Analysis of general switched-capacitor networksusing indefinite admittance matrix

E. Hokenek and Prof. G.S. Moschytz

Indexing terms: Admittance matrices, Networks, Switched-capacitor filters

Abstract: In a previous publication it was shown how switched-capacitor (s.c.) networks can be analysedusing nodal-charge equations. The result was a description of s.c. networks as time-variant sampled-data net-works that led to a 4-port equivalent-circuit representation in the Z-domain. The 4-port representation hasbeen expanded by considering six basic building blocks for the design of a general passive or active s.c. net-work. In this paper, an indefinite admittance matrix (i.a.m.) method for the analysis of s.c. networks is pro-posed which uses the 4-terminal admittance matrix of a capacitor in the s.c. network. An algorithm andexamples of the proposed method are given.

1 Introduction

Recent advances in m.o.s. integrated-circuit technology andin the implementation of analogue m.o.s. sampled-datafilters, commonly called switched-capacitor filters, haveestablished a need for an algorithm that can perform theexact analysis of such networks from a given nodaldescription.1'3 Several analysis techniques so far evolvedare either too complicated or are restricted by the net-work topology; for example, by the need for a sample-and-hold circuit at the input. Other exact analysis methods,using the conservation of charge principle, are useful forrelatively simple networks, but become cumbersome withincreasing network complexity.

In References 1 and 2, s.c. networks were analysed usingnodal-charge equations. This made possible a closed-formdescription of these networks as time-variant sampled-datanetworks. Physically, the topology of an s.c. network ischanged between two states by operating the switches attwo (even and odd) switching times. It was shown thatany s.c. network can be represented by a 4-port obtainedby transformation of the time-variant network equationsinto the Z-domain. In this 4-port, even and odd time slotscorrespond to an input/output combination. Furthermore,an analysis method was presented that yields the corre-sponding 4-port equations.

In this paper, a method for the direct analysis of s.c. net-works is proposed which is based on the method presentedin References 1 and 2. Using the indefinite admittancematrix (i.a.m.), this method enables an analytical comput-ation of s.c. networks to be made directly without consider-ing the equivalent 4-port circuits. Similarly, as in traditionalnetwork theory, the i.a.m. of s.c. networks can be simplifiedand the corresponding network functions in z evaluated.The algorithm used is based on the matrix analysis of con-strained networks.8

it easier for the reader to understand the extension of thismethod to the use of the i.a.m. presented later.

The relationship between the current and the voltage ofa capacitor in a continuous system is defined by

= Cdvjt)

dt (0

Now consider the capacitor as a part of a discrete-time sys-tem. Assuming an instantaneous surge-like charging of thecapacitors, it can be assumed that the voltage on thecapacitor remains constant during the sampling period r.Therefore, the continuous time derivative in eqn. 1 at atime instant t — nT can be replaced by a finite difference,namely

where Av = v[nr] — v[(n — l ) r ] and At — r.Writing n instead of m in eqn. 2, we obtain

(2)

i{ri) = Cv{ri) — v{n —

(3)

Let us now. consider a capacitor 'embedded' in an s.c. net-work, as shown in Fig. 1. The sampling period of this net-work is the sum of two nonoverlapping clock phases, one ofwhich is defined as 'even', the other as 'odd'. The switchesin the network which are closed in the even clock phase aredenoted as 'even' switches (Se), the others as 'odd' switches(S°). By operating the switches in two clock phases, thetopology of the s.c. network is changed between two states;these are designated as 'even' and 'odd', depending on theclock phase n. The electrical signals occurring during evenclock phases are signified as even (e.g. ie, ve) and odd (e.g.i°, if), respectively. Using these designations for even and

2 S.C. network analysis

In what follows, a brief description is given of the analysisapproach reported in detail elsewhere.1'2'4 This will make

Paper 539G, first received 18th June and in revised form 16thNovember 1979Dr. Hosticka and Prof. Moschytz are with the Institut furFernmeidetechnik, ETH-Zentrum, CH-8092 Zurich, Switzerland

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

ve(n) ie(n)

S v (n-1) S i (n-1)

Fig. 1 Capacitor embedded in s.c. network

21

0143-7089/80/010021 +13 $01-50/0

odd clock phases in eqn. 3, the even and odd currents canbe described as follows:

ie(n) = Cve(n)-if(n-\)

/°(n-l) = Cif3(n-\)-ve(n~2)

(4a)

(4b)

Using the z-transformation, eqns. 4a and 4b can berewritten as follows:

Ie(z\ — — iVe(z\—z~xV°(zVt

I°(z) = ~{V°(z)-z-lVe(z)}T

or in the form of an admittance matrix:

7e \ (Ge G°\(V(

7 ~ \G° C

where

C CGe = - and G° = -z~l -

T T

(5a)

(5 b)

(6a)

(6b)

Eqn. 6 corresponds to a 2-port representation of a capacitorin an s.c. network. The 2-port links the even and odd signalpaths corresponding to the even and odd states of the net-work; it is therefore called a 'link two-port', or l.t.p.2

