A singly-terminated highpass filter designed using the abovemethod is given in Reference [11] as an example of anotherspecial type of transfer function.

-1.5.16k

frequency, Hz

Fig. 7 Computed performance of new (upper curve) and standarddesigns

4 Conclusions

It has been shown that a method of lossy ladder synthesiswhich is based on a partial knowledge of the driving-pointimpedance is feasible and advantageous. Three types of filtercan be designed using the technique. Doubly- and singly-resistively-terminated filters are discussed together with filterswhose terminations are impedances. Filters with both lossyand lossless resonant sections are possible, and zeros of trans-mission on, or to the left of, the imaginary axis in the s-planeare realisable.

The method is reduced to an algorithm, which is easy toprogram for a computer.

Two examples are given, and a reference to a third, inwhich it is demonstrated that good results are obtainable inpractice.

5 Acknowledgment

Acknowledgment is made to the Director of Research ofBritish Telecom, part of the Post Office, for permission topublish this paper.

6 References

1 PRESTCOTT, A.J.: 'Loss-compensated active gyrator using differen-tial-input operational amplifiers', Electron. Lett, 1966, 2, (7),pp. 283-284

2 ROLLETT, J.M.: 'Economical RC active lossy ladder filters, ibid.,1973, 9, (3), pp. 70-72

3 DESOER, C.A: 'Notes commenting on Darlington's design pro-cedure for networks, made of uniformly dissipative coils (d0 +5)and uniformly dissipative capacitors (d0 —6)', IRE Trans., 1959,CT-6, pp. 397-398

4 NIGHTINGALE, C: 'Elliptic type transfer functions and the dualityof points and lines'. IEE Colloquium on electronic filters, Digest 37,1977, pp. 4/1-4

5 BASKOV, Ye. I., and LEBEDEV, A.T.: 'Optimisation of thecharacteristics of electrical filters using lossy components', Tele-commun & Radio Eng., 1969, 23, pp. 49-55

6 LEE, H.B., CARVEY, P., and EVANS, D.: 'Program refines circuitfrom rough design data', Electron., 1970, 43, pp. 58-65

7 NIGHTINGALE, C, and ROLLETT, J.M.: 'Exact synthesis ofactive lowpass frequency-dependent negative-resistance filters',Electron. Lett. 1974, 10, (3), pp. 34-35

8 SAAL, R., and ULBRICH, E.: 'On the design of filters by synthesis',IRE Trans., 1958, CT-4, pp. 284-327

9 BROYDEN, C.G.: 'A new method of solving nonlinear simultaneousequations', Comput. J., 1969, 12, pp. 94-99

10 NIGHTINGALE, C, and ROLLETT, J.M.: 'A method for designingdoubly-terminated all-pole ladder filters with reactive componentshaving freely-assignable losses', Electron. Lett., 1972, 8, (18), pp.461-463

11 NIGHTINGALE, C: 'A class of transfer functions having finitestopband attenuation in which both passband and stopband exhibitequiripple behaviour', ibid., 1974, 10, (8), pp. 381-383

12 BRUTON, L.T.: 'Network transfer functions using the concept offrequency-dependent negative resistance', IEEE Trans., 1969,CT-16, pp. 406-408

Design of 4th-order monolithic crystal filtersK.K. Tuladhar, B.Sc, B.Sc.(Eng.), Ph.D.(Eng.), and B.D. Cox, C.Eng., M.I.E.E.

Indexing terms: Crystal filters, Narrowband approximation, 4th-order A T cut resonators

Abstract: Simple direct formulas are given for the design of 4th-order AT-cut resonators or monolithic dualfilters. It is shown that the matching transformers or annulling coils are not required to achieve a desiredimpedance.

1 Introduction

To date, polylithic filters, which use a number of 4th-orderAT-cut resonators or monolithic duals are synthesised for achosen motional inductance of the resonator. This, in turn,produces a natural impedance of the filter. This naturalimpedance is not often the desired, or preferred, impedance.The desired, or preferred impedance is achieved by using eithera pair of matching transformers or a pair of coils to annul thestray capacitance at the input and output terminals. Thistechnique is satisfactory for the filter which uses two or more4th-order resonators. However, for the single 4th-order res-onator filter, this technique is not satisfactory for the follow-ing two reasons. First, two matching transformers or twoannulling coils are required, and secondly these transformersor coils and the 4th-order resonator have to be packed into a

Paper 1418G, first received 23rd January and in final form 11th May1981Dr. Tuladhar is, and Mr. Cox was formerly, with Cathodeon CrystalsLtd., High Street, Linton, Cambridge CB1 6JU, England

bigger can. The second factor makes the size of the filter large,hence it is difficult to miniaturise. The best solution to thisproblem is to design a 4th-order resonator which is matchedto the desired, or preferred, impedance. The method ofdesigning such a dual will now be given.

