High-pass/low-pass phase shifters

Updated January 5, 2007

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Updated for January 2007! We show the exact calculations for tee and pi lumped element filters to achieve any phase shift for any frequency and characteristic impedance. Yet another example of microwave engineering knowledge that you'll only get here at Microwaves101.com, because 1. other sites and books think you already know this or 2. they don't know it themselves!

If a constant phase shift is desired over a wide frequency range, the switched line phase shifter isn't going to cut it. A high-pass/low-pass phase shifter can provide near constant phase shift over an octave or more. By "high-pass/low-pass" we refer to the fact that one arm forms a high-pass filter while the opposite arm forms a low-pass filter. A second advantage of the high-pass/low-pass phase shifter is that it offers a very compact layout because lumped elements are typically used instead of delay lines. This is an important consideration for "low frequency" designs (i.e. below X-band) because delay transmission lines can be quite large. The cutoff frequencies of the two filter networks obviously must be outside of the phase shift band for this scheme to work.

The figure below shows predicted phase response for a high-pass/low-pass 90-degree phase shifter optimized for C-band operation. The values of the lumped elements in this simulation are C1=1.40 pF, L1=2.09 nH, C2=0.40 pF, and L2=0.63 nH (referring to Figure 4). Switch insertion phase contributions were neglected in this simulation. Note that between 4 and 8 GHz the phase shift (green trace) stays within -90 to -99 degrees, a nearly flat response that is not possible from a switched line phase shifter. High-pass/low-pass phase shifters usually require two control signals. By our Microwaves101 convention, the high-pass arm provides the reference state, as it provides the least-negative insertion phase.

Guess what? In practice, the distinction between switched-line and high-pass/low-pass phase shifters is blurred. Lumped capacitors are employed in the filter networks, but the inductors are really just transmission lines of high characteristic impedance.

High-pass/low-pass phase shifter topology

High-pass/low-pass phase shifter, C-band 90 degree bit

Calculating exact lumped element values

In the preceding example, an optimizer was used to determine the L and C values. But you can use simple math to come up with values that will give you the exact result at one frequency, then optimize the values to your heart's content if you like!

First, here annotated schematics to reduce confusion. Note that you have two choices for each filter, the tee or the pi; the choice might be driven by whichever one offers a more compact layout, or perhaps one choice will violate a design rule for minimum capacitor size. Also note that you could use more lumped elements than the three shown, but nine times out of ten this will git 'er done!

And here's the equations. Note that phase is in radians when you use Excel. Also, for a phase bit with phase shift, use /2 in the equations to obtain half of the phase shift from each network. We've lost the derivation for these equations, the first person that shows us how they were derived will win a nice crisp twenty dollar bill!

One of these days we'll offer a spreadsheet download to perform the calculations and plot the phase response over frequency (it's not that hard to do, we just don't have the time to do it right now!!!)

Before we post a spreadsheet, for now we'll paste in the values for four different bits at 10 GHz (which of course we calculated in a spreadsheet!) You can scale the L and C values with 1/frequency. Each filter give you half of the desired phase shift, i.e. the LP filter for 22.5 but gives 11.25 degrees phase.

For yucks we used the equations to create a 90 bit centered at 6 GHz to compare to the preceding example. The calculated high pass cap C1 was 1.281 and the high-pass inductor L1 was 1.876, while the low pass cap C2 was 0.375 pF and the low pass inductor L2 was 0.549 nH. These are pretty close to what we ended up with.

In many real phase shifters you'll find that more than 50% of the phase shift is done in the low pass network and less than 50% in done in the high -pass. Why? It just seems to give better bandwidth!

More to come soon!