Circuit realisation of products of grounded multiportmatrices, using multiport immittance convertors

Georgi A. Nenov, Dozent, C.Sc, Dipl. Ing.

Indexing term: Circuit theory and design

Abstract: A method for circuit realisation of products of arbitrary grounded multiport matrices is proposed.The synthesised structures consist of multiport immittance convertors (MICs) and of grounded multiports,the matrices of which have to be multiplied. As active elements in the MIC circuits, conventional operationalamplifiers (OAs) and transconductance amplifiers (TAs) are used.

1 Introduction

It is well known that the direct realisation of a product of 2 m-port-network matrices is possible using only cascade con-nection of the corresponding 2m-ports. The transmissionmatrix in this case is a product of the transmission matricesof the 2m-ports.

Recently, however, a few papers concerning circuit reali-sation of products of 2-port immittance and (or) hybridmatrices have been published [1—3]. The results obtainedare undoubtedly positive, but they impose two essentialrestricitions; namely: (i) only products of 2-port matricescan be realised, and (ii) all the 2-port matrices in the commonstructure, except one, should be ungrounded [2]. These re-strictions follow from the practical requirement that, as activeelements in the structure, only OAs with differential inputsand with grounded outputs are to be used.

In the paper presented, a method for circuit realisationof products of arbitrary grounded r-port matrices is proposed.The aforementioned restrictions are overcome by the uti-lisation of operational amplifiers (OAs) and transconductanceamplifiers (TAs) in the synthesised MIC structure. For thispurpose, the circuit elements generalised nullator (ng) andgeneralised norator (Ng) are introduced, and they are used forthe multiport variables conversion.

The results obtained are a precondition for the synthesisof grounded multiports, with matrices which are presentedas products of corresponding transmission, hybrid andimmittance matrices.

2 Analysis of the multiport matrix product

The grounded r-port Mo is shown in Fig. 1. It consists ofan {n + 1) r-port immittance convertor (MIC) and of n ground-ed r-portsMj ,M2,. . . ,Mk, . . . ,Mn.

Let us suppose that, for this structure, the followingexpression is valid:

where M(k), k = 0, 1, 2, . . . , « , are transmission, hybridor immittance (r x r)-matrices of the grounded r-ports Mo,Mt,:..,Mk,... ,Mn;t

{k), k = 0, 1, 2,...,n, are con-nection (r x r)-matrices of the converting grounded r-portsTo, T1,\..,Tk,...,Tn*. The r-ports Tk form the MICin Fig. 1. The matrices in expr. 1 satisfy the equations:

(2)

T h e connection matrices t have mathematical meaning and ther-ports 7fe corresponding to them may not exist as separate circuits.

Paper 2128 G, first received 11th September 1981 and in revised form7th July 1982The author is with the Higher Institure of Chemical Technology,Bourgas, 8010 Bourgas, Bulgaria

s = 1 , 2 , . . . , « - !

ke{Q,s, n)

(3)

(4)

(5)

Here, W^k) is a vector of dependent variables of the r-portMk and Wfk) is a vector of independent variables of thesame r-port. Each of the variables in W^ and W/fe) is avoltage or a current. For example, if_M(te) is an (r x r) ad-mittance matrix, the vector W$° =~ [/?/ /<&> ...I$>]t

consists of the port currents of the multiport Mk and thevector Wlk)=[VWVtfK..V}V]t consists of the portvoltages of the multiport Mk (t denotes transposition). Ifat least one of the matrices M(fe) is a transmission 2 m-portmatrix, r = 2m is even, but if expr. 1 contains no trans-mission matrices, r is even or odd.

r1 \ * * / I \ w/

0—L—V,(0)

(0)

(0)

(0)

r(0)

0-Vr(0)

« 0

MIC

i d ) V|O)

1 ( 1 )

r(1) r(1)

-—0-Vr|o)

2(2)-0-

0-

(2)r

0

M2

02(n) Vz(

If(n) r(n)

-——0",(n)

i I LJFig. 1 Common strucutre for realisation of the matrix product

When we assume that the reference directions for positivecurrents are against to the port terminals of the multiportsMo, Mx,...,Mn, each of the connection matrices tik) hasthe form:

(6)

(7)

10 0143-7089/83/010010 + 05 $01.50/0 IEEPROC, Vol. 130, Pt. G, No. 1, FEBRUARY 1983

C —

af

cf =

r(k)

diag(dg>p+<r

V%>/V<$

t-p+q + 2

u) — 1dj I1 a)

bf = VV/lfr"

s=\, 2 , . . . , w - l ; y = l , 2 , . . . , / , / + 1,

Here, af\ bf\ cjk) and df> are real and

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

,l + p,l + p

vff

(20)

where R Y and /? d' are the sets of the independent and thedependent voltages, respectively, and R\ and Rd are the setsof the independent and dependent currents, respectively.The abbreviation 'perm' (i.e. 'permutation') in eqn. 6 meansthat it is possible to arrange the submatrices of eqn. 7 in eqn.6 in a different way, depending on arranging the variables inthe vectors W\k) and W$\

For the multiport immittance covertor (MIC) in Fig. 1,the following equation is valid:

W'd =

Here:

K = ... w\f}t

if = d i a g ( f ( 0 ) ' r ( 1 ) ' . . . r ( " ) ' )

and

diag (l<n) - l(b

n) Un) -

diag ( I f - I P - 1<S) 1? ) .

