R Nodal AnalysisofSwitched-Capacitor Netwotks. Time"Freq~ency, andz-Domain Modified..

186

Time" Freq~ency, and z-Domain Modified..

Nodal

.

Analysis of Switched-Capacitor Netwotks.

JOOS VANDEW ALLE. MEMBER,IEEE,HUGO J. DE MAN, AM) JAN RABAEY

,

Manuscript rcceived November J2. 1979; rcviscd September S. 1980. Tbc authors are with thc Katholiekc Universiteit Leuven, Heverlee, Be1gium.

I. lNnODUcnON

R

ECENTLY much attenûon has been devoted to switched-capacitor nctWorks (SC netWorks). A topic of cajor concern is tbe construcûon of simple, efficient, and general simulation programs for such networks. One particular successful approach [1] in this direcûon is tbc adaptation of the modiIied nodal analysis [2] to SC net-works. lt requires only minor modifications of existing programs based on MNA in order to analyze SC networks efficiently. Here we describe a uniIying framework for this time-domain approach and for approaches based on

z-AbJt1'Ga-A rlgorous l.Dd~ derindoa of tbe modif1ed DOdal

cquadoas DeccIed ror a dme. rrcquency and z-domab anaJysis of

multi-pbase ~adtor DetWOrts Is pYesI. A.lso . ~ eqWTaieut

cin:uit for switd»ed-op"dtor DetWorks Is derhed. Thls t.edmJque a.UowsU) ha.Ddle COIldnUOUScoup~ piece'lrise-collSWlt mpua as "'eU as COIldnU-OU! Inputs. 'These resuJts c::an he used to adapt e:dsting dn:uit ana.Iysls progn.ms, based oa mod11"ledDOdaI ana.Iysis, ror clfideDt simu.ladoa oC switched<ap8dtor DCtWOrics.

,

T I ~t ~2 11 '2 'J ~ ~ ~ t.N.1 : 1 t~.,

_'---'

!"-2 !'..J 181

i

~1"" 41,..1 '2*' '2H0212"'J...

'

'_

...pIIOMt

'--" .. :>reM?

Fig. 1. The timing ÎDstanccs Cld phasc:s oC UI.N-phase SC netWork.

CCCCQ!cr C i C i

---v, Yj _'eII S

i~--Ï.

'"I "i á:u~ IIIroooswitcftS

-I

"I~_{or _lil}~ .!ft

Ieg.so.ru

.

i Q.! j "1 - "1

ctlCt9'

~

·

I ·

.

~

lQo;.CDt'II\'I:III..t ~,

~

':- m "0 ''''''0 j;.... cIICr9' c:".t.clI..t eIIcrg. sour~!14

:Jo

E:

-I

a

l

J

: :

][

.;

1

[

0

]

jo'oo~_o !ft r. 0 0..", 0 " 400"" 0 i ~ ~ ~

;:;~

!~ "; 1 ~ o

l

' J

.

000, 0.. ~ 0 IR 0 0 0011 0 Vrn

.

0 " 0 0 0 0'''' 0!ll.. q "'qro-'I-,~ro:;.o ~cf 0 lIoW" 0 0' 0 0 Clo 0

Fis. 2.. Col1tnDutiOI1or "stamp" ror tbe different components ol &SC .

netWork to tbe MNA equations (1).. .

transforms. This wi1l provide additional insight in the ;ystem-theoretic:aspcc:tSof SC networks. It tums out that :here cxists an casily computablc r-domain transfer matrix wbic:hcompletely charactetÏzes the .signa! handIing capa-biIitiestrom thc N phases at the input to thc N phases at :he output. Many of oUt results c:an bc obtamed by uhpting (3]-{7]..[9]. [10] to thc MNA framework and to nultiphasc SC networks. Howcver wc prefer thc direct iepvation beca.usc of its conceptual simplicity and .thc iirect link with the computer implementation.

Il. TIME-DoMAJNMNA

Wc consider .arbitrary linear networks containing ideal iWitches, capacitors~ independent voltage and charge iOurces(VS and QS). and fout' types of dependent sources YCVs, QCVS. VCQS. QCQs). Tbe switches are con-:roUedby Boolean clock variables ~/(t)-O. or 1. ~J(t),..O :resp.. ~,(t)-l) corresponds to an open (resp., closed) iWÏtchat time t if this switch is griven by clock i.

I

.---..

!he time is partitioned into time slots 6.",-(t t"'+I] such that the doek signals (and hence the network).do not vary in 6 We assume that the cloek signals are T-periodic:,te., 4t,(t+/T)-cp,(t) Vt,i (Fig. I). Each period has N time slots. Tbc umon of. the .time slots Ale'AIc+N'6.1c+2N is ca1led phase k. All time slots of pluse k have the same duration and the cloek values are the same. i.e.. 4t,(t)=-CPl1c'for aU t in phase k. We a1so assume that there are no resistors in the networks. Hence, there are no transients in the waveCorms.In practice this is not a severe restriction since aU SC networks are designed to reach their equilibrium state quickly enough otherwisc their operation would be unreliabte.

In Theorem 1 we show how the time-domain equations of a SC network can bc set up in the MNA framcwork [2]. It is useful to keep in mind that MNA is an extension of the nodat equations by adding some branch relations and that each component contributes a pattern (called stamp) given in Fig. 2 to the equation (1).

.,

1"

Theorem J: Thc abovc dcfiDed linear T. periodic switched c:.apacitornetWork is described in tbc ÛJIledo-mam by tbc equaûons for I EAk+llt

where 0(1) (resp.. u(t» is tbc vector of tbc voltage re-sponses at tbe nodes (resp.. voltage sources in some se1ected branches) and wherc q(t) (tesp.. w(t» is tbc vcctor of tbc charges transfcrred in some selccted branches (resp., injected by charge sources in tbe nodes) betWeen tk+11Iand I for tEAk+l" and where tbe contribuûons to Ak' Bk' .Ck. Dk' EIc' w(t). and u(t) are constructcd as

follows.