3 4-terminal admittance matrix of a capacitor in s.c.networks

If an s.c. network originally has n nodes, then, using theanalysis introduced in Section 2, the number of nodes inthe analogue 4-port is 2n. Assuming that the nodes 1,2,. . . , n are in the even-phase path, and n + 1, n + 2, . . . ,2« in the odd-phase path of the analogue 4-port, then thetwo paths are symmetrical with respect to the nodes. Henceevery capacitor between the k and m nodes in the originals.c. network is transformed into an l.t.p. between the nodepairs (k,m) and (n + k, n + m) in the analogue 4-port.This is shown in Fig. 2. As a consequence, the 2-portvoltages in eqn. 6 can be expressed by the node voltages inthe analogue 4-port, namely

(la)

(1b)

Ve = Vk-Vn

v° = v +fe —where

4 ~ -*m

Thus the 4-terminal admittance matrix, obtained from the2-port admittance matrix of eqn. 6, is given by

= - /n+m (1c)

This is the general 4-terminal admittance-matrix represent-ation of a capacitor in an s.c. network. The l.t.p. circuitcorresponding to the [Yc] matrix in eqn. 8 is shown inFig. 3. This l.t.p. circuit is equivalent to the l.t.p.s presentedin Reference 2. Note that the l.t.p. contains only fournodes; these correspond to the four terminals of the 2-port.Thus the 4-terminal admittance matrix [Yc] of eqn. 8 isequivalent to the indefinite admittance matrix of an l.t.p.Accordingly, such features as the zero-sum matrix and theequality of all first cofactors8 are, of course, present.

4 Indefinite admittance matrix of active s.c. networks

When two terminals of a multipole network are switchedtogether, the corresponding operation in the indefiniteadmittance matrix of the multipole is called node (or pole)

k Ve m

vo odd path

s.c. networkT= sampling period

Fig. 2 Transformation of capacitors into linked 2-ports

n-k

m I

Fig. 3 Linked 2-port circuit corresponding to [ Yc] of eqn. 8

(h \

m

k

' Ge

Gl

-G(

m

-Gl

Gl

G'

-G<

n + m

G°\

G°

Ge

Ge I

- [Yc]

'Vk \

(8)

22 IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

contraction.8 Consider, for example, the 4-pole networkshown in Fig. 4c. The corresponding i.a.m. is given by

1 2 3 4

[Y] =u yn 723 y™

*\ yyi y^ 734

\ V41 742 743 744)

1

(9)

where the numbers on the side of the matrix refer to thenode designations. Switching the two. nodes 3 and 4together, as shown in Fig. 46, we have the following twonew terminal conditions:

V3 = K4 = (10a)

(106)

As a consequence of eqns. 10a and 106, the columns 3 and4, and the rows 3 and 4, respectively, are added. The result-ing contracted matrix [Y]' therefore results as

[Y]' =

1 2 3'1

; l t 7l2 7l3 + 7l4

;21 722 723+724

+ 741 732+742 733+734 + 743+744

1

2

3'

(11)Note that the contracted matrix is of a lower order by onethan the original. Had m pole pairs been switched together,the order of the corresponding contracted i.a.m. wouldhave been reduced by m.

Consider now an n-node s.c. network. The equivalent 4-port network has 2n nodes, where the first n nodes in theeven path are symmetrical to the n + 1 to 2n nodes in theodd path. Consequently, the i.a.m. of the 4-port network isof order 2n. Activating an 'even' switch Se betweenterminals i and / connects together terminals i and / in theeven path of the equivalent 4-port network, as shown inFig. 5a. This results in the contraction of nodes / and / inthe corresponding i.a.m. By contrast, activating an 'odd'switch S° between terminals i and / connects togetherterminals n + i and n + j in the odd path of the 4-port net-work (see Fig. 56), resulting in the contraction of nodesn + i and n+j in the corresponding i.a.m. As a conse-quence, the algorithm necessary for the i.a.m. analysis of ans.c. network can proceed as follows:

(a) Considering only the capacitors, set up the i.a.m.[Y] using a [Yc] matrix as given in eqn. 8 for each capa-

Fig. 4 Multipole network

a 4-poleb 3-pole contraction of a

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

citor. For an s.c. network with n nodes, the i.a.m. will be oforder 2n. If a capacitor is connected between terminals kand m in the s.c. network, the elements of the [Yc] matrixare ordered accordingly in the k,m, n + k and n + m rowsand columns of the i.a.m.

(6) Assuming me even switches Se and mo odd switchesS°, [Y] is contracted to [Y]'; the order of [Y]' is 2n -(me + mo). A switch Se between terminals / and/ results inthe contraction of rows and columns / and / ; a switch S°between terminals / and / in the contraction of rows andcolumns n + i and n + j .

