Package Inductance and RLC Circuit Analysis. New Today. Bondwires. Package Parasitics. Model for Invertor with Inductance. Inductance (Inductantie)

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1 Package Inucance an ircui Analyi New Toay FATO Jawel, ik a beken al fenomeen Tenmine oner mahemaici En in wikung aangelege kringen Mijn naam, al u he ween wil i i Ik zi in allerlei berekeningen En aan he eine hef ik mij weer op U vraag nie, enk ik, om verhanelingen Maar wen a ik mezelf nu een onpop Welnu ik ben e worel ui Ik funcioneer, ofchoon ik nie bea Denk aar maar een langurig over na (Ui: Dr. P en Marjolein Kool, Wi en nauurlyriek me chemich upplemen, Ameram: Nijgh en van Dimar, h. 5 Inucive effec, for wiching (ranien behavior Applicaion of omplex Number elaion beween complex exponenial an harmonic (inuoial funcion Damping facor, ampe naural frequency, unampe naural frequency Ocillaory behavior (oplingering oncep of overampe, criically ampe, unerampe an unampe repone /7/4 7 inucance /7/4 7 inucance Package Paraiic Bull Microproceor hip Nee raniion from ubmicron onchip worl o PB (prine ircui Boar worl elaively large package paraiic Paraiic = unwane bu nonavoiable elecrical effec Inucance of package mu ofen be accoune for Onchip inerconnec inucance alo becoming imporan Bonwire: connecion from ilicon chip o package pin Bonwire /7/4 7 inucance 3 /7/4 7 inucance 4 Moel for Inveror wih Inucance Inucance (Inucanie 5 V power Many inereing properie an we eermine pee an oher properie of uch yem? i v v = v in in p n groun v ou Aion of o an migh reul in ocillaing behavior! Nee complex number o eal wih i. i v v i = an exhibi ual role of i an v aue many inereing an ueful properie of elecrical circui an circui analyi We will apply hi righ now! /7/4 7 inucance 5 /7/4 7 inucance 6

2 D Equivalen ircui D value of capacior curren: vc v = c c = i = D moel for i open circui Dual! D value of inucor volage: = = v = D moel for i hor circui /7/4 7 inucance 7 = V A V Fir Orer ircui i v : τ = : τ = / F: waveform IV: iniial value FV: final value. Ue KV for circui wih > : V i =. earrange: = i V 3. ompare o cae: v c = vc V 4. emember general formula: F = (IV FV e (/τ FV 5. Wrie own reul (by inpecion: V ( V i ( i ( e = /7/4 7 inucance 8 acae Inveror wih Inucance 5V power p Package an power upply Simplify ircui Driving gae Inerconnec p in oa gae power 5V n power groun Package an power upply groun in in Driving gae Inerconnec p n in oa gae in 5V V groun n i in v c in Prooypical erie circui Principal ubjec of hi chaper /7/4 7 inucance /7/4 7 inucance ircui Differenial Equaion V v c v c = i Sae variable: v, i Noe: KV for inucor volage i ual from K for capacior curren v i v v c i =i v KV aroun loop V v v v = V i v = /7/4 7 inucance ircui Differenial Equaion K & KV v c = i V V i v = ewrie vc = i V = v i v = V i i v c /7/4 7 inucance

3 v = V i Two couple fir orer D.E. Nex: fin v c an i Saify D.E. Saify iniial conion v c (, i ( Proceure: Aume general oluion Deermine eay ae oluion Fin naural frequencie (, Deermine coefficien of ran. par General Proceure General meho ienical o an cae /7/4 7 inucance 3 v Aume General Soluion ( = v ( V = V e V e V ( = i ( I = I e I e I i V in Deermine Seay Sae Soluion i v c V = v c ( = I = i ( = V in /7/4 7 inucance Fin Naural Frequencie v = V i e = x = = haraceriic Equaion of erie circui Sign of Dicriminan = Noe: c a = ; b = ; c = = ± a b c = b ± = D criminan Four ae epenng on ign of criminan D > like an overampe D = = criically ampe 3 D < an unerampe 4 = e( i = unampe b 4 ac a Will be explaine horly Nex: Suy cae,, 3 an 4 /7/4 7 inucance 5 /7/4 7 inucance 6 Damping Overampe criically ampe unerampe unampe Wha oe i mean? Many yem vibrae/ocillae/cycle (more or le repeiive behavior mechanical yem, elecrical, economic, Damping i he mechanim ha rie o uppre ocillaion an force he yem ino able eayae chokemper of a car emper of Eramu brige Brige can ocillae! Tacoma Narrow Brige Tacoma Narrow Brige, November 7, 4 /7/4 7 inucance 7 /7/4 7 inucance 8 3

