Rat-race couplers

Revised February 21, 2006

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Applications of rat-race couplers are numerous, and include mixers and phase shifters. The rat-race gets its name from its circular shape, shown below. The circumference is 1.5 wavelengths. For an equal-split rat-race coupler, the impedance of the entire ring is fixed at 1.41xZ₀, or 70.7 ohms for a 50 ohm system. For an input signal V_in, the outputs at ports 2 and 4 (thanks, Tom!) are equal in magnitude, but 180 degrees out of phase.

The coupling of the two arms is shown in the figure below, for an ideal rat-race coupler centered at 10 GHz (10,000 MHz). An equal power split of 3 dB occurs at only the center frequency. The 1-dB bandwidth of the coupled port (S41) is shown by the markers to be 3760 MHz, or 37.6 percent.

The graph below illustrates the impedance match of the same ideal rat-race coupler, at ports 1 and 4. By symmetry, the impedance match at port 3 is the same as at port 1 (S11=S33). For better than 2.0:1 VSWR (14 dB return loss), a bandwidth of 4280 MHz (42.8%) is obtained.

Below the phase difference between arms 2 and 4 is plotted. At the center frequency. a perfect 180 degree difference is observed. The bandwidth that better than +/- 10 degrees is maintained is 3200 MHz, or 32%.

Unequal-split rat-race couplers

In order to provide an unequal split, the impedances of the four arms are varied in pairs, as shown below.

Equations for the Z_0A and Z_0B line impedances, as a function of the power split PA/PB, are given below:

Z_0A and Z_0B are graphed below versus the power split express in dB (coupling ratio) for a 50-0hm system. Click here for info on how to think in dB.

The graph below shows the frequency response for a rat-race coupler where PA/PB=0.25. This corresponds to a 50-ohm power divider where the power out of port 2 (P_A) is six dB below the power out of port 4 (P_B). Solving the above equations for the line impedances yields Z_0A=111.6 ohms, and Z_0B=55.9 ohms. Note that in many real-life cases, this coupler may prove impractical because a line impedance as high as 111.6 ohms may be difficult to accurately achieve in a 50-ohm system.

The graph below shows the frequency response for a rat-race coupler where PA/PB=4.0. This corresponds to a power divider where the power out of port 2 (P_A)is six dB higher than the power out of port 4 (P_B). The line impedances are opposite to the case where PA/PB=0.25; here Z_0A=55.9 ohms, and Z_0B= 111.6 ohms.

Check out our all-new unequal-split power divider calculator, it handles Wilkinsons, rat-races and branch-line couplers!