(c) All further steps follow exactly those of conventionali.a.m. analysis, whereby the special nature of the equivalent4-port s.c. network must be kept in mind. Essentially, thismeans taking the symmetry of the even and odd networkpath into account. Thus, for example, if the contractedi.a.m. [Y]' is to be converted to a definite admittancematrix (d.a.m.) by grounding terminal /, then conventionali.a.m. theory requires the elimination of row and column/, whereas our 4-port theory requires the elimination of thetwo rows and columns/ and n + / . This is obvious, since thenetwork is to be grounded both in the even and the oddpath. The order of the corresponding contracted d.a.m.,which we denote by [y]), is then reduced by an additionaltwo, i.e. in place of the order given under (6) we now have2(n— l) — (me + mo).

At this point an example should be instructive. Considerthe s.c. network shown in Fig. 6. The corresponding [Y]matrix is given by

s.c. network11

even

odd

path

path

I1

s c. network1

l

even

odd

»_j ,

path

path

J

n.j

Fig. 5 n-node s.c. network

a Effect of S between terminals i, jb Effect of S° between terminals /, j

Fig. 6 Passive s.c. network corresponding to eqn. 12

23

m =

1

2

3

4

5

6

7

8

1

Gt0

0

-Gt

Gf

0

0

2

0

Gf0

0CO

G3

0

c°

3

0

0

Gf

-Gf

0

0

/"* O

CO

— G2

4

-Gf

-Gf

/^o

C 0

i G° ';=1 1

5

Gf

0

0

Gt

0

0

-Gf

6

0

G3

0

c°

0

Gf

0

-Gf

7

0

0

G\

~G2

0

0

G|

-Gl

Note the symmetry, which is characteristic of the i.a.m. ofa passive network. Denoting the four quadrants of [Y] by

m =n.n M Jn,n + 1

1-* Jn+l ,n L-* Jn+l ,n + l

(13)

where the symmetry implied by the arrows in eqn. 13indicates that

[Y]n,n = [Y]n + 1,n + l (14a)

[Y]n,n + 1 = [Y]n+1,n (146)

Each of the submatrices has the structure characteristic ofan i.a.m., i.e. a positive diagonal with the sum of admit-tances connected to node / (which we shall denote by G,-,)and negative elements corresponding to admittances con-nected between nodes / and / (which we shall denote byGu and Gj£, respectively). Thus the form of the submatricesin eqn. 13 is as follows:

4 n

r22

r33

r44 .

05a)

[Y]' = 4

(12)

and

4 n

1

2

= 3

4

r22COGu

'33

r44

(15b)

Note that eqn. 15b can be derived directly from eqn. 15aby changing each Ge to G°. In fact, since

(16)G° = -z~lGe

(see eqn. 6b), we need derive only the submatrix [Y]nn ineqn. 13, and then obtain the complete matrix [Y] asfollows:

m =[Y]nn

[Ge]nn

[Y)nn

(17)

1

2

4

5

6

7

1

ce • ceGi -r-G2

0

-(Gf + Gf)

CO

Gi

0CO

G2

2

0

Gl

-Gl

0CO

G3

0

Following step (b) above, we now contract [Y] to take theswitches Se and S° into account. The former, which is con-nected between terminals 1 and 3, requires contraction ofrows and columns 1 and 3. We obtain

G2°)

(18)

Lr2

5

Gf

0

G°

Gt

0

0

Gt

6

0CO

G3

c°

0

Gl0

-Gf

7CO

0

-G 2 °

0

0

Gf

-Gl

8

— (G?+(CO^3

-Gt

-Gl

-Gl3

1=1

24 IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

where me (the number of even switches) is equal to one inthis example. Note that the contraction for the evenswitches affects only the three submatrices [^]n,n>[^]«,n+i ancJ ITL+i.n- Contracting for the mo oddswitches S°, which affects all but submatrix [ y ] n n , thenresults in

M m e m o = [Y\' =

1

2 Geii=i

0

- . 2 Gei1

Gf

G2

2

2

0

Gl

-Gl

0

— G3

42

i = l '

-Gl3

1=1 '

~~^i

3

1=1

5

Gf

0

-Gf

Gf

0

6

G?

G3°3

1=2

0

3 c

8

2

2 Gi=i '

-Gf3

1=2

1 G\1=1

(19)

Notice that, after each contraction step (i.e. in eqns. 18 and19), the zero-sum-matrrx feature must still be valid, since norows or columns are actually removed from [Y]. This is auseful test for possible errors during the contraction process.Finally, grounding the network of Fig. 6, say at terminal 4,we must remove rows and columns 4 and 8, which results inthe following d.a.m.:

btt =

Ox + Lr:

0

Gf

G°2

2

0

G 3

0

G3°

0

Gf

0

(20)

Naturally, the zero-sum-matrix condition is now no longervalid. Notice that the order of [y]'4 (which is four) corre-sponds to our formula 2(n — l)— (me + mo), since thenumber of nodes n = 4 and me — mo = 1.

