# R

jmRC -R

Eq 25

The A6 matrix is proportional to Aa. If we feed the section of the network with a linear combination of fa and y/^, the section suppresses y/a relative to % by a factor of

Eq 26

The HA5WH network has the properties that the magnitude of the ratio given in Eq 26 is always less than one for positive frequencies and is exactly zero for <u=l/(-RC), The first property says that additional network sections can only improve the relative 90° phase shift of the outputs. The second says that we can set the frequencies of exact 90° phase shift by selecting the R-C values of single network sections. These two properties greatly simplify the design and optimization of the network.

The sideband suppression at a single frequency is given for an n section network,with R-C values in section i given by Ri and Cj, as

Suppression = 20^1og10

Eq 27

A simple method of picking the R-C values for each section is to use a computer to plot the above result, and adjust n and Rfii to achieve the required suppression. This is, in fact, the obvious technique to use if you are trying to design with a set of parts already in your junk box. However, the form of the suppression makes it easy to select optimum values, as seen in the next section.

### Optimizing the Sideband Suppression

The optimum values of Rfii can be easily calculated using elliptic functions. Typically, we want the worst-case suppression to be the highest possible. This leads us to the equal ripple or Chebychev approximation. The mathematics are straightforward and given in detail by Saraga.3 For an upper and lower frequency offu and/} respectively, the R;C, values for an «-section network are,

2nft

Eq 28

given by simply choosing all the Rfi^ values to be the same and equal to

2*ffufi

(see note 3). Also, if maximum suppression is needed at a particular frequency (for example if you wanted to use audio tones in a single-sideband transmitter to produce frequency shift keying), it is simple to select /¿(C, values appropriate for these frequencies and then optimize the other network sections.

Effects of Amplitude Variations and Component Tolerances

So far, I have only looked at the relative phase shift of the two outputs. To have a high-quality audio signal, the network must have a flat amplitude output. Usually, this is handled by constructing, respectively, an all-pass network. Since the HA5WH network is not an all-pass form, we must examine its attenuation as a function of frequency. In Figs 3, 4, and 5, I have plotted the sideband suppression and the amplitude and phase variations of one of the output signals, for the optimal 4, 6, and 8-section filters designed for the frequency range 300 to 3000 Hz with equal value resistors. The network sections are ordered from largest RC value to smallest, as in the original HA5WH design. As shown, the amplitude variations are less than ±1 dB, the phase variation is smooth, and the sideband suppression is of the equal ripple form—as expected.

One of the main selling points given in the Handbook description of this network is the claim that lo^-tolerance components can be used to obtain a high-performance network. From the analysis in the previous section, if cyclic symmetry is maintained, the network will perform is the complete elliptic integral of the first kind, and dn(u,&) is a Jacobi elliptic function.4'5

One of the FORTRAN programs calculates the iijC, values given the upper and lower frequencies and the n value. In Table 1, I give some calculated values for some networks of interest to hams, and their theoretical sideband suppression. These theoretical results will, of course, be best cases assuming perfect components.

In passing, I note that Saraga's Taylor approximation is

Eq 28