffcents n calculatons and smulatons to be able to accurately reproduce the leakage nductance k leak ( ) leak k Generally use at least four sgnfcant dgts when the couplng coeffcent s close to (k s known to more sgnfcant dgts than the measured nductances t was dered from)

34 Couplng Polarty Conentons SIDE # 33 Polarty dots ndcate that all wndngs whch hae dots of the same style wll hae matchng oltage polartes between the dotted and un-dotted termnals when one wndng s dren and all of the other wndngs are open-crcuted By conenton, poste current flows nto the termnals that are labeled as beng poste regardless of the polarty dot postons Voltage reference polartes can assgned n whateer manner s conenent by changng the polarty of the couplng coeffcent k Pn-for-pn equalent electrcal behaor

35 Three-legged Transformer Couplng Coeffcent Dot Conenton SIDE # 34 Three types of polarty dots are requred for a three-legged transformer There are three couplng coeffcents The trple product of the three couplng coeffcents s negate It s smplest to make all three of the couplng coeffcents negate k k3k3

36 Equalent Crcuts For Two Wndngs SIDE # 35 Many possble crcuts, but only three parameters are requred My faorte s the Cantleer model: two nductors and one deal transformer k a :N e b Ideal a leak b a Ne b Measure the nductance at wndng wth wndng shorted Measure the self nductance of wndng and subtract a Measure the self nductance of wndng and calculate aboe equaton

37 Cantleer Model Deraton k a :N e b Ideal If wndng s shorted, then the nductance measured at wndng s smply a because the deal transformer shorts out b. Thus a s equal to the leakage nductance measured at wndng. To conert the coupled nductor model a leak ( ) to the cantleer model, recall that leak k Therefore a ( ) k

38 k a :N e b Ideal By obseraton, b must equal the open-crcut nductance of wndng mnus a b a To conert the coupled nductor model to ( ) the cantleer model, recall that a k ( ) Substtute b k Therefore b k

39 k a :N e b Ideal The nductance of wndng,, s equal to b reflected through the deal transformer N e b Sole for the turns rato Recall b k Thus, n terms of the coupled nductor model Ne b N e k

40 Two eakage Inductance Model SIDE # 39 Two eakage Inductance model allows the actual turns rato to be used, but t adds unnecessary complexty These leakage nductances are not equal to the preously-defned conentonal leakage nductances measured by shortng wndngs l + Mag + N l Mag N Mag N N k l Mag N :N Ideal l Mag N N leak leak N N

41 SIDE # 4 Two eakage Inductance Model Deraton Start wth the fundamental VI equatons + jω jω jω jω N Mag N jω jω + Wrte down the open-crcut prmary oltage due to a current n the secondary By nspecton, the mutual nductance s Mag N N

42 SIDE # 4 Sole for Mag Mag N N Recall l + Mag + N l Mag N Sole for the two leakage nductances l Mag l Mag N Recall the equaton for leak from the deraton of the Self and eakage Inductance model By symmetry, leak leak N

43 SIDE # 4 Sole for leak leak Recall Mag N N N N Therefore Mag leak Also, Mag N N leak

44 Equalent Crcuts For More Than Two Wndngs Many possble crcuts, but only N(N+)/ parameters are requred My faorte s the extended cantleer model Exact correspondence to the couplng coeffcent model SIDE # 43 R. W. Erckson and D. Maksmoc, A multple-wndng magnetcs model hang drectly measurable parameters, n Proc. IEEE Power Electroncs Specalsts Conf., May 998, pp Extended cantleer model wth parastcs: K. D. T. Ngo, S. Srnas, and P. Nakmahachalasnt, Broadband extended cantleer model for mult-wndng magnetcs, IEEE Trans. Power Electron., ol. 6, pp , July K.D.T. Ngo and A. Gangupomu, "Improed method to extract the short-crcut parameters of the BECM," Power Electroncs etters, IEEE, ol., no., pp. 7-8, March 3

