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Figure 6.8 Magic T-junction and its field patterns.

divided into ports 2 and 4 equally and in the same phase, due to symmetry. The electric field distribution has an even symmetry with respect to the middle line of port 3, whereas the field of the TE10 mode has an odd symmetry. Therefore, a wave cannot couple from port 1 to port 3. Also, a wave applied to port 3 (D) is divided into ports 2 and 4 with equal amplitude, but now the output waves have an opposite phase.

6.1.5 Microstrip Directional Couplers

Let us consider two parallel microstrip lines that are placed so close to each other that the fields of the lines couple to each other. Then a wave propagating in one line can excite a wave in the other line. The fields of a coupled line can be presented as a superposition of an even mode and an odd mode, shown in Figure 6.9. In case of the even mode the currents of the lines are equal and in the same direction; in case of the odd mode they are equal but in the opposite directions. A coupled line may be represented with an even-mode characteristic impedance Z 0e and an odd-mode characteristic impedance Z 0o , which are characteristic impedances of one of the strip

Figure 6.9 Fields of (a) the even mode and (b) the odd mode in a coupled microstrip line.

conductors relative to the ground when the coupled line is operated in the even mode or odd mode, respectively. The graph in Figure 6.10 gives these impedances when the relative permittivity of the substrate is 10.

The microstrip lines shown in Figure 6.11(a) are coupled to each other over a length of l . Let us assume that a signal source is at port 1 and other ports are terminated with loads having an impedance of Z 0 . It can be proven that all the ports are matched if

Figure 6.10 Even-mode and odd-mode characteristic impedances of coupled microstrip lines; er = 10.

Figure 6.11 Directional couplers based on coupled microstrip lines: (a) single-element coupler; and (b) multielement coupler.

126 Radio Engineering for Wireless Communication and Sensor Applications

When l = l g /4 the coupling to port 4 reaches its maximum value and no signal is coupled to port 3. Thus, this device operates as a directional coupler. (Note that the coupled port is on the same end as the input port, unlike in Figures 6.6 and 6.7.) The characteristic impedances of the coupled line depend on Z 0 and on the voltage coupling coefficient K = | V4 /V1 | as

Equations (6.20) through (6.22) are valid if the even-mode and oddmode waves propagate at the same speed. In a coupled microstrip line, this is not exactly true, and the directivity is worse than in an ideal case. The structure of Figure 6.11(a) is best suited for realizing a weak coupling (large C ). The lines would be impractically close to each other to achieve a strong coupling. The single-element coupler has a bandwidth that may be too narrow for some applications. A broadband directional coupler is obtained by connecting quarter-wave sections with appropriate couplings, as in Figure 6.11(b).

The directional coupler shown in Figure 6.12 is called the Lange coupler and is suitable for realizing strong couplings as 3 dB to 6 dB. It is made of several coupled lines bonded together with thin wires.

The ring coupler illustrated in Figure 6.13(a) is a 180° hybrid. Its scattering matrix is obtained by multiplying the matrix of (6.17) by −j , if

Figure 6.12 Lange coupler.