(21)

(22)

(23)

(24)

(25)

(26)

(27)

for s = 1, 2 , . . . , « — 1. The size of the unity matrices l£fc),l(b

fe), l£fe) and 1^} in eans. 25 to 27 coincides with the size ofthe matrices a , b^k , c(fe) and d^k\ respectively. Conse-quently, the matrices r(0),r(n)/ and f(s)' can be obtained fromthe corresponding matrices t®\ t^ and r(s) after change ofthe signs of some of the submatrices a(fe), A(fe), c(k) and d(k)

(k = 0, s, n) in them. This change follows from the assumptionthat the positive current reference directions for MIC areagainst its port terminals and, accordingly, these directions

are opposite to the corresponding reference directions for ther-ports Mk.

Because, in the common case, each pair of elements Wtj andW^ in the vectors W\ and W'd does not correspond to the sameport, matrix T in eqn. 24 usually is not a hybrid matrix [4].By using appropriate rearrangement of the elements in W'itand Wd, however, matrix T can be transformed to hybridmatrix//. This possibility is shown later, in the example.

3 Generalised nullator and generalised norator

The idealised 1-ports 'nullator' and 'norator', shown in Fig. 2,are widely utilised in network analysis and network synthesis[6]. It is well known that the nullator (n) satisfies theconditions:

I2 = 0Vx = V2 h = 0

whereas, for the norator (TV):

Vx — arbitrary V2 — arbitrary

11 = —/2 — arbitrary

Obviously, exprs. 28 and 29 can be written, respectively, as:

V2IVX = 1

fx = 0,12 = 0 (/2 / / , - indefinite)

and

(28)

(29)

(30)

Vx —arbitrary V2 — arbitrary ( V2 / Vx —indefinite)

h\h = " I (31)

Let us generalise exprs. 30 and 31, as follows:

V2lVt = a

A = 0,/2 = 0(/2//!-indefinite),

and

(32)

Vx -arbitrary V2 — arbitrary (V2/Vx -indefinite)]

hlh = d j(33)

where a and d are real. Then exprs. 32 correspond to a general-ised nullator («g) and exprs. 33 correspond to a generalisednorator (Ng). The circuit realisations of generalised nullatorsand generalised norators using nullators, norators and resistors,and the corresponding relationships, are given in Table 1. Thepractical implementations of the proposed generalised nullatornorator structures are based on utilisation of the nullor equiva-lent circuits of OA and TA [5] in Fig. 3. If, however, someelements in the circuits la, 2a, 3b and 4b in Table 1 arereplaced by the circuits in Fig. 3b to d, the remainder, in thiscase ungrounded norator, cannot be realised using the circuitin Fig. 3a. That is why only four relatively simple 0A:TA

o"n"

y 1 - N "

Fig. 2 Nullator and norator

IEEPROC, Vol. 130, Pt. G, No. 1, FEBRUARY 1983 11

Table 1: Nullator-norator circuits of generalised nullators and norators

Element Nullor structures Relationships

a = - /?:/ /?!/, = 0;/2 = 0/ 2 / / , -indefinite

V2IVX =aa = R2/R{

/ , - 0 ; / 2 0

/ 2 / / i -indefinite

V, -atbitraryV2 -arbitraryV2/Vi -indefinite/ , / / , -dd = RJR2

V, -arbitraryV2 -arbitraryV2 / \ / , -indefinite

c/= RXIR2

9m y, v

•0

Fig. 3 Nullor OA- and TA-equivalent circuits

V\ =b = c = -gm

structures of ng and Ng, given in Fig. 4, will be used further.Here the circuits (a) (b) (c) and (d) of Fig. 4 correspond tocircuits \b, 2b, 3a and 4a in Table 1, respectively. It is im-portant to point out that, for 0 = 1 , the ng circuits 2a and bin Table 1 are equivalent to a series nullator and, for d = — 1,the Ng circuits 3a and b in the same Table are equivalent toa series norator.

4 Synthesis of MIC

To synthesise the multiport immittance convertor (MIC), thematrix T must be realised in accordance with relationships8 to 19. These relationships should be implemented in thecircuit independently of one another. For example, the

12

I, . 9m =«

c _J_ d

Fig. 4 OA: TA realisations of generalised nullators and norators

IEEPROC, Vol. 130, Pt. G, No. 1, FEBRUARY 1983

circuit which states the relationship between 2-port voltagesshould not affect the corresponding port currents, or viceversa. Obviously, realisation of the relationships 8, 12 and 16needs generalised nullators, whereas realisation of relationships11,15 and 19 needs generalised norators, and hence

a e {a)h)) d E {d)k)} (34)

The circuit implementation of the relationships 9, 10, 13,14, 17 and 18, which correspond to voltage-to-current andcurrent-to-voltage conversion, must also satisfy the precedingindependent conditions. For this purpose, the nullor TA-circuits, shown in Fig. 3c and d are suitable, and hence

b e [b)k)) c E {c)k))

In Table 1, Fig. 3 and Fig. 4, we assume:

VUV2 SR

(35)

(36)

In the synthesis of the practical MIC circuit, every free OAinput of a generalised nullator and every free grounded OAoutput of a generalised norator form an operational amplifier.