45 Magnetc Deces Can Be Approxmately Modeled Wth Magnetc Crcuts Reluctance paths are not as well defned as electrc crcut paths Accuracy s mproed by usng more reluctance elements n the model SIDE # 44 R R3 R4 R5 R R3 R4 R5 R R6 R3 R R7 R7 R5 R8 R6 R9 R6 R R4 MMF R3 R7 R5 R8 MMF R9 MMF3 R4 R8 R9 R R R6 Wndng Wndng Wndng 3 R7 R8 R9 R R R Hgh-leakage Transformer wth two E-Cores

46 Wndng Self Inductance SIDE # 45 Replace the MMF source for each wndng not beng consdered wth a short crcut Determne the Théenn equalent reluctance, R th,at the MMF source representng the wndng for whch the self nductance s beng calculated Inductance turns squared dded by the Théenn equalent reluctance R R3 R4 R5 R R6 R5 MMF Rth MMF MMF MMF3 N N N R7 R8 R9 N 3 R3 R6 R7 R8 R9 R R R4 R short short N R th N x turns of wndng x

47 eakage Inductances SIDE # 46 Replace the MMF source for each wndng not beng consdered wth a short crcut Replace the MMF source for the shorted wndng wth an open crcut Determne Théenn equalent reluctance, R th, at the MMF source for the wndng where the leakage nductance s to be determned Inductance turns squared dded by the Théenn equalent reluctance R R3 R4 R5 R R6 R5 MMF Rth MMF N MMF MMF3 N R7 R9 N R8 3 open R4 R3 R6 short R7 R8 R9 R R R N leak3 N R th N x turns of wndng x

48 Reluctance Modelng References SIDE # 47 R. W. Erckson and D. Maksmoc, Fundamentals of Power Electroncs, nd Ed., Norwell, MA: Kluwer Academc Publshers, S.-A. El-Hamamsey and Erc I. Chang, Magnetcs Modelng for Computer- Aded Desgn of Power Electroncs Crcuts, PESC 989 Record, pp G. W. udwg, and S.-A. El Hamamsy Coupled Inductance and Reluctance Models of Magnetc Components, IEEE Trans. on Power Electroncs, Vol. 6, No., Aprl 99, pp. 4-5 E. Coln Cherry, The Dualty between Interlnked Electrc and Magnetc Crcuts and the Formaton of Transformer Equalent Crcuts, Proceedngs of the Physcal Socety, ol. 6 part, secton B, no. 35 B, Feb. 949, pp. - MIT Staff, Magnetc Crcuts and Transformers. Cambrdge, MA: MIT Press, 943

49 Energy Storage SIDE # 48 The magnetc energy stored n one nductor s: WM The energy stored n a set of N coupled wndngs s: W MN T [][ ][] The energy stored n a set of coupled wndngs s: W M + [ ] ( + )

50 Stablty of a Set of Coupled Inductors SIDE # 49 As set of coupled nductors s passe f the magnetc energy storage s non-negate for any set of currents For two coupled nductors, ths s guaranteed f: A set of three coupled nductors s passe f the followng condtons are met: k + k3 + k3 kk3k3 k k 3 k 3 Ylmaz Tokad and Myrl B. Reed, Crtera and Tests for Realzablty of the Inductance Matrx, Trans. AIEE, Part I, Communcatons and Electroncs, Vol. 78, Jan. 96, pp k

51 Stablty of a Set of Coupled Inductors SIDE # 5 When there are more than three wndngs, the couplng coeffcent matrx defned below can be used to determne stablty K k k N k k N k k N N k qr k rq A set of coupled nductors s passe f and only f all of the egenalues of K are non-negate There are N egenalues for a set of N wndngs Egenalue calculatons are bult-n functons of programs lke Mathcad and Matlab

