What is a low-pass filter?

What is a low-pass filter?

The short answer is that a low-pass filter (lpf) allows low frequencies to pass while blocking high frequencies. The classic example is the bass tones in an audio system. The low-pass filter allows the low tones to pass to the large woofer whereas a high-pass filter (hpf) allows the high frequency tones to pass to the tweeter. There may even be a band-pass filter to allow the midrange (voice) signals to pass to a midrange speaker.

In this example, the filters are used to mitigate the physics associated with the loudspeakers. The low tones are passed to a large transducer capable of moving large volumes of air to provide the thunderous gut-punching bass while the high tones are better produced by smaller devices.

To fully understand the implications and operation of filters, we need a working mental model to differentiate between the time and the frequency domains. We must master the language used to describe frequency varying signals so that we can design and then evaluate the performance of a filter.

This article will point to the foundations. It presents the simulation results for an ideal 1st order low-pass RC filter, and then conclude with a hazard that students may encounter. Specifically, the problems encountered when a filter is used in a non-ideal environment with non-ideal components.

What are the foundations upon which filters rest?

The traditional electronics class introduces filters somewhere between discrete transistor amplifiers and op amps. It’s a shame that we don’t have more time to explore the rich language associated with frequency domain analysis. In fact, many students will attempt to describe filters without ever having used a spectrum analyzer. On a related and perhaps equally frustrating situation, you may be enrolled in a digital logic class exploring how to apply a low pass (antialiasing filter) to an ADC. The situation may clear up when you finally complete your Signals and Systems class. Until then please continue with this article.

Tech Tip: A low-pass filter is universally used before an Analog to Digital Converter (ADC). The low-pass filter serves as an anti-aliasing filter preventing an objectionable form of distortion. The cause of this distortion is related to the Nyquist sampling theorem which tells us that the highest ADC input frequency must be less than half the sampling frequency. For example, a system designed for voice frequency may have a sampling rate of 6 kHz. Any tones higher than 3 kHz will result in signal distortion.

No doubt, this is a complex Gordian Knot and likely the very reason you are visiting this page. Filters are a challenging concept because their application calls upon many related but essential concepts such as frequency domain analysis, bandwidth, resonance, harmonics, Fourier series, and decibels. Each is complex on its own but likely leave you feeling like you are drinking from a fire hose.

A complete solution is beyond the scope of this article. Instead. please allow me to direct you to a few useful videos. Video 1 is a demonstration of the spectrum encountered in a kitchen. It’s a near perfect introduction to sound and sets that stage for ways to think about filters. The second video is part of my lecture series for communications systems class. It provides multiple ways of thinking about spectrum. It builds on your earlier Circuits I and Circuits II phasor calculations and explores bandwidth using an over-the-air radio broadcast. Finally, it presents a simulation involving a low-pass filter, high-pass, bandpass, and notch filters. If you have time and interest, you can find the next video in the series that takes a deep dive into the Fourier Series.

Video 1: One of my favorite videos exploring the sound spectrum demonstrating the complexity and richness of an everyday event in a kitchen.

Video 2: This is a much longer video which is part of a lecture series completed long before I joined the DigiKey team.

Ideal low-pass filter calculations

The textbook definition for a low pass filter is presented as:

f_c = \dfrac{1}{2\pi RC}

Let R = 1.0 kΩ and C = 0.1 uF. The result is a -3dB cutoff about 1.6 kHz. This is supported by the online DigiKey low-pass filter calculator as shown in Figure 1. The results are further supported in Figure 2 which presents the results of a Multisim Live frequency sweep. Note that Figure 2 is presented using a linear scale. In this image, the amplitude is seen to drop from 1.0 V on the input to 0.71 V on the output amplitude.

Figure 1: Image of the DigiKey RC low-pass filter calculator with R = 1 kΩ and C = 1 uF. The resulting cutoff frequency is about 1.6 kHz.

Figure 2 Result of a MultiSim Live frequency sweep showing the filter output amplitude has fallen from 1.0 to 0.71 volts at the cutoff frequency.

Tech Tip: The term cutoff is often misinterpreted. In the context of a low-pass filter, the cutoff frequency is the point where the output signal is 70.7% of the input. This may be expressed in decibel form as the -3 dB point. It’s important to realize that the signal is significantly reduced at the cutoff. The actual attenuation gradually starts at a lower frequency.

Problems when we depart from the ideal low-pass filter

In the previous section, we were careful to use the term “ideal” when describing the filter. This term applies to the filter itself and to the way in which the filter is used. As for the filter, the term “ideal” implies that the components behave in an ideal way (see the Tech Tip). As for the output, we assume the filter is disconnected from any load. Stated another way, we assume the output of the filter is looking into an infinite impedance.

Tech Tip: Components such as capacitors and inductors are far from ideal. One of my favorite labs involves a RL low pass filter. For lower frequencies the circuit behaves as expected. At higher frequencies it deviates wildly from the expected performance. The problem is that an inductor operated at high frequency starts to look like a capacitor due to the inter-winding capacitance.

A classic mistake is to feed the output of a filter into an inverting op-amp. Suppose this amplifier was designed for a gain of 10 using 1.0 kΩ and 10 kΩ resistor pair. Considering the op-amps virtual ground, the filter would see a 1 kΩ load. The filter output amplitude is no longer unity for low frequencies. In fact, for our chosen resistors, we have formed a voltage divider with the output cut in half for low frequencies. Our cutoff frequency has also shifted. Without explanation, we should dust off Thévenin’s theorem and get busy with the AC analysis.

Figure 3 Result of a MultiSim Live frequency sweep showing signicant deviation when the filter is loaded with a 1kΩ load

This shift in frequency has serious implications. For example, a person who does not understand this ideal distinction may design a crossover for the woofer in a speaker system. We can see that they will be seriously disappointed with the results as the frequency and attenuation levels will be nowhere near ideal. To make matters worse, the loudspeaker itself isn’t particularly well behaved. For example, it has a physical resonance that changes depending on the dampening of the enclosure. This results in an electrically reactive component that changes the load on the filter.

Conclusion

This is a fascinating point of study where theory encounters the real world. The low-pass filter is anything but simple. Please keep these thoughts in mind as you design circuits. Perhaps there is a reason many people call it an art.

Best Wishes,

APDahlen

About the author

Aaron Dahlen, LCDR USCG (Ret.), serves as an application engineer at DigiKey. He has a unique electronics and automation foundation built over a 27-year military career as a technician and engineer which was further enhanced by 12 years of teaching (interwoven). With an MSEE degree from Minnesota State University, Mankato, Dahlen has taught in an ABET accredited EE program, served as the program coordinator for an EET program, and taught component-level repair to military electronics technicians. Dahlen has returned to his Northern Minnesota home and thoroughly enjoys researching and writing articles such as this. LinkedIn | Aaron Dahlen - Application Engineer - DigiKey

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