Construction of theoTime Domain MN A Equations (J)

1) .Choose tbe unknown variables of tbe netWork as .f~nows. The voltages at a11tbc nodes (except tbe referencc

voltagc) consûtutc tbe unknown voltage vcctor o(t). Tbe

. charges in switches, voltagc sourccs, tbc controllcd . and

branches of VCYS. QCQS. and QCVS's, and controlling

. branches of QCQS's constitute tbc unknown charge vector

q(I). Thc voltage input u(i) (resp.. c;hargeinput w(t» has tbc same sizc as q(t) (resp.. 0(1».

2) Sct up (1) with Ale' Bk' Ck' Pk' Ek' u(t). and w(t) zero.

3) Fot each component of the circuit idcntify tbc stamp of Fig. 2. Obse.~e tbat the stamp of a switch includes tbc Boolcan variabie of tbc doek wmch controls tbc switch. If tbc component is connec:ted to tbc rcference node delctc tbc corresponding row and column in tbc stamp. Using tbc indexcs of tbc rows and columns in tbc stamp add tbe contrÏoution to tbc appropriate entrics in tbe matrices Ak' Bk' Ck' Dk of tbc left-hand side of (1), and to Ek' u(t), and

w(t) of tbe right-hand side of (1).

ProoI: Tbe charge w(1) injcc:tedin tbe nodes betWecn tt+l. and tEAk+l" is cqual to tbc net charge flowing away from this node jn tbe otber branches or using Fig. 2:

Tbe remáiDder cquaûons of (1) are preciscly a vector formulaûon of tbe constituûve equations in some se1ected

branches. 0

Oearly equations of (1) can be set up ea.si1yby com-puter using tbc stamps and compare favorably witb tbe involved topologie or numerical procedures of [3]-[5], [7]. [9], [10]. Otber issues are compared in Scction V. These

equ.aûons can be wily cxtended to include nonpenodic networks and nonlinear voltage controlled capacitors and voltage sources [1]. However the subsequent derivations depend on linearity and periodicity and hence wc have assu:ned these tbroughout.

Observe that tbc matrices Ak =EJcand do not depend on. k since aD stamps of. Fig. 2 satisfy this propeny. In Section V we wil1 see tbat other stamps wbjch do not satisfy this propeny are often more efficie::lt.There w.e

---IEEEnANSAC'I10HSOH caaJn'S ANI) SYnzws. VOL.c.u-2!, NO. 3, WAAO!198

discuss also tbc improvement such stamps can bring to tb. above construction. Tbc following corollary gives ajustifi cation rrom (1) tbat tbc voltage transfer in a SC netWod is not aIfec:ted,by a relative change in al! capacitor valucs Corollary J: By mu1ûplying aU capacitances and ar controlling factors of VCQS by exand by multiplying aJj controlling factors of QCVS by I/a, al! charges of. a SC netWork are mulûplied by ex and all voltages remam unchanged.

.ProoI: 11lis corollary follows directly from (1).

0

In order to solve (1) wc dccompose tbc input signals and tbc output signals into pieccwise-constant contribu-tions and tbe remaindcr waveforms and obtain thc

follow-ingrcsu1t.

.

Theorem 2: The abovc dcfmcd liDcar T-periodic SC netWorkis descnèed in tbc time domain by

00 (3.b) where (4.a.) (4.b) lor tEAIe/+N (2)

and analogous dccompositions for q(r). w(t). a.n~ u(t).

.

ProoI:Substitutc(4) in (1) and cvaluateit at

t-tk-+IN+1in order to obtain (302.).Subtracting (302.)from (1) generates (3.b). Ccnverscly adding (302.) and (3.b) and

substituting (4) ~ves (1).

0

In words tbc Theorem 2 says tbe following. Dccompose tbc signals in10 a pieccwise-constant waveform such tbat tbc amplitude in each ÛJIleslot is cqual 10 thc ~aluc at tbe end of this time slot and a. .remaindcr waveform.. The output is tben tbc sum of a piecewise-constant waveform obtained by solving tbe N-periodic implicit liDear alge-braic and difference cquaûon (3.a) and a rcmainder out-put obtained by solving tbc liDear algebraic cquations (3.b) (capaciûve reedthrough).

Roughly spea.king tbc existence and uniqueness of tbe solutions of (3) is guaranteed ror almost a11networks of interest. Fot passive netWorks tbere are even topologic conditions. In tbc case tbat tbc independent voltage and charge sourccs are tbc only active components, it can be shov.n tbat (3) is always solVlJbleif in time slot k tbere is no C'.ltset of charge sources and open switches and na loop of voltage sources an~ closed switches. This topological condition is trivially saûsfied by any practica.l circuit since this condition is eitber violated if tbe excitation is un-acceptable or if one of tbe switches can be reversed in this time slot.

I

"i

!

lIl. z-DOMAINMNA

In tbis section we use z-u-ansform techmques to solve (3). We make again use of tbc linearity of the capacitors and the controlled sources. Since (3) is time-varying the techniques of the z-traDSform are not readily applicable. Fommately (3) is periodie and hence we can adapt a metbod of Jury [6. p. 57]. We partition the sequence of values at the end of each time slot Ol' 02' 03' OH_I' OH' .OH+I' .. into N different sequences each having tbe same

phase: Ol' 0H+ I' t7zH+I"

..

and~, 0H+2' t7zH+2"

. '. ett...

until OH'02.'1'03H"" (Fig. 1). We take tbc standard z-transform of tbese N sequences.

co

Yk(z):' 2:{Ok+/H}- L ok+/IIZ-/ (5) 1-0

:where k=a 1,2,"', N. Analogously we obtain Qk(Z), Wk(z), and Uk(z). Remember also [6] that if 0,

-

0. i <;0

.~(Ok+(/_I)H}-Z-IVk(Z). (6)

I

"0 Theorem'03:Tbc above descnöed linear T-periodic SC

netWork

~

descnöed in tbc z-domain by

189

Tbc submatrices Gkn. Hkro' Kkro' Lkn allow a very simple

interpretation. Up to this point we consider tbc SC net-work as a discrete device which transforms tbe input

sequences of samples at

ç.

i-l,2,...

of tbc voltage

sources and charge sources into tbe output sequcnces of samples of voltages or charges at 1,-. ;.. 1,2,. . '. IC only nonzero voltages sourccs are applied during time sIo1$ 11,n + N, n + 2N,. .. and if tbe output node voltages are only observcd during time sIo1$k, k + N. k + 2N.. .. tben Hkn(z) relatcs tbe z-transform of this input sequence to this output sequence, i.e.. p;.(z)-Hkro(z)lT,.(z) or it telates inputs at phase 11to outputS at phase k. In otber words HbCz) is tbc z-transform of hkn(mT), m-O. 1,"', i.c., tbc node voltage responses obscrvcd during time slo1$ k, k + N, k + 2N,' .. to unit input voltages applied during time slot n. Oearly ~n(z) is tbe z-transform of tbc node voltage response during time slo1$ i, i + N. i + 2N,.

