Computing with diode model

Transcription

1 ECE 570 Session 5 C 752E Comuer Aided Engineering for negraed Circuis Comuing wih diode model Objecie: nroduce conces in numerical circui analsis Ouline: 1. Model of an examle circui wih a diode 2. Ouline of ical numerical roblems in circui simulaion 3. Discree ariable echniques 4. Secral echniques 10

2 1. Model of an examle circui wih a diode We shall examine a simle diode configuraion (a simlisic model of halfwa recifier) on he lef and is equialen circui on he righ. r E G E r i D Q D he resisor r in he equialen circui reresens he inernal resisance, r G, of he source and he diode resisance r s. For simlici we assume ha Q D reresen he diode juncion charge sorage. he equaion for he diode olage wrien using he equialen circui is nv d c E = s e 1 d r 11

3 c dq d where D PN juncion diode model. = is he incremenal juncion caaciance defined in he secion describing he 12

4 2. Ouline of ical numerical rocedures in simulaion of a circui. DC analsis Sead sae soluion of circui equaions wih all deriaies equaed o zero. n he circui his means ha caaciors are oen and inducors shored. n he examle circui he DC analsis is defined b he nonlinear algebraic equaion nv e 0 = s e 1 his equaion is soled ia linear aroximaion and ieraions (NR). r ransien analsis Ofen used, er inoled, exensie in CPU ime. Discree ariable echniques Variables are comued in discree imes. Secral echniques Series exansion of circui ariables 13

5 3. Discree ariable echniques (ime marching mehods) Basic mehods (mus be imlici): Procedures 1.ime discreizaion. single se [Backward Euler (BE), raezoidal (R)] mulise [Gear s]. he circui ariables are comued in he discree imes [ 0, 1, 2,..., M ]. Examle of circui equaion in discree ariables (diode circui discreized using BE) c( ) k 1 = e 1 r k 1 nv E k 1 k k 1 k 1 s k 1 k k = 0,1,2,3,... M 14

6 where k is he numerical aroximaion o ( k ). he iniial condiion (saring oin),, is usuall defined hrough DC analsis. he equaion is imlici because he o unknown, k 1, has o be comued from he nonlinear equaion. he difference, k 1 k, is he numerical inegraion se size Linearizaion. o simlif he noaion we shall designae = k 1, and hus consider he roblem of finding a soluion for one se onl ( k 1 ). Wih his noaion he circui equaion becomes k nv E k 1 c = s e 1 k 1. k r he nonlinear elemens are linearized using alor exansion around a emorar oeraing oin, (saring oin), and resuling linear nework is soled o deermine he imroed alue, 1. 15

7 should be oined here ha in he linearizaion rocedure we consider onl firs wo erms of he exansion so a single soluion of he resuling equaions ma no be adequae and ieraions are usuall required Linearizaion of VCCS in he diode model. o illusrae he rocedure we al he linearizaion o he curren source (VCCS) of he examle diode circui. he resuling relaion in he ieraie form wih he ieraion coun,, is or else i = nv e 1 s nv e D = s 1 nv = i nv e 1 s nv e s nv e D s k nv D nv D G 1. 16

8 his righ hand side of his relaion is used o relace he VCCS equaion and shows ha he linearizaion can be inerreed as a relacemen of he original nonlinear VCCS b an indeenden, consan curren source,, defined b D = nv e 1 s nv e and conneced in arallel wih he conducance, D s k nv G = D D G, which is also a consan s nv of he alue deermined b he known olage,, (known as a saring oin or as a resul of reious ieraion). he circui inerreaion of VCCS linearizaion is illusraed on he nex age. e nv 17

9 he linearized circui reresenaion of VCCS D D i D linear equialen 1 G D Dl 3.3. Linearizaion of caacior relaions in he diode model. o discuss he caacior linearizaion rocedure is conenien o rewrie he discreized circui equaion in he form nv E k 1 g g k = s e 1 r 18

