nodal analysis of switched-capacitor networks - isi

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-26, NO. 2, FEBRUARY 1979

Nodal Analysis o f Switched-Capacitor 93

Networks

Absstract-Switched-capacitor (SC) networks comprise capacitors interconnected by an array of perlodlcally operated switches. Such networks are particularly attractive in light of the high circuit density possible with MOS integrated circuit technology and hybrid integrated circuits using thin-film and silicon technology.

The paper describes the analysis of SC networks by using nodal charge equations. It is showu that SC networks are time-variant sampled-data networks, which cau be viewed as taudeh connected four-ports in the r-domain. One pair of ports is viewed as a signal path corresponding to the even time slots, the other pair of ports as a path corresponding to the odd time slots of the periodically operated switches.

In a subsequent publication the authors will show how four-port equivalent clrcults in the z-domain of six basic building blocks can be used for the description of any SC network. This method allows the direct use of traditional network analysis tools lie the transmissiou matrix for de,riving transfer functions. The method ultimately leads to a two-port analysis of SC networks in which conventional two-port theory can be applied.

I. INTRODUCTION

R APID ADVANCES in MOS integrated circuit technology permit an ever increasing circuit density per

silicon chip. So far, this technology has been used mainly by the designers of digital systems, although MOS technology allows the implementation of capacitor arrays for use in analog devices as well [ 11, [2], [lo]. Consequently, interest has recently focused on the concept of switched- capacitor (SC) filters which comprise only capacitors, interconnected by an array of periodically operated switches [3]-[9]. This approach can provide the filter functions previously obtained with LC or active-RC filters. W ith the elimination of resistors (which require a large silicon area, have poor temperature and linearity characteristics, and, furthermore, their temperature coef- ficient is difficult to match with that of capacitors when realized with MOS technology), the use of this technology for the design of analog active filters may soon become feasible.

MOS switches are already available; also toggle switches can be obtained by connecting two MOSFET’s, as shown in Fig. 1 [ 12]-[ 151. The two gates shown are pulsed with a two-phase nonoverlapping clock at a frequency Q2, =2~/ T. Contrary to the resistors, MOS capacitors, at this moment, already have close to ideal characteristics; temperature coefficients may be as low as

Manuscript received March 10, 1978; revised July 13, 1978. An oral presentation of this paper was given at the International Symposium on Circuits and Systems, New York, May 18, 1978.

C. F. Kurth is with Bell Laboratories, North Andover, MA 01845. G. S. Moschytz was with Bell Laboratories, North Andover, MA. He

is now with the Swiss Federal Institute of Technology, Zurich, Switzer- land.

J -T’

(c) (d)

Fig. 1. Switched-capacitor network.

10 ppm/“C or less, and the loss factor can be kept sufficiently small. Furthermore, capacitor ratios can be held to less than 0.1 percent using standard MOS processing techniques. By applying MOS processing techniques to SC networks, the realization of the “filter on a chip” may .therefore be at hand. Another technological alterna- tive is the combination of beam-leaded MOS switch arrays with thin-film capacitors deposited on a ceramic substrate to provide a hybrid integrated package resem- bling presently manufactured hybrid integrated RC-active filters. The absence of resistors on the substrate and the presence of only silicon chips (i.e., switches and opamps) and thin-film (or chip) capacitors may permit filter packages that are both smaller and less costly than those manufactured with present techniques.

W ith this in mind, we have examined how circuits comprising only switches and capacitors may be analyzed. It turns out that traditional two-port theory cannot be applied directly to such circuits. However, by introducing some new concepts it can be shown that, ultimately, classical two-port theory can indeed be used to advantage.

In the following the analysis of all-capacitor networks will first be reviewed by using the concept of nodal charge equations. After the inclusion of switches, it is demonstrated that in all cases, charge equations similar to Kirchhoff’s current equations apply except that the storage properties of the capacitors must be taken into account. This leads to the description of SC networks as time-variant sampled-data networks. In a subsequent paper it will be shown how the method can be used to treat general and more complex bilateral SC networks without opamp isolation. Equivalent circuits in the z-

CARL F. KURTH, FELLOW, IEEE, AND GEORGE S. MOSCHYTZ, FELLOW, IEEE

009%4094/79/0200-0093$00.75 01979 IEEE

94 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-26, NO. 2, FEBRUARY 1979

domain will be derived and design techniques for such circuits will be given and compared with laboratory measurements.

II. ANALYSIS OF CAPACITOR NETWORKS

A. The Continuous Case

Consider the capacitor C shown in Fig. 2. The incre- mental charge q(t) stored at any given time t can be expressed by the current i(t) and the voltage u(t), namely

q(t)=L’i(t)dt=C*u(t)-C-n(O) (1)

where u(0) represents the voltage across C at time t = 0. The analysis can be extended to a general capacitor network with (m + 1) nodes as shown in Fig. 3. Consider- ing the current flowing into each node and the voltage from each of the m nodes to the (m + 1)th reference node, a set of m nodal charge equations can be obtained and expressed in matrix form

[q(f)]=[jo’i(t)dt]=[C].[u(I)]-[C].[u(O)]. (2)

Note that when describing the capacitor network in terms of charges and voltages, the matrix [C] contains only real-valued time-invariant elements [ 161, [ 171.