52 Consstency Checks SIDE # 5 The egenalue test can let you know f there are measurement errors, but t won t help dentfy the errors The ratos of magnetzng nductances on the same core leg should be approxmately equal to the square of the turns ratos Set up test smulatons to erfy that the models match the test condtons Test leakage nductances by shortng one wndng and applyng a sgnal to the other wndng (Compute the nductance ndcated by the oltage, current and frequency) Check for couplng polarty errors, especally f there are wndngs on multple core legs Compare test data wth multple wndngs shorted to smulatons or calculatons wth multple wndngs shorted

53 Inerse Inductance Matrx SIDE # 5 The nerse nductance matrx, Γ, s the recprocal of the nductance matrx Γ Each dagonal element s equal to the recprocal of the nductance of the correspondng wndng when all of the of the other wndngs are shorted Example applcaton: You hae a four-wndng transformer. What s the nductance at wndng when wndngs 3 and 4 are shorted? Set up an nductance matrx, x, by extractng all of the elements from the total nductance matrx that apply only to wndngs, 3 and 4

54 Inerse Inductance Matrx Example SIDE # 53 Remoe the outlned elements from to get x x Use a computer program to compute the nerse of x Γx x Γx s the nductance at wndng when wndngs 3 and 4 are shorted

55 Modfed Node Analyss SIDE # 54 Spce uses Modfed Node Analyss (MNA) to set up the crcut equatons Ths type of analyss s well suted for handlng crcuts wth coupled wndngs A good descrpton of the method s found n:. O. Chua, near and Nonlnear Crcuts. New York: McGraw-Hll, 987, pp

56 Modfed Node Analyss Example SIDE # 55 Assgn node numbers The reference node s node Assgn currents for each branch Compute mutual nductances from the self nductances and couplng coeffcents k

57 SIDE # 56 Wrte a KC equaton usng the node-to-datum oltages as arables for each node other than the datum node, unless the node has a fxed oltage wth respect to the datum node. In that case, wrte an equaton that assgns the node oltage to the fxed alue.

58 SIDE # 57 Node s Node jωc ( ) C ( ) jω + s 3 jωc jω R R Node Current arables must be mantaned for each coupled nductor Other current arables can be replaced by equalent expressons + C

59 SIDE # 58 Wrte an equaton for the oltage dropped across each coupled nductor wth the node arables on one sde and the nductance and current terms on the other sde jω jω 3 jω jω Note that the mnus sgns are due to the fact that was assgned to be flowng out of the dotted end of the wndng nstead of nto t

60 SIDE # 59 Rearrange the node equatons to facltate wrtng matrx equatons ( ) C s + s jωc jω + jω C R 3 R + jωc 3 jω jω jω + jω 3 jω jω 3 jω + jω

61 SIDE # 6 Conert the node equatons nto matrx form U X G + jω jω jω jω jω jω 3 s C R C s C + jω jω 3 + C R jω ω j + jω ω j

62 Modfed Node Analyss Example Conclusons The matrx equaton can be soled numercally n programs lke Mathcad or Matlab The approach s straghtforward and can be used wth many wndngs If you want to hae a symbolc soluton, smple cases can be handled wth equalent crcuts (the cantleer model s easest for two wndngs) Equalent crcut equatons can get ery messy wth more than two wndngs Is there a better way to fnd symbolc solutons by usng Dr. R. D. Mddlebrook s Desgn-Orented Analyss technques? (I hope to fnd out) See Dr. Mddlebrook s web ste: for nformaton on hs analyss technques SIDE # 6

63 Couplng Stablty Example Usng Mathcad SIDE # 6 Fgure. Three coupled wndngs wth resste loads. Suppose we hae a three-wndng nductor wth a resstor termnatng each wndng as s shown n Fg.. If we externally force an ntal condton of currents n the wndngs, and then let the nductor "coast" on ts own, the currents should all exponentally decay to zero as the stored energy s dsspated. If the couplng coeffcent matrx s not poste defnte, howeer, at least one of the egenalues for the system wll be not be negate.

