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,
(19) |
The input to the HA5WH network contains only the last two eigenvectors written above. That is
V_{in} = + , | (20) |
With the input as in Eq. 23, the output will in general be
V_{out} = g_{a} + g_{b}, | (21) |
V_{A} = (1 - j)g_{a} + (1 + j)g_{b} | |||
V_{B} = (1 - j)jg_{a} - (1 + j)jg_{b} | (22) |
20 | (23) |
The analysis so far shows how the HA5WH network can be motivated. The C_{4} 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 C_{6} , or a 45^{ o } shift from C_{8} , etc.
The first step in selecting the component values is to calculate
the eigenvalues of the four M matrices. By direct multiplication,
I get
= = + 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
. | (26) |
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 R_{i} and C_{i} , 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 R_{i}C_{i} 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.