B. The Time-Sampled Case

Let us now assume that the capacitors of the network are not charged continuously but in surges of i(t)&t - wr), where i(t) at t = n7 is the value of the current surge after every interval 7. This can be linearly related to the charge as follows:

/ ‘=O” i(t)8(t-n7)dt=i(wr)*~,,=q(m) (3)

1=-m

The nodal charge equations become

[q(n)]=[70-i(n)]=[C].[v(n)]-[C]-[4n-l)]. (4

Solving for the currents i(n) we obtain

iQ+ [ $p)- [ ;].4n-1) (5)

where i(n) and u(n) are vectors. The matrix equation (5) corresponds to Kirchhoff’s

current law as applied to an all-capacitor network with sampled input currents. These equations will be referred to as nodal charge equations: they form the basis for the analysis that is to follow. The currents i(n) can be considered to charge the capacitor network instantaneously at time m. Since no resistors are assumed in the network there is no dispersion of the charging process and u(n) can be assumed to change instantaneously in steps. The matrix [C/T,,] of (5) contains only real-valued and frequency-independent elements. For convenience 7,, will be dropped in further derivations; thus all capacitors are

Fig. 2. Charges and voltages on a capacitor.

Fig. 3. Capacitor network.

Fig. 4. Capacitive two-port.

Fig. 5. Resistive two-port.

normalized to 7,,= 1 s and have the dimension of a con- ductance.

Example: As an illustrative example for the use of the nodal charge equations consider the capacitor network of Fig. 4. Applying the analysis to nodes 1 and 2, respectively, we obtain

i,(n) [ I[

= (G+cJ -c3 u,(n) b(n) - c3 G+ G) * dn) I[ 1 If the- network were resistive instead of capacitive, as shown in Fig. 5, we would obtain the very similar current equations: i,(t) [ I[ G,+G, -G, u,(t)

i2(t) = -G, G,+G, * u*(t) ’ I[ 1 (7)

KURTH AND MOSCHYTZ: SWITCHED-CAPACITOR NETWORKS

Conclusions: By deriving the nodal charge equations for a capacitive network one obtains a matrix with real-valued and frequency-independent elements comparable to the y-matrix of a resistive network; the important difference between the two is that the capacitive network has storage capability as represented by the vector v(n - 1) in (5) and (6).

III. THE ANALYSIS OF SC NETWORKS

A. Properties of SC Networks in the Time Domain

To appreciate the significance of adding switches’ to an all-capacitor network let us consider two basic connec- tions for such switches. .

I) Switches Across Capacitors: Consider the capacitor with a switch across it, as shown in Fig. 6(a). The 50-percent duty-cycle switch S is closed every other r seconds as indicated in Fig. 6(b). Assuming that S is open at time n7 we obtain the charge equation

i(n)=C.v(n)-C.v(n-1). (8)

However, since S was closed at time (n - l)r, the capacitor was discharged during that time interval, i.e., its storage property had been removed. Thus v(n - 1) = 0 and (8) becomes

i(n) = C*v(n). (9)

In comparison with (7) we see that a capacitor with a switch across it, as in Fig. 6, behaves exactly like an equivalent resistor whose value is

PI l R-4 [‘I= C [F] ’ 2) Switches Between Capacitors: By placing switches

between capacitors the topology of the network is being changed every r seconds, and at the same time charge is being instantaneously exchanged between capacitors. Consider for example the network in Fig. 7(a). It consists of a toggle switch and three capacitors. Assuming that at odd times n the switch is on the left, one obtains the network shown in Fig. 7(b). Applying the charge equations to the network in this state results in the matrix equation

n odd i,(n) [ I[ = (C,+Co) 0 h(n) 0 c* fdn) v*(n) 1

Similarly, for even times n, the network has the topology shown in Fig. 7(c) and the charge equations become

‘Here and in all that follows 50-percent duty-cycle switches are assumed, where on and off time is 7 seconds, respectively.

l+ v(n) -; I (0) (b)

Fig. 6. Switched capacitor.

n: odd I (n) L* - +

v,(n) I-- TCO

o -------&)

cz/r 2 -

1 -

0 - 0

(b)

n:even 1,(n)

:,z-

0 2’ + -- Cl

/, co Q/-Y v&n)

- - 0 - 0

(c)

Fig. 7. SC two-port.

n even i,(n ) c, [ II. 0 v,(n) iz(n) = 0 (G+ Cd ’ 4n> I[ 1

Instead of the time-invariant elements in the matrix equation of (5) the general charge equations for an SC network will have time-variant coefficients and the general matrix equation will have the form

i(n) = [ C(n)].v(n) - [ C(n - l)]~(n- 1). (13)

Depending on which time m- is considered, (13) will have different elements in its matrices. Our objective is now to combine equations of the kind given by (11) and (12) into one, such that it will be valid both for even and odd n, in short, for all times n7. The charge equations (11) and (12) describing Fig. 7 differ because capacitor C, is connected to node 1 for odd, and to node 2 for even times. This time-variant topology can be taken into account by introducing the following two time-dependent “switching


TABLE I USEFUL Z-TRANSFORMS

. -

Time Series z-Transform

1) f(n)

2) a"f(n)

3) t-1)” f(n)

4) (-1)” f(n-1)

5) (-1)" f(n-k)

6) *e(n) 1 + = t-11" 2

7) AO(n) 1 - = t-11” 2

8) Ae(n)-f(n)

9) A'(n)-f(n)

10) Ae(n) f(n-1)

11) A'(n) f(n-1)

-.

m

F(z) = 2 i=O

fi z-i

F(z/a)-

F(-2)

--z -lF(-z)

(-l)k z-~F(-z)

1 -2 1-z

-1 z 1 - z-2

1 l F(z) = F(z)+F(-z) 1 - z-2 2

2-l

1 - z-2 l F(z) = F(z)-F(-z)

2

-' (F(z) - F(-z))

-' (F(z) t F(-z))

: asterisk: stands for complex convolution in [23].

functions” [ 17]-[22]:

1+(-l)” 1 forn even A’(n)= 2 = 0: for n odd

l-(-l)” A 0(n)= 2

= 0, for n even 1, for n odd.