2 Design procedure

The design procedure requires centre frequency fo, bandwidthBW, desired or preferred impedance Rs, Cs and Rh Cx of the4th-order resonator filters.

The design procedure is as follows:(a) Consider a 2nd-order lowpass prototype, Fig. 1A,

out

a= /L/C

Fig. 1A Two conventional 2nd-order lowpass prototypes

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where the T ' of frequency independent reactances representsan impedance inverter. The values for L and a can readily beobtained for a chosen allpole 2nd-order lowpass prototype,either from the tabulated results [1] or from well knownexplicit formulas [2].

(b) By introducing an extra frequency independent reac-tance jb, the lowpass prototype given in Fig. 1A can be suit-ably modified to Fig. IB. It should be noted that these twonetworks, Figs. 1A and B, are identical, and the naturalimpedance of the filter is defined as the impedance of thefilter when b - 0.

(c) As the bandwidth BW and centre frequency fo areknown, the resulting 4th-order network obtained by perform-ing lowpass to bandpass transformation is given in Fig. 2A,where

fo = I/^TTVZ/C7)

L' = LR/(2nBW)

R is the scaling factor and can be determined.

'out

Fig. 1B Modified 2ndorder lowpass prototype using networktransformation given in Fig. 1C

-jb -jb

r—O-O—= 1U = ( U

Fig. 1C Growth of frequency-independent reactances and usefulnetwork transformation

(d) Now, using a narrowband approximation, the impedanceof the network in Fig. 2A can be equated to the desired, orpreferred impedance of the 4th-order resonator filter inFig. 2B.

Thus,

(l+b2)-R = Rs = Rt

and

(l+b2)-R/b = 1/(2TT/OCT)

(Ub2)R

^ _ «

r)jQR - ^ out

Fig. 2A 4thorder bandpass filter circuit obtained by performinglowpass to bandpass transformation to Fig. IB circuit

cm cm "-m ^m u m

cs| + cc7Dlitni1 MR.Vout

Fig. 2B 4th-order resonator bandpass filter matched to desiredimpedance

where

Or = Cs + Co = Ci + Co

b = 2nfoCTRs

and

R = RsKl+4ir2f2C^R2s)

Because Rs >R, it should be noted that the matching can onlybe to a higher resistive termination.

(e) Hence b and R can now be computed, as / o , CT,RS areknown. Again, using narrowband approximation, the 4th-orderresonator parameters can now be derived, as follows:Antisymmetric frequency

fa =

= fo + (a-b)BW/2LSymmetric frequency

= fo-(a + b)BW/2L/s =

and motional inductance Lm = l!.(f) Now, by knowing Lm,fa,fs of the 4th-order resonator,

its physical design can easily be carried out, using publishedresults [3 -6] .

3 Design example

A 4th-order resonator filter at 10.7 MHz will be designed todemonstrate the design procedure described previously. Thechosen design is based on a 1.0 dB ripple Chebyshev lowpassprototype, and the values of L and a, normalised to 3 dBpoints, are 2.218435 and 1.630866, respectively.

Suppose it is required to match the 4th-order resonatorfilter to 1.9 k£2 and 4.3 pF and the minimum passband band-width required is 9 kHz. A typical Co of the 4th-order res-onator is 1.8pF. Therefore, the values of b and/? are, respect-ively, 0.779197 and 1182.22. Hence l '=46.379mH. Theresulting values of fa and fs are 10.701728 MHz and10.695111MHz.

The computed magnitude response for the final design isgiven in Fig. 3. Curve A of Fig. 3 corresponds to ideal losslesscase, whereas curve B of Fig. 3 corresponds to nonideallossy case producing an insertion loss of 0.15 dB for theresonator Q equal to 60 x 103.

4 ConclusionsSimple design formulas, which permit the rapid design of the4th-order resonator or monolithic dual filter have beenpresented. Unlike conventional designs, the troublesomematching transformers or annulling coils are not required toachieve the desired, or preferred impedance.

frequency, kHz.from 107 MHz (scale 2)-25 -20 -15 -10 -5 0 5 10 15 20 25

-5 -U -3 -2 -1 0 1 2 3 U 5frequency, kHz.from 107 MHz (scale 1)

Fig. 3 Computed magnitude response of 4th-order resonator filter

IEEPROC, Vol. 128, Pt. G, No. 4, AUGUST 1981 235

5 Acknowledgment

Part of this work was carried out when one of the authors(KKT) was seconded to Philips Research Laboratories, Redhill,Surrey, England, and some help given there by R.F. Milsomis acknowledged.

