1956 IRE TRANSACTIONS ON CIRCUIT THEORY

Reviews of Current Literature.

295

This section of each issue is devoted to a few rather extensive reviews of recent papers related to circuit theory and pub- lished in journals other than the PROCEEDINGS. Suggested titles and reprints of papers for review are invited by the com- mittee.

-J. T. Bangert, Chairman Reviews and Abstracts Committee, PGCT Bell Telephone Labs. Inc., Murray Hill, New Jersey

Committee Members C. G. Aurell Chalmers Institute of Technology Gothenburg, Sweden

V. Belevitch 97 Rue Gabrielle Brussels, Belgium

N. H. Choksy Johns Hopkins University Baltimore 18, Maryland

W. H. Kautz Stanford Research Institute Stanford, California

D. G. Lampard National Standards Laboratory Sidney, Australia

Y. Oono Kyushu University Hukuoka, Japan

H. J. Orchard 22 Glenwood Ave., Kingsbury London, N.W. 9, England

J. F. Peters Nordwestdeutscher Rundfunk Hamburg 13, Germany

J. Rybner Royal Technical Institute of

Denmark Copenhagen, Denmark

The Continuous Delay-Line Synthesizer as a System Analogue- J. H. Westcott. (IEE Monograph, no. 176M; May, 1956.)

In this paper, Westcott first discusses several ways of synthesizing waveforms, the essential equipment being a tapped delay line and one or more integrators; he then considers the accuracy of the approximation for some cases typical of servo systems. This review will be confined to the general synthesis aspects: An attempt will be made to relate the paper to other work, particularly on z transforms, and also to indicate some generalizations which suggest themselves.

The synthesized waveform i??(t) is assumed to coincide with the true waveform H(t) at regular intervals; in this review, unit intervals are taken for simplicity. Thus, writing H, for H(n), .

IT(t) = c H”f(t - n) 0) n

where.f(t) is an interpolating pulse which, in order that A(n) = H(n), must have a z transform equal to unity. As is well-known, H(2) can always be adequately approximated in this way if its spectrum is negligible forf > 4. Given this condition, the goodness of the approx- imation can be related to the spectrum of f(t); the ideal interpolating pulse, sin rrt/rrt, is distortionless up to f = $. Westcott does not for- mulate explicitly the various interpolating pulses he makes use of but the frequency function (the “sampling weighting function”) which he takes as a criterion of performance is, in fact, the spectrum of the interpolating pulse. Westcott also uses “midinterval” approxima- tions where

R(t) = c IL--l,af(t - n + 3. n (2)

Interpolating pulses for the normal case, (I), have discontinuities (in the pulse or its derivatives) at integral values of t; for (2) the discon- tinuities are halfway between.

The procedure will be illustrated for a triangular interpolating pulse;f(O) = 1, and the base of the triangle extends from t = -1 to t = 1. The Fourier spectrum of IT(t) is, in general, from (1)

F{I-zT) = c H,z-~F{~} (3) 7t. where z = exp jw. For the triangular pulse

F(f) = (1 - z-‘)2/szz-’ (4)

where s = jw. An alternative expression for FJ?, can be deduced by considering the way the waveform is synthesized. A unit step is

applied to the delay line. The sum of the-outputs from the taps is then passed through an integrator. Hence

(5)

where, arL is the amplitude of the contribution from the nth tap. From (3), (4), and (5) the relation between the an and the ordinates of H(t) can be deduced. We have

a, = IL,+, - 2H, + H,-, = 6’H,

in central-difference notation. In this scheme of Westcott’s, in which only integrators are used

outside the delay line, the line is performing two functions: It is con- trolling the amplitude of the interpolating pulses, and, in combina- tion with the integrators, it is providing the right shape of interpolat- ing pulse. There might in practice be something to be said for partly separating these functions, by associating the integrators with a sepa- rate tapped delay line, the combination having, for the triangular pulse, a transform (1 -z-1)/s. Then

a, = H,,, - H, = 6Hn+,,2, This form might be easier to adjust, since there would be less inter- action between the effects of different tapping points.

Westcott confines himself to synthesizing step-like waveforms, a step-function being used for the imput. To synthesize impulse-like waveforms, it would be better to use an impulse for input. For a tri- angular interpolating pulse, we would then need a double integrator, or we could use a separate line to get (1 - z-~)~/s~, leading to

O?.-

a,, = 6’H,,

a, = H,+, respectively.

The interpolating pulses which Westcott uses are in fact the first few of two general classes (normal and midinterval) which can be ob- tained as follows. For the normal case

FLf,(t) I = z&-~)/s’,

u,(z-1) being a rational function. Hence, using Z( ) to indicate a .a transform, we have

296 IRE TRANSACTIONS ON CIRCUIT THEORY December

1 = z{fr} = t&-ljZ(l,lsr} so that

2&1) = l/Z{ l/s’). The z transforms of l/i’ are already available,il2 so we can obtain the f?(t). For the midinterval case

F ( gv( 2) ) = v,(z-‘)x--1’2/sr where 21, = 1/2(x-“‘/S’}.