So far, we have considered only passive s.c. networks. Ifactive elements are present, they appear twice in the equiv-alent 4-port network,2 namely once in the even and once inthe odd path. If operational amplifiers are used, the con-tracted admittance matrix [Y]'(grounded or not) can befurther reduced by the rules of constrained networks due toNathan.8 Thus, for example, the presence either of a finite-gain, an infinite-gain or a differential-input amplifier willdecrease the order of |T]'by two; that is, by the reductionof a row and column in both the even and the odd networkpaths, respectively. The final order of [Y]', after v ampli-fiers have been taken into account, will therefore be2(n —v) — Qne + mo). If the network has been grounded,the order of the corresponding d.a.m. will be 2{n —v—\) —(me + mo). Having reached this point of matrix reduction,conventional matrix techniques modified for use withequivalent 4-port networks can be used to obtain therequired network functions. This is briefly described inwhat follows.

5 Network functions of active s.c. networks

One important reason for using the i.a.m. for the analysis ofs.c. networks is that any network immittance or transferfunction can be readily derived, once the i.a.m. has beenobtained. Since the latter can be obtained very easily fors.c. networks, as has been shown above, it may be presumedthat i.a.m. theory will become a very useful tool for theanalysis of s.c. networks.

5.1 Transfer impedance functions

Consider the multipole network shown in Fig. 7. It can beshown8 that the transfer impedance Ztr = Vkm\li} (i.e. thevoltage drop across terminals k and m caused by the currentsource /,;) is given by

tr -km _ Vu

— m)km

(21)

where Y) and Y%m are the first and second cofactors of thei.a.m. [Y], respectively, and

sgn.y =+ 1

- 1

y>0

y<0

multipolenetwork

km

Fig. 7 Multipole network

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980 25

Because of the equicofactor property of the i.a.m., anyother cofactor Y% can be used instead of YJ in the denomi-nator of eqn. 21. At this point, it is important to note that,in the second cofactor Yxlm in eqn. 21, row/ is deleted,becausestraint

the current /,- is already determined by the con-

h — hj — lj (22)

Furthermore, column m is deleted because the voltage atterminal m is used as a reference with respect to the voltageat terminal k. The remaining rows and columns in bothcofactors of eqn. 21 are deleted when Vk is computed as afunction of/,- by the use of Cramer's rule.8

Assume now that the multipole network in Fig. 7 is ans.c. network. The corresponding 4-port network is thenshown in Fig. 8a, where only the terminals of interest areshown. For the sake of simplicity, capital letters are used todenote the odd terminals of the 4-port, i.e.

n + i = I n + k = K

n + m = M(23)

The two designations will be used interchangeably in whatfollows, depending on which appears advantageous to thereader.

In the general s.c. 4-port network, each network func-tion occurs fourfold.2'14 Thus, for example, instead of onetransfer impedance, as in eqn. 21, there will now be four, asindicated by Zee, Zeo, Zoe and Z°° in Fig. 8a. To computethese transfer impedances, we note that the constraint ofeqn. 22 now implies two constraints, namely

h = hi = —Ij

and (24)

deleted. It readily follows that the four transfer impedancesof Fig. 8c are given by

*h

Z eo _tr -

yijJf. .-. f1 x l kmM

= sgn(z-/)sgn(*-/tt)—yj-

= sgn(i-j)s&\(K-M)

{25a)

KMm

oe _ vkmZ oetr -

JIJ= sgn(/— — m)

//,=o

/,-,=o

yjJYJJ

(25b)

(25c)

(25d)

Thus, for the s.c. network in Fig. 6, we obtain from eqn. 19the following expressions:

= y* = sgn(l-4)sgn(2-4)8

248Vl48/ ;

7/48' 4 8 D(z)

(26a)

eo = - 2 . = Sgn(l-4)sgn(6-8)

where

D(z) = (1 -z~2

1486484848 D(z)

(26b)

il -z~2)+ C2[Ct + C3 + C3]}(26c)

f° and Zfr do not exist for this circuit.

Likewise, both terminals m and M will serve as referencenodes for the voltages Vkm and VKM, respectively. Thus,wherever row / was deleted in eqn. 21 because of the con-straint of eqn 22, both rows / and J will be deleted in the4-port network. Similarly, instead of deleting only columnm, as in eqn. 21, both columns m and M must now be

: I ,

vkm«m

KM

KM

°M

Fig. 8 4-port network corresponding to Fig. 7

a Transfer impedancesb Voltage transfer functions

26

5.2 Driving-point-impedance functions

The driving-point impedance Zdp of a general multipole net-work can be obtained directly from eqn. 21 by letting i = kand/ = m. Thus

V-— l lT

Hi

vj* j

(27)

In our 4-port s.c. network, the two driving-point impedancesalso follow directly from the expressions given in eqn. 25;namely, at the 'even' port we obtain

(28a)

(28b)

(29a)

and at the 'odd' port

hoFor the circuit of Fig. 6, we obtain

Zee 'Itdp - "~ -

\tl*\4

y4S' 4S D(z)

and

7 ° o — LJI — J 584 'Ldp — r — T , 4 8 —

V5S4/ ;

