Analyzing a phase-shift network shows how its performance varies with component tolerances.
by Kevin Schmidt, W9CF
The phasing method of single-sideband generation or detection requires two signals with a 90° relative phase shift over the audio frequency range. The phasing method has never been very popular, particularly once relatively inexpensive filters became available. In the future, presumably, digital signal-processing techniques will perform the necessary audio phase shifting or directly generate the radio frequency single-sideband signal. Why then should you be interested in audio phase-shift networks? Perhaps because they are relatively low cost, easy to build and are fun to play with. In addition, the techniques that I describe here are useful for efficient analysis of other cascaded networks.
For many years, the ARRL Handbook has included a circuit for an audio phase-shift network designed by HA5WH,11 have not located the original reference for this network. The Handbook claims that the circuit gives approximately 60 dB of opposite sideband suppression using 10% tolerance components. This flies in the face of the usual result that you need 1% components to get around 40-dB suppression. In this article, I will analyze and give design equations for this type of network. Unfortunately, this analysis shows that using 10% tolerance components can lead to poor sideband suppression. With ideal compo-
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nents the network can give excellent performance, and by using either high-tolerance components or well-matched lower-tolerance components, the network still can give good performance.
In the following sections I give the general formula for the sideband suppression in terms of the phase and amplitude errors in the phasing network; derive an efficient method of analyzing a general network of the HA5WH type; give the analysis of an ideal realization of the network; describe the optimization of the network in terms of easily calculated elliptic functions; and, give the effects of component tolerances. The result is a set of simple design equations for the ideal network and an estimate of the sensitivity to component tolerances. A set of FORTRAN programs that implement the methods described are available for downloading.
The Effects of Phasing Errors on Sideband Suppression
The phasing method generates a single-sideband signal, given mathematically as cos((a>c ± co„)t), where the + (or -) sign gives the upper (or lower) sideband, and coc = 2nfc where fc is the carrier frequency. Similarly, coa = 2nfa where fa is the audio modulating frequency. The cosine can be written as cos ((coc ±<a(I)i) = cos (®ci)cos (®ai)+ sin(a>ci)sin(ft>Q<)
the basic equation of the phasing method. The multiplications on the right-hand side are accomplished using balanced modulators, and the two audio frequencies (as well as the two radio frequencies) must be 90° out of phase and
Fig 1—The schematic diagram of the HA5WH wide-band phase-shift network.
of equal amplitude. I will assume that the radio frequencies are exactly 90° out of phase, and of equal amplitude. Using the usual complex notation with yAe-'<0al to be one audio signal, and VBe-'a'at to be the other, the result of using a nonideal phasing network will be
and the sideband suppression (or enhancement) is given by
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