4 //4 7 inucance Tacoma Narrow Brige ae : Overampe Serie epone Solve for v = V i e = x = = haraceriic Equaion of erie circui = = ± Now conier cae of > > D> = 5.8. Two real naural frequencie No exra fficulie compare o or cae Sricly ecaying ranien oluion /7/4 7 inucance /7/4 7 inucance ae : riically Dampe epone Now conier cae of = ± D= = > = = = : amping facor Houon, we have a problem 5.8. Becaue = = we woul have v ( = Ve Ve V = ( V V e V Then, V an V are no inepenen becaue hey refer o ame exponenial erm. Acually, V V can be replace by V. emember: for a econorer yem, we acually nee wo inepenen exponenial oluion an wo conan o aju for wo iniial conion. Similarly for i /7/4 7 inucance Secon Inepenen Soluion o D.E. e = e e = e e Mofie oluion: v c = V e V e V c Prouc rule v u u v = u v Derivaive (almo proporional o funcion i = I e I e I D = naural frequency Mofie oluion compare o or cae Overhoo in ranien oluion = e e More formal erivaion in D.E. coure /7/4 7 inucance riically Dampe Waveform v c = V e V e V c 8 7 V 5 6 V 5 V c ae 3: Unerampe epone Now conier cae of < > < Houon, we have anoher problem We nee quareroo of negaive number = ± Soluion: complex number See TIO omplexe ekenwijze FATO Jawel, ik a beken al fenomeen Tenmine oner mahemaici En in wikung aangelege kringen Mijn naam, al u he ween wil i i Ik zi in allerlei berekeningen En aan he eine hef ik mij weer op U vraag nie, enk ik, om verhanelingen Maar wen a ik mezelf nu een onpop Welnu ik ben e worel ui Ik funcioneer, ofchoon ik nie bea Denk aar maar een langurig over na D< (Ui: Dr. P en Marjolein Kool, Wi en nauurlyriek me chemich upplemen, Ameram: Nijgh en van Dimar, /7/4 7 inucance 3 /7/4 7 inucance 4 4

5 Imaginary Number We nee quareroo of negaive number We offer number of which he quare i onvenionally enoe a i: i = In EE, i i reerve for curren, ue j inea (j = Hence: x = j x = j x = j x Now, we have real number, e.g.,,.343, An imaginary number, e.g. 3j Their combinaion i calle a complex number omplex Number omplex number i um of real an imaginary number, e.g. 3j Aion keep real an imaginary par eparae: (3j (3j = j(33 = 6j Muliplicaion ue j = : (3j x (3j = 3j 3j j = 86j All normal rule for aion, ubracion, muliplicaion, viion, bu real an imaginary par are kep eparae, excep for j, which become /7/4 7 inucance 5 /7/4 7 inucance 6 b = Im(z omplex Plane an Polar Form imaginary axi bj z=abj e jϕ = coϕ jinϕ z = a bj real axi a = e(z r = z ϕ = arg(z z = r(coϕ jinϕ = r e jϕ Muliplicaion ofen eaier uing polar coornae: z z = z z arg(z z = arg(z arg(z (mo π Euler ieniy will be imporan = ± Unerampe epone Now conier cae of < > < ω e( = = ω = ± ± Damping facor ω Unampe naural frequency Woulbe naural frequency when = D< /7/4 7 inucance 7 /7/4 7 inucance 8 = omplex Naural Frequencie ± ω < = ± j ω = ± ( ( ω = ± j ω ω ω Im( = ω Dampe naural frequency = ± j ω > Main reul: wo complex naural frequencie = jω = jω = omplex Arihmeic for Unerampe epone an wo complex naural frequencie Alo nee complex coefficien in linear combinaion of boh inepenen oluion for v c an i eul: Familiar general oluion, bu wih complex arihmeic Bu we can have complex volage an curren, can we? v ( = Φ e Φ e V i ( = Γ e Γ e I i, Φi, Γi are complex /7/4 7 inucance /7/4 7 inucance 3 5

6 v ( = Φ e Φ e V i ( = Γ e Γ e I jω jω = Γ e Γ e I = e ( Γ( coω j inω Γ ( coω j inω I = e (( Γ Γ coω j( Γ Γ inω I Similar for v Apply Euler Ieniy Dilemma can be olve by exploiing Euler Ieniy e jϕ = coϕ jinϕ = jω = jω /7/4 7 inucance 3 eal or omplex? i = e ( (( Γ Γ coω j( Γ Γ inω I ook complicae complex Bu phyical ignal are imply real Soluion: Γ an Γ can no be choen inepenenly Im(Γ Γ = Im( j (Γ Γ = Γ an Γ mu be complex conjugae: e(γ = e(γ an Im(Γ = Im(Γ Γ = Γ ike an e: Γ = a jb an Γ = a jb Then: Γ Γ = a an j(γ Γ = b Finally, le: I = a an I = b learly eal! i ( = e ( Icoω I inω I Similar for v (See book /7/4 7 inucance 3 Γ an Γ are omplex onjugae If we wouln know ha Γ an Γ are complex conjugae, we can cover i a follow: Γ = a jb Γ = c j Unerampe epone Damping facor Dampe naural frequency i ( = e ( Icoω I inω I v ( = e ( Vcoω V inω V Γ Γ = a c j( b Im( Γ Γ = = b Γ Γ = a c j( b e( Γ Γ = a c = a = c Γ = a jb = Γ Exponenially ampe Sinuoial Waveform Seay Sae V i an I i again follow from iniial conion an erivaive a (our meho (or ubiuion of above general oluion in original DE (book /7/4 7 inucance 33 /7/4 7 inucance 34 Unerampe Waveform v ( = e ( Vcoω V inω V /7/4 7 inucance 35 ω V V V c ae 4: Unampe Serie epone When = =, ω =ω = ± = ± jω i ( = Icoω I inω I Ocillaing behavior, wihou amping, ocillaion coninue forever Perpeuum Mobile? = ± ω v ( = e ( V coω V inω V =, ω =ω v ( = Vcoω V inω V /7/4 7 inucance 36 6

