The next figure shows the response of an ideal branchline coupler where the each side is a quarter wavelength at 10,000 MHz (10 GHz). The first graph shows the losses from the input to the two output arms. S21 is the transmission loss from the top port to the upper right port, S31 is from the input to the lower right port. Using the ideal transmission line impedances shown above provides a equal 3 dB split at the center frequency. The markers have been aligned to show the 1-dB bandwidth of the coupler, which is 2580 MHz or 25.8%.
The second graph shows that the bandwidth where the device has better than 14 dB return loss (1.5:1 VSWR) is 2080 MHz, or 20.8%. The isolation (power coupled to the terminated port) is also plotted here and is very nearly equal to the return loss.
The next plot shows the phase difference between the two outputs (ideally 90 degrees, remember?) For +/-10 degrees the bandwidth is about 4300 MHz, or 43%.
Double-box branchline coupler
New for February 2006: we've added a separate page on the double-box coupler, which explores this topic further!
As with the Wilkinson power splitter, the bandwidth of a branchline coupler can be improved by adding sections. The next figure shows a "double-box" branchline coupler with its ideal impedances. We've never seen this in a text book, have you? Microwaves101 rules!
In a 50-ohm system, the line impedances of the end vertical segments work out to be 120.71 ohms, and the center vertical segment 70.71 ohms. In practice, it may be hard to accurately achieve the 120.71 ohm impedance lines accurately.
Update September 2017: the lower right port should show +90 degrees, not -90 degrees. Thanks to Chih-Jung! If you think about it, the path goes through three -90 degree legs, or -270 degrees, which is +90 in microwave-land. One of these days we will fix it.
Ideal double-box branchline coupler
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The next three figures show the frequency response of the ideal double-box branchline coupler, centered at 10,000 MHz (10 GHz). In this case, the 1-dB response of the coupled arm is 35%, the 14 dB return loss band (1.5:1 VSWR) is 41%, and the +/-10 degrees phase difference is 50%. However, the tradeoff for the extra bandwidth in real life will be added loss of the second box section, not to mention the added size.
Power split of ideal double-box branchline coupler
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Return loss (blue) and isolation (red) of ideal double-box
branchline coupler
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Phase response of ideal double-box branchline coupler
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As a final illustration of improving the bandwidth of the branchline coupler, a double-box structure was tuned to increase the frequency bands for 1-dB coupling and 14 dB return loss. Note that the equal 3 dB power split at the center frequency must be somewhat corrupted as a tradeoff. The arm impedances of this coupler are now 38 ohms for the series arms and 100 ohms and 65 ohms as shown for the three shunt arms, as shown below. The figures speak to the results. The 1-dB band is now 55.6%, while the 14 dB return loss band is 58.4%.
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Double-box hybrid coupler tuned for more bandwidth
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Power split of tuned double-box hybrid coupler
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Return loss (blue) and isolation (red) of tuned
double-box branchline coupler
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Lumped element branchline coupler
New for March 2007: here's a page on lumped element Wilkinsons!
Lumped elements can be used to approximate transmission lines in a branchline coupler. The structure for a quarter wave transmission line can be realized with a pair of shunt capacitors of equal value, separated by a series inductor (a "pi" network). By optimizing the inductor and capacitor values, different line impedances can be "faked". Why would you want to use shunt elements? So you make branchline couplers at lower microwave frequencies (UHF through S-band) and not have to deal with huge transmission line lengths.
Below we present a lumped-element quadrature coupler which was optimized to work at 100 MHz. The lumped elements are ideal; if this was a real design that we were getting paid for we would have included all of the parasitic elements into the capacitor and inductor models, which is necessary if you want a true prediction of how the circuit will perform on the bench. Note that if you want to scale the design to a different frequency, all you have to do is scale the element values with inversely with frequency (use half the values for a 200 MHz design).
Schematic representation of 80 to 100 MHz lumped-element
quadrature coupler
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Below are the power splits of port 2 and port 3. Note that we managed to get them within maybe 1.5 dB of each other at the center frequency. Maybe you could do better, Mr. Smart Guy.
Below is the phase difference between the paths 1-2 and 1-3. Note that 90 degrees is achieved with less than 1 degree of error from 80 to 100 MHz. In real life it won't be as good because you will have to allow for tolerances on the parts.
Finally, we show the port matching (S11) and the isolation (S23). Note they are similar, this is true of most quadrature coupler designs.
Here's a branchline coupler in MMIC representation, with the quarter wavelength arms changed into lumped inductive and capacitive elements. It's a double-box structure on four-mil GaAs. It worked great at S-band, with less than 1 dB resistive loss in spite of all those spiral inductors! Such a structure can make an efficient power combiner for the IF output of a higher-frequency image-rejection mixer.
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Lumped-element branchline coupler (MMIC representation)
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Unequal-split branchline couplers
By varying the impedances of the opposite arms in a branchline coupler, unequal power splits can be obtained, as shown in the figure below.
Unequal branchline power splitter
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The equations for the line impedances Z0A and Z0B are given below, as functions of the power split PA/PB and the system impedance Z0.
The plot below shows the characteristic impedances Z0A and Z0B, for a fifty ohm system, as a function of the coupling ratio PA/PB expressed in dB. Note that two very different topologies result when PA is greater than PB (Z0A and Z0B are higher impedance than in an equal-split branchline) and when PA is less than PB (Z0A and Z0B are lower impedance than in an equal-split branchline).
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Line impedances of unequal-split branchline coupler
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The chart above does not completely tell the story of the tradeoffs made when you select which port provides the most power. Check out the power split responses for PA/PB=0.25 and PA/PB=4.0 below. The bandwidth for PA/PB=4.0 is far superior.
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Unequal-split branchline frequency response, PA/PB=0.25
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Unequal-split branchline frequency response, PA/PB=4.0
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Check out our unequal-split power divider calculator, it handles Wilkinsons, rat-races and branch-line couplers!