Darling Ton Synthesis Revisited

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14 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 1, JANUARY 1999

Darlington Synthesis RevisitedHerbert J. Carlin, Life Fellow, IEEE

(Invited Paper)

AbstractAn approach to Darlingtons synthesis is describedwhich is based entirely on a single Foster reactance (LPR)function. The procedure stems from the remarkable result thatany section producing an arbitrary transmission zero within itsclass (Brune, type C, type D, reciprocal, or nonreciprocal), canbe extracted from any prescribed LPR function, leaving an LPRremainder of reduced degree. The nonreciprocal sections play aprominent role in the discussion.

I. SOME HISTORICAL NOTES

ALMOST 60 years have passed since Sidney Darlingtonpublished his landmark contribution on insertion losssynthesis [1]. The circumstances for the genesis of Sids

article were somewhat unusual. The work was, of course,

carried out at the Bell Telephone Laboratories in Murray Hill,

NJ, but the paper itself is Darlingtons doctoral dissertation

for the Ph.D. degree granted at Columbia University, New

York, NY. Rather than being associated with an engineering

department, the research was under the formal jurisdiction

of the Faculty of Pure Science. The word formal is used

advisedly here. Columbias Physics Department was world

class, but assuredly there was no one in residence competent

to pass judgment on Sids work. Outside referees were called

on to evaluate the dissertation, and in fact these included the

now legendary Ronald M. Foster. A considerable number of

years later, Ronald told me that reviewing Sids dissertationwas a formidable task. Anyone who reads the article, even

today, can understand that as an understatement.

The idea for image parameter filters is based on the phe-

nomena of cutoff in a ladder structure made up of an infinite

cascade of identical sections, each composed of finite, lossless

elements. For example, a never-ending (series), (shunt)

ladder propagates signals up to a finite cutoff frequency,

beyond which there is no further transmission of signal energy.

Since fabricators of hardware insist on components of finite

size, cutoff, as exhibited by an infinite structure, was a concept

that spawned a veritable cottage industry of technical paper

production devoted to methods for achieving similar cutoff

characteristics in a ladder of finite extent with resistive ter-minations. Ultimately, highly sophisticated (and complicated)

analytic methods of analysis and design were constructed

including procedures for image parameter filters with elliptic

function response.

Sid Darlington started afresh by asking a straightforward

question that defined the real problem and bypassed all the

Manuscript received April 4, 1998.The author is with the School of Electrical Engineering, Cornell University,

Ithaca, NY 14853 USA.Publisher Item Identifier S 1057-7122(99)00538-3.

paraphernalia of cutoff in an unterminated infinite structure.

How do you build a finite two-port of reactors operating

between resistive loads, which transmits frequencies in a

passband and reflects them in an adjacent stopband? As

pointed out by Kuh and Sandberg, in the introduction to this

Darlington Memorial Issue, the answer came in two parts.

First, find a rational transfer function which approximates

the desired frequency response and is physically realizable.

Then synthesize the response function by a doubly terminated,

lossless ladder structure.

The opening article for this memorial issue also mentions

that Darlingtons insertion loss method was slow to catchon. But at the present time it is the universally accepted

synthesis cornerstone for the design of lossless analog filters,

equalizers, matching and frequency shaping networks, and

similar components. Perhaps Sids somewhat casual approach

to the derivation of the ladder sections needed for complete

realization of the overall two-port delayed acceptance. It is

not unlikely that the realizability of the Brune, type C, and

(particularly) of the type D sections was perfectly obvious to

him but not nearly so to others. Once his procedure became

established, dozens of new articles and textbook discussions

appeared presenting proofs for the extraction of the various

ladder sections. But it is probably fair to say that even at

the present time there are few demonstrations to be foundof the validity of the entire synthesis process, which are

trustworthy, complete, and not overly complicated. The point

of view of this paper is that difficulties arise in most of

the published material because the positive real (PR) input

dissipative driving point impedance to the terminated two-

port is usually taken as the vantage point for synthesis. The

approach here is to base the entire realization process on

the single lossless Foster reactance (lossless positive real or

LPR) input impedance function to the unterminatedstructure.

A Foster function is simpler to work with than a general

dissipative PR function, and its employment helps avoid many

of the difficulties usually encountered in establishing the

Darlington synthesis. Essentially this is due to the remarkable

result that if any transmission zero producing section (Brune,

Type C, Type D, reciprocal, or nonreciprocal), set for an

arbitrary zero within its class, is extracted from any LPR

function, then the remainder is a Foster function of reduced

degree, or a short or open circuit.

Sid was eclectic in his interests which ranged from every

aspect of electrical science to mountain climbing. I remember

once playing hookey with him from an IEEE meeting in San

Francisco. We were walking near Fishermans Wharf when Sid

10577122/99$10.00 1999 IEEE


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CARLIN: DARLINGTON SYNTHESIS REVISITED 17

(a) (b)

Fig. 1. (a) Brune section (M > 0 ) and type C reciprocal section ( M


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18 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 1, JANUARY 1999

Fig. 3. Ladder realization with reciprocal type D section for Example 2.

square. The matrix is compact at all poles, except that

includes the excess pole . Find the cascade representation

of .

First we find the D section for the complex zeros of

transmission. Extract from to force the

remainder to satisfy . Thus Equating

real and imaginary parts, , . (These are

guaranteed to always be positive.) If the resulting remainder

admittance is expanded in partial fractions

The impedance is an nonpassive shunt element (in

parallel with ) with zeros at the .