..

to tbe same input, andocan tbus be derived from tbc same netWork analyses as Hkn(z). Tbus one column of tbc transfer matrix of (8) can be detmnined from one net-work analysis. 0

CorolltJry2: Any above defmed SC netWork is

com-,here tbc missing entries are zero.

Proof: By writing (3.a) in full for k= 1.2,..., N we btain an augmented set of time-invariant 2lgebraic and ifference equations on which (5), (6) can he applied in

'der to obtai.n(1). 0

As an interesting alternaûve to solving (3.a) in tbc time )mam. one canequivalently solvc tbc z-domain equa-)ns (1). In terms of tbc inverse of tbc matrix in (7). 1ich is cilled tbc r-dDmaintransfer matrix, tbc equations

I~mc

pletely characterized by tbc z-domain transfer matrix of (8).

Proof: Observe tbat the inverse of tbc matrix of (3.b) can be easily obtai.ned from tbc z-domain transfer matrix

(8) at z- co: .

[

_~:~=!_

]

_

[

~~k~~)

j

_ a..k~(

~

~

][

_~:~!!_ ]

.'

for t E~k+'H

q.(I) Ku(co): Lu(co) U.(I)

.

GIH :. 11.u G2H I H21

...

I

...

G/'Il GH2 ... G..,HI HHI

_~---

Hm :.. HHH

Ku

KI2

...

KIH I Lu

LI2

...

LIH

K21

Kn

. . .

KIH I L21

Ln

. . .

LZH

. .. I

.41

-

E -11 B

1% I

1

Y1

Wi

-1;

.42

I _{112 W;} -E:s .43 I_I B, JIj W3

.

.

_.

.

.

.

I

.

.

I

.

-EH

.4H

I BH VH WH (7)

---j---

---

..

_---I

Cl I Dl

.

Ql . UI

c;

I D2 Q2

.

C;

I D3 0 Q3 I

..

.

.

I

.

. I

.

.

CH I_{I DH JL QH J L .uH} (9)

0

,

WI

W1

...

WH (8) ---1. UI U2 , . . UH YI Y2

. .

y,

.- I

-Ql Q2

...

QH

,90

-cirèuit reduces to tb~sc oi [41,-[71via some t.rWförmatiom (tbC: gc:neralizcd circu1ator corrcspönds to =Üic:gyrator of tbc-link two port of [4D. We MVe ápplicchhiS:a!gorithm to .tbe circuit of Fig. 4(a) Wi~-c1oé:k sigIïa1s of F~ 4(b) and'

obtaîn tbé equivalent -clrciûtof Fis 4(c}.~ =--: :$:[

,-.- -- -- .~~~_7.:F~Q~c;_D~~~~~~_~~~.~A

WC descn"bcin this sccûon now tbc. -praëtiëally UScf1Ü frcqucncy properties can be dcrived from tbe time-domain or z-domain analysis of tbc previoussccûom; In many appllcations such as-in filiering óne is Ïnnii:llyinterestcd in tbc frequency componen1$of tbe inpur.3Jid-output voltage : wavefórms. Sc -we win onfy dcrive :thc .cxptessions for voI~geS. Sine: a SC ~etWor1fä a-timc;.V~g :netWorkin gën;ral many frèquenëieS appear- ä.t - ihë -output for a sinuSoidal excitation. - ;-- _. - .: .::. -:- - -:. :::: ~

. :. ~re~ -4: Tbc abovè desètib~ -liDcär.T-pèriodic SC-. netWorkgeneratcs fór a voltage-cxcitäSC-.tion-u(t)"witb Four-ier transfop:n U(w) a .voliage rcsponSc:"'f/) with Fouricr -:.-:-_-~_.:_:~__~~orm V(~rgivcn by:-:7--:_': :::-:: :~~ ~~:.

- ... +co.. . __ .- -.:..:::::.: .~-='Yci;Y';;'~

Z ~X{w~G)":;ÜJJÛ(~-Ii~J.; ..' (11)

- ""-00 .

_0- _"_

... ---.-.

-- - - . -~ ---- - - - . - .---- . - .- _{-. .---}." .-- -' . .

-- - -- - .

. - - _. .

=: -::.:: ::' .F.iz:..3~~.1lie symbol of. '" i~ érc:ul&tor (6).With comtantG.

- ---

- -.

- - -....

-.

--

-.

".- - - _.. c _ _ --:..""

- ,:,,; -I: 3. Tbc Fourier:tra.nsforni. V",(w) of ö",(t) which is tbc voltage

.

-

tI_{(t ) obscrveëiduring phasc k and zero outside is.}

I~I

.

-

-

.

-

I ..". +_."':

The $C DetWork;;r(;) .wi~-~bc:k signaIs (b) hu ADequivalent ~

'y,(

-'

~

₎

_ ..;:

_X

-

_{(w w-n", )}

_Ui(

_{",-nw )}

₍₁₂₎

QJ'CU1t(c). '.: . k, 41" k, 1 1

. ,,--co

- :' J =' . '" -~1'

One can wonder nextwhethcr 1h~o-emu".a ûme~aria.nt _w.1lq(w1~~1rjx,~(;(!1]e Iransmissionfimaion-,

"\ ...,. - . ,

netWorkof impedanCe1S

which is descnDcdby' (7) ,and

.:

'

.

