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Tapered 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 tapered 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

  Here we will use most famous method while analyzing the tapered hybrid, that is even-odd mode  analysis.

Even-Odd mode Analysis of the Tapered coupled Line Hybrid

    The tapered coupled line 180 degree hybrid can provide any power division ratio with a bandwidth of a decade or more. This hybrid is also referred to as asymmetric tapered coupled line hybrid.

Figure 1: Schematic diagram of the tapered coupled line hybrid
 
 
 
 

Figure 2: The variation of characteristic impedance

The schematic circuit of this coupler is seen above. The ports have been numbered to correspond functionally to the ports of the 180 degree hybrids according with to general considerations about 180 degree hybrids. The coupler consists of two coupled lines with tapering characteristic impedances over the length 0<z<L. At z=0 the lines are weakly coupled so that Z_oe(z)=Z_oo(z)= Z_o, while at z=L the coupling is such that Z_oe(L)=kZ_o, where 0<=k<=1 is a coupling factor which we will relate to the voltage coupling factor. the even mode of the coupled line thus matches a load impedance of Z_o/k (at z=L) to Zo, while the odd mode matches a load of kZ_o to Z_o; note that Z_oe(z)Z_oo(z)=Z_o² for all z. The Klopfenstein taper is generally used for these tapered matching lines. For L<z<2L, the lines are uncoupled and both have a characteristic impedance Z_o; these lines are required for phase compensation of the coupled line section The length of each section, q=BL, must be the same , and should be electrically long to provide a good impedance match over the desired bandwidth. First consider an incident voltage wave of amplitude V applied to port 4, the difference input. This excitation can be reduced to the superposition of an even-mode excitation and an odd-mode excitation, as shown in Figures 3,4.

Figure 3: Even-mode excitation

Figure 4: Odd-mode excitation

At the junctions of the coupled and uncoupled line (z=L), the reflection coefficients seen by the even or odd modes of the tapered lines are

Then at z=0 these coefficients are transformed to

,

.

Then by superposition the scattering parameters of ports 2 and 4 are as follows:

S₄₄ is equal to

and S₂₄ is equal to

.

By symmetry, we also have that S₂₂=0 and S₄₂= S_24.

To evaluate the transmission coefficients into ports 1 and 3, we will use the ABCD parameters for the equivalent circuits shown in Figure 5 and 6, where the tapered matching sections have been assumed to be ideal, and replaced with transformers.

Figure 5: Even-mode case

Figure 6: Odd-mode case

The ABCD matrix of the transmission line-transformer-transmission line cascade can be found by multiplying the three individual ABCD matrices for these components, but it is easier to use the fact that the transmission line sections affect only the phase of the transmission coefficients. The ABCD matrix of the transformer is

[ 0;0 1/],

for the even mode, and

[1/ 0;0 ],

for the odd mode. Then the even- and odd-mode transmission coefficients are

T_e=T_o=2/(k+1)e^-2jq,

Since T=2/(a+B/Z_o+CZ_o+d)=2/(k+1) for both modes; the e^-2jq

Factor accounts for the phase delay of the two transmission line sections. We can then evaluate the following S parameters:

S₃₄ becomes equal to

S₁₄ becomes equal to

.

The voltage coupling factor from port 4 to port 3 is then

b=| S₃₄ |=, 0<b<1

while the voltage coupling factor from port 4 to port 2 is

a=| S₂₄ |=, 0<a<1.

Power conservation is verified by the fact that

| S₂₄ |²+| S₃₄ |²=a²+b²=1.

If we now apply even- and odd-mode excitations at ports 1 and 3, so that superposition yields an incident voltage wave at port 1, we can derive the remaining scattering parameters. With a phase reference at the input ports, the even- and odd-mode reflection coefficients at port one will be

,

.

Then we can calculate the following S parameters:

S₁₁ is

,

S₃₁ becomes equal to

and they are equal to ae^-2jq.

From symmetry, we also have that S₃₃=0, S₁₃=S₃₁, and that S₁₄=S₃₂, S₁₂=S₃₄. The tapered coupled line 180^o hybrid thus has the following scattering matrix:

So finally S matrix has the following form

e^-2jq.