5 Example

Realise a grounded 4-port with hybrid matrix which is aproduct of a transmission, an admittance and an impedancematrices, according to expr. 1, for

= 4

and

Mio) = h

(37)

(38)

= a

Mi2) = y

(39)

(40)

= z

(41)

From exprs. 6 to 19, we can write

for

fl(0) = diag(l 1),

,<O =

for

b{i) = diag (1 \),d{1) = diag (1 1) ,

r(2) = diag ( 1 1 1 1 ) = a(2)

IEEPROC. Vol. 130, Pt. G, No. 1, FEBRUARY 1983

(42)

(43)

(44)

r ( 3 ) =for (45)

Using exprs. 38 to 45, and according to exprs. 21 to 27, oneobtains:

(46)

(47)

(48)

KggT = diag ( 1 1 - 1 - 1 - 1 - 1 1 1

1 1 1 1 - 1 - 1 - 1 1 )

Having in mind the transforming structures in Figs. 3 and 4and matrix 48, we obtain the MIC circuit, given in Fig. 5.

.0)

Fig. 5 Synthesised multiport immittance convenor (MIC)

Substituting this circuit in the structure in Fig. 1, for r = 4,n = 3, we find, after analysis, that the hybrid matrix withrespect to the port terminals l(0), 2(0), 3(0) and 4(0) is

h = My . . /(3> (49)

If, instead of the ordering in exprs. 46 and 47, the MIC-portvariables are ordered as follows:

K = v®

1? /Si3) K g K g /S43)],the vectors 50 and 51 satisfy the equation

K =

(50)

(51)

(52)

13

where

H =

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

- 1

0

0

matrix

0

0

0

0

0

0 -

0

0

0

0

0

0

0

0

- 1

0

of the MIC

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

- 1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

- 1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

- 1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

References

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

- 1

0

0

0

0

0

0

0

0

0

0

0

0

(53)

6 Conclusions

A method for circuit realisation of products of arbitrarygrounded multiport matrices is proposed. It consists of inde-pendent conversion of the variables (voltages and currents) ofthe multiports by using generalised nuUators (ng), generalisednorators (Ng) and voltage-to-current and current-to-voltagetransforming circuits. The practical circuit implementation ofa matrix product consists in multiport immittance convertor(MIC) synthesis, using operational amplifiers (OAs), trans-conductance amplifiers (TAs) and resistors.

The theoretical results obtained are applicable to thesynthesis of grounded active multiports with preliminarygiven transmission, hybrid or immittance matrices, presentedas matrix products.

1 POSPISIL, J., and MOOS, P.: 'Nonconventional two-port matrixdecompositions and their realizations'. Proceedings of the SSCT-77,Kladno, 1977, pp. 250-255

2 MOOS, P.: 'Utilization of nullors in the realization of sum ordifference decomposition of cascade matrix'. Proceedings of the 2ndconference on electronic circuits, Praha, 1976, pp. 219-220

3 MOOS, P., and POSPISIL, J.: Nonconventional two-port matrixdecompositions and their realizations', Int. J. Circuit Theory &Appl., 1980, 8, pp. 13-18

4 SILVA, M.S.: 'Multiport convertors and invertors', ibid 19786, pp. 243-252

5 LENK, J.D.: 'Manual for operational amplifier users' (Reston,Virginia, 1976)

6 HEINLEIN, W.E., and HOLMES, W.H.: 'Active filters for integratedcircuits' (R. Oldenbourg Verlag, Munchen-Wien, 1974)

Georgi A. Nenov was born in Bourgas,Bulgaria in 1939. He received the degreesof Electrical Engineer and Candidate ofTechnical Sciences from the HigherInstitute of Mechanical and ElectricalEngineering of Sofia, Bulgaria, in 1962and 1973, respectively. From 1963 to1966 he was an assistant in the HigherInstitute of Mechanical and Electrical \Engineering in Varna, Bulgaria. In 1966he joined the Institute of InstrumentDesign in Sofia. From 1974 to 1980 he worked in theBulgarian Academy of Sciences and in the Institute of Radio-electronics, Sofia. He is now an Associate Professor of Elec-trical Engineering and Electronics in the Higher Institute ofChemical Technology in Bourgas. His research interests are inthe field of active filters, digital filters, switched-capacitorfilters and network analysis and synthesis.

14 IEEPROC, Vol. 130, Pt. G, No. 1, FEBRUARY 1983