6 References

1 ZVEREV, A.I.: 'Handbook of filter synthesis' (Wiley, New York,1967)

2 RHODES, J.D.: 'Theory of electrical filters' (Wiley, London, 1976)3 SYKES, R.A., SMITH, W.S., and SPENCER, W.J.: 'Monolithici crystal filters', IEEE International convention record, 20th-23rd

March, 1967, pp. 78-934 BEAVER, W.D.: 'Theory and design of the monolithic crystal

filter'. Proceedings of 21st Annual frequency control symposium,1967, pp. 179-199

5 REDWOOD, M., and REILLY, N.H.C.: 'New method of providingcoupling between resonators in an electromechanical filter',Electron. Lett., 1966, 2, (6), pp. 220-222

6 REILLY, N.H.C.: 'Wave propagation analysis of the evanescentmode piezoelectric filter'. Ph.D. thesis, University of London, 1969

Waveform correctors for use in reduced bandwidthvideo systems in the local network

P. Challener, B.Sc.Indexing terms: Cables, Television, Waveform correctors

Abstract: The paper describes the construction and limitations of a type of waveform corrector suitable foruse on video systems in the local telephony network. Component values are included for route lengths up to1.5 km.

1 Introduction

A problem which arises when transmitting video signals overcable pairs is linear waveform distortion owing to the ampli-tude/frequency and phase/frequency characteristics of thecable. It is usual to reduce the effects of these non-idealfrequency-domain characteristics by means of equalisingnetworks sited at repeating amplifiers along the cable route.However, because video signals consist essentially of wave-forms, and it is not always simple to transform waveformtolerances into the frequency domain, most correctionmethods for video systems allow the equaliser components tobe adjusted to satisfy a time-domain waveform test. Such atime-domain approach is known as waveform correction and isdistinct from the term equalisation, which implies a purelyfrequency-domain approach. The waveform most commonlyused for waveform correction is the pulse and bar, and distor-tions of this waveform can be expressed in terms of ^-ratingfactors [1]. Such factors provide objective measures of wave-form distortions, having the property that a given magnitudecorresponds to about the same level of subjective impairmentin the received picture, irrespective of the exact nature of theeffect produced.

The simplest method of correcting a video signal trans-mitted over a cable pair is to use a tandem connection of1st-order 'nonresonant' equalisers. Combination of suchnetworks in ladder form further simplifies realisation of therequired characteristics, and it has been shown that there isalso a reduction in the range of component values required[2]. A practical amplifier arrangement utilising these ladder

Vout

IFig. 1 Amplifier circuit for use with ladder correctors

= 1/(1 +Rf/Z)

Paper 1436G, first received 30 th January and in revised form 4th J u n e1981The author is with the Visual Telecommunications Division of BritishTelecom Research Laboratories, Martlesham Heath, Ipswich IPS 7RE,England

structures is shown in Fig. 1 and is used throughout thispaper, but other forms are described in the literature [3]. Incommercially available equipment, the amplifier itself isusually a discrete component device capable of a high gain-bandwidth product.

Because networks such as that shown in Fig. 1 cannotproduce phase-shifts in excess of 90° or attenuation slopes inexcess of 6dB/octave, there are limits on the total length ofline which can be successfully corrected with a single ladder;consequently, most correcting amplifiers allow several separateladder equalisers to be cascaded to provide the requiredcharacteristics.

2 Television standards for the local network

Most closed-circuit TV links offered by British Telecom atpresent utilise special cables such as polythene quad cable orcoaxial cable to transmit 5.5 MHz TV at 625 lines/picture.These cables are costly to instal but offer good transmissionproperties for video signals without allowing a large amountof crosstalk to occur. Because the characteristics of suchcables are well known and readily repeatable, it is possible toutilise fixed 'block' equalisers for waveform correction withoccasional 'mop-up' correctors aligned on a waveform test toensure satisfactory waveform fidelity over the entire link.However, it is known that the local telephony network alsois capable of supporting a limited video service, restricted insize by crosstalk, and this exploitation of existing plantcould have a substantial cost advantage over the use of specialcables for many applications. In order to obtain a reasonablerepeater spacing in the presence of crosstalk, it is thought thata 1 MHz upper frequency should be imposed on the videosignal, and by restricting the system to 313 lines/picture, areasonable balance between vertical and horizontal pictureresolution can be achieved in this bandwidth. In this way, arepeater spacing of 1.5 km can be readily achieved on mosttypes of local cable.

Equalisation of video signals on local cables is difficult,because cable routes seldom consist of single gauge conductorsover an entire link, and it is often difficult to predict theircharacteristics with any accuracy. Hence it is not possible touse block equalisers alone to provide waveform correction.Furthermore, the characteristic impedance of each part ofthe cable will rarely be consistent over the entire link and thiscauses problems in defining the terminations as well as giving

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