48

T\C2 + \C3 +

(29b)IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

5.3 Voltage transfer functions

The voltage transfer function of the general multipole inFig. 7 can be obtained by dividing the transfer impedanceZtr in eqn. 21 by the driving-point impedance Zdp in eqn.27, as follows:

tin — = sgn (m - n) sgn (i - j) —^ (30)1U

To obtain the corresponding voltage transfer functions forthe s.c. 4-port network, we consider Fig. 8b. Notice thatany one of the four transfer functions now occurring areobtained by driving one terminal pair from a voltage sourcewhile short-circuiting the other.14 This is apparent from thedescription of the network by the voltage transfer matrix

VKM

where

Vr

u

km

Hee

Heo

V/j=0

eo _ VKM

(31*)

Heo =

riOe _ ' km V,KM

Vjj=0(3\b)

Remember that short-circuiting two terminals / and /' in amultipole corresponds to reduction of the i.a.m. by con-traction. Hence, before considering the 4-port equivalentof eqn. 30, we must contract the i.a.m.; by row and column/ and J when driving from the 'even' terminal pair / and /and, conversely, by row and column / and / when drivingfrom the 'odd' terminal pair / and J. Indicating this newcontraction (which has nothing to do with the contractiondue to the switches as described earlier, and which ispresumed already carried out) by a ~, we obtain from eqns.30 and 31 by contraction of terminals / and J in [Y]:

02a)

= sgn(i -/) sgn(K -M)

and, by contraction of terminals / and /,

jjoe 'km

yijjKMm

(32b)

(32c)

KM

Vn=0

(326?)

For our circuit in Fig. 6, we obtain, for example,

Ree = IT = ^ C 1 ~4)sgn(2-4)148

l+(C3/C2)( l -z-2)

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

(33a)

y

l+(C3/C2)(l-z-2)and

Hoe =

(33b)

(33c)

It follows from the theory presented in Reference 2 thatthe capacitor Cx does not appear in the transfer functionsof eqns. 33a and 33Z>, because C\ is not preceded by an oddswitch S°. In the case of grounded networks in which acommon terminal serves as a reference both for the drivingsource and the output signal (e.g. j = m in Fig. 7), thecorresponding definite admittance matrix contains neitherthe rows nor columns / and m. In the corresponding 4-portequivalent network, the rows and columns / and M are alsodeleted, and the simplified network functions may be givendirectly in terms of the d.a.m. These, together with thei.a.m. expressions, are summarised in Table 1.

Table 1: Simplified network functions in terms of d.a.m. and i.a.m.

—Ok

—om

d.a.m. Ok

7eeZtr

Ztr

ztr

-,ooZtr

7eezdp

zdp

Hee

Heo

Hoe

H°°

sgn ( / - / ) sgn (k -m) Y^M/YJJ

sgn {i -j) sgn {K -M) Y^JYJj

sgn (I-J) sgn (k-m)YIkJJlMIYi

jJj

sgn (I-J) sgn (K-M) Y^JYJ^

YUJIYiJ

YUiIYjJ

sgn (/ -j) sgn (Ar -m) Y^MIY^

sgn (/-/) sgn (K-M) Y^JYIJJ

sgn (I-J) sgn (Ar - m) ?£%„/?$

sgn (/ - J) sgn (K-M) Y^JY^f

vlkly

YK/Y

yniy

VKIV

y\lv

y\ly

vli/viiil i il

YKllVil

vSitrli

yhiyW

6 Some additional features of s.c.-network matrices

6.1 Second-order cofactors

Consider once more the general form of the i.a.m. for theequivalent 4-port of an s.c. network. From eqn. 13 wefound it to be

[Y] =l*\n,n Jn,n

(34)

The individual submatrices are closely related, as indicatedby eqn. 17. Since each of the submatrices is itself an i.a.m.,each has the property of being an equicofactor matrix, i.e.all first-order cofactors of each submatrix are equal.8 As aconsequence, an arbitrary row or column can be deleted

27

from each quadrant (or submatrix) of eqn. 34, and theresulting second-order cofactors will all be equal. Hence wecan state that all second-order cofactors of the i.a.m. [Y]of a 4-port equivalent s.c. network are equal, provided thatany two deleted rows or columns are not in the same quad-rant of [Y].

6.2 Even and odd admittance matrix

Consider a general active s.c. network characterised by thefollowing parameters:

n = number of nodesme = number of even switchesmo = number of odd switchesv — number of amplifiers

The definite admittance matrix of this network will havethe form

= [y) (35)

As indicated already (in Section 4), the orders of the vari-ous network matrices will then be as summarised in Table 2.

If we are interested only in network functions involvingeven input terminals, the odd terminals must be suppressed.8

The resulting 'even matrix' is then defined by

h = [y)ev€

where

[y]e = [yn

(36a)

(36b)

Similarly, suppressing the even input terminals, we obtainthe 'odd matrix':

where

(37a)

07b)

Note that we must use the definite admittance matrix toperform this suppression. The corresponding operationusing the i.a.m. cannot be carried out, since [Y22] and [Yn]are zero-sum matrices; their determinants are thereforeequal to zero, and an inversion is not possible.