(144

(14b)

With these switching functions, the two equations (11) and (12) can be combined into the form of (13), namely

(15)

Equation (15) is the time-variant matrix equation required to describe the network of Fig. 7(a). As will be seen later any SC network can be described by a matrix equation of this kind, where the time variance is expressed by the switching functions A ‘(n) and A “(n).

Conclusions: SC networks can be described using the analysis methods of sampled-data systems with time- variant coefficients, whereby a similarity exists to the nodal analysis of resistive networks [23]-[29].

B. Properties of SC Networks in the z-Domain

For design purposes it is desirable to describe SC networks as two-ports in the frequency domain. For sampled-data systems, the link to the frequency domain can

be established with the z-transformation. In the following, the example in Fig. 7(a) will be used to describe some typical properties of SC networks in the z-domain. A generalization of these properties and of the ensuing analysis methods will be presented in Section IV.

For the z-transformation of the nodal charge equations in (15) some basic z-transforms are required, which are listed in Table I. Derivations are given in the Appendix. Applying formulas (8)-( I 1) to (15) we obtain

Z,(z)=C,V,(z)(,l-z-‘)+c, V,(z)- V1(-z> 2

-c 0

Vz/;(z)+ V2(-z) z-1

2

12(z)=c2v2(z)(1-z-1)+co v~(z)+2v~(-z)

-c 0

V,(z)- Vd-z) z-, 2

(164

(16b)

The problem is that both V,(z) and y(-z) appear in these equations, whereas to obtain the z-transformed [r]- matrix for the network only, v;(z) should occur. The problem can be resolved by examining the z-transforms of the terms V,(z)+ K( - z). With formula 1 in Table I it becomes apparent that

vi(z)+ f-3-z) 2

=v +v z-*+ 02 ** . = v(z) (17a)


Fig. 8. Four-port analogy of SC network in z-domain.

where c.e(z) implies the even terms of v,(z) and

V;(z)- lq-z) 2 = v,z -‘+v3z-3+v5z-5

+ = . . . = y(z) (17b) where vi”(z) implies the odd terms of v;(z). Using this concept of even and odd terms of a time series and associating it with the switching functions A ‘(n), A “(n), the equations in Table II may be derived as explained in the Appendix. W ith these relations all functions in z may now be broken up into even and odd terms. W ith formulas 1) through 6) and 9) of Table II, (17a) and (17b) become

zI’+zp=c,(v~+vp)(l-z-‘)+c,vp-c,V;z-’

(18) and

z;+z;=qv;+ v;)(l-z-‘)+cOv;-cOvpz-’

(19) whereby, for simplicity, V,(z) = Vi and Zi(z) = Ii. Separat- ing the even and odd current components results in the following matrix equation:

Cl - c,z-’

- c,z-’ =

I-

q+co

0 - c,z-’

0 0

The even and odd parts of the products have been sep- arated as follows:

and

Ev(Vi’+ vi”)(l-z-l)= ~i’-z-‘~? (214

Od(tj’+ 7)(1-z-‘)= I$‘-z-‘T @ lb) where Ev and Od stand for even and odd part, respectively. Equation (20) can be interpreted as the current equation of the four-port shown in Fig. 8. The two input and output ports are associated with the signal components in the time domain at even and odd time slots of the periodically operated switches. Equation (20) corresponds to a y-matrix representation of a 2m-port network, where for this example m =2.

97

Fig. 9. Basic branch in SC network.

Similarly any general SC network can be modeled as a am-port network with the nodal charge equations in the z-domain written in the general form

I=[y]V.

Note that the elements of this [y]-matrix either consist of terms C (which correspond to resistors R,, = l/C) or Cz - ’ which correspond to resistors with delay (or storage) properties. In the example it is noteworthy that the sym- metry of the matrix indicates reciprocity of the network, which is plausible even from a physical point of view, since the network is passive.

IV. GENERALIZED NODAL ANALYSIS OF SC NETWORKS

The analysis demonstrated so far by an example, will now be derived more rigorously using the nodal charge equations to analyze a general capacitor network connected to an array of periodically operated switches. Assume that no branch between two nodes in an SC network is more complex than a capacitor with a switch in parallel as shown in Fig. 9. For C= 0 the branch reduces to a simple switch. W ith the switch continuously open the branch reduces to a nonswitched capacitor. W ith this assumption any SC network can be broken down into a simple capacitor network and an array of switches as shown in Fig. 10. To avoid ambiguity each switch S has a superscript e or o to indicate that it is closed during even or odd time intervals, respectively.

