Realization of a Constant Phase Difference 



By SIDNEY DARLINGTON 



This paper bears on the problem of splitting a signal into two parts of like am- 

 plitudes but different phases. Constant phase differences are utilized in such cir- 

 cuits as Hartley single sideband modulators. The networks considered here are 

 pairs of constant-resistance phase-shifting networks connected in parallel at one 

 end. The first part of the paper shows how to compute the best approximation 

 to a constant phase difference obtainable over a prescribed frequency range 

 with a network of prescribed complexity. The latter part shows how to design 

 networks producing the best approximation. 



A PERENNIAL problem is that of designing a circuit to split a signal 

 into two parts which are the same in amplitude but which differ in 

 phase by a constant amount. A 90-degree phase difference is needed, for 

 example, in the single sideband modulation system due to R. V. L. Hartley.^ 

 It is well known that it is not possible to obtain exactly equal amplitudes 

 and exactly constant phase differences at all frequencies except in the 

 trivial special case of a 180-degree phase difference. Various methods have 

 been devised, however, for approximating these characteristics over finite 

 frequency ranges. The most obvious method is to use a pair of constant 

 resistance phase shifting sections in parallel at one end and with separate 

 terminations at the other end^ as indicated in Fig. 1, 



This paper is devoted to the problem of obtaining approximately constant 

 phase differences under the specific assumption that pairs of constant re- 

 sistance phase shifting networks are to be used. The paper has been written 

 with two objects in mind. The first is the development of a method for 

 determining the best approximation to a constant phase difference which 

 can be obtained over a prescribed frequency range with a pair of phase 

 shifting networks of a prescribed total complexity. The second object is 

 the description of a straightforward design procedure by means of which 

 the networks can be designed to give this best possible approximation. 



The problem under consideration is typical of those usually described 

 as problems in network synthesis. In other words, a network of a prescribed 

 general type is to be designed to approximate as closely as possible an ideal 

 operating characteristic of a prescribed form. The same procedure will be 

 followed as that appropriate for most such problems. The procedure begins 

 with the development of a mathematical expression representing the most 



^U. S. Patent 1,666,206, 4/17/28, Modulation System. 



* Another common method uses reactance shunt branches between effectively infi- 

 nite impedances, such as the plate and grid impedances of screen grid tubes. 



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