Theory and Design Procedures

The schematic circuit representation of a single-section coupler is shown in Fig. 3.1 where the port 1 is the input port, port 2 is the through port, port 3 is the coupled port and port 4 is the isolated port. Because the network is symmetrical about the plan P-P’, the even- and odd-mode analysis could be adopted. The 4-ports network shown in Fig. 3.1 could be analyzed easily by decomposed into even- and odd-mode excitation as shown in Fig. 3.2(a) and (b), respectively. In Fig. 3.2(a) and (b), the Z0e and Z0o are the characteristic impedances and θe and θo are the transmission phases of the even and odd mode, respectively.

Fig. 3.1 Schematic circuit representation of a single-section coupler.

Z^0e

Z0o

Fig. 3.2 Schematic circuit expression of coupler with (a) even mode excitation and (b) odd mode excitation.

Using these equivalent circuits, the ABCD matrices of the even- and odd-mode excitation can be expressed as And, the corresponding even- and odd-mode scattering matrices can be obtained.

[ ] ^{11 21}

0 0

Finally, after composition of these modal parameters, the various elements in the 4-port scattering matrix are given as follows:

11 11

Following conditions should be satisfied for perfect port match and isolation.

11e 11o Therefore, (3.10) gives the necessary condition for TEM backward-wave directional coupler.

When the propagation wave is non TEM wave (θe≠θo), only Z0e=Z0o can satisfied the matching condition at all ports. However, there is no meaning for a coupler to set Z0e=Z0o. In other words, theoretically, only TEM mode propagation structure can realize a matched coupler. However, in practical application for a single-section coupler even the phase different between even- and odd-mode is over 30% the return loss is still in acceptable value of 15dB.

From (3.9) and (3.10) assuming (θ=θe=θo), we obtain

From equation (3.11) and (3.12) we know that the coupling (S31) and the through (S21) of the

In equation (3.13) K is called the coupling factor which defines as the coupling at the central frequency of a coupler. Therefore, the maximal coupling between port 1 and 3 (or port 2 and 4) occurs whenθ = and the maximal coupling values are given by ^π₂

Thus, the scattering matrix of a backward-wave coupler with quarter-wave length can be represented as follows:

Although the above described theory is only valid for a single-section direction coupler, it is the basic building block of a multi-section directional coupler. Each section of multi-section directional coupler is a single-section direction coupler with designed coupling factor so that a good single-section directional coupler is the most important part to build a multi-section directional coupler.

Fig. 3.3 A N-section cascaded symmetrical quadrature hybrid.

In order to obtain a near-constant coupling over a wider frequency range, a multi-section coupler structure is the most commonly used method. The multi-section coupler cascades several sections of coupler with various coupling factors. Each section is quarter wave length long at center frequency. By properly choosing the even- and odd-mode impedances of each section, the multi-section coupler could have a Chebyshev or Butterworth type of response over a broad bandwidth. The multi-section coupler could be either symmetrical or asymmetrical. Shown in Fig. 3.3 is a symmetrical multi-section coupler. For a 90⁰ quadrature coupler, the circuit should be symmetric. An N-section directional coupler contains N different single-section coupler in a cascaded manner. Each single section is denoted by the even- and odd-mode characteristic impedances. As shown in Fig.2.2b, the i-th section in symmetrical coupler will be identical to the (N+1-i)th section. The signal goes out from the through port is 90⁰ out of phase to that of the coupled port ( S∠ 31= S∠ 21+90 degree). This phase relationship is independent of frequency. Because of this property, the 3-dB symmetrical directional couplers are widely used in several microwave circuits such balanced amplifier, balanced mixer, phase shifter, power divider, and beam-forming network for array antennas and for direction-finding antennas. The asymmetrical multi-section coupler does not have the mentioned phase property as symmetrical one and is not as popular as symmetrical multi-section coupler. Here, only the symmetrical multi-section coupler will be discussed.

An analytical design expression of a multi-section directional coupler is very cumbersome even the order is small. However, synthesis of multi-section symmetrical TEM coupler using insertion loss method has been presented by Cristal and Young [15]. Moreover, the design tables for equal-ripple and maximally flat responses of multi-section symmetrical coupler up to nine sections are generated in [15]. Therefore, the design procedures of multi-section cascaded coupler are described as follow.

1) According to the specification for desired coupler, utilize the design table to obtain the required number of sections and the even- and odd-mode characteristic impedances of each section. Here, a 5-section symmetrical 3dB quadrature hybrid is chosen as an example.

2) Usually, the outmost sections (section 1 and 5) are the most loosely coupled sections. The conventional parallel-coupled microstrip lines are suitable for these sections. For this kind of coupled lines, commercial circuit simulators such as ADS, or Microwave Office can get the physical dimensions easily.

3) Use Fig. 2.3 (a) and (b) to obtain the physical dimensions of the central section (section 3) which is the most critical one.

4) Last, to design the inner sections (section 2 and 4), designers should properly choose the width and distance of the strips on the main substrate by the aid of Fig.

2.2 (a) and (b) to minimize the discontinuities between sections (especially between the outmost sections).

5) Finally, the whole multi-section quadrature hybrid performances are verified by 3-D EM simulator HFSS.

3.3 Ultra-Broadband Multi-Section Quadrature Hybrid Using Modified