0 0

-c,z-' 0

c*+c0 - c,z-'

-c,z-' c,

(20)

The capacitor network is described by the nodal charge equations given by (5) namely

i(n) = [C]w(n) - [C]w(n-1). (22)

“Charge” Equilibrium Equilibrium injection at instant “n” at instant “n - 1”

at “n” “Memory” The voltage relations of the switch network can be described separately at even and odd time instances. Since the relation is not unique a reference voltage has to be picked. For the example in Fig. 10 we obtain, with v, as reference, for n = odd

2)’ = 2)’ v,=o

98 WEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-26, NO. 2, FEBRUARY 1979

i3 1

any time instant n

/ Switch Network

Y ‘O%

iL1.zi SC- Nbixk SO Se

Fig. 10. SC network as a superposition of a C-network and a switch network.

and v3=v,

or in matrix form

I,+[; i, Fi].[;q+

Similarly for n = even v, = 0, v2=v1 v3=v3

01 I 1 02 f 03

01 02

03

Using the previously introduced switching functions (14) the voltage relations in the switching networks can be expressed in general terms at any time instant n

v(n)={A~(n)~[S”]+A’(n)~[S”]}w(n). (23)

This relation also has to be valid at the instant n - 1, hence

u(n- l)= {A”(n - 1). [S”] +A’(n - 1). [S’] l-,(n- 1).

(24) The matrices [SO] and [Se] are sparse matrices which

have only ones or zeros as elements; they build the link between the nodal voltages at even and odd time instants. Equations (23) and (24) can now be introduced into the nodal charge equations (22) of the capacitor network. The result is a matrix representation of the nodal charge equation in the time domain for the entire SC network at

i(n) = [ C]. {A “(n). [S”] + A’(n)* [S’] }w(n)

-[C].{A”(n-l).[S”]

+A’(n-l).[S’]}.u(n-1). (25)

Equation (25) corresponds to the previously derived equation (13). The matrices [C(n)] and [ C(n - l)] are now expressed by a product of two matrices, the first of which represents the nonswitched capacitor network, the second the time-variant switching network. The advantage of this representation is that the matrix [C] of the -nonswitched capacitor network and the matrices [SO] and [Se] can be established independently, and by inspection.

With formulas (1) to (4) of Table II, the z-transform of (25) can now be obtained directly, namely

Z(z)=[C]*{ [s”]+-o(z)+ [S’]*V’(z>}

-[c]*{z-‘[s”]*v”(z)+z-‘[S’]*V’(z)}.

(26)

I(z), V(z), and V’(z) are vectors of the z-transforms of the currents and voltages at the m-nodes of the SC network, i.e.,

ve<z>

I. v;(z) V’(z)= .

i . K(z) VP(z) V,“(z)

V”(z)= : . (27)

Kx4 The current vector can also be split into an even- and odd-element vector, namely

Z(z) = P(z) + P(z). (28) Reordering, the entire matrix equation (26) into even and odd parts we obtain the matrix equation


TABLE II USEFUL RELATIONS FOR THE ANALYSIS OF SC NETWORKS-

99

1) Ae(“) f(n) I F(z) + F(-2) 2 = Fe(z)

2) A’(“) f(n) f F(z) - F(-z) 2 = F’(z)

3) Ae("-1) f("-1) 5 z-l Fe(z)

4) A’(“-1) f(“-1) 5 z-l F’(z)

5) Ae(") f("-1) = A"("-1) f (n-1) f z-lF'(z)

6) A'(") f("-1) = Ae("-1) f("-1) z, z-lFe(z)

7) Ae(n) + A'(") = 1 5 Ae(z) + A'(z) E -&--- 1 - z-l

8) Ae(") f(n) + A'(") f(n) = fe(") + fO(") = f(n)

9) fe(") + fO(") = f(n) 2 Fe(z) + F’(z) = F(z)

I SC - NETWORK I

Fig. 11. (a) SC network of dimension m in the time domain. (b) SC network of dimension 2m in the z-domain.

Equation (29) can be considered as the general nodal charge equation for an SC network in the z-domain; it corresponds to the time domain equation (13). If the network has m nodes and one ground node as a reference, the matrix relation in (29) has a dimension of 2m x 2m. It can be interpreted as the nodal equation for a 2m-port. Each pair of ports corresponds to the even and odd current and voltage components appearing at any given node of the SC network in the time domain. This is illustrated in Fig. 11.

Similarly for n even _

i;=i,+i,

l;=o

L3= i,

and

The two groups of switches which close at even and odd The matrices [I’] and [I”] are incidence matrices which times, respectively, will reduce the actual number of nodes link the two different sets of currents together. Since no

associated with the switches by a factor of two. Consider, for example, the closing of a single switch: the pair of nodes associated with that switch will be reduced to a single node. Consequently at this instant the sum of the two .previously independent currents will flow into the combined node. To demonstrate this consider again the example of ‘Fig. 10. At switching time the combined currents l’ can be expressed by the previously assumed fictitious currents i as follows:

For n = odd

or in matrix form

A i, = i, + i,

l\ = i, ^ i,=O


it i3 ‘2

'-i'"' Tco Tc2 0 I 1 1 o

(4

. -

ij t3 I2 Cl co CP

(4

Fig. 12. (a) SC network of Fig. 7(a). @) Switch network of SC network in Fig. 12(a). (c) Capacitor network of SC network in Fig. 12(a).

memory is involved the equations can be transformed into the z-domain simply by substituting the instantaneous values at even and odd times by even and odd parts Z’(z) and Z”(z) of the transformed currents. This leads to a

yields the final expression (31).

*[ [S’] -z-y S”] V=(z)

-z-‘[Y]’ [S”] I[ 1 P(z) . (31)

Example: To illustrate the general analysis discussed above, the nodal equations for the example in Fig. 7(a) will be derived again using (31). In Fig. 12(a), the config- uration of Fig. 7(a) is redrawn. As ,can be seen, a third node is introduced which allows the separation of the capacitor network and the switch network as shown in Fig. 12(b) and (c).