:

.:

N

whetber such a netWor1~caD.be derived immediately from :: X(~,:n)- .~ Xt(""n)

tbc SC netWork.Tbc key idca m obtaiIrlngsuch an cquiva- -:

~

.

k-l

.'

tent netWork is 10 convert tbc N phases of any branch into with tbc Irart!rnissionfunction Xk( w, S1)of phase k given by

N different branches. Th.is convcrtS tbc Q,i!ferc:ntinstanccs N

of t~ into different Ioaûons in space. We need tbc

Xt(~;rn.-~k(~)e_-J""t..~

~

eJfJtl.'Hkl(eiOT)

:

following intrinsic N-pon ca11ed a genoalized circuJm01' -- - - . __ l-~___ _.

wjth ëönstanl G which is defmed by (Fig. 3) - +

[

(Xt(w

-

n)

-

(Xt(w)

]

exp

[J(

a

-

w)t ",+,] HIek(00)

Q,.O _~z-I- Vl 1

:--:'

_- _

- --, -- -- - .

(i3)

- -~_.

i

- -

Q . .

i

:: :

-

G 0 - .: - 13: -;-witli Ek; üeruie~:by (8)' and:: - -:=-'.-. : .:-:

. -:::

-..-.--

Q.' -

-G

0

.;

::

A

.: ~~::~.!~:_ -

'.-

'..=

(Xt(p)-2{sin[P(/",+,-tk)/2]

~_:.--qff_ -

.

-G:O VNJ -- 'cxp(jp(tt+l-tk~/2n/Tp. (14) (10) ,ç._Th

..

.

_d

_ca1cu1 '

Proo;: e denvation of (11) IS one by fu-st at-CoroStructionof the Equivalent Circuit

~,

of a sc NelWork ing thc Fourier transform of tbe pieccwise-COJ].Stant

con-~ tribuûon ö",(I)in Ö",(/)using (8) and V",(w)using Tbcorem

2 and tben summing over aU phases k-l,2 , N. Tbc response is first calculated for a sinusoidal excitation and afterwards extended to arbitrary excitaûon. Tbc details

are given in tbe Appendix.

0

Oearly tbc fust term in (13) takes into account tbe piecewise-constant signals and tbe second term tbe con-tinuous coupling. From (11), (12) it follows tbat a sinusoidal input u(t) generates an output which has a line spectrum whosc lines are W1a part. Mosüy onc is only interested in tbe contribution at tbc same frequency as the inpuL This is among others the case il tbe bandwidtb of 1) For cach of tbc N phascs (i.c., N time sIo1$in onc

period) a netWork is drawn witb tbc switches in ~ correct .posiûon for this time stoL

2) Tne N netWorks are interconnected by generalized cirC'Jlatorsas follows.For each capacitor C; in tbe original circuit we need a circ~ator witb constant Cl' Port 1 of this ~.rculator is connected to tbc corresponding capacitor of tbc fust circuit, port 2 to .that of the second circuit and so on. Repcat for each capacitor.

It is easy to check that tbe resulting network is de-scribed by (7). For 2-phase SC netWorks this equivalent

u(t) is smaller tbaD hal! tbc sampling frequency. For such cases we defme tbe°.foilowing practical frequency-dc/TIIJin tTansjer functions H(w) (resp., HIe(w» [5]. Apply a sinusoidal excitation u(t)= UeJw1and compute or measure tbc frcquency component at tbc same pulsation w in tbc output o(t) (resp.. o,,(t», tben H(w) (resp., H,,(w» relates tbc phasor U to that of tbc output o(t) (resp., t\(t» at pulsation w. Theorem 4 allows to compute H and Hle easi1y.

Corollary3: Under the assumptions of Theorem 4 N H.t(w)-cx.t(w)e-jwt...

L

eJwl'.'H.t/(eiwT) 1-1 [ I,,+,-/.t . ] + . T -CX.t(w) Hu(C:C) N H(w)-

L

B.t(w) ,\:-1 (15) (16)

'v{here cxk'is given. by (14).

.. Proofl.Take only the term witb n

-

0 in the sum (11)

and (12). .

0

v.

ExAMPu A}o.1)CoMPUTATIONAI.AsPECTS

In this section we fU'Stillustrat~ the above derivations on our cxample of Fig. 4. Then we discuss the computer implemc:itations via. time domain analysis (303.)and via. the z-domam analysis (7) and compare both brieny with other tcchDiques[3]-[5], [7], [91 [10]. TIle analysis via the time domain has been. implemeuted in the DIANA.

?rogram and is further descnèed in [1].

.

TIle matrices of °ülébasic time-domain cqua.tions (3.a) for tbe circuit of Fig. 4(20)are obtained using tbe construc..

tion .givenin section I (A.t -E.t)

<;

. -C:z 0 0 0 I -If/tk 0 0 1 I

-<;

<; +C.

-C.

0

0 I -~'k

-'hk

0 0

I

_

o

-C.

C.

0

0 I

0

-c?1k 1

0

o

0

0

Ct

0:

1

0

0

0

o

0

0

0

C, I

~._---

0

1

0

Q.

-If/uc -~tk

0 1 0 I 0 0 0 0 I

o

110

0

00

o

0:

0

0

00

o

Ol

0

0

00

. . 1 1 2

o

o

1

2

-J1

o

191 These equations are equivalent with tbe time-domain equations for tbe multipbase SC networks via topological [5], [10] or numerical [9] procedures. The number of equations in [5], [9], [10] may, however, be smaller. This corresponds to a preliminary elimination of some varia. bles. Clearly any SC network analysis. approach in the time domain must be equivalent with our MNA equations (3.a) tbc same way as for netWorks without switches [2]. Some straightforward eliminations can, without any com-plication, be included in our approach by. using a macro-model, whic4 we call a cornposite branch. A frequently used composite branch is tbat of tbe subnetwork ofFig. 5. By using the basic derivation it would need a 5 X5 matrix and as variables 1,),.1,))'1,)""0/. and qs. Ir node / is not connccted to any other braDch and Ü one is not interested in fJ,and qs this subnetwork has a stamp:

Using the stamps of such .compositë branches the con-struction of the time-domain equations CaD.be rephra.sed.

