#### Rat-race coupler

Figure 1: Simple rat-race coupler

Figure 1: Simple rat-race coupler

#### Rat-race coupler

A rat-race coupler is a very simple design of a directional coupler. It can be used as a power splitter, for example in mixer stages, as a transmit/receive switch, or as a transformer from differential inputs or outputs to ground related levels.

A rat-race coupler can be constructed in various wired technologies. Solutions from coaxial cables, from stripline or waveguide sections, are known. A ring coupler made of waveguide sections consists of a rectangular waveguide bent into a circle with a mean circumference of exactly 1.5 λ. It has four ports, each of which is located a quarter wavelength apart around one half of the ring (Fig. 1). They cause a phase shift of 90° from one port to the next. The other half of the ring has a length of three-quarters wavelength. It causes a phase shift of 270°.

Figure 2: Rat-race coupler made of striplines, here for matching the differential output of a 24 GHz radar chip to a patch antenna.

Figure 2: Rat-race coupler made of striplines, here for matching the differential output of a 24 GHz radar chip to a patch antenna.

The feed lines of the waveguide are E-type T-junctions. In cable variants, the impedance of the feed line must split in the ring for optimal matching. The ring should therefore have an impedance with a factor of √2 compared to the impedance of the ports. For the solution with coaxial cable sections, which is interesting for radio amateurs, this can be approximated with 75 Ω (ring) and 50 Ω cables (feed lines) with sufficient accuracy.

The full scattering matrix for an ideal −3 dB rat-race is:

(1)

A signal fed into the rat-race coupler at port 1 is divided into two parts, one of which is phase-shifted by 180°. One part moves clockwise through the ring, the other part moves counterclockwise. To obtain an output signal in one of the connections, both power components must again be applied in counterphase as they were fed in. This condition is automatically fulfilled for the opposite port 4 because both components have completed the same delay line length.

Figure 3: Response of a T-connection for in-phase and out-of-phase fields

Figure 3: Response of a T-connection for in-phase and out-of-phase fields

In the case of port 2, one part received an additional phase shift of 90°, the other part also received an effective phase shift of 90° through the detour of 5/4λ. However, this port 2 is exactly λ/2 away from port 4, so that the output signal is inverted.

At port 3 the phase relations are λ/2 and the other part on the longer way 1λ, so that a phase difference of also 180° occurs on the detour lines, which equalizes the phase difference at the feed-in. Therefore this port has no output signal (see Fig. 3, right part) but must be equipped with a terminating resistor for the line adjustment (in Fig. 2: R20).