7 eview of Naural Frequencie Digial Syem Swiching Spee The naural frequencie can be hown in complex plane Im Im Im Im e (x e e e ompare wiching pee wih an wihou upply line inucance Overampe riically Dampe Unerampe Unampe 5. ompleely worke example Bu alo ee oher example in book Fully worke example 5.7 /7/4 7 inucance 37 /7/4 7 inucance 38 Digial Syem Swiching Spee ompare an ae ou = Ω in = pf = nh V = 5V ou = Ω in = pf = nh V = 5V Deermine higholow waveform here Aume iniially high for long ime Wihou Wih Ω v? nh pf Ω v? pf ompare an cae /7/4 7 inucance 3 /7/4 7 inucance 4 Wihou Ω τ = Ω x pf =. ns v( = 5 pf v( = v = FV (IV FVe /τ = 5e /7/4 7 inucance Sraegy: Solving Secon Orer circui. Wrie own fferenial equaion (eermine ae var. Wrie own characeriic equaion an olve for 3. Aume general oluion, epenng on i, a in cheme below (A, B an can be curren or volage or Overampe: wo fferen, real riically ampe: = = 4. Solve for eay ae oluion, an conan (ue I.. an ime erivaive a 5. heck oluion! Ae Be Ae Be Unerampe: an = e( ω = Im ( complex conjugae e ( Aco ( ω Bco ( ω (unampe when = /7/4 7 inucance 4 7

8 Deermine (omplex Naural Frequencie Wrie Aume Soluion nh Ω v? i pf v = V i = 5 ± 8.66 j Uni: / hange econ ino nano econ (for convenience = 5 ± j Uni: /n = ± = ± = 5 ± 8.66 j Unerampe! = 5 Damping facor ω = 8.66 Dampe Naural Frequency ( = e ( V co( ω V in( ω V ( = e ( Ico( ω I in( ω I v i /7/4 7 inucance 43 /7/4 7 inucance 44 Seay Sae an Iniial Value Deermine onan nh Ω v? i pf ( = e ( V co( ω V in( ω V ( = e ( Ico( ω I in( ω I v i Fir: eayae value V =v ( = I =i ( = v ( = i ( = v ( = 5 i ( = Nex: Deermine conan (V, V, I, I From iniial conion From erivaive a (book ue ubiuion in D.E. /7/4 7 inucance 45 /7/4 7 inucance Ue Iniial onion (an SeaySae Value v ( = e ( V co( ω V in( ω V v ( = 5 = e ( V co( V in( V ( V V = V = V = 5 i ( = e ( Ico( ω I in( ω I i ( = = e ( Ico( I in( I ( I I = I = I = v ( = e ( 5co( ω V in( ω i ( e = I in ( ω.. 3. Prouc ule: Example: hain ule: Example: Analyi Tool eminer u e y y u = x u x x v u v = u v Analyi in = e in in e = e co e in u( x = ω x in( ωx = in( u ( ωx u x = co( u ω = ω co( ωx ombine: e in( ω = ωe co( ω e in( ω eview! Deermine e co ( ω /7/4 7 inucance 47 /7/4 7 inucance 48 8

9 Ue TimeDerivaive a for Aume Soluion v ( = e ( 5co( ω V in( ω v ( e = ( 5co( ω V in( ω e ( 5ω in( ω Vω co( ω Noe: Ue: co( = in( = e = Ue TimeDerivaive a for D.E. v v = ( = v ( i( = V i Known! = 5 = Hence: v ( = 5 Vω = ( = v = /7/4 7 inucance 4 /7/4 7 inucance 5 ombine TimeDerivaive Info a Inverer Pair Waveform v ( = 5 Vω = ( = v = V ω = 5 V =.8 earlier: ω = 8.66 = 5 Final reul ( in n: 5 v ( = e ( 5co( in( i =. 577e in( Final reul ( in n: 5 v ( = e ( 5co( in( i =. 577e in( Ue he preceng nh Ω v? pf v when = proceure for i. /7/4 7 inucance 5 /7/4 7 inucance 5 Summary: Solving Secon Orer circui. Wrie own fferenial equaion (eermine ae var. Wrie own characeriic equaion an olve for 3. Aume general oluion, epenng on i, a in cheme below (A, B an can be curren or volage or Overampe: wo fferen, real Ae Be riically ampe: = = 4. Solve for eay ae oluion, an conan (ue I.. an ime erivaive a 5. heck oluion! Ae Be Unerampe: an = e( ω = Im ( complex conjugae e ( Aco ( ω Bco ( ω (unampe when = /7/4 7 inucance 53