10 g ( ) where = k 1 c k reresens he nonlinear conducance. he linearizaion of caacior is erformed relacing he nonlinear relaion g b dg g( 1) = g( ) ( 1 ). d Aling his exansion o he lef side of he discreized circui equaion and reaining consan and linear erms onl we obain nv E k 1 c Gc 1 = s e 1 where he equialen curren source, relaions: = c, and conducance, r Gc, are defined b he following 19

11 and dg = g ( ) ( ) d c k k = dg Gc = g d = k. 20

12 hese linearizaion rocedures can be inerreed in erms of an equialen circui linear c D 1 Gc equialen c which is comosed of conducance, c source alues are udaed a each ieraion. G, and curren source, c. he conducance and curren 21

13 3.4. he linearized equialen circui of he examle circui wih a juncion diode. Using he linearized VCCS and caacior circuis we can finall consruc he equialen linear circui for he inroduced here examle of nonlinear nework wih a diode. Such a nework is used in he circui ieraie soluion rocedure and is ofen called comanion nework. r Ek 1 1 GD Dl Gc c should be emhasized here ha all elemens of equialen circui (comanion nework) are consan he circui model is in he form of linear algebraic equaions. Howeer, he coefficiens are udaed from ieraion o ieraion. 22

14 4. Secral echniques Linearizaion in he case of secral echniques is based on NewonKanoroich aroach and resuls in a linear circui wih ime aring coefficiens. he equialen linear nework has ime aring elemens which is er disinc from he resul of linearizaion used in conjuncion wih ime marching mehods where all elemens are fixed. he circui model is in he form of linear differenial equaions wih ime aring coefficiens. his major disincion sems from he fac ha he linearizaion is erformed no around a fixed = ) eoling in ime. oeraing oin ( ), bu around a rajecor, ( 4.1. Linearizaion of caacior relaions. he linearizaion of lef side of he circui equaion (inoling he linearizaion of caacior) nv c d = E s e 1 d r 23

15 requires an addiional exlanaion. should be noed ha and such ha he lef side is inerreed as a funcion of wo ariables d z d = are reaed indeendenl ϕ, z. A linear exansion of a funcion wih wo argumens around, (, z ), can be obained as follows ϕ ϕ ϕ = (, z ) ϕ (, z ) z z z he funcion, ϕ (, z), in he case of ineres has a er simle form argumens is z z c z, which showing all d d c = c d = d = c z z. 24

16 Exanding his funcion according o he discussed rule and arranging he erms we obain he following linear aroximaion d c d d ( ) dc d ( ) dc d ( ) c ( ) ( ) ( ) d d d d d ( ) ( ) We shall inroduce he ime aring conducance G c ( ) G c = dc d ( ) d d ( ) c. and he ime aring curren source 25

17 where = dc c = d ( ) d d ( ). or else showing he indeenden ariable exlicil =. 26

18 Using he inroduced noaion for he gien funcions we can wrie he linearized relaion in a comac form d d c c Gc c d d which in he circui inerreaion means ha he equialen circui for he nonlinear caacior is a arallel connecion of ime aring caaciance, conducance, and curren source: linear c G ( ) equialen circui (ime aring elemens) c c ( ) c ( ) 27

19 4.2. Linearizaion of V characerisic. he linearizaion of he righ side of he circui equaion is sraighforward and ields d c Gc c = d nv s nv s nv E = s e 1 e e nv nv r. D D G 28

20 nroducing he ime deenden curren source defined b he formula = nv e 1 s nv e D s nv and ime deenden conducance defined as ( ) s nv GD e nv we can wrie he linearized circui equaion in he following comac form = 29

21 d c Gc c = d E ( ) = D GD r which forms a linear differenial equaion wih ime aring coefficiens. 30

22 4.3. Linearized equialen circui of diode nework. he linearized dnamic circui for simulaion of diode nework using secral echniques is shown in he schemaics below. he circui is dnamic as i conains he caacior, c ( ) circui a waeform, comonens.. o sole he, needs o be sulied o deermine all ime deenden circui r E D ( ) G D ( ) G ( ) c c c ( ) 31