Thursday, May 28, 2020

# Synthesis of Phase Difference Networks

Parent Category: 2014 HFE

By Mark Mell

This article presents a systematic method to overcome the complexities in the traditional synthesis of phase difference networks. The solution presented reduces the synthesis to simple mathematics.  A step by step procedure in the form of an example is given. The network is generated and the results analyzed and verified consistent with the transfer function.

Phase difference networks are two independent frequency selective circuits where the phase displacement between the two approximates a constant value. These networks find use in many applications from beamforming to image rejection mixers. One extremely useful case is when the two networks are all-pass, i.e. where no amplitude variation is introduced.

A closed form solution for these networks has previously been published (1), but a simpler, more direct approach is possible by imposing a restriction on the pole and zero locations of the resulting networks.  Making use of this restriction reduces the complexity of the synthesis considerably bringing it into the realm of simple mathematics.

Approximation and Synthesis

Like most synthesis problems, the approximation and synthesis is greatly simplified if the network pole and zero locations are normalized about 1 radian. For the purpose of this article only a second order network will be considered as it is the most practical case. In a passive synthesis the second order network easily facilitates the absorption of distributed parameters, which the first order cannot. Second order networks also result in minimal phase error over wide bandwidths making higher order networks almost unnecessary.

Table 1 lists the generalized transfer function H(s) of an all-pass network along with the phase function θ(s) in terms of the real and imaginary components of the transfer function. The basic synthesis problem then becomes finding suitable pole and zero locations for the two independent networks (listed in Table 1 as θ1(s) and θθ2(s)) such that the subtraction of the two equates to an error function approximating the desired phase displacement. Table 1

An all-pass network by necessity contains poles and zeroes of equal magnitude with opposite polarities thereby assigning each to their respective halves of the s-plane. Each second order network considered here can then be described by two independent variables, one for each of the two pole/zero pairs comprising the network.

Pole/Zero Location

The pole/zero location restriction that greatly reduces the complexity of the synthesis is now employed as shown in Table 2, that is equating the poles and zeros of the second network to be the reciprocals of the first.   When employing this restriction the simplification is self evident as the approximation problem becomes the solution of two variables as opposed to four.

The synthesis starts by substituting the two pole/zero pairs into the generalized transfer function and then breaking down the resulting expression into real and imaginary components. These components can then be used to describe the phase function associated with each network. Table 2 contains the results from performing these substitutions. Table 2

Since our primary objective is to approximate a constant phase displacement between two networks, it is to our advantage to take the derivative of the phase displacement function. Beyond simplification, the resulting expression describes the phase error from which we’ll extract the inflection points.  In taking the derivative the nominal phase displacement will be discarded. This constant will need to be reconstructed later in the process. After taking the derivative and implementing some convenient substitutions, a representative error function is shown in Table 3. Table 3

The numerator of the resulting function is of principal concern as its roots determine the frequency locations of the inflection points. The distribution of polynomial coefficients of this function is rather convenient as its roots occur at ±λ, ±1, and ±1/λ radians. Thus control of the network bandwidth is possible through the selection of a single independent variable λ.

Table 4 details the generation of a sixth order polynomial in λ along with the steps to normalize the inflection point function providing for the equivalence of the coefficients of both equations. The resulting expression contains three variables: ω0, ωω1, and λ. Table 4

Selecting a value for λ close to 1 will result in a network with minimal phase error across a narrow bandwidth. As λ is moved away from 1 the bandwidth increases as does the resulting error.

If ω0 and λ are selected as independent variables the calculation of dependent variable ω1 will result in an optimum phase displacement network. The actual phase difference is at this point is unknown, the result of taking the derivative and discarding the constant.

The nominal phase uncertainty can easily be overcome through the use of a simple recursive process employing linear interpolation. A sample routine written for Mathcad is given in Appendix A.

An Example

The phase plot shown in Figure 1 is the result of setting θ equal to 90 degrees and lambda to π-1. This configuration yields a 1 degree phase error over nearly a 10:1 bandwidth.  As is evident from the curve, the inflection points are located as expected at 1/λ, 1, and λ. Figure 1

The resulting pole locations for these networks always reside on the negative real axis, so technically the circuit could be realized as the cascade of two single pole sections. In keeping with our original premise, one second order network will be used in the synthesis. Table 5 details the equivalences between 2 single pole all-pass networks in cascade and 1 two pole all-pass network.

Calculating Normalized Element Values

Table 6 provides the equations necessary for calculating the normalized element values of passive two pole all-pass networks where the first network is calculated directly from ω0 and ω1 with the second network from the reciprocal of the two. Note that the bridging inductor L3X in the resulting circuit is center tapped so if modeling be sure to include the associated mutual inductance in series with L4X . The calculated values assume a perfect coefficient of coupling but any leakage inductance can be subtracted from L4X when implementing actual components.  Likewise the effects of distributed capacitance of L3X can be accounted for by reducing C3X by that amount. Table 6

Table 7 list the normalized element values for the example documented with Figure 2 the results of a frequency domain analysis of the derived circuits. Table 7 Figure 2

Conclusion

In summary, a design method has been presented for deriving phase difference networks consisting of two 2-pole all-pass sections. The design starts by selecting a value for λ about 1 radian to yield a given bandwidth. Then by selecting a starting value for ω0 the corresponding value for ω1 can be determined using the equivalence in Table 4. Through analysis of the resulting transfer function the nominal phase displacement can be determined and then through linear interpolation the desired offset can be achieved.  The MathCAD program in Appendix A can be used directly or as a flow diagram in duplicating these steps.  As the final step the individual networks can be designed using the equations given in Table 6.

Mark Mell holds a BSEE from Merrimack College and has worked in the design and synthesis of filters since 1980. He is currently a Senior Engineer at Crane Aerospace & Electronics where his most recent work includes the design of microwave multifunction assemblies, down converters and detector log video amplifiers. He can be reached at: mark.mell@crane-eg.com. Note: Enter phase shift as lagging, i.e. negative magnitudes.

Appendix A

References

1.  Darlington, S., “Realization of a Constant Phase Difference,” Bell System Technical Journal, Vol. 29, January 1950, pp. 94 - 104.

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