As with the i.a.m. and its submatrices, as given by eqn.17, the d.a.m. in eqn. 35 will have the following form:

[y] = (38)

where the [Gfj\ are submatrices with real elements, each ofwhich has the dimension C/T. Hence, according to eqns. 36and 37, the even and odd matrices will have the followingform:

and

= [Ge22] -z-2[Ge

2l)[Geu]-l[Ge

l2}

(39a)

(39b)

If the s.c. network is passive, then the initial i.a.m. (i.e.before contraction due to the switches) will have the formgiven by eqn. 17. In this case, the even and odd i.a.m. canbe obtained from expressions analogous to eqns. 36b and31b, resulting in

[Y]e = [y

where

p = \-z

= p[Ge] = p[C/r]

-2

(40)

(41)

This result is not surprising, since the i.a.m. of a continuouscapacitance network has the form

[11 = s[C] (42)

and the p-transform (whereby s is replaced by p) has beenshown to be valid when considering only even-to-even orany other of the four possible timing combinations impliedin Fig. 8 (see References 2 and 14).

6.3 Order of s.c. network functions

The general transfer functions of an s.c. network as definedby eqn. 3\b can be expressed as the ratio of two poly-nomials in z, namely

H(e,o)(e,o)=

Q

I0=0

N(z)D(z)

(43)

By taking into account the dimensions of the various sub-matrices making up these polynomials, it is possible to giveupper bounds on their order, i.e. on the number of polesand zeros of the transfer function. As derived in Appendix11, the order depends on the number of nodes n, the num-ber of amplifiers v and the number of even and odd switchesme and mo, respectively.

For me>mo, we obtain

P < Q < 2(n - v - me - 2) (44a)

Table 2: Orders of network matrices

Matrix

i.a.m. [ / ]d.a.m. [y]

Submatrices of [ / ][Yln,n[ Y]n,n+l[Y]n+l,nf Y]n+isn+i

S u b m a t r i c e s o f [ y ]

1/..J

[/„]

Order of matrix

[2(n —v) — me — mo] X [2{n — v) — me — mo][2{n — v — 1) — me —mo] X [2{n — v — 1) — me — mo]

Hn — v) —me] X [(n — v) —me][(n — v) —me] X [(n — v) —mo][ in — v) — mo] X [ (n — v) — me][in — v) -mo] X {(n — v) —mo]

[(n — v — 1) -me] X [{n — v — 1) - m e ][(n-v-l) -me] X[in—v--\) -mo][(n-v- 1 ) - m o ] X[{n-v-"\) - me][(n-v-l) -mo] X [(n -v 1 ) - m o ]

28 1EEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

and for me < mo

(n-v-mo-2) (446)

In many practical cases, the upper bound for Q coincideswith the actual order of the denominator D(z).

Eqn. 47 defines a resistor in the even path, namely betweenthe nodes k and m, where Re

eq - T/C.Similarly, a capacitor bridged by a shunt even switch

defines a resistor in the odd path, namely between thenodes n + k and n + m, where R°Q = T/C (see Fig. 106).

7 Some illustrative examples

To demonstrate the analysis procedure outlined above,some illustrative examples are included.

7.1 Grounded capacitor

Consider the grounded capacitor shown in Fig. 9. Fromeqn. 8 we have

[Yc] =

k

m

n + k

n + m

k

-G'

G°

m

-G

Ge

n + k

G°

-G°

Ge

G° -

n + m

-G°\

G°

-G

Ge 1(45)

Suppose that node m is grounded. Then, according toSection 4, the rows and columns m and n + m must beeliminated. Thus the [yc] m matrix is obtained as follows:

k n + k

k [Ge G

k\G° G(46)

This, of course, corresponds directly to eqn. 6. It can beapplied directly for the derivation of the d.a.m.

s.c. network

s.c. network

Vf even path

odd path

e1

k

1even

odd

m

1path

path

o n.m

Fig. 10 Shunt-bridged-capacitor network

a Shunt odd switchb Shunt even switch

7.2 Resistor

We consider now a capacitor bridged by a shunt odd switchas shown in Fig. 10a. Starting out with the correspondingsubmatrix in eqn. 45, the switch S° implies contraction ofnodes n + k and n + m. Thus the matrix simplifies to

k

k (Ge

m \-Gl

m

-G'

Ge(47)

7.3 Integrator

Consider the network shown in Fig. 11 (see Reference 3).To demonstrate our analysis method in detail, we shall gothrough this example step by step.

TFig. 9 Grounded-capacitor network Fig. 11 Integrator network

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980 29

(a) Using the [Yc] matrix, we obtain the initial i.a.m.

