Theoretical Information About Tapered Coupled Line Hybrid

Theoretical Information About Branch-line Couplers

Generally branch-line couplers are 3dB, four ports directional couplers having a 90° phase difference between its two output ports named through and coupled arms. Branch-line couplers(also named as Quadrature Hybrid) are often made in microstrip or stripline form.

1.DESIGN OF BRANCH -LINE COUPLER:

The geometry of the branch-line coupler is shown in Figure1. A branch-line coupler is made by two main transmission lines shunt-connected by two secondary (branch lines). As it can be seen from the figure, it has a symmetrical four port. First port is named as Input port, second and third ports are Output ports and the fourth port is the Isolated port. The second port is also named as direct or through port and the third port is named as coupled port. It is obvious that due to the symmetry of the coupler any of these ports can be used as the input port but at that time the output ports and isolated port changes accordingly. When we analysis the scattering matrix of this coupler we will see also the result of that symmetry in scattering matrix.

Considering the dimensions of the coupler the length of the branch line and series line isgenerally chosen as the one fourth of the design wavelength . As it is shown in Figure 1, if wename the length of series and stub transmission lines as L then L can be find as following:

At that point we will se the calculation of the other dimension parameter of transmissionlines; w/d ratio. We generally design branch-line couplers in two forms: Microstrip line and Stripline.Geometry of the microstrip line and stripline can be seen from Figure2.

According to the impedance choice of the series and stub microstrip transmission lines wecan calculate the w/d ratios of the those lines in microstrip form by using the following formulas:

Given εr and Z0

Considering the Stripline branch-line coupler design, we can calculate w/d ratios for each (stub and series) transmission line in the branch-line coupler with following calculations:

2.ANALYSIS OF BRANCH-LINE COUPLER

2.1.Even-odd mode analysis and S-parameters

In the analysis of the branch-line coupler we consider the scattering matrix of the coupler.In order to find them we use even-odd .mode analysis. In both mode we divide the branch-linecoupler symmetrically as in the Figure 3.

Generally considering that we give V voltage to the Input port. In the even odd mode analysis we consider it we give that V voltage in even mode of it to Input port and rest to theIsolated port and for the odd mode we give Input port of it and to the isolated port 1/2 of it.Furthermore, while making even-odd mode analysis, for the even mode we think that the stubs of the divided circuit are open circuited and for the odd mode they are short circuited. For thisanalysis, if we consider the superposition of the incoming voltage, it results as V voltage to the Input and 0 voltage to the Isolated port. Furthermore we have for each mode incident and reflectedwaves, for even mode it is illustrated in the Figure 4. As it is seen we have an incident wave of the actual voltage and at first stub we have a reflection having a reflection coefficient Γe and at second port a transmitted signal having transmission coefficient Te. Considering the contribution of the even mode to the port waves for first port we have 1/2VΓe, for second port we have 1/2VTe, for third port 1/2VTe, and for the fourth port 1/2VΓe.

In addition, for odd mode incident and reflected waves are illustrated in the Figure 5. As itis seen we have an incident wave of the actual voltage at first port and 1/2 of it at fourth port as incoming wave. Also at first stub we have a reflection having a reflection coefficient Γo and at second port a transmitted signal having transmission coefficient To. Considering the contribution of the odd mode to the port waves for first port we have 1/2VΓo, for second port we have 1/2VTo, for third port -1/2VTo, and for the fourth port --1/2VΓo. At this point, we express the emerging waveat each port of the branch-line coupler as the superposition of the even and odd mode waves as following:

B1=(1/2Γe+1/2Γo)V 1.7

B2=(1/2Te+1/2To)V 1.8

B3=(1/2Te-1/2To)V 1.9

B4=(1/2Γe-1/2Γo)V 1.10

The ABCD matrix is used to find the overall transmission and reflection characteristics of the network. Having YA=1/ZA and YB=1/ZB we have the ABCD matrix of even and odd mode. Foreven mode ABCD parameters are as following:

Since we have l=λ/4 (and work with our design frequency), βl=(2π/λ)*(λ/4)=π/2

Therefore cosβl=0 and sinβl=1 and the ABCD matrix is following:

For the odd mode ABCD matrix:

Since we have l=λ/4 and so βl=(2π/λ)*(λ/4)=π/2

So cosβl=0 and sinβl=1 and the ABCD matrix is following

At that point we can find Γe, Γo, Te, To by using following equations:

Then solving above equations with parameters of even and odd mode ABCD matrixes at centerfrequency where ƒ=νp/λ=νp/4 :

At this time we can say that B1/V=S11, B2/V=S12, B3/V =S13 and B4/V =S14. Therefore S-Parameters are as following:

And the scattering matrix of Branch-line coupler is

2.2. Matching Condition

Looking above equations if we consider the matching condition;

thenS11 and

S14 becomes zero. In that matching case; the power entering port1is evenly divided between ports2 and 3 with a 90° phase shift between these output ports. No power is coupled to port 4 (isolatedport). Therefore, the isolation and directivity of that matched coupler, which will be mentioned infollowing part, is very high (for perfect case infinity), at center frequency.

2.3.Coupling, Directivity, Isolation and Power-split Ratio

As it can be seen from the matrix above that scattering matrix of branch-line coupler issymmetric and the each row of it is just the transpose of its each column.

Considering the coupling which is the ratio of power at port 1 to power at port 3, directivity which isthe ratio of power at port 3 to power at port 4 and the isolation which is the ratio of power at port 1 to power at port 4 of the branch-line coupler:

Coupling = = 10log(P1/P3) = -20log |S13| dB 1.31

Directivity = = 10log(P3/P4) = 20log (|S13|/|S14|) dB 1.32

Isolation = = 10log(P1/P4) = -20log|S14| dB 1.33

The power split ratio (P) which is used to express the coupling of the branch-line coupler in terms of the ratio of powers to the coupled (port 3) and direct ports (port 2) :

= 10log(P3/P2)=-20log (|S13|/|S12|) 1.34

2.4.Behaviour of S-parameters verses frequency

In order to define the behaviour of the s-parameters with the frequency change we followthe following way. Let us consider ABCD matrixes of even and odd mode expressed in (1.12) and (1.15), respectively. With those matrixes, in order to calculate s-parameters in center frequency we have taken β value as π/2 and therefore cosβ was 0 and sinβ was 1 (β 2π /λ and λ=νp/ƒ). In thiscase since we will observe the dependence of s-parameters to the frequency we will take sinβ andcosβ as they are and calculate s-parameters with them.

Solving (1.12) and (1.15), then ABCD matrixes are:

Solving for Γe, Γo, Te, T o :

Putting x for cosβ and y for sinβ in the equations;

At this point, if we use (1.25), (1.26), (1.27), (1.28), then we get all the necessary s-parameters in our hand. After finding s-parameters, we can find magnitude of s-parameters and plot the magnitude verses frequency plot. This simulation program can plot the magnitude of s-parameters vs. frequency plot.

References:

1. Fooks, E. H. Microwave engineering using microstrip circuits, Prentice Hall New York 19902. Pozar, David M. Microwave Engineering Second Edition, Wiley, New York 1998

Theoretical Information About Tapered Coupled Line Hybrid

Tapered Hybrid junction is a four-port network with a 180 degree phase shift between two output ports but it can also bethat output ports are in phase. The 180 degree tapered hybrid can be constructed in several forms such as planar form olike 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 moretapered 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 thegeneral considerations about 180 degree hybrids. The coupler consists of two coupled lines with tapering characteristilines are weakly coupled so that Zoe(z)=Zo (z)= Zo, while at z=L the coupling is such that Zoe(L)=kZo, where 0<=k<=voltage coupling factor. the even mode of the coupled line thus matches a load impedance of Zo/k (at z=L) to Zo, whilethat Zoe(z)Zo (z)=Zo

2 for all z. The Klopfenstein taper is generally used for these tapered matching lines. For L<z<2L, the linesimpedance Zo; these lines are required for phase compensation of the coupled line section The length of each section, qlong to provide a good impedance match over the desired bandwidth. First consider an incident voltage wave of amplitudeexcitation can be reduced to the superposition of an even-mode excitation and an odd-mode excitation, as shown in Figures

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 th

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= S24.

To evaluate the transmission coefficients into ports 1 and 3, we will use the ABCD parameters for the equivalent circuitmatching 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 individualit is easier to use the fact that the transmission line sections affect only the phase of the transmission coefficients. The ABCD

[ 0;0 1/ ],

for the even mode, and

[1/ 0;0 ],

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

Te=To=2 /(k+1)e-2jq,

Since T=2/(a+B/Zo+CZo+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 waveparameters. 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 180o hybrid thus

So finally S matrix has the following form

e-2jq.