3 4- 5

o

o

o

o

o

o

o

o

""+IN

(17)

,

F

'- '.-191- IEU nANSAcnONS ON eDtCIJITS AND SYS'1"D(S,VOL c:.u-28. NO. 3. WA1a 1981 . ~ - .. _... ., - . -. -.- . _.. .

'---'"

. - -' - - . .'~;.~;~:."_:.~<":-:~>~: . ~._ _ ~:..':;:" :~._~ - :~:~~:S. t:~~~~~~C_~~~.~~--".~.;~;_~::."<~:'~.-:.:~~~~' ~_~~'~':~~

~.-~c_-: .' .TheTC ~a.r~J~S UDknowu-:vmables, siD:ce DO mtcri1a1 nooe voftàgës-of -á êOri1Pó~te branch' appea.r And SiIÏcc 'so~e branch

~'_'..~~"~::-~~ges- arc -eliminated; In oUt C2.SCwc'-only:havc

S

ëquatioÎlS' --" - -.

L

2

3

~

U .. -1 2. 3 . _.. __<10,--;'-

-..- . ... -- . .-..- ...-C14f11k<PU:-l

+<;

C1<Plk~lk-I-<;

- _ .0 - .. C1~lk91k-l-<; ..<;~lktlk-.1~<; 7C;~k~k-1 +C~_ C;~k~k-l-'C4

"-"_ -_~'. . -' -t

-

0 --'

. -

C;~k<h-;-I-C4.

.

= .--:':c;~k~Ll

- -:...

.

+~

i

---

~-O

- - - :-.- --:-::::

- _

0

- - - - -.--- - -"- - - _

0

--

-

_'1_'

.

". ~.:.-~_.:=': " .-0. -~ . - ~ n_~s ; .0 ::: ~ :=::::::. __'-::_::: _.'~..::-.:=...:-: :';:0..-: _-; ;..~ . ..

::. :':':

~

tlia:rW:ith.cÖmpoSili:~ränëh~îhè-~uf~ Ák and E" for .!he N ph~..and the~kft-and rigm-hàJid side of (3a)

.=.:::~-=:càn:.b~~téré]if. :Exp~ .:S1iC?W:II--:mit=üi~:u,sc of $Ome simp1e com~te brand:1cs leads te' s~ts .of ~quä.tions of

..::~ : ~::sizl:s:co:Ç.ariiblë tó.:u1osè ëf[SJf [91 'f1Ql;Tht. cq~tions in t1lc'z;ao-Ii1.ain1iav(á slzc 3-tUneS-that Qf (19) sincc there are 3

~:'=:: '~?n...~~~~;:~~:~:~.~;~~~~~\/~'-.~;'~~~~~~.~

-- --

_ -:::;< ::=.=:;_.

:=,=<

._,,<-,,:..-."

]

'""'"~-: _...&"'" ."~'-_,,",.-4 ..~_{- ... .} I _.-I_{. C2} ' _..-I _{C2 0 I 0 11} "1'

::=~~

= _.;~ .~;i~~J.,.~.~:.~_:~

:::::<:15

).:.) ~_.~:~(~Cc!L__!<-'~Ç2~~~~!:-~~c..7:Ç~)L(L:::~~ ..::.::o::{. 0 :u .'__ _ ...0 .: ::::. =G;. "':'; ~':(;..-~-:. .~.-=; ',,:,~ :;.:::.;;:.-~:~ _c_" 0-- .--c'z-"c. . :"z-Ic.. .~_ .__0 _____.

'_' __ --

: ...". _ _ _ _ _ _:1. _ _ _ _ _ _ _ _ _ _ _ .1., __ _ _ _ _ _1_._ _ _t :iJ-__ _ _ ._r~ CI+C2c:-. 0 I C1+C2 -C2 0 r -. -- _

-

1 10 11 - ot-~

--- - -

C2 -C2-C)-c. '41 -C2 c2+C)+C. -c.q o. . I 0 I 0 .'Ot . 0 o c. -c.t 0 -ç. ~ 1 :.. : . ': :

-

I I 1 Ot

- - - -

T ~cl:c-;

- - c2- - - ö T- ë,-;~ - - - - --c; - - - - ö - ~I- - - -1- - - =-1-0-- i'

o

I

C2

-

-~ -_c.

~

I :-C2 ~

-

C2+c. -c. I 0 I 0 I 0 0 I 0 --; -c).j.'c. -'41 O. _-c. c)+c. I I I 1 0

- - - ~T- - -.- - - - -- -

~ T

- - - -.- - - -1 - -1 - ï- -

--o -,. .J .. '.- 0 . 0 0 0 0 1 0 0 I . 1_. I I I

- - - -

~

- - - - -

- - -

~

- -.- - - -

- - - -.- - -.- -1 - -1 - -1

--. 0 -1&- I' . - - o' . I I . 0 0- I ~ : . 0 - . = I 0 I 0

.

I 0

- - - -

- --- - .-. - -

-

. . .1 1 I

-.-:__ - __

: I' =I ~0 -:. :

-"

) -. I I I O' I !'" I__J ..0._ '0--1010' I .-

o

.0

o

--.ö---"k+IN' o ---YII 0 V21 C V"

_---

₀ --YI2 ₀

n

V= 0 Vn 0 (20) --- ._--0 VI) 0 Yn

-0 r! 'vn

---0 .. .---t QII UI Q21

---.---

0 ,.., QI2 _u2

r

Q=

---

₀ ----u) :1 Q" Q:o :

Icl I I Cdl! I 193 I/~

Fig.6.

.Computer an.aJysisRSUltsof tbe SC l1etWorXof Mg. 1. (a}-(c) Responses 011impuhe:sin pba.se1.2 and 3. (d}-(e) Frcquency-do1:nam tra::ls!erfunctiOI1S~(w) of oba.scsk-l. 2. a.nd 3. (f) Fr~cy. domai:l tnnsfer functiol1 Ji<3%I(w)cd ali.uing tmI1 ~(w.

w-(2f1IT».