The capacitor network yields the [C] matrix

I I c =

c, 0 0 c, 0 0 0 0 I

(32) c;

and from the switch network we obtain the voltage rela- general‘rklation tions

(30) n=odd 7I7 This equation links the current vector pf the previous result in (29) to the new current vector Z(z). Introducing

v:= v:

the new currents Z ultimately reduces the dimension of the n = even

I 1 =o I 0 0

S” 10 1 0 0 1 (33)

matrix equation drastically; for each current at one end of a switch, which is added to a current at the other end (at closing time), a line of zeros will be introduced into the

(34)

matrix equation (30). A corresponding number of voltages at one end of the switch becomes redundant, and a square matrix relation between the new currents i and the re-

Substituting the submatrices (32) (33), and (34) into (29) the SC network can be described by the following

maining voltages is maintained. Combining (29) and (30) relation:

Multiplying the matrices in (35) yields

v; V;

v; vp . (35)

vz”

vi

(36)


The current relations turn out to be

n=odd n = even

i; = i, + i, 1 i, = i,

I&= i, f2=i,+i,

I;=0 A i,=O

or, expressed in the form of (30)

Substituting (37) into (36) according to (31) and multiplying the two matrices in (36) and (37) yields

i; 0 = -0

11 c,z-’

i; 0

ii 0

0

-cot-'

- c*z-'

0

101

culty led to the idea of analyzing simple building blocks with which any complex SC networks can be designed. This extension of the present theory will be described in a subsequent paper. In what follows, the four-port representation of SC networks will be explained, since this represents an important and integral part of the analysis and synthesis of SC networks.

V. FOUR-PORT REPRESENTATION OF SC NETWORKS

As indicated above, (31) corresponds to a y-matrix representation of a network with 2m-ports. Consequently, as for passive time-invariant networks, any two independent nodes can be declared as input and output ports with respect to ground and the network can be described using the relations between these two ports. Selecting two nodes in an SC network and declaring them as input and output ports with respect to ground corresponds to declaring two pairs of ports as input and output ports in the z-domain, namely: the even and odd input, and even and odd output pair. The result is a four-port representation of the SC network in the z-domain as already shown in Fig. 8. In so

0 0 0 v;

c,+co 0 0-v;

0 G 0 v;

0 0 0 v;

(38)

As can be seen columns and rows three and six are zero; consequently they can be dropped. This reduces (38) to the following:

i; Cl 0 - c,z-’ 0 V,e i; 0 c,+ co - c,z-' -c,z-' v; = -0

- c,z-’ - c,z-’ 0 by’ (39

11 c,+co

i; 0 - c,z-' 0 c2 vi

W ith the currents Z in (20) physically equivalent to the doing all other nodes become irrelevant and the currents i in (39), the two expressions are identical as can associated voltages and currents can be eliminated by be shown by reordering rows and columns in (39). The setting the currents flowing into such nodes equal to zero. results are the same, but the methods differ. Equation (20) This will always result in a reduction of the general was derived almost directly by writing the nodal charge 2m-port equation (31) into a four-port y-matrix repre- equations by inspection; (39) resulted from a systematic, sentation as follows: general procedure. It is obvious that the general method requires more steps and will not necessarily be the fastest method for an explicit algebraic analysis of complex SC networks. Its merit lies in the systematic approach that it provides to set up (31) by inspection for any given network. This suggests the use of a computer for a

pjj=[;;::j ;;.;#ijJ]. ;jol,-

numerical analysis in the frequency domain, i.e., a point by point computation with z - ’ = e-jw’ and a subsequent For simplicity the “*” over the currents has been numerical matrix inversion. As it turns out, the first dropped and the currents I” and I’ correspond to the method also becomes rather involved, since the nodal physical currents flowing into the ports, with the orienta- charge equations of more complex SC networks are not tion as shown in Fig. 8. Equation (40) corresponds to a easily derivable by mere inspection. This analytical diffi- two-port y-matrix equation. The submatrices [Y,,], [Y,,],


[ Y2,], and [Y,,] are 2 X2 matrices; their elements are related to the components in the SC network.

As in the analysis of passive time-invariant two-ports it is also here desirable to establish a transmission matrix. This will allow the convenient analysis of cascades of these four-ports. Converting (40) into a transmission matrix form yields

where the submatrices [A], [B], [Cl, and [D] are related to the submatrices in (40) as follows:

[A]=-[ y2*]-‘*[ y221

[B]=-[ y2,1-’

[C]=+[ Y12]-[ L].[ y2,1-‘.

[D]=-[ y,,].[y2,]-‘.

(424 (42b)

[ Y22] (42c)

(424 The negative signs of the currents I,’ and 1; in equation indicate that the currents at the output ports flow into the network, as shown in Fig. 8.

