Filter Q Factor Explained

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.

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 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.

It is important to note that this approach works for narrowband filters. However, when f₁ and f₂ are widely separated, which usually means two octaves or more between f₁ and f₂, this results in a wideband with the filter often constructed by combining a high pass filter for f₁ and a low pass filter for f₂. 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.

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.

Source: Knowles Precision Devices