Most hams probably are unfamiliar with group theory, however, the results can be easily verified without using group theory. The right eigenvectors and the eigenvalues of a matrix M are defined by finding the solutions to the equations,
|M = .||(18)|
The cyclic eigenvectors in our basis, are those that change by a constant phase between the elements, with the same phase change between the last and first elements. This gives,
The input to the HA5WH network contains only the last two eigenvectors written above. That is
|Vin = + ,||(20)|
With the input as in Eq. 23, the output will in general be
|Vout = ga + gb,||(21)|
|VA = (1 - j)ga + (1 + j)gb|
|VB = (1 - j)jga - (1 + j)jgb||(22)|
The analysis so far shows how the HA5WH network can be motivated. The C4 eigenvectors have equal amplitudes for the 4 voltages, and have a phase shift between adjacent ports of 0 o , + 90 o , 180 o , and 270 o , this last is equivalent to a phase shift of - 90 o . We want to choose the network drive, connections, and component values to select out one of the two 90 o phase shifted eigenvectors. As an aside, the same ideas could be used to design a 60 o relative phase shift by using a network invariant under the group C6 , or a 45 o shift from C8 , etc.
The first step in selecting the component values is to calculate
the eigenvalues of the four M matrices. By direct multiplication,
|= = + jC,|
|= - - C, = - + C,|
|= - C, = + C,|
|= = - - jC,||(24)|
The effect of one of the A matrices, when a single eigenvector is input, is given by replacing the M matrices in Eq. 11 by their eigenvalues. After a little algebra, I get,
|A a =|
|A b =||(25)|
The A b matrix is proportional to A a. If we feed the section of the network with a linear combination of and , the section suppresses relative to by a factor of
The HA5WH network has the properties that the magnitude of the ratio given in Eq. 30 is always less than 1 for positive frequencies, and it is exactly zero for = 1/(RC) . The first property says that additional network sections can only improve the relative 90 o phase shift of the outputs. The second says that we can set the frequencies of exact 90 o phase shift by selecting the RC 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 RC values in section i given by Ri and Ci , as
|Suppression = 20||(27)|
A simple method of picking the RC values for each section is to use a computer to plot the above result, and adjust n and RiCi 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.