## 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 and an odd-mode characteristic impedance Z0o, 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.

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 Zo. It can be proven that all the ports are matched if 20 40 60

100 120 140 160 180

Z0e/Q

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

20 40 60

100 120 140 160 180

Z0e/Q

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

XJ4 y4 IJ4 As/4 Aj/4

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

When l = Ag/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 Zo and on the voltage coupling coefficient K = | V4/V11 as

Equations (6.20) through (6.22) are valid if the even-mode and odd-mode 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. Figure 6.13 Hybrids: (a) ring hybrid; and (b) branch-line hybrid.

V4

c

1

Figure 6.13 Hybrids: (a) ring hybrid; and (b) branch-line hybrid.

Ii 4

the reference planes are at the T-junctions. The characteristic impedances of the ports are Z 0 and that of the 3Ag/2-long ring is /2Z0. A wave applied to port 1 (2) does not couple to port 3 because the two paths have a difference in length of Ag — Ag/2 = Ag/2, but it couples to ports 2 and 4 in the same phase. A wave applied to port 2 (A) couples to ports 1 and 3 in an opposite phase.

The branch-line coupler shown in Figure 6.13(b) is a 90° hybrid. The Ag/4-long branches have characteristic impedances of Z0 and Z0/^/2. For example, a wave applied to port 1 couples to ports 2 and 4 with a phase difference of 90°, and port 3 is isolated from port 1. Both ring and branchline couplers can be modified so that the ratio of output powers differs from 1.

### Example 6.1

Design a 50-ft microstrip directional coupler operating at 1 GHz with a coupling of C = 15 dB. The properties of the substrate are: er = 10, h = 0.254 mm, t = 5 /m. Use the structure presented in Figure 6.11(a).

### Solution

The voltage-coupling coefficient corresponding to a 15-dB coupling is K = a/ 1/1015 = 0.1778. From (6.12) and (6.13) we get Z0e = 59.8ft and Z0o = 41.8ft. We use the graph of Figure 6.10 for er = 10, and read w/h = 0.9 and s/h = 0.9. Thus, both the width of strips forming the coupled line and their separation is 0.23 mm. According to (3.85), the effective relative permittivityjs erf = 6.69. The length of the coupled line section is A/4 = c/(4fA£reff) = 29.0 mm. The strip width of 50-ft lines is 0.234 mm.