# Directional Couplers - simulation by YAdocs

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```									                                             Directional Couplers [1]

THE QUADRATURE (90›) HYBRID

The Hybrid coupler is often made of microstrip or stripline as shown in Figure 1. The microstrip form is
also pictured in Figure 2. These couplers are 3 dB directional couplers with a 90› phase difference between the
outputs of the through and coupled lines. It is also known as a branch-line hybrid.

Figure 1. Geometry of a Quadrature Hybrid coupler. [1]

Figure 2. Photograph of a microstrip Quadrature Hybrid coupler. [1]
Referring to the geometry of Figure 1, the ideal branch-line coupler, with all ports matched and power
entering port 1, has evenly divided outputs at ports 2 and 3 with a 90› phase difference between them. The isolated
port (port 4) has no power exiting it. The resulting [S] matrix is then given by

0 j 1 0
        
− 1  j 0 0 1
(1)         [S ] =
2 1 0 0 j 
        
 0 1 j 0
Because of this symmetry, we can use any port at the input port. The output ports will be on the opposite
side of the junction from the input port and the remaining port will be the isolated port. This symmetry is reflected
in the scattering matrix, as each row can be obtained as a transposition of the first row.

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The even and odd mode analysis of Figure 1 is carried out in detail in [1]. Because of the length
requirements on the lines, the [S] matrix is not frequency independent. This easily illustrated by doing Example 7.5.
EXAMPLE 7.5 Design and performance of a Quadrature Hybrid [1]
Design a 50 Ω branch-line quadrature hybrid junction, and plot the S parameter magnitudes from 0.5 f0 to 1.5 f0
where f0 is the design frequency.
Solution
The design is trivial. The lengths of the lines are λ/4 at the design frequency and the impedances are 50 Ω and 35.35
Ω. We now use Serenade Design Suite to simulate the coupler over the frequency range of interest. A quadrature
hybrid is one of the microstrip circuit elements available. We choose that geometry with a substrate that is 31 mils
thick and a dielectric constant of 2.2. The transmission line tool is used to get the physical lengths and widths of the
transmission lines. The circuit diagram and analysis results follow in Figures 3 and 4. Notice that in the microstrip
form, the center frequency is at .99 f0 where with ideal lines the response is as shown in Figure 7.25 of [1]. Can you
explain this?

Figure 3. Serenade Design Suite schematic of a branch-line coupler using microstrip.

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Figure 4. Simulation results for the microstrip branch-line coupler.
Coupled Line Couplers
We begin this section with a review of coupled line theory. Figure 5 shows some coupled transmission line
geometries that can be represented by the equivalent circuit shown in figure 6. There are two modes that can be
excited for this geomentry - even and odd. They are depicted in Figure 7.

Figure 5.

Figure 6. A three-conductor coupled transmission line and its equivalent circuit. [1]

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Figure 7. Even- and od-mode excitations for a coupled line, and the resulting equivalent capacitance networks.
(a) Even-mode excitation. (b) Odd-mode excitation.
Analysis of the equivalent circuits of Figure 7 result in the following equations for the even- (Ce) and odd-
mode (Co) capacitances:
( 2)         C e = C11 = C 22
(3)          C o = C11 + 2C12
These in turn give the following values for the even- and odd-mode impedances
L        LC e        1
( 4)         Z 0e =       =            =
Ce       Ce         vC e
L         LC o        1
(5)          Z 0o =      =             =
Co        Co         vC o
where v is the phase velocity of the wave. The even- and odd-mode capacitances can be determined analytically for
coaxial, parallel plate, or stripline. Quasi-TEM lines, such as microstrip require approximate analysis or numerical
techniques. Imperical equations for design of coupled lines can be found in [2]. These equations are used in a
Mathcad document to do Example 7.7 in [1]. We can also use the Transmission Line Tool in Serenade Design Suite
to get the dimensions of the coupled stripline. The screenshot for Example 7.7 is shown in Figure 9. Notice that the
dimensions are slighly different for the two methods. Can you explain this?
We can now simulate this design over the bandwidth of interest using Serenade Design Suite. We take the
design from the Mathcad document. Figures 10 and 11 show the schematic and results for coupling and isolation
respectively.

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Figure 8. Mathcad document used to design a 20 dB stripline coupled line coupler.

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Figure 9. Transmission Line Tool for design of the 20 dB stripline coupled line coupler.

Figure 10. Serenade Design Suite Schematic for the 20dB stripline coupled line coupler.

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Figure 11. Serenade Design Suite simulation results for the 20dB stripline coupled line coupler.
As Figure 11 shows, the coupling and isolation for the coupled line coupler are frequency dependent
because of the λ/4 length requirement. As with matching sections, the bandwidth can be increased by using multiple
sections of coupled lines. The details of the design are shown in Section 7.6 of [1]. A Mathcad document was
created for the solution of Example 7.7 in [1]. The design procedure yields even- and od-mode impedances for the
three coupled line sections. They are:
Z0e1 = 50.629 ohm      Z0e2 = 56.695 ohm       Z0e3 := Z0e1
Z0o1 = 49.379 ohm      Z0o2 = 44.096 ohm       Z0o3 := Z0o1

From this information, we can design the coupler using equations from [2] or by using the Transmission Line Tool
in Serenade Design Suite. Simulation of this three section coupler yields the results shown in Figure 12.

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Figure 12. Simulation results for the 3-section 20dB stripline coupled line coupler.

[1] David M. Pozar, Microwave Engineering, John Wiley & Sons, Inc., New York, 1998.
[2] Inder Bahl and Prakash Bhartia, Microwave Solid State Circuit Design, John WIley & Sons, New York, 1988.

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