The z-domain transfer mAtrix is tbc inverse of the matrix in (20) and is not included' for brevity. TIle frequency-do~ c:har2.cterisûcs Xk( "', n) (a1i2.sing effcctS and .frequency-domain transfer function H"(,,,» can then be dcrivcd easily from (13)-(16). Usually onc is oDly inter-estcd in the input-output relationship. In our example tbc relation betWeen 11(branch 3) and ~ (node 3) is desircd. Hencc we need thc entries HW>(z) with k, /-1,2,3 which relate Ut' U2,U3to JIj., Jlj2' and VJ3'These can be

com-bincd in (13) to obtain X~32>(

"'.n) withk- 1.2,3.

.

. .Tbc pnética1 computer implementation of tbc time and

fr'equency responses is now bric;f1ydescribed, and applied . on our cxample.

COmpulalionof the Piecewise-ConstantResponseto a Piece-wise-Constant Voltage beitanon IIJ/.lljJ'IIjJ in

Branch j

I) Set up tbc equations (3a) and use as many compositc branches as poSSlole.

2) For cach phase reorder' thc equations and fmd tbc LU decomposition (sce [2D.

3) Start wîth zero iDitial conditions and solve (3a) with sparse matriX techDiques for time slot 1.2,3.

. . '.

Each time the right-hand side of (3a) iS obtained from tbc previous time slot.

This procedure is implemented as an option in tbc DIANA program. lt generates for tbc C:.rcuitof Fig. 4 tbc. responses given in Fig. 6(a.,b. c) on impulses in phase 1,2,

and 3, reSpec'tively.

.

Computalion of X~/. J>(CoJ,!2)or mi. j)( "') k =- I, . . . , N ~ia

the Time Domain (3.a)

I) Apply a unit voltage excitaûon in branch j during phase k-I,2,..., N and compute for each k tbc piece-wise-<:onstant voltage resPonse X~j) at tbc node i as descnoed above.

2) Decompose each X~j) into N signals h<1f(mT) and apply tbc discrete Fourier transform on these N2 signals in order te obtain H~~j)(ejCT)for k, /-1,2, N. .

3) Combine the rcsu1ts as in (13) or (16). Remembcr from tbc z-transform [6] tha.t

H~~j)(cc)-h<1k1(O). (21) Observe here that the number of resonses to be computed in [I} can be reduccd using tbc -adjoint'SC netWork [8}Ü only one value of k iS of interest. By applying this ap-proach on our examplc of Fig. 4 we obtain thc plots for tbc frequcncy-domam transfer function and thc a1i2.sing

effectas givenin Fig. 6.

.

-Computalionof X~,.j)("" a) k-l,2,..., N via the z-. Domain (7)

1) Set up tbc equatiOJl (7) using the stamps..

2) For each value of a of interest set z_eJCT and' ~/-eJQtI.I. /-1,2,"', N and

~

other inputs zero. Solve (7) for tbese values using sparse matrix tcclmiques in order te obtain

N

. L ejCtl.'H~~J1(ejCT),

1-1

fork-I,2,...,

N.

3) Solve (7) for z...cc, ~/=ÖJ:I with /-I,2,...,N. Combine tbe results as in (13) or (15).

Let us now compare the two techDiques. Oearly (3a) is an N times smaller set of equations tban (7) but it has to bc solved more often. Also tbc time-domain techDiquc requires some FITs which can bc time consuming and can be inaccurate for very selcctive mters. en the other hand, tbc time-domain techmque allows a. mixed mode simulation of SC circuits along with tbc digital control circuitry as implemented in DIANA [I}. [11}.

Moreover, it allows a smdy of the nonidca1 cffects of

,

-0

194 lEE!! T1I»ISAcnONS 0101CDlCUJTS ANC SYS'1'ZMS.VOL. CAS-2!. NO. 3. MAAOI 1981

sc c'.rcuits: offs~t and leakä:gëCiI:rttrn-drift; l'l"Onlineu--which-bas--onlycontrib~tions a.t,tPe pulsa.tions Co)-O+nCo)8' distortion. and doek feedthrough [1], [11]. In order to derive Vk(Co)we define a bloek wavc!orm

Tbc compuutions presented in [3]-[5], [7], [9], [10] all Sk(t) which is 1 in ~.t and zero outside. Using Theorem 2 use tbc z-dom2.Ïnapproach and in scme form or anotber and (9) we have

rcq~ to solve sets of equaûons equivalent to (7). Some + co

.

efficiency can be obtained by-making \!Sc Gf tbc strUëlill'"e Vk(Ü..-:':"::2:-' -""'k.,N(l-}oW

of tbc problem in [5], [9]. lt is bowever believed tbat tbc 1-

-

cÖ - . - ---, ' for.:nulation (7) allows a clcar view on-alllSSUesinvolved _. + co + co

in using tbe :-domain tccbDique. Furiber ide3$ in this .

-

2:

S.t+IN(t)Ck'+I1" +

~

Sk+IN(t)C.(t)

~:_ . _{tl b} " . _cd.

1-- co

1-- co

. I.:liectlon are curren y emg mvestlgat

-.

'.

Concm:üng otber useful désigu-cliaractëriSûcs of SC-=-~-. --~!'k(I).rk(I)+HkA:(O:~ (e.g., noisc performance, sensiûvity, group delay) we just . -. .. + co - . '~_....

I:entioD. bere tbat tbe adjoint SC netWork allows [8]

.

2: J'k+IN(t)(eJOT -e)QI.., I)

compuuûona! savings whicb are even, ~~~e_impt.essive 1- -00 .

than for RLC ciI-cuits. - -

~

wbere rk(t)-1 for Ik -lk+1 <t <:0. From (A.2) we bave

, ,- '_, '_ lönhe-fust-t~rm -cf tbc right-!!~ side of (A3):

V. CoNCLUSIONS.. .' ~ N

It is shown tbat tbc widely Uscd_MNATCëb:Dique can be 5t~t-).'r-it)} -2'ITak(Co) ~eJOT)

· e.2.SÜyaQ.2.pted to analyze SC netWorkS. It provides us ~so ...: -'. -:: :0: : .. .-

.

