# Filter Q Factor Explained

- Posted by doEEEt Media Group
- On August 11, 2022
- 0

Knowles Precision Devices perform a deep dive on the different ways you can think about Q factor for the components going into the filter or filter.

As an RF engineer, you frequently hear the term “quality factor,” or Q factor, used as a shorthand figure of merit (FOM) for RF filters.

In short, the Q factor is expressed as the ratio of stored versus lost energy per oscillation cycle.

More specifically, the Q factor generally describes specifications such as the steepness of skirts or the selectivity and how low the insertion loss is. Overall losses through a resonator increase as the Q factor drops and will increase more rapidly with frequency for lower values of resonator Q.

However, understanding how the Q factor is determined is a bit more intricate. Let’s start by looking back to the example bandpass filter specification we showed in this article.

In this example, the X axis shows the operating frequency of the bandpass filter, while the Y axis shows the power allowed through the filter in decibels. We can mark the following characteristics of this filter on this graph:

- Center frequency – The geometric or arithmetic mean of the bandpass filter’s upper and lower cutoff frequencies or 3dB points.
- Bandwidth – Usually taken from the 3dB points on either side of the center frequency.
- Insertion loss – Drawn here as the loss at the center frequency. Generally, when someone says high Q about insertion loss, this usually means low insertion loss.
- Selectivity – This measures a filter’s ability to pass or reject specific frequencies closer to the band of interest. People usually mean this when they talk about ‘steep skirts’ or a ‘sharp response.’ Generally, high Q means high selectivity.

**Understanding the Different Components of Q Factor**

There are three types of Q – loaded (Q_{L}), unloaded Q (Q_{u}), and external Q (Q_{e}) that make up the Q factor. Q_{L} is measured by looking at a plot of a filter’s performance. The standard definition of QL is as a FOM for bandwidth calculated with the following equation:

Q_{L} is driven by what goes on inside the filter, which is the Q_{u}, and the way that the device is coupled to the external world, which is the Q_{e}:

Q_{L} is a convenient way to talk about a filter’s performance as plotted. But when it comes to what makes a filter works the way it does, it’s best to look at the Q_{U} of the resonators the filter is built up from. Now let’s look at three ways to define the Q factor using the different Q types.

**Three Ways to Define Q Factor**

As mentioned, there are a few different ways to define the Q factor, depending on the context of the discussion. This includes the following:

- Bandpass Q Factor – This talks about the width of a filter. Sometimes this is Q
_{L}, as discussed above, but with wide filters, it is tricky to use Bandpass Q Factor - Component Q Factor – Addresses individual inductor or capacitor Q
- Pole Q Factor – This tells us about the performance of different parts of a filter response and is more abstract and based on Pole Zero Plots.

While component and bandpass Q are the two most common types of Q factors referenced, let’s further explore the context of all three to understand better what someone may mean when they say a filter or component has “high Q.”

**Bandpass Q Factor**

When connecting components to create a resonant circuit, we need to look at QL, which for bandpass filters refers to selectivity, as shown in Figure 2.

If the resonant circuit has Bandpass properties, we can define Q_{L} with the following formula:

It is important to note that this approach works for narrowband filters. However, when f_{1} and f_{2} are widely separated, which usually means two octaves or more between f_{1} and f_{2}, this results in a wideband with the filter often constructed by combining a high pass filter for f_{1} and a low pass filter for f_{2}. In this situation, it might make sense to think about the pole quality factor, which we will discuss later in this post, or to look at the performance of the high pass and low pass sections and their Q_{L}.

**Component Q Factor**

As mentioned, the component Q factor looks at just the component, such as the inductor or capacitor, in isolation from the rest of the circuit. Components have Q_{u} related to the component values and loss. Since inductance and capacitance provide an opposition to AC that is measured in terms of reactance, let’s look at Q in terms of how the component behaves under reactance.

For a reactance with no loss:

For an Inductive reactance, Q increases with frequency and decreases with loss:

For capacitive reactance, Q decreases with frequency and with loss:

More specifically, the Q of an individual reactive component depends on the frequency at which it is evaluated, typically the resonant frequency of the circuit used in. The formula for Q depends on whether we imagine the R to be in series with or in parallel with the reactance. The following formulas can be used to calculate Q_{u}:

**Pole Q Factor**

For more complex systems such as wider filters, we can look at the pole Q factor, which tells us about the performance of different parts of the filter response. A filter has a transfer function *H(s)* which tells us what an output signal will look like for a given input signal.

Filter Transfer Functions are expressed in terms of the complex variable ‘s’ because some problems are much easier to solve in the Laplace domain than in the time domain. The output signal Y(s) can be converted back into real numbers, and we can see how a filter’s performance is determined by the transfer function H(s) structure. We can find the values for s when the transfer function either gets large because the denominator heads to zero or gets small because the numerator heads to zero.

When heads to zero, we call these values of s ‘zeros’ because the transfer function tends to get smaller. When heads to zero, we call these values of s “poles” because the transfer function tends to get larger. In the Pole Zero Plot in Figure 3, you can see that the poles are marked with an X.

In this plot, two poles are complex conjugate pairs. The length of the arrow from the origin to the X is the frequency ωp. The distance along the real axis can be written in terms of the Q factor:

Poles and zeros come from analyzing the system as a whole. Poles close to the y axis enhance amplitude response, making that part of the filter “sharp,” which is one of the ways the Q factor drives selectivity. Additionally, based on this plot, a high Q for a pole means low sigma. Since earlier we said this real component has to do with damping, which in turn is related to the energy loss of a resonator, this makes sense.

**Source: **Knowles Precision Devices

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