[II = 56

7

8

9

10

G°2

~G°2 G°2

G\

-G°x

G°x ~G\

(b) Considering the two switches by contraction, we have

[11me,mo

1

3

4

5

6

8

9

10

1

G\

G°x G

G°x

(c) In the case of the infinite-gain o.a. (i.e. Nathan'smethod), we have

G°2

-G°2

G\

~G°2

10

G2

~G\

~G\

G\

— G2

G 2

10

G°x

— G, —'

G\

(48)

(49)

L-* \me,mo

i

3

5

6

8

10

1

G\

~G\

~G°x

10

- G

(50)

This is the final i.a.m. of the ne twork . Note that row andcolumn 6 are zero, so that node 6 can be eliminated. Thereason for this is that the switch Se at the input is even.Thus terminal 1 is permanently connected to the inputvoltage source in the even path and permanently discon-nected from it in the odd path . 2 Thus, if the type of inputswitch is known , the corresponding node (even or odd) canalways be eliminated.

id) Selecting node 5 as reference, the d.a.m. of the net-work is obta ined:

[y]s= 3

G\

G°x -

G°2

G\

(51)

(e) Using the expressions in Table 1 for the transferfunctions of the grounded network, we obtain

30 IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

y\i

y\l

1-z"2 '

(52)

C, 1-z"(53)

Note that row and column 6 were already eliminated insteps b and c, for the reason given above, and thereforehave no further significance. The total transfer functionresults as:3

H(z) =V4+V9

C, 1 -z(54)

where H°\z) = 0 and Hoo(z) = 0.

7.4 Active filter

The network in Fig. 12 was first published in Reference 3.The final d.a.m. of the network can be derived followingthe same steps as in the preceding example. We obtain

1

1

5

6

11

13

7

— ce

11

- G ?

G°

14

-G\

-G\

(55)

The evaluation of the transfer function yields

18 Vx

(56)

This result agrees with that given in Reference 3. Since theeven and odd transfer functions were not there defined,only the result given in eqn. 56 can be found there. Withour analysis, the even-to-odd transfer function readilyfollows as

Heo(z) =

,18(14)8

.,18 (z2)2 ~

(57)

This yields the total transfer function

H(z) = K14

(z2)2 - k(\

(58)

where Hoe = 0 and Hoo=0.Note that the order of H(z), i.e. the number of poles, is

four. This corresponds to the upper bound given in eqn. 44,since n = 8, v = 1 and me = mo = 3. The number of nodesin eqn. 44 corresponds to that of the i.a.m., so that theground node must also be included (i.e. n is equal to 8,not 7).

7.5 Frequency-dependent negative resistance

The considered network in Fig. 13 is based on Reference15. The final d.a.m. of the network can be shown to be asfollows:

1

3

5

9

12

13

15

1

G\

G?

\

3

Gl

Gi+G!

G?c°

CO J_ /~"°C/4 t LT5

8

ce

ce

-G°

G°

~Gf

12

G?

13 15

G° G4° + G5°

Gt -Gi

-G Gil

(59)

r rrFig. 12 Active filter

IEEPROC. Vol. 127, Pt. G, No. 1, FEBRUARY 1980

out

3

C5

Se1C3

8

Se 5 S°

Fig. 13 Frequency-dependent negative resistance

31

The evaluation of the transfer function yields

tree/ \ ^3(10)^ ' ,,1(10)

• 1(10)

jieor \ ^12(10)11 V°' ~ v K l 0 ) ~

^1(10)

The total transfer function results

fcz-'O

ytz"2

Z)(z)

Arz"1

D(z)

[-z-1)

(60)

(61)

D(z)

that it facilitates the analytical calculation of any networkfunction from any pair of terminals to any other pair interms of the capacitor values of s.c. networks. Hence thisoffers great advantages for network optimisation anddesign, as well as for the calculation of network sensitivities.A computer program for the analysis of s.c. networks forsensitivity computation has been developed; it is based onthe i.a.m. method described in this paper. The advantagesof the analysis method are very apparent in c.a.d. appli-cations, as will be described in a forthcoming descriptionof the analysis program.

where

k =C3(C4+C5)

D(z) = (1 -z-y + {(C4 + Csy\C2CsC? - C4)}

x( l -z"2) + A:

8 Computer analysis of s.c. networks

An implementation of this method by computer is readilyfeasible, and is presently being completed using the mixed-nodal tableau methods described in References 12 and 13.The details of this implementation will be published shortly.An input listing of this program is given as follows:

SC-FDNR(FIG. 13)

INPUT LISTING

Explanation

SIN 1 0 ODDS12 1 2 EVENS23 2 3 ODDS39 3 9 ODDS34 3 4 EVEN -> 'even' switch between terminals 3 and 4S47 4 7 ODDS78 7 8 EVENS56 5 6 EVENS58 5 8 ODD -> 'odd' switch between terminals 5 and 8SOUT 3 OUT ODDCl 2 0 82-6NC2 3 0 8-23NC3 8 9 81-8N -> capacitor C3 between terminals 8 and 9 =

81-8nFC4(6 0 15-13NC5 6 7 15-07NVIN 1 0 1A 1 4 5 8CLOCK 1-6K•FREQ TF OCT 20 100 400PLOT TF V(OUT)•END

Fig. 14 shows the output plot for the f.d.n.r. lowpassdepicted in Fig. 13.

