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Fig. 1 gives the circuit diagram of the HA5WH network as shown
in the ARRL Handbook.
Figure 1:
The schematic diagram of the HA5WH wideband phase shift network.

Given this circuit, it is easy to analyze
the network numerically using a mesh or nodal analysis. The
disadvantage of this brute force approach is that it gives
no insight into why the network works, or how changes in the
network affect its performance. I will therefore describe a
method that is both more efficient numerically, and by using
the symmetry of the ideal network, leads to simple design
equations.
Clearly, the network consists of 6 sections each with 4 input
connections and 4 output connections. One of these sections is
shown in fig. 2.
Figure 2:
The schematic diagram of 1 section of an HA5WH network.

I have labeled the input voltages and currents
V_{1} , V_{2} , V_{3} , V_{4} , I_{1} , I_{2} , I_{3} , I_{4} . The
corresponding output voltages and currents are labeled
V'_{1} , V'_{2} , V'_{3} , V'_{4} , I'_{1} , I'_{2} , I'_{3} , I'_{4} .
A straightforward nodal analysis of this network gives the
8 linear equations represented by the matrix equation
  (5)

where V , V' , I , I' are length 4 vectors, and the M_{ij}
are 4 by 4 matrices. Eq. 5 compactly represents the
8 equations that are the requirements of current conservation
at each of the nodes of the network section. The M_{ij}
matrices are

M_{11} =



M_{12} =



M_{21} =



M_{22} =
 (6)

In exact analogy with cascading twoport networks using ABCD matrices,
to cascade these network sections, I define a new matrix equation,
  (7)

Solving for the A_{ij} matrices gives,
A_{11}

=

 M^{  1}_{12}M_{11},
 
A_{12}

=

M^{  1}_{12}
 
A_{21}

=

M_{21}  M_{22}M^{  1}_{12}M_{11}
 
A_{22}

=

M_{22}M^{  1}_{12}
 (8) 
where M_{12}^{ 1} is the inverse of the matrix M_{12} .
Labeling the 8 by 8 matrices for each of the n sections of the
network by A^{ (1)}, A^{ (2)}, ... A^{ (n)}, the matrix relating
the input to the output of the full network is , made
up of the four 4 by 4 matrices
,
  (9)

where is the matrix product
A^{ (1)}A^{ (2)}A^{ (3)}...A^{ (n)}.
The handbook circuit drives 4 resistors on the 4 output connections.
Labeling these as R^{ (out)}_{1} , R^{ (out)}_{2} , R^{ (out)}_{3} , R^{ (out)}_{4} ,
and defining a 4 by 4 load matrix L,

L =
 (10)

I can write the relationship between the output voltage and current as,

(I_{out}) = L(V_{out}).
 (11)

Solving for I_{out} , and back substituting gives the final network
matrix
equation relating the 4 output voltages to the 4 input voltages,

(V_{out}) = (1  L)^{ 1}(  )(V_{in}),
 (12)

where 1 in Eq. 15 stands for the unit matrix

 (13)

If the output resistors are large compared to the other circuit impedances,
L can be taken to be zero. In that case the equations simplify to,

(V_{out}) = (  )(V_{in}).
 (14)

The handbook network has (V_{in}) proportional to

(V_{in})
 (15)

After calculating the and L matrices from the circuit values,
the output signals need to be combined as,
V_{out,1}  V_{out,3} = V_{A}


 
V_{out,2}  V_{out,4} = V_{B}


 (16) 
and the sideband suppression is given by Eq. 3. The relative
amplitude and phase of the signals can also be calculated. Most phase
shift networks are based on all pass networks so that the amplitude
of all signals are equally attenuated. The HA5WH network is not an all pass
network. Ideally, we want both good sideband suppression and we
want V_{A} and V_{B} to be constant in amplitude and phase across
the passband of the audio circuit.
I have written a Fortran program to implement the analysis of this
section. It is given in the Appendix. If the matrices
that are inverted
become singular, the above analysis breaks down at the singular points.
For example, M_{12} becomes singular when

= .
 (17)

Near these points, roundoff error in the calculations will be large.
For the analysis done here, this is not a big problem. However,
analysis on networks with many sections or near singular points
will require more numerical care than I have taken in the program
in the appendix, or the use of a the standard formulation where the full
set of network equations are solved at once.
Next: 4 Analysis of the
Up: Originally published in August
Previous: 2 The Effects of