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Ring Hybrid junction is a four-port network with a 180 degree phase shift between two output ports but it can also be operated so that output ports are in phase. The 180 degree ring hybrid can be constructed in several forms such as planar form or other forms like wave guide forms. In this toolkit we shall only deal with planar forms, i.e. microstrip and stirpline forms (see figures below for planar forms)

  Here we will use most famous method while analyzing the ring hybrid, that is even-odd mode  analysis. However, before going through the detail of even-odd mode analysis, the relation between impedance and width over depth ratio for transmission lines will be explored, which plays a very crucial role in our ring hybrid design process.

Relation between impedance and width over depth ratio for transmission lines

   For ring hybrid design in our analysis one must necessarily drop in the panel, where the impedance values are derived from entered width over depth ratios or vice versa. Because we think that those geometric information about the line from which we construct our ring hybrid junction has much importance. In that panel user has two options. S/he might enter impedance values and obtain width over depth ratios from them or vice versa. The formulas used for these calculations are different for stripline and microstrip.  Let's analyze the case as follows:

ε_r : dielectric constant
Z : impedance value
W : width of the line
d : depth of the substrate

A-For stripline:
A-1 If impedance(Z) value is entered

A-2 If width over depth ratio(W/d) is entered

B- For microstrip:
B-1 If impedance(Z) is entered

B-2 If width over depth ratio(W/d) is entered

  so to sum up using those formulas W/d ratios are returned to user when he entered impedance values and impedance values are returned to user when he entered the W/d ratios for transmission lines that form the ring hybrid junction.
 

Even-Odd Mode Analysis of the Ring Hybrid

 Above you see a detailed image of the ring hybrid junction and the variables related to it are also shown and described on the image.
  First consider a unit amplitude wave incident at port 1, the sum port, of the ring hybrid. at the ring junction this unit wave will divide into two components, which both arrive in phase at ports 2 and 3, and 180 degree out of phase at port 4. Using the even-odd mode analysis technique we can decompose this case into a superposition of two simpler circuits which are considerably easy to analyze. When we give excitations to the port 1 as in the following figures;
 
 

For even-mode excitation.

for odd mode excitation.

  Then from those pictures we see that the amplitude of scattering waves from the ring will be

Then we can evaluate the required reflection and transmission coefficients defined as in the figures above, showing even and odd mode cases, using the ABCD matrix for the even and odd mode two-port circuits in those figures. Then from there the results are equal to matrix multiplication of three matrices as follows

so the result is

 
  for the even mode excitation.

so the result is

for the odd-mode excitation.

  Then with the aid of a conversion table from ABCD matrix to reflection and transmission coefficients

so using these results

  which shows that the input port is matched, poet 4 is isolated, and the input power is evenly divided in phase between ports 2 and 3. These results also form the first row and first column of the scattering matrix.

  Now consider a unity amplitude wave incident at port 4 (difference port)of the ring hybrid junction. The two wave components on the ring will arrive in phase at ports and 3, with a net phase difference of 180 degree between these ports. The two wave components will be 180 degree out of phase at port1. This case can be decomposed into a superposition of the two simpler circuits and excitations shown in figures below

for the even mode
 
 

for the odd mode case.

  Then the amplitudes of the scattered waves will become

then the ABCD matrices for the even and odd mode circuit of the figures above

so the result is

for the even-mode excitation case.

so the result is

for the odd-mode excitation.

Then using  a conversion table

so using these results then put those into places to obtain

  which shows that the input port is matched, port 1 is isolated, and the input power is evenly divided into ports 2 and 3 with a phase difference of 180 degree. These results form the fourth row and column of the scattering matrix and the other remaining elements can be found from symmetry considerations.