9 Conclusions

A direct method for the analysis of s.c. networks based onthe theory of indefinite admittance matrices has been pre-sented. One of the important advantages of this method is

32

30r

20

^ 10

- l 0 u

50 r

-50

-100

-150100 200

frequency, Hz400

Fig. 14 Output plot for f.d.n.r. lowpass of Fig. 13

IEEPROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980

10 References

1KURTH, C.F., and MOSCHYTZ, G.S.: 'Nodal analysis ofswitched-capacitor networks', IEEE Trans., 1979, CAS-26

2KURTH, C.F., and MOSCHYTZ, G.S.: 'Two-port analysis ofswitched-capacitor networks using four-port equivalent circuitsin the z-domain', iMi., 1979, CAS-26

3 HOSTICKA, B.J., BRODERSEN, R.W., and GRAY, P.R.: 'MOSsampled-data recursive filters using switched-capacitor inte-grators', IEEE J. Solid-State Circuits, 1977, SC-12, pp. 600-608

4 HOSTICKA, B.J., and MOSCHYTZ, G.S.: 'Practical design ofswitched-capacitor networks for integrated circuit implement-ation', IEEJ. Electron. Circ. & Syst., 1979, 3, pp. 76-88

5 KURTH, C.F.: 'Steady-state analysis of sinusoidal time-variantnetworks applied to equivalent circuits for transmission net-works', IEEE Trans., 1977, CAS-24

6 CAVES, J.T.: 'Samples analog filtering using switched-capacitorsas resistor equivalents', IEEE J. Solid-State Circuits, 1977, SC-12

7 HODGES, D.A., GRAY, P.R., and BRODERSEN, R.W.: 'Poten-tial of MOS technologies for analog integrated circuits', ibid.,1978, SC-13

8 MOSCHYTZ, G.S.: 'Linear integrated networks: fundamentals'(Van Nostrand - Reinhold, 1974)

9 MOSCHYTZ, G.S.: 'Linear integrated networks: design' (VanNostrand - Reinhold, 1975)

10 JURY, E.I.: 'Sampled-data control systems' (Wiley, 1958)11 WEINBERG, L.: 'Network analysis and synthesis' (McGraw-Hill,

1962)12VLACH, J., SINGHAL,K., MOSCHYTZ, G.S., and HORN, P.:

'Computer design of higher order active networks, part I:Analysis and experimental results', Proceedings IEEE ISCAS,Phoenix, 1977

13 VLACH, J.: 'Computer design of higher order active networks,part II: Optimization and study of various topologies'. Proceed-ings IEEE ISCAS, Phoenix, 1977

14 MOSCHYTZ, G.S., and HOSTICKA, B.J.: 'The transmissionmatrix of switched-capacitor networks and their application inactive filter design', (to be published)

15 HOSTICKA, B.J., and MOSCHYTZ, G.S.: 'FDNR transformationfor switched-capacitor networks', to be presented at ISCAS,Tokyo, 1979

11 Appendix

Note that each of the transfer functions of a grounded s.c.2-port, as given in Table 1, has the same denominator,namely the second-order cofactor y//. Thus, with eqn. 43,we can write

D(z) = r' = yU (63)= 0

Considering either of the two transfer functions Hee orHeo, we have, with eqn. 39,

where

[Xe] = [Gf2][Ge22]-l[Ge

21]

Similarly, considering H°° and Hoe,

y\\ = det [yi]0 = det {

where

(65)

{} (66)

(67)

The order of the matrices [Gfi]} and [Xe]\ is (n — v —me-2) [see Table 2], and that of [G|2]/ and [X°]J is(n — v—mo—2). Thus the determinants in eqn. 64 andeqn. 66 can only be equal (and yield polynomials of orderQ as specified by eqn. 63) if the matrix of higher order con-tains a corresponding number of zero elements.

It can be shown that for j3 = O in eqn. 63, we obtainfrom eqn. 64

bQz~Q = d e t { - z 2 |

and, from eqn. 66,

bQz~Q = d e t { -

It follows that either

bQZ-Q = z-2in-v

or

bQz-« = z_ _-2(n-z>-mo-2

(68)

(69)

(70)

(71)

H = det [y\Y = det {[G?l]\-z-2[X']i} (64)

where one of the two determinants in eqns. 70 and 71 will bezero, depending on whether me ^ mo.If det {— [X°]/} = 0,i.e. me>mo, then

Q<2(n-v-me-2) (72)

and if det {— [Xe]]} = 0, i.e. mo>me, then

Q<2{n-v-mo-2) (73)

Consequently, it follows that

P<Q<2(n-v-me-2) ioxme>m0 (74)

and

P<Q<2(n-v-mo-2) for mo>me (75)

ErratumPANDEY, K.: 'Second generation teletext and viewdatadecoders', Proc. IEE, 1979, 126, (12), pp. 1367-1374

The following acknowledgment should appear underFig. 6 on p. 1371:

Reprinted from Electronics, August 18th 1977.Copyright © McGraw-Hill, Inc., 1977. All rights reserved.

ETC61G

IEE PROC, Vol. 127, Pt. G, No. 1, FEBRUARY 1980 33