64 SIDE # 63 As s shown below, ths produces an unstable stuaton where more energy can be extracted from the nductor than was ntally stored there. When the ntal energy s dsspated, the stored energy becomes negate, and the nductor contnues to deler power. Ths, of course, s mpossble. Therefore, a set of couplng coeffcents n whch the couplng coeffcent matrx s not poste defnte descrbes a coupled nductor that s not physcally realzable. Because Spce and other crcut smulators allow such nductors to be specfed, t s nstructe to examne what happens when nonphyscal sets of couplng coeffcents are specfed.

65 SIDE # 64 We frst examne a physcally-realzable case. After that, we show what happens when the couplng coeffcents are mproperly chosen. Enter the alues of the nductances and couplng coeffcents : μh k :.96 : μh k 3 :.99 3 : μh k 3 :.98

66 SIDE # 65 Construct the couplng coeffcent matrx. K : k k 3 k k 3 k 3 k 3 Compute the egenalues of the couplng coeffcent matrx. egenals ( K) All of the egenalues are poste. As s shown below, ths leads to negate egenalues for the tme response of the system.

67 SIDE # 66 Construct the nductance matrx. : K, K 3, 3 K, K, 3 3 K 3, 3 K 3, 3 3 Ealuate the nductance matrx μh Enter the alues of the resstances R : Ω R : Ω R 3 : Ω

68 SIDE # 67 Enter the ntal alues of the currents I : A I : A I 3 : A Defne a current ector I I I I 3 Defne an ntal condton ector I I I I 3 : I A We can thnk of ths as a stuaton n whch the current n wndng s externally forced to be Ampere, and then the crcut s left to dsspate the stored energy startng at tme t.

69 SIDE # 68 Compute the stored energy T I I. 5 J The tme response of the crcut of Fg. can be descrbed by the followng equaton I R I R I 3 R 3 d I () d t Compute the nerse nductance matrx Γ : Γ μh

70 SIDE # 69 Compute the egenalues of Γ. egenals ( Γ) μh Wrte () n terms of the nerse nductance matrx d dt I I R Γ I R () I 3 R 3 Wrte () n a way that shows I n the rght sde d dt I Γ R R R 3 I (3)

71 SIDE # 7 R Defne G G: Γ R (4) R 3 Substtute G nto (3) d dt I GI (5) Compute the egenalues of G λ : egenals ( G) λ s -

72 SIDE # 7 Compute the egenectors of G Λ : egenecs ( G) Λ s - The soluton to (5) wll hae the form I Λ c c c 3 e λ t e λ t (6) e λ 3 t

73 SIDE # 7 The alues of the constants c, c and c 3 can be determned from the ntal condtons I Λ c c c 3 c c c 3 : Λ I c c c As Check to see f the ntal condtons are satsfed Λ c c c 3 A I A

74 SIDE # 73 Compute a coeffcent matrx c c c 3 C : Λ C (Note that the mnmum ndex alue for matrces n ths document s set at, not zero.) A Defne current functons I () t C e λ t C e λ + t C e λ 3 : + t,,, 3 I () t C e λ t C e λ + t C e λ 3 : + t,,, 3 I 3 () t C e λ t C e λ + t C e λ 3 : + t 3, 3, 3, 3

75 SIDE # 74 Check the current alues at a few ponts I ( s ) A I. μs ( ).83A I μs ( ).3A I ( ) A I. μs ( ). A I μs ( ).4A I 3 ( ) A I 3. μs ( ).4A I 3 μs ( ).33A

76 Plot the current alues t :, 9 s.. 7 s SIDE # 75, I () t.5 I () t I 3 () t Fg.. Wndng Currents. t I drops quckly as the current bulds up n the other two wndngs. All of the currents then decay.