Again, for demonstration purposes the four-port transmission matrix for the example in Fig. 7(a) can be calcu- lated. From the y-matrix in (20) the submatrices for (40) can be extracted as follows:

1 II lY II 12

c, -c,z-’

lY II - - 1

1-i

- -coz-’ c,z’ 0 c,+co 0 0 1 y2, =

0 0

- coz-’ 0 1

1 II y22 = c2+co - c,z-’

-c,z-’ 1 c, * Unfortunately [Y,,] and [Y,,] are singular, which means that their inverses’[ Y,,]-’ and [Y,,]-’ do not exist and the transmission matrix for this network cannot be established. A plausible physical explanation for this is related to the fact that at even and odd times one or the other of the two switches in the network is open, thus, no continuous transmission path exsits. One can overcome this problem, by introducing two capacitors Co, and Co2 across the switches as shown in Fig. 13. They can be considered as the parasitic leakage capacitors of the switches and will be generally small. The important point is, that this “practical trick” avoids the singularity in [Y,,] and [Y,,] and that the transmission matrix can now be derived. The new [Y]

Fig. 13. Two-port of Fig. 7(a) with leakage capacitors across the switches.

matrices with the capacitors Co, and Co2 are given by the following equations:

Although none of these are singular, considerable mathematical computations are still required in order to obtain the transmission matrix of the entire four-port, as, for example, the matrix multiplications in (42a) and (42~). This may be left to the interested reader. In a subsequent paper the previously mentioned building block approach will be described. Using this approach the network in Fig. 13 can then be analyzed as a cascade of five four-ports, as indicated in Fig. 14. Each four-port represents an equivalent circuit of a building block in the z-domain. The “practical trick” of introducing parasitic capacitors is no longer required to make the network analyzable. As will be seen the four-port equivalent circuit approach provides a much better and quicker insight into the performance of SC networks. It also allows the four-port analysis to be reduced to a two-port analysis that resembles conventional two-port theory almost perfectly [29].

APPENDIX

Since a significant part of the analysis presented here is based on the relations listed in Table II, some comments about their derivation may be justified. The backbone for transforming nodal charge equations of SC networks into the z-domain are (1) and (2) in Table II.

As defined in (14a) and (14b)

1+(-l)” A’(n)= 2

Aqn)= 1-t- 1)” 2 . @a

These two functions can be represented as time series as shown in Figs. 15 and 16. The time series (- 1)” as shown in Fig. 17 is the cause for only even or odd samples

KURTH AND MOSCHYIZ: SWITCHED-CAPACITOR NETWORKS 103

Fig. 14. Cascade of SC building blocks and a four-port equivalent circuit in z-domain.

0 1 2 3 4 5 6 7 6 9 10 II ‘2--+ n

Fig. 15. Even time series A=(n).

t

no(n)

ti-

0 1 2 3 4 5 6 7 6 9 10 If (2-n

Fig. 16. Odd time series A’(n).

z- p lane for F (1)

Fig. 18. Rotation of z-plane by 180” due to modulation of time series with (- 1)“.

Fig. 19. Shift of spectrum by l/29, due to modulation with (- 1)“.

Fig. 17. T ime series of (- 1)“.

appearing in A ‘(n) and A”(n), respectively. Any random time series f(n) multiplied by A ’ or A ’ can be interpreted as the sum (or difference) of the random time series by itself and its modulated version (- l)‘f(n).

That is

or

A “(n)*f(n) = g(n) - f( - l)?(n). (A4)

As the z-transform of (- I)“-f(n) can be expressed rather easily (see 3) of Table I), its interpretation in the z-domain, or the effect of this modulation on the spectrum of the random time series f(n), merits some further attention.

The z-transform

W ) can be interpreted as a 180” rotation of the z-plane for F(z), by realizing that - z = z*ej” = z’, and with 217/G, = r, z’= &w+(“~/2)). I’ is the new variable after rotation. This is demonstrated for a second-order system in Fig. 18. It means that a modulation by (- 1)” (or alternating the sign) of a time series f(n) corresponds to a rotation of 180” in the z-plane, or more important, to a shift of the spectrum of f(n) by half the sampling frequency 52,. This is shown in Fig. 19. This shift of a spectrum by half the sampling frequency via sign alternation is frequently used in digital signal processors or digital filters.

From the time series in Figs. 15 and 16 it can be seen that either the odd or the even samples are missing. Thus by multiplying a time series f(n) with A e or A ’ the odd or

104 , IEEE TRANSACTTONS ON CIRCUITS AND SYSTEMS,VOL. CAS-26, NO. 2, FEBRUARY 1979

even terms of f(n) are eliminated, as was also shown in (17a) and (17b). This is why the z-transform of the (A3) and (A4) can be considered the even or odd parts, F’(z) or F”(z), respectively, of F(z). With equation (A5) the following relations can be derived:

Z[k’(n)-j’(n)] =$F(z)-;F(-z)=F’(z). (A7)

By inspecting Fig. 19 it can be concluded that F’(z) and F”(z) contain the original spectrum plus (or minus) the spectrum of F( - z), which is shifted by half the sampling frequency.

This is important to know, since it can lead to aliasing when Q2, is not high enough. The summation of the two spectra is equivalent to adding energy around 1/2G,, 3/2Q2,, etc., or sampling with half the sampling rate.

The rest of the transforms in Table II are based on the two transforms described above. They can be derived by applying known rules of the z-transformation. .

For example, the transform 3) in Table II can be derived, with A ‘(n) -f(n) = v(n) and Z[y(n)] = Y(z), Z[y(n - l)] = z-‘Y(z). Thus Z[A’(n - 1) .f(n - l)] = z-‘Z[A’(n)-f(n)]=z -‘F’(z).

ACKNOWLEDGMENT

The authors appreciate E. I. Jury’s thorough study of the manuscript and his subsequent comments.