Thus, for Westcott’s midinterval straight line approximation

It seems that the frequency spectra of the.f&) and the g,(t) approach the ideal in a maximally flat way. Some of the power series expan- sions can be obtained from Table I of a recent paper.3

The best of t,hc approximations discussed by Westcott is his “curved approximation” for which the interpolating pulse is thef&) of this reviewi the spectrum is

F(f ) = $1 - 2-Y (sin ~/TV 3 -= s”z y1 + 2-y cos af

which becomes infinite at f = f, corresponding to the fact that fa(t) includes an undamped oscillation. The corresponding midinterval case, for which

8(1 - 2-y

is in fact better and the spectrum is finite. The adjustment relation is

Han+l + Ban + a,-,)

if the line is followed by a double integrator

if the line is followed by a network with transform (1 -z-1)z/sz. It is worth noting that precisely the same hardware is used for

normal and midinterval cases. The difference lies purely in the selec- tion of points at which the synthesized waveform is’made to coincide with the desired waveform. Practical adjustment procedures would not necessarily stick rigidly to this point-by-point comparison, and might well lead to results which do not correspond to either case.

TV. E. THOMSON Post Office Research Station

London N.W.2, England

1 D. F. Lawden, “A General Theory of Sampling Servos,” IEE Monograph, no. 4; 1951.

1 J. G. Truxal, “Numerical analysis for network theory,” IRE TRANS., vol. CT-l, pp. 49-60; September, 1954.

3 R. Boxer and S. Thaler, “A simplified method of solving linear and nonlinear systems, PROC. IRE, vol. 44, pp. 89-100; January, 1956.

4 At one point, Westcott calls this a mldinterval approximation, but this may be a slip.

Analysis of Linear n-Port Networks-I. Cederbaum. IEE Mono- graph, no. 163R; January, 1956.

The external behavior of a linear n-terminal-pair network is completely specified by n linear simultaneous equations involving the voltages and currents at the terminal-pairs (or ports). Altogether,

there are ( 1

c different sets of independent equations, reflecting the

various possible ways in which the independent variables can be chosen. Each such set is equivalent to, and can be deduced from, any other set.

If the coefficients in one set are known, it is possible, by algebraic manipulation, to obtain the coefficients for any of the others. As with so many problems of this kind, without proper organization, the alge- braic manipulation becomes classified as “straightforward but la- borious.” This paper by Cederbaum supplies us with the necessary organizing machinery for the restricted cases in which one and only

one variable at each terminal-pair is chosen as an independent vari- able. There are then only 2” different possible sets of equations.

For the sake of argument the author assumes that, in the known set, all the independent variables are currents and consequently all the coefficients have the dimensions of an impedance. It is first recog- nized that, in any other set of equations, each coefficient will be the quotient of two cofactors taken from the determinant of coefficients in the known set. For any particular set the denominator cofactor is the same for all coefficients and is obtained from a consideration of what variables are being interchanged.

The essence of the symbolic notation introduced by the author is a met,hod whereby the different numerat,or cofactors can be found quickly and easily. Solely from a knowledge of what vnriables are being transposed, one writes down first a matrix of index pairs. Each index pair is interpreted as an operator and the whole matrix is then arranged to operate on the denominator cofactor. The result is a matrix of numerator coefficients.

Rules are given for the different ways that an index pair can mod- ify the cofactor on which it opcrgtes, depending upon the numbers defining the index pair and the cofactor. The order of the cofactor may remain unchanged or else it may be increased or decreased by unity.

The value of-a new notation of this kind cannot be assessed accu- rately without some experience in applying it t,o practical problems up to the stage where the user has acquired considerable familiarity. The author gives a few simple examples of its use to illustrate what can he done and it remains for readers to give it a try. A first reading certainly suggest that it may be very helpful.

H. J. ORCHARD Post Office Research Station

London, N.W.2, England

Inter-reciprocity Applied to Electrical Networks-J. L. Bordewij k. Appl. Sci. Res. (Netherlands) vol. 6, sec.B, no. l-2, pp. l-74; 1956.

Networks satisfying the reciprocity theorem have’ symmetric Z (impedance), Y(admittance) and #(scattering) matrices. If V,, and I, are the voltage and current one-column matrices at the terminal pairs of a 12 port, and if Vb and Zb are similar matrices for another electrical state, reciprocity requires (in matrix notation, and denot- ing transposition by a prime)

I:v* = r;v, (1)

which amounts to the symmetry of the bilinear form I’, Z Zb. It is convenient to consider (1) as the definition of reciprocity, since this avoids committing oneself with any specific n port ma,trix.

For general (nonreciprocal) n ports,

ILZI, = ILZ’I, ‘(2) is an identity, for (2) is a scalar, and this still gives (1) if one defines

Vb = ZbIb; v, = Z’I,,

i.e., if one relates two different electrical states (a, b) of two different networks (of matrices Z and Z’ respectively). The network Z’ is called the transpose of the network Z and the relations (1) and (2) determine an interreciprocity between the two networks. The paper shows how to construct the transposo of a nonreciprocal network and shows that this new concept is quite useful to simplify the derivations of many properties of amplifiers. In fact, this method is very similar, and yields exactly the same advantages, as the use of adjoint linear systems in many branches of mnthemat,ical physics, although this identity does not seem to have been recognized.

The construction of the transpose of a network (within equiva- lence) is obtained by transposing all nonreciprocal components four- poles, since reciprocal components (including pure connections) are their own transposes. The elementary nonreciprocal components are gyrators and unilateral devices such as ideal triodes or transistors. Transposition merely introduces a phase reversal in a gyrator. For unilateral devices, the following classification is proposed:

1) Ideal triode (pentode): II = 0; I, = SV,. 2) Dual triode (transistor with low rc with emitter grounded):

VI = 0; Vz = RI,. 3) Ideal transistor: Vr = 0; Zz = (Y II. 4) Dual transistor (cathode follower triode): Ii = 0; VZ = 0 VI.

Transposition then merely reverses input and output in (1) and (2),