Formulation of Circuit Equations

Transcription

1 ECE 570 Sesson 2 IC 752E Computer Aded Engneerng for Integrated Crcuts Formulaton of Crcut Equatons Bascs of crcut modelng 1. Notaton 2. Crcut elements 3. Krchoff laws 4. ableau formulaton 5. Modfed nodal analyss Supplemental readng: Vladmrescu, Chpt. 2 Ruehl, Chpt. 2 (by K. Snghal and J. Vlach) 1

2 1. Notaton t ndependent arable (tme) x ndependent arable (space), applcable n dstrbuted elements Lumped elements Varables ( t) = t dependent arable (current) = t dependent arable (oltage) q = q t dependent arable (charge) Φ = Φ dependent arable (flux). Value of a arable at a specfc moment of tme: ( 0) ( t1) ( 0) q( 0) ( 0), alues of current at t = 0 and 1,, Φ alues of oltage, charge, and flux at 0 t = t respectely, t =. 2

3 Parameters such as R, L, C can be functons of dependent arables. hs feature prodes mechansm for mplementaton of nonlnear elements. 3

4 Dstrbuted elements (used n modelng nterconnectons) Notaton = x, t current as dep. ar. functon of poston and tme (, ) = x t oltage as dep. ar. functon of poston and tme. Value of dependent arable at specfc argument alues (, 0) x ( x, 0) ntal dstrbutons of current and oltage n a lne (, ) d t ( d, t) tme aryng current and oltage at the poston x = d. 4

5 (, ) a t 1 ( a, t ) 1 alues of current and oltage at the poston x = a at tme t 1. Parameters: R, L, C, G are specfed n proper unts ( Ω, H, F, Ω 1 Ω ) per unt of length. In some stuatons (case of nonunform lossy lnes) these parameters can depend on poston and frequency: Example: R = or R (, ) R x ω =. R ω 5

6 2. Basc Crcut Elements (deal components) I. Lumped elements A. Resstor Lnear Symbol Voltage controlled Current controlled R 1 = R = R 6

7 Nonlnear resstor Symbol Voltage controlled Current controlled R = = ( ) 7

8 B. Capactor Lnear dq d q = C; = = C C dt dt : 0 IC Nonlnear C dq dq d d q = q( ) ; = = = C ( ) dt d dt dt IC C ( ) : 0 ; ncremental capactance 8

9 C. Inductor Lnear d dt L = L ; IC : ( 0 ) Nonlnear L d Φ d d = = L IC d dt dt L ( ) d Φ = d ; : ( 0 ) n crem en tal n d u ctan ce 9

10 D. Dependent (controlled) sources Symbol Lnear Nonlnear VCVS c k k = E = k k c k k c CCCS c k k = F = k k c k k c 10

11 VCCS c k k = g = k k c k k c CCVS c k k = R = k k c k k c 11

12 E. Independent sources exctatons (external, tme dependent) Voltage sources V = V t E = E t Current sources I = I t 12

13 II. Dstrbuted elements F. ransmsson lnes nterconnectons Lossless m... m... m m 0 D x 13

14 14 Vector notaton: 1 2 m x t x t x t x t,,,, = = = = = = = = 1 2 m x t x t x t x t,,,, = = = = = = = = Model of nterconnectons (lossless transmsson lnes) 0 0 L IC x x t C IC x x t ; :, ; :, = = = = = = = = L matrx of m m nductances (PUL) C matrx of m m capactances (PUL) D length of nterconnectons

15 Model of lossy lnes (n frequency doman) dv = ( R jω L) I dx di = ( G jωc ) V dx R L C G where { } { } (, ω ) (, ) (, ω ) (, ) V = V x = F x t I = I x = F x t = R ω matrx of resstances representng the conductor losses = L ω matrx of nductances = C ω matrx of capactances = G ω matrx of conductances representng the delectrc losses. Specfcaton: matrces R, L, C, G and lne length D. 15

16 3. Krchoff laws Krchoff current law KCL of currents nto a node = 0 Krchoff oltage law KVL of oltages n a loop = 0 16

17 4. ableau formulaton Equatons determnng the topology KCL: A = 0 ; A ncdence matrx, branch currents KVL: = A n n ; branch oltages, nodal oltages Consttute equatons (descrbe the physcs of the elements) CE: Y Z = W ; Y "admttance" matrx, Z "mpedance" matrx, W ector of sources 17

18 Note that CE are mxed: Y K w 1 1 j = K2 Z 2 w E Y Z W currents oltages Y1 admttance matrx, Z2 mpedance matrx, K1, K2 dmensonless matrces w ector of current sources, w ector of oltage sources j E 18

19 Matrx formulaton Sparse ableau Analyss (SA) Vector of unknowns: Model equatons: = n x A KCL, dm Z = dmy = n b 0 1 A 0 = KVL, dm A = n b Z Y 0 n W CE, b # of branches x Model n the compact form: S w S x = w 19

20 Modfed tableau Obtaned from the SA a the elmnaton of branch oltages usng KVL: = whch yelds A n KCL: A = 0 YA Z = W CE: or n the matrx form n A 0 0 = Z YA n W 20

21 5. Modfed Nodal Analyss (MNA) Based on separaton of currents nto: I 1 branch currents of elements hang an admttance representaton (these currents are elmnated from the equatons) I 2 branch currents of elements whch do not hae an admttance representaton (ncludes branch currents of oltage sources and those that requred for output) J ndependent current sources. hus we partton the ector of currents accordngly: I1 I = 2 J 21

22 he ncdence matrx s parttoned such that KCL ( A = 0) can be wrtten as I1 A A A 1 2 J I 2 = 0 A1 I1 A2 I2 = AJ J J (1) he ector of branch oltages s parttoned approprately: = ( V1 V2 V J ) such that the KVL ( = A n ) can be wrtten n the form = = = V1 A1 V1 A1 n V2 = A2 n or else V2 = A2 n V J A J VJ = AJ n (2) 22

23 Consttute equatons are also wrtten n the parttoned form: 1. for elements wth admttance representaton YV 1 1 = I1 (3) 2. for elements wthout admttance representaton K1V 1 K2V2 Z2I2 = W2 (4) Note: the ector W 2 contans entres representng the oltage sources. Substtuton of (3) and (2a) nto (1) yelds Substtuton of (2b) nto (4) yelds AY A A I = A J (5) n 2 2 J K A K A Z I = W (6) 1 1 n 2 2 n he equatons (5) and (6) defne the modfed nodal analyss (MNA) wth n, I2 as unknowns. 23

24 In the matrx form the MNA s defned as follows: AY A A n AJ J KCL = K A K A Z I W CE MNA matrx nterpretaton of block matrces A Y A Z K A A 2 2 =Y MNA admttance matrx, mpedance matrx of mpedance defned branches K A =C dmensonless matrx n consttute equaton for "nonadmttance" branches, ncdence matrx for "nonadmttance" branches 24

25 MNA equatons wth block matrces n J Y A2 A J = C Z I W he entres of the MNA matrx are determned a nspecton of the net lst followng some smple rules, whch are sutable for computer mplementaton. 25