Phase Delay vs. Group Delay

When you master phase, you become like a God, capable of performing wonders that mere mortals can only dream of. Wonders like making laser beams (using phase engineered quarter-wave reflectors), communicate tremendous information great distances through thin air (all modern communication formats use both amplitude and phase), and create amazing products (balanced amplifiers, balanced mixers, phased array antennas, Mach Zender modulators, the list never ends).

BUT…phase is the hardest thing to understand in microwaves, RF, and photonics. It is hard to measure, hard to visualize, and makes some very confusing homework problems that kept me in the late night coffeeshops of Champaign-Urbana well past my bedtime.

In this post we will make a dent in the universe of phase understanding by clarifying the difference between phase and group delay, and in the process explain why you can’t match phase with variable line lengths. When you buy a phase shifter, it is sometimes what I would call a real phase shifter, and sometimes what I would refer to as a ‘group delay shifter’. The trombone type variable delay lines (we like the ones from sage) are actually variable time delay elements, and not phase shifters.

A group delay (or time) shift is easy to understand: it is how long the pulse (or wave) takes to arrive at your measuring receiver. Differential delay is therefore the difference in how long it takes for two pulses or waves to arrive. In passive components it is just the distance divided by the speed of light (or whatever your wave is) at your frequency in your material.

Phase is much more difficult. It is the integral of group delay over frequency (plus an offset), or differently the group delay is the derivative of the phase vs. frequency. This is why filters can be used as time delays; the edges of the filter have significant phase variation that leads to significant group delay variations over a narrow bandwidth (this is called Kramers-Kronig relation).

A variable length delay line, therefore, can only change the phase by changing the group delay. But by changing the group delay, you are changing the integral (slope) of the phase vs. frequency. This means that the phase change will be different at different frequencies. This is very different than what you get from a quadrature hybrid coupler, or a balun, where the phase shift is constant across frequencies. The difference is shown below. First is a plot of the phase difference between the two outputs of a BAL-0520 Balun (180°), a QH-0226 quad hybrid (90°), a coupler plus two 37.5° Schiffman phase shifters we developed as a custom (165°), a PD-0220 wilkinson power divider (0°), and a PD-0220 with an extra .570″ adapter on one side (variable).

As you can see, the phase is flat across the bandwidth of the device for everything except the PD-0220 with the extra delay line (adapter). This has a rapidly changing phase across frequencies. If we take the derivative of this we should get the group delay, but instead I measured the differential group delay with the PNA-X.

Here you can see that the differential group delay between outputs for each of the devices is 0, except for the power divider with the adapter, which has a flat constant group delay (ignore the big hump, I think that is from the calculation the PNA is doing with the phase flip).

So what is the lesson? You can phase match two outputs using a variable delay line, but only at a single frequency. Otherwise you have to do it with a coupler, a balun, a Schiffman, or some other true variable phase circuit.