I:-!:

;itb

~

genen! e~uivalent cü:~t ~~_~d?i~0.~~fi~s{g1iiin.:~~_.~~~~.: ::- -/)(O;,.,;...wi;.~}-.~+oo __

~e S1g:W bandling capabilines o(~~~etWorks.-Tbe:z~ ".. ~: =-..:..-:::-e.--=. - _. _:..

~

ö(Co) 0 nCo)8)' (A.4)

domam transfer matrix turns out to bë a kcy tooI in 11--co . ....

obÏäliilir2f~~aracteristics 'andis in-fa~ a:.lD'brid..-Tbc fO\!per transform of tbc second term in (A.3) c:anbe

~

~f

ihe equ19ä1ent. circuit:.:Two:pi11çti~'~~\11a:.~~ ob~~~~. ~y:~d'~èchDiqüës. '2:$0: :: :- ~-o :':':-~~: :-=.:.

tiqns-'óf~cSe:r-rèq~ncy.: cbära.ctenstics..:~~~'\'~f~~m.:

t

~~-.~.~.: ~- ~:.:-:".:o_= :"

1

-':':: :.:"::.o~-o.::~. ::. ..::.~:: 0": ::::-- .~ tbfTêSûft!;:~&=~asëd..On;a:'tepea.1c.d.~lUJion:~.(~):.Û1:~~ ~~.':~!s~~-;;nr _:"~"~_"_ . o:::-o::~,'= ~'~"" .-.;. .

tbe:~ciómàiJi:~

iiS'impmé:irtatibn~~DJANAbas-. .J...;"-;'--- -:- '-: - -

~ '-='. oU:.'::- . :"'1. ?:=-::::.:~:-°2lfOw~à1yzö~cc~ully- -man~~c.J;Ït~~m:!.ï;Ik: ., -:::. ..:: :.o.:~: -::-::: .:; ''::'-"-:+00 0

~~:'öt11ër:j$-'öä.Scd ön:-.tbe:SölutiOn:::of:.ro~rOl'~~ -2'ITak(Co)-O)el(~:-W)I.:-~

~

.}(If-O-nCo),)

(A~l

eXëÏtittöm:ind~::tCJ:bc.=-:~ii:îpUtatiQna11y~~~;. _'. - 0 -,._~_ ,".. -,)

'~1Y~j;

~; _~:'~-,-::"~ :. '~: é:~~_::'

~

"

0::'

~

an~'::~~~o~;;~~:;, ::~:

::"~'_:;:C~~~: ,',~':~";::

~

.__ 00-- - - ÁPPENDIX-.. ..0.

~

{ -~! (t)eJ01..' I

}

'-,' -' -- - - '_ .

::.: :~~:_~~~'_~__-:::hO~F OF THEO~-~-; -. ~o~~ ~.~:~~~~_~1~~~_ =~/~~~. .-~ .-", ~~~ _-:-~-:- ~~. ~-/:~ ,~~:

Wë ëaIc:\ime 1mt tbe spectrum of tbe response for a ~ '

-C

' ~ J(Q-W)I ~ '.' ',: --; _._~;

~i~:.excitaûC?Il: lI(t)=.U~~:. _~t :ok{t) 1?e_~e: r~- ,==2~a.t -"J,e. -. ..1 ,.-~oo ö(Co)-O-nCo)8)' (A.6J.

5pQp.se~p'led at tbe end of ilie time s~ots-~k'-~k~N"'" ;.;" Remembcr that Hkl(z) is tbc z-transfonn ot" hk~(mT), m=O, 1 ! tben w~ have in general from (8) for this

$.~oJ~ ëXóta.ûOD. -: :::-: : :=: =. '= (A.3) _. =-..co _. _. ~(Ï)~ :":o~ ~;S(t':"'nT-tk+1) - - . - - ~ --- ~: ',,:,~

TIiis implies tbat tbc spectrum y~(Co) for tbe m;Î1soida.1

exci~û~~ t:'~t)~ UeJo1is _ _ _ ' _ _ __ _ _ _ ' .. - -'.- -. . ~- .-,,- " _ - '. + co _' _ .. , ." .::-:. -. ~(~)--xk(~,n)U2~

_-

_.

_.

-~-.-8.~~-;'Q~~~~~j:_ (A.7)~ 11- -CG . - . .

.

-- ----

--.--'.

:

:~-:--.~

-.

1

-~ l_ ~_h~~(mT)UeJOt("-"')T~II"I;

)]

: . :..~. .' {-l.\m-O_

-(

--~

"--00

.Ö(I-nT-th'1~ejO"T

)(

~1

~

ejOII-IHk/(eJOT)U

)

.

(A.I)

-

Tbc Îincarity óf tbe SC network tben implics that tbc

response to an arbitrary input

1 +00

.

1I(1)--

_f

U(O)ejOldo,

2'IT -00

bas a spcctrum

+co +00

yk(Co)

J

_-co Xk(Co),J2)U(S2)

2:

ö(Co)-S2-nCo)8)d1'2.

. II--CO

This implies (12).

o

T!le Fourier tIansÏorm of tbis sinusoidally modulated ;)Ulse train is +oc Vè(w)= 2: Co)sö(w-S2-nCo)s) N

.2:

ej('Jt'.I-w'..')j{..t/(ejnT)U (A.2) 1-1 ACJQ.lO'WLEDGMENT

Tbc autbors appreciate tbc improvements suggested by L. Claesen and tbc reviewcrs and tbe ÎD.tercsûngdiscus-sions with Y. P. Tsividis and F. Brglez.

~~CES

H. De Man. J: Rabacy, G. Amout. and 1. Vandewalle. MPrac:::cal imple=tatio~ of &gencra1çompl.ltcr aidcd design tcc:h.tliqucfor switcl1cd Qpacitor circuitS." IEEE J. Solid,SlJUe Circ-Jiu, voL SC-15, pp. 190-200, Apr. 1980.

C.W. Ho, A. Ruch1i. anei P. Br=. "'The oO<illicd:od.d ap-proa.ch 10 netWork analysis," IEEE Tuuu. CÏTaliu Syzt.. voL CAS-22. pp. 504-509, June 1975.