77 SIDE # 76 Defne a current ector to facltate calculatng the stored energy. It () : I () t I () t I 3 () t Defne a functon to compute the stored energy. Wt () : ( It ())T It ()

78 SIDE # 77 t :, 7 s.. 5 s 5. 6 Wt () Fg. 3. Stored energy, J. t The stored energy s dsspated as expected.

79 SIDE # 78 We can now examne a case where the couplng coeffcent are mproperly specfed. Are relately small change n one of the couple coeffcents s all that s requred to create an unstable confguraton. Enter the alues of the nductances and couplng coeffcents : μh k :.96 : μh k 3 :.99 3 : μh k 3 :.99 (Preously.98)

80 SIDE # 79 Construct the couplng coeffcent matrx. K : k k 3 k k 3 k 3 k 3 Compute the egenalues of the couplng coeffcent matrx. egenals ( K) One of the egenalues s negate. As s shown below on, ths makes one of the egenalues for the tme response of the system poste.

81 SIDE # 8 Construct the nductance matrx. : K, K 3, 3 K, K, 3 3 K 3, 3 K 3, 3 3 Ealuate the nductance matrx μh Enter the alues of the resstances R : Ω R : Ω R3 : Ω

82 SIDE # 8 Enter the ntal alues of the currents I : A I : A I 3 : A Defne an ntal condton ector I I I I 3 : I A Compute the stored energy T I I. 5 J

83 SIDE # 8 Compute the nerse nductance matrx Γ : Γ μh Compute the egenalues of Γ. egenals ( Γ) μh

84 SIDE # 83 R Defne G G: Γ R (4) R 3 Compute the egenalues of G λ : egenals ( G) λ s - One of the egenalues s poste, so the system s unstable.

85 SIDE # 84 Compute the egenectors of G Λ : egenecs ( G) Λ s - The soluton to (5) wll hae the form I Λ c c c 3 e λ t e λ t (6) e λ 3 t

86 SIDE # 85 The alues of the constants c, c and c 3 can be determned from the ntal condtons I Λ c c c 3 c c c 3 : Λ I c c c As Check to see f the ntal condtons are satsfed Λ c c c 3 A I A

87 SIDE # 86 Compute a coeffcent matrx c C : Λ C c c A Defne current functons I () t C e λ t C e λ t + C e λ 3 t : +,, 3, I () t C e λ t C e λ t + C e λ 3 t : +,, 3, I 3 () t C e λ t C e λ t + C e λ 3 t : + 3, 3, 33,

88 SIDE # 87 Check the current alues at a few ponts I ( s ) A I. μs I ( s ) A I. μs I 3 ( s ) A I 3. μs ( ) 4 5 A I. μs ( ) 3 5 ( ) 7 A I. μs 5 A I 3. μs ( ) A ( ) ( ) 6.6 A 6 A

89 Plot the current alues t :, s s SIDE # 88 4 I () t I () t I 3 () t Fg. 4. Wndng Currents. t I rses nstead of decayng. The other currents start at zero and then buld up.

90 SIDE # 89 Defne a current ector to facltate calculatng the stored energy. It () : I () t I () t I 3 () t Defne a functon to compute the stored energy. Wt () : ( It ())T It ()

91 SIDE # Wt () t Fg. 5. Stored energy. The stored energy decays to zero, but t then goes negate as our magnary nductor pumps out energy at a rapdly-ncreasng rate. One of the thngs that we can learn from ths example s that estmatng alues for couplng coeffcents can easly produce a nonphyscal crcut. Tryng to smulate such a crcut may produce frustraton as one tres to fgure out why the crcut won't conerge.

92 Acknowledgement Thanks to my manager, Dr. Dad J. Chrste, for reewng the presentaton and prodng helpful comments and suggestons Also, thanks to Dr. James H. Spreen for hs contnung gudance

93 SIDE # 9 NASDAQ: AEIS Adanced Energy Industres, Inc. All Rghts Resered.