Expanding

The shunting b branch consists of the active series resonator

in series with the antiresonant passive circuit , all in

parallel with .

The negative coil can be eliminated by

separating into two parts, , such that the Tee

of inductors satisfies . Then

the resultant Tee is the equivalent of a pair of perfectly

coupled coils with self and mutual inductances .

We find . To eliminate the negative capacitor, let

, and form a Tee of capacitors with elastances

satisfying , so that . The

elastance Tee has as equivalent a capacitor followed by

an ideal transformer of turns ratio

The , arm, modeled by the type D section, produces the

complex zeros of transmission. The remainder in the output

arm, connected across the type D section, has impedance

. Finally, a third-order zero

of transmission at infinity is required. Therefore, the output

port is placed across (this has no effect on ), which

results in a double zero of transmission at infinity due to

. Then, an inductance , realizing the excess poleat , is placed in series with the output port, to raise the zero

multiplicity to . Note that a total of seven reactors have been

used. This is the same as the degree of , plus one more for

the excess pole. The circuit is shown in Fig. 3. The Type D

section is equivalent to that given by Fig. 2(a). This is shown

by introducing an output transformer (or changing the load

value) to cancel the transformer across . Additionally, this

requires a change in the value of the coupled coil elements. We

also need an ideal transformer across the parallel resonant

circuit, which in turn allows to be replaced by a capacitor

across the input of a pair of perfectly coupled coils.


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CARLIN: DARLINGTON SYNTHESIS REVISITED 21

constant

a simple pole with positive residue, and similarly for a

pole at .

A pole occurs in for finite when .

Therefore, in the neighborhood of ,

We have by (26), . In the neighborhood of ,

, and , with

since is LPR, so , .

Referring to (22)

where is a squared pure reactance. Thus in

the neighborhood of the pole

a simple pole with positive residue. Conclusion: is

LPR.

Finally, we note that because of the presence of

in , the remainder is raised by in degree. But

at and at , so that both

and have the same set of four zeros with quadrantal

symmetry. Cancellation results in a degree reduction of

, a net reduction of . Therefore, the final condition is

verified for the realizability of the nonreciprocal section

extraction from any LPR .

Once the nonreciprocal extractions have been established,

it is not difficult to provide an alternate (and mercifully brief)

proof of the cascade synthesis of reciprocal type C and D

sections. Consider a pair of nonreciprocal type C or type D

sections, the members of the pair to be removed in succession

(always realizable). For the first of the pair choose ;

the second is set to have (i.e., the gyrator ratios

for the two sections are opposite in sign though generally

unequal) with an RHP zero of transmission. A combined

pair has overall transfer functions ,

and . It is easily shown that

. Therefore, reciprocity holds, and each pair is clearly

lossless. Furthermore, the overall transfer impedance has

its RHP transmission zero(s) paired with the LHP image(s). A

composite section pair evidently forms a reciprocal type C or

D section followed by a realizable remainder, and the result

is proved.

REFERENCES

[1] S. Darlington, Synthesis of reactance 4-poles which produce prescribedinsertion loss characteristics, J. Math. Phys., vol. 18, no. 4, pp.257353, Sept. 1939.

[2] H. J. Carlin and P. P. Civalleri, Wideband Circuit Design. Boca Raton,FL: CRC, 1997.

[3] V. Belevitch, Classical Network Theory. San Francisco, CA: HoldenDay, 1968.

[4] P. I. Richards, Resistor transmission lines, Proc. IRE, vol. 36, pp.217220, Feb. 1948.

[5] D. Hazony, Elements of Network Synthesis. New York: Reinhold,1963.

Herbert J. Carlin (M47SM50F56LF83) wasborn in New York City. He received the Bachelorsand Masters degrees from Columbia University,New York, NY, and the D.E.E. degree from thePolytechnic Institute of Brooklyn (now the Poly-technic University of New York) in 1947.

After five years at Westinghouse Company, hejoined the Microwave Research Institute (MRI) ofthe Polytechnic Institute of Brooklyn and pursuedresearch in microwave circuits and network theoryunder Ernst Weber and Ronald M. Foster. He later

became Chairman of MRI. In 1966, he joined the Cornell College ofEngineering, Ithaca, NY, as the J. Preston Levis Professor of Engineering, andwas Director of the School of Electrical Engineering until 1975. He retiredfrom Cornell as Professor Emeritus in 1989 but continued his research under

programs sponsored by the National Science Foundation (NSF) and CRS(the Italian Research Agency). He has published more than 100 technicalpapers and is the coauthor with the late Anthony Giordano of the book

Network Theory (1964), and coauthor, with P. P. Civalleri of the bookWideband Circuit Design (Boca Raton, FL: CRC, 1997). He was awardeda Senior Research Fellowship by the National Science Foundation (1966),was a Visiting Professor at MIT (19721973), and has held numerous visitingappointments abroad at universities and research institutions in Italy, Ireland,France, Switzerland, Great Britain, Canada, Israel, Japan, and China.

Dr. Carlin is a member of the Electromagnetics Academy (Boston), formerChairman of the IEEE Circuits and Systems Society (CAS), and has receivedthe IEEE Centennial Medal (1984).

Darling Ton Synthesis Revisited

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