# Group Delay Engineering in RF Filters

Wave propagation always creates a phase shift in a harmonic signal, and this is normally seen in an S-parameter plot or a Bode plot of the system’s transfer function. But due to frequency-dependent circuit behavior and dispersion in material constants, the phase shift seen by an RF signal creates something called group delay. For an interconnect, we might refer to the group velocity, but for filters this creates a phase shift that varies over frequency.

Filters are one type of circuit that have a group delay that is not always easily predicted. While you might think the ultra high frequency range is the hardest to work in, it is actually the middle frequency range that can be quite difficult. Mid-range frequencies are at the borderline where impedance matching is needed on most components, but the required filter circuit suffers heavily from parasitics. Also, you might expect passbands to be free of group delay dispersion, but this is almost never the case.

So to help you design RF filters properly in the range where they can be most difficult, we will outline a systems analysis approach that looks at filters at the system level. Multiple factors contribute to the group delay seen in a filter circuit:

• The type and size of components

• Trace and pad connections to components

• The order of the filter circuit

• The type of filter circuit

## What is Group Delay in an RF Filter?

Group delay in RF filters follows a very simple concept: it refers to the potential for waves with different frequencies to experience different phase shifts. This same concept is used to understand transmission lines and waveguides. However, the group delay in filters refers to the different phase shifts experienced by different frequency components in a signal. If different frequency components experience different phase shifts, then the filter might distort a signal or create the appearance of additional noise where it does not exist.

Group delay has a simple equation that defines it which is based on a derivative:

Basically, if you know the phase shift for a filter circuit as a function of frequency, then you can calculate the group delay. Group delay does not tell you how a single frequency component travels through a circuit, instead it tells you how the entire signal over a large bandwidth travels through a circuit. The group delay is shown as a curve which typically slopes upwards or downwards in your required frequency range.

An example group delay plot for a filter circuit is shown below. This group delay plot illustrates the non-ideal behavior of a real filter: it would be preferable that the filter have a horizontal line for the group delay, but real filters may only exhibit this behavior over narrow frequency ranges. Because these plots are composed of numerical data, you would just be taking finite differences between points in the plot.

Group delay is most often described in units of picoseconds or nanoseconds depending on the frequency range. Upward or downward sloping group delay curves indicate whether lower or higher frequencies accumulate larger or smaller phase shifts.

## What Determines Group Delay in a Filter?

A filter can be composed of many elements, and linear filters will be composed entirely of passive elements, or possibly active elements operating in the linear range. Group delay is influenced by multiple factors that relate to the type of filter, the order of the filter, the level of passband ripple, and parasitics in the filter components.

 Reactive elements Reactive elements contribute to phase shift Component case size Case size creates parasitics that modify the phase response Filter order The filter order determines the passband rolloff Filter topology The filter order determines the passband ripple

In the table above, it should be clear that it is difficult to generalize how a given filter affects group delay over very large frequency ranges. High pass and low pass filters can be seen to exhibit non-ideal behavior as you get close to the passband edge. In RF devices, we normally prefer higher order bandpass filters. This is quite important because it significantly attenuates unwanted frequencies from entering an active device and interfering with the desired signal.

Why should we worry about group delay dispersion in filters? For a filter operating at a single frequency, group delay dispersion does not really matter unless that one frequency must be precisely phase matched to another oscillator. But in modern RF systems, many devices are operating with a modulated signal over a range of frequencies. How we reach a flat group delay condition requires understanding all of these factors and how the arise in different frequency ranges.

## Group Delay Dispersion in Different Ranges

The process of engineering group delay dispersion is not simple and it may rely on numerical techniques to converge on a design that meets a specification. The process also depends on whether your signal is at low or high frequencies. This has to do with the practical range of frequencies seen in a PCB and their comparison to the frequencies where parasitics becom noticeable in a filter circuit.

### Very Low Frequencies and Very High Frequencies

At very low frequencies and very high frequencies, we have essentially the same approach to building a filter circuit, the difference is where the filter circuit is placed. At low frequencies, such as below approximately 1 GHz, the filter circuit can be built entirely on the PCB. The filter would provide two functions:

• Impedance matching in the RF bandwidth

• Attenuating anything outside of the passband

When placed on the PCB and operating at low frequencies, the circuit will be relatively unaffected by parasitics because larger component values are typically required.

At very high frequencies, such as above approximately 10 GHz, RF systems that need a filter will typically find the filter on a component die. The other possibility is finding them on a module which will be connected to the system through a coaxial cable. The reason for this is that parasitics will greatly dominate the behavior of the circuits in the passband. To reduce this, you would then put the circuit in a very small area, which requires me to be on the semiconductor die or as a printed circuit in a shielded module.

### Mid-Range Frequencies (1 to 10 GHz)

In the mid-range frequencies is where impedance matching can be very challenging. This has less to do with the filter design and more to do with excessive parasitic capacitance and parasitic inductance around the filter in the PCB layout. At these frequencies, the pads and traces create large deviations in the capability to impedance match and interface, and it is often the case that an impedance matching network has to be manually tuned to reach optimal matching.

In order to design and model an RF filter at midrange frequencies, the circuit being designed and simulated would not simply be a regular filter circuit. Instead, include transmission line sections between components to account for parasitics between components. It is these transmission lines sections and the parasitics they create to the reference that cause discrete RF circuits to be so difficult to design.

## Some Methods for Engineering Group Delay

In any of these frequency ranges, your job as a designer is to engineer the passband you want and attempt to reach an optimal group delay curve for your passband. SPICE can be used to engineer group delay at very low and very high frequencies because the parasitics can be accounted for easily or ignored. The typical approach will be the following process with AC sweep simulations:

1. Sweep through the desired bandwidth with an AC sweep to determine voltage and current at each port

2. Determine the input and output impedances to see if they match target values

3. Use these values to calculate the S-parameters at the desired input port

4. Calculate the group delay from the S21 phase within the frequency range

5. Use a parameter sweep to iterate through component values and monitor the group delay graph

Eventually, you may find that you converge to a reasonably flat group delay curve. The ideal group delay curve will be flat in the passband and could have any other value outside the passband.

In mid-range frequencies, the presence of parasitics make SPICE difficult to use in front-end engineering. Usually you will need to design a filter topology, place it in the PCB layout, complete routing, and then add the parasitics into a SPICE simulator. Once the parasitics are in the SPICE simulation, the group delay for the filter can be calculated. This is why a 3D simulator may be preferred for these kinds of circuits: they can determine the parasitics directly and include them in any calculation of the S-parameters within the passband.

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