M. 1.. Liou &ndY. 1.. Kuo, ME::uctanaJ)"SÏsoCswitchcd c:a.pa.citor circuia wiLb arbiuuy inputs." IEEE TTtJM.CimJiu Syzt.. voL CAS-26, pp. 213-213. April 1979.

C. F. Kp.nh and G. S. MoschytZ, '"Two-pon analysis oCswitchcd-capa.c:ilOrnetWork using lour-pon cquival=t circuia in tbe :. c1omain."IEEE TrtJnS.CimJiu Syzt.. voL CAS-26, pp. 166-180, Mar. 1979.

Y. P. Tsividis, MAnalysis of switchcd ca.paci:ivenetWorks," IEEE

TrtJnS.Orasiu SY3r..voL CAS-26. pp. 935-947, Nov. 1979.

:EoL Jury, TMory anti .ÁppüCDlionof Ik :-Tran.iomr .'rfe<.Jwd. New Yodc: WiJcy, 1964.

Y. I- Kuo, M. I- Liou, and 1. W. }(~~...h. '"Equivalent circuit çproach te tbe cc:puter-aided analysis of switched c:a.pa.c:itor c:irc:uiu."IEEE TTtJM.Ciradu Syzt.. vol. CA5-26, pp. 708-714,

Sept. 1979. .

1.

Vandcwane, H. De Mm and 1. 1W:Iacy,11Ie a.djointswitcbcd ca.pacitor netWork anei ia applica.tions," in Prot:. IEEE Symp.

CJmtiu anti SY3Unu, pp. 1031-1034, (HoustO!!.1'X). 1980.

F. Brglc:z."'SCOP-A switèed-capacitOr optimi:z.atienprogram." in Proc. IEEE Symp. Cirt:-Jiuanti Syszenu, pp. 985-988, 1980.

q. C. Fang &nd Y. P. Tsividis, MMocillicdnodal analysis wiû

Î::Ij)rovednumcrical meLbodsfor switcl1edcz.pa.citivenetWorXs."in

Prot:. IEEE Symp. Cirt:-Jiuand SY3Zoru,pp. 977-980 (Houston,

'IX). 1980.

Ho De Man. 1. lUbacy, G. Amout, mei 1. Vandcwane, '"DIANA as a. mixed mode simulatOrfor MOSLSI sa.mpled-d:tta circuitS." in Proc. IEEE Symp. Cirt:-Jiuand SY3Zenu,pp. 435-438, (HoustoD., 'IX). 1980.

+

J0C8 VIIDdewaDe (S71-M'77) was bom in

Kortriji:. Bclgium, on August 31. 194. He ~ =ived tbc :Eo:Eodegree and tbc Ph.D. d.egr= in applied science:s fn:m tbc: ~Lboliw UIIi-vcrsitcit !.#uvc:n, Heverlee:. Bclgium. in 1971 mei

1976, r:spccüvc:!y.

From 1m te 1976 he was Assistant a.t tbe E...S.A.T. uboratory, ~tholiw Universiteit !.#uv=. From 1976 te 1978 he was a Rcse:arr:h A»oc:iate mei Crom 1u1y 1978 10 Ju1y 1979 he WIS Vtsiáng A.sSÏstant Professor a.t tbc: Univcr-sity oe Ca1i!onIia. Bcrlcclcy. Sincc: July 1979 he is ba.ck witb tbc E..S.A.T.

195 La.horatery of tbc ~Lbolicke Universiteit LcuvCD.whc:rehe is Assistant !':ofcssor. His research intcrcsu are mainly in maLbematic:a.1sys1Cm tbeery md ia appliQtions in c:im1it tbeary anei çontroL

+

Hugo J. De Man WISbont in Boom. Belgium. on September 19, 1940. He recc:ivedLbcclcctri-cal =gincc:ing degrcc and tbe Ph.D. dcgrcc in applied sciencc:Irom tbe ~tbolieke Universiteit Lcl.lve!!.Heverlee, Bclgium. in 1~ anti 1968, respccûvcly.

In 1968 he became a. mc:mberol tbe sta!f oe tbe I.aboratory CorPhysics anei Electronics ol Scmiccuductoa a.t tbc: Univ=ty ol !.#uvc:n,

worldng CDint.cgrated circuit tcc!mology. From 1969 10 1971hc: was at tbc:Elcctronic Rcscuch ubontOry, University of Cali!ornia.,Bc:kclcy, IS a.nESRo-NASA Post Doct.oral Research Fenow, working en ccmputer.aided dcvicc:anti cir-c:uit design. In 1971 he returned te tbc University oC !.#uveu as I. Rcsc:a.n:hAssoc:iateoCtbc NFWO (Belgian NationaJ Sc:iencc:Fo\Cda-tion). In 1974he becamc:a. Professor at LbeUniversity oe LcuvCD.Dunne tbe winter quancr oC1974-1975 he WISa Vuiting A»oc:iate Prolessor at tbc Unive:$ÏtyoCCali!ornia.,Bc:rkclcy.H3s ac:ual field oCrcsc:a:chis tbc desig::tol intt:grat.edcircuits mei ccmput.eNided design.

+

J&DRaba.e1was bont in Vc:umc:,Bclsium. CD August 15, 1955. He reccivcd tbc: EL dcgrcc Crem tbe K.a.tboliekc: Univc:niteit Leuven, Heverlee, Belgium. in 1978. He is woridng tQ. wuds tbc:Ph.D. d.egrccen tbe tbeery anei appli-ca.tion ol swiu:.hedcapacitOr mtcrs in high qua.1-ity aUdio systems.

In 1978he obtaincd m LW.O.N.I- lcllowship which aJlows him te work as a Rcsc:archAssis-tant at tbe La.boratOry E...S.A.T..~tboliekc Universiteit Leuv=. In 1980,he obtaÛled a fcl-10wshipas N.F.W. O.-a.spirant. .' t ...--,u u ... ... [1) [2] (3] [4) [51 [6] (7)

..

(8) (9) [10} [11]