REFERENCES

[II

PI

I31

[41

I51

VI

I71

181

[91

[lOI

[ill

1121

[131

[141

[151

B. V. Gokhale and J. Ziedenbergs, “MOS capacitance under punchthrough conditions,” in IEEE Int. Electron D&ices Meet. Dig., pp. 529-532, 1974. A. G. Fortino and J. S. Nadan, “An efficient numerical method for the small-signal AC analysis of MOS capacitors,” IEEE Trans. Electron Devices, vol. ED-24, pp. 1137-l 147, Sept. 1977. R. Boite and J. P. V. Thiran, “Synthesis of filters with capaci-- tances, switches, and regeneration devices,” IEEE Trans. Circuit Theory, vol. CT-15, pp. 447-454, Dec. 1968. W. Poschenrieder, “Frequency filters with periodically operated switches,” in Proc. NTG Symp., Stuttgart, Germany, pp. 220-237, 1966. S. Cantarano and G. V. Pallottino, “Approximate results for networks containing periodically operated switches,” Proc. IEEE, vol. 56, p. 2070, Nov. 1969. A. Fettweis, “On the theory of periodically reverse-switched capacitor networks,” Arch. Elek. Ubertragutzg, vol. 24, no. 12, pp. 539-544, Dec. 1970. J. A. McKinney, “The periodically reverse-switched capacitor,” IEEE Trans. Circuit Theory, vol. CT-15, pp. 288-290, Sept. 1968. M. L. Liou, “Exact analysis of linear circuits containing periodically operated switches with applications,” IEEE Trans. Circuit Theov, vol. CT-19, pp. 146154, Mar. 1972. R. Orfei and G. V. Pallottino, “Networks containing periodically operated switches: A state space approach,” Int. J. Circuit Theory and.Appl., vol. 1, pp. 373-386, 1973. L. Baker and S. D. Bedrosian, “Amplification of resistance concepts in switched networks,” in Proc. IEEE Int. Symp. Circuits Syst. 342-346, Apr. 1974. - “RC immitances and periodically switched networks,” Elec-- tron. ‘L&t., II, vol. 11, pp. 622-623, Dec. 1975. J. L. McCreary and P. R. Gray, “Al l -MOS charge redistribution analog-to-digital conversion techniques,” IEEE J. Solid-State Circuits, Vol. SC-IO, vol. 6, pp. 371-379, Dec. 1975. J. T. Caver et al., “Sampled analog filtering using switched capacitors as resistor equivalents,” IEEE J. Solid-State Circuits, vol. SC-12, pp. 592-599, Dec. 1977. R. M. Crawford, MOSFET in Circuit Design. New York: McGraw-Hill, 1967. W. M. Penney and L. Lau, Eds., MOS Integrated Circuits. New York: Van Nostrand-Reinhold, 1972.

VI

I171

1181

[191

1201

PII

PI

1231

~241

~251

PI

1271

@ I

1291

C. A. Desoer and E. S. Kuh, Basic Circuit Theory. New York: McGraw-Hill, 1969. -, “Steady state transmission through a network containing a single time-variant element,” IRE Trans. Circuit Theory, pp. 249- 252, Sept. 1959. A. Fettweis, “Steady state analysis of circuits containing a periodically operated switch, IRE Trans. Circuit Theory, vol. CT-l, pp. 17-21, 1955. - “Theory of Resonant- Transfer Circuits,” in Network and Switihing Theory. New York: Academic, 1968. ~ “Realization of General Network Functions Using the Reso- nant:Transfer Principle,” AEUE, Electron. and Commun., vol. 25, pp. 295-303, 1977. C. F. Kurth, “Spectral matrix for the analysis of time-variant networks,” Elec. Commun., vol. 39, no. 2, pp. 277-292, 1964. -, “Poles and zeros in time-variant networks,” (in German), AEUE, vol. 17, pp. 547-563, Dec. 1963. E. I. Jury? Theory and Applications of the I- Transform Method, New York: Wiley, 1964. (Republished by Kreiger, 1973.) A. G. J. Holt and C. Pule, “Relation of resampled switched sampled analogue filters to digital filters,” in Int. J. Electron., vol. 28, no. 4, pp. 331-340, 1970. D. L. Fried, “Analog sampled-date filters,” IEEE J. Solid-State Circuits, vol. SC-7, pp. 302-304, Aug. 1972. C. F. Kurth, “Steady-state analysis of sinusoidal time-variant networks applied to equivalent circuits for transmission networks,” IEEE Trans. Circuits Syst., vol. CAS-24, pp. 610-624, Nov. 1977. B. J. Hosticka, R. W. Broderson, and P. R. Gray, “MOS sampled- date recursive filters using state variable techniques,” in Proc. Int. Symp. Circuits Sysr., Phoenix, AZ, pp. 525-529, Apr. 1977. I. A. Young, D. A. Hodges, and P. R. Gray, “Analog NMOS sampled-data recursive filters, ” in 1977 IEEE Solid-State Circuits Conf., pp. 156-157, Feb. 1977. G. S. Moschytz, Linear Integrated Nehvorks: Design. New York: Van Nostrand-Reinhold, 1975, ch. 2.

+

Carl F. Kurth (Sh4’64-F’79) was born in Germany. He received the Grad.Ing. degree in electrical engineering from Polytechnikum Mittweida, Mittweida, Germany, in 195 1.

From 1951 to 1960, he was with VEB Radio und Femmeldetechnik, Leipzig and Berlin, Germany, where he headed the networks and modulator laboratory in Leipzig, from 1954 to 1960. From 1960 to 1964, he was with Standard Electrik Lorenz AG, Stuttgart, Germany, in charge of the exploratory laboratory for trans-

mission technology. After a short period of teaching at Lehigh Univer- sity, Bethlehem, PA, in 1965, he joined Bell Telephone Laboratories, North Andover, MA, where he is presently a Supervisor in the Transmis- sion Technology Laboratory working in the area of digital signal processing. He taught courses in network synthesis and feedback analysis at Bell Laboratories. He has published papers in the fields of networks and transmission system analysis. He authored patents in both the United States and Germany.

Mr. Kurth served as the Chairman of the IEEE committee on Network Applications and Standards from 1973 to 1976. He also served as an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS and on the Administrative Committee of the Circuits and Systems Society, of which he is currently President-Elect.

+

George S. Moschytz (M’65-SM’77-F’78) received the M.S. and Ph.D. degrees in electrical engineering from the Swiss Federal Institute of Technology, Zurich, Switzerland, in 1958 and 1960, respectively.

From 1960 to 1962 he was with RCA Labora- tories Ltd., Zurich, where he worked on color TV signal transmission problems. In 1963 he, joined Bell Telephone Laboratories, Inc. Holmdel, NJ, where as Supervisor of the Active Filter Group in the Data Communications

Laboratory he investigated methods of synthesizing hybrid-integrated

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-26, NO. 2, FEBRUARY 1979 105

linear and digital circuits for use in data transmission equipment. From MA. He is author of Linear Integrated Network: Fundamentals (New 1972 to 1973 he was on a leave of absence from Bell Laboratories and York: Van Nostrand-Reinhold, 1974) and Linear Integrated Networks: was Visiting Professor at the Swiss Federal Institute of Technology. Design (New York: Van Nostrand-Reinhold, 1975), has authored Since 1973 he has been Professor of Electrical Engineering and Director numerous papers in the field of ‘active network theory, design and of the Institute of Telecommunications at the Swiss Federal Institute of sensitivity, and holds several patents in these areas. His present interests Technology. During the summer of 1977 he worked in the Transmission are in the field of active and digital filters and signal processing. Networks Department at Bell Telephone Laboratories, North Andover, Dr. Moschytz is a member of the Swiss Electrotechnical Society.

‘Notes on vt-Dimensional System Theory DANTE C. YOULA, FELLOW, IEEE, AND G. GNAVI

Abstract-This paper makes three observations witb regard to several issues of a fundamental nature that apparently must arise in any general theory of linear n-dimensional systems. It is shown, by means of three specific interrelated counterexamples, that certain decomposit ion techniques which have proven to be basic for n= 1 and 2 are no longer applicable for n > 3. In fact, for n > 3, at least three equally meaningful but inequivalent notions of polynomial coprbneness emerge, namely, zero- coprimeness (ZC), minorcoprimeness (MC), and factor-coprimeness (FC). ‘Theorem 1 and 3 clarify the differences (and similarities) between these concepts, and Theorem 2 gives tbe ZC and MC properties a useful system formulation. (Unfortunately, FC, which in our opinion is destined to play a major role, has thus far eluded the same kind of characterization.) ‘Ibeo- rem 4 reveals tbat the structure of 2-variable elementary polynomial matrices is completely captured by the ZC concept. However, there ls reason to believe that ZC is insufficient for n > 3 but a counterexample is not at hand. ‘Ibe matter is therefore unresolved.

I. INTRODUCTION

T HIS PAPER contains three observations with regard to the structure of n-dimensional linear systems plus

a certain amount of peripheral discussion concerning their implications for System Theory in general.

In the first, 0,, we show by means of a specific counterexample that the recent important result due to Morf, Levy, and Kung [l] on the feasibility of primitive factori- zation for polynomial matrices ‘in two variables does not generalize to three or more variables (n > 3).

Manuscript received April 6, 1978. This work was supported by the Rome Air Development Center under Contract F30602-78-C-0048. G. Gnavi was supported by a fellowship grant from the Consejo National de Investigaciones Cienti’ficas y Technicas, Argentina.

D. C. Youla is with the Polytechnic Institute of New York, Farming- dale, NY 11735.

G. Gnavi is with the Polytechnic Institute of New York, Farmingdale, NY 11735, on leave from the Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina.

In the second, O,, we explore several possibilities for extending the useful concept of coprimeness of l-variable polynomial matrices to the n-dimensional case. It is found that from the point of view of synthesis, the situation changes profoundly for n > 3 and that the theory is beset with many of the same difficulties encountered in algebraic geometry.

In the third, O,, we illustrate the true importance of the ZC concept by employing it to give a complete characterization of 2-variable elementary polynomial matrices and a partial characterization for n > 3.

0,: Let A(zi,z2; . . ,z,,)=A(z) denote an m X m polynomial matrix in the n variables z;, i= I-+n, and let d(z) = det A(z) go.’ Suppose that d(z) = di(z)d2(z) where d,(z) and d*(z) are both polynomials. Then, for n > 3 it is not always possible to find two polynomial m x m matrices A ,(z) and AZ(z) such that det A;(z) = d(z), i = 1,2 and

A(z)=AdzMz)- (1)

Proof (by counterexample): Let 9 denote the collec-, tion of all polynomials f(z,,z2,z3) in three variables such that f(t3, t4, t’)EO. Clearly, ?? contains

f,=z;-z,z,

f,= 3 z2z3-zI

f3 = z; - z:z,. (2)

Actually, fi, f2, and f3 generate ?? in the sense that

‘det A(r) = determinant of A(z).

0098-4094/79/0200-0105$00.75 01979 IEEE

nodal analysis of switched-capacitor networks - isi

Documents