Purpose and Characteristics of Resistor-Capacitor (RC) Low-Pass Filters

Let’s take a look at the most common means of dealing with EMC problems – RC filtering. This article introduces the concept of filtering and details the purpose and properties of resistor-capacitor (RC) low-pass filters.

Let’s take a look at the most common means of dealing with EMC problems – RC filtering. This article introduces the concept of filtering and details the purpose and properties of resistor-capacitor (RC) low-pass filters.

1. Time domain and frequency domain

When we look at an electrical signal on an oscilloscope, we see a line that represents the change in voltage over time. At any given moment, the signal has only one voltage value. What we see on the oscilloscope is the time domain representation of the signal.

A typical oscilloscope is intuitive, but it is also somewhat limited in that it does not directly Display the frequency content of a signal. In contrast to the time-domain representation, the frequency-domain representation (also known as the spectrum) conveys information about a signal by identifying the various frequency components that are present simultaneously.

Frequency domain representation of sine waves (top) and square waves (bottom)

2. What is a filter

A filter is a circuit that removes or “filters out” a specific range of frequency components. In other words, it separates the spectrum of the signal into frequency components that will pass and frequency components that will be blocked.

Let’s assume we have an audio signal consisting of a perfect 5 kHz sine wave. We know what a sine wave looks like in the time domain, in the frequency domain we only see frequency “spikes” at 5 kHz. Now let’s assume we activate a 500 kHz oscillator to introduce high frequency noise into the audio signal. The signal you see on the oscilloscope is still just a sequence of voltages, with a value at each moment in time, but the signal will look different because its time domain changes must now reflect the 5 kHz sine wave and high frequency noise fluctuations. However, in the frequency domain, the sine wave and noise are separate frequency components present simultaneously in the one signal. Sine waves and noise occupy different parts of the frequency domain representation of the signal (as shown in the figure below), which means that we can filter out noise by directing the signal through a circuit that passes low frequencies and blocks high frequencies.

3. Types of Filters

If a filter passes low frequencies and blocks high frequencies, it is called a low pass filter. If it blocks low frequencies and passes high frequencies, it’s a high pass filter. There are also bandpass filters, which pass only a relatively narrow range of frequencies, and bandstop filters, which block only a relatively narrow range of frequencies.

Filters can also be classified according to the type of components used to implement the circuit. Passive filters use resistors, capacitors, and inductors; these components do not have the ability to provide amplification, so passive filters can only maintain or reduce the amplitude of the input signal. On the other hand, an active filter can both filter the signal and apply gain because it includes active components such as transistors or op amps.

Active low-pass filter based on the popular Sallen-Key topology

4. RC low-pass filter

To create a passive low pass filter, we need to combine resistive elements with reactive elements. In other words, we need a circuit consisting of resistors and capacitors or inductors. Theoretically, the resistor-Inductor (RL) low-pass topology is comparable to the resistor-capacitor (RC) low-pass topology in terms of filtering capabilities. But in practice, the resistor-capacitor version is more common, so the rest of this article will focus on RC low-pass filters.

RC low pass filter

An RC low-pass response can be created by placing a resistor in series with the signal path and a capacitor in parallel with the load as shown. In the diagram, the load is a single component, but in a real circuit it can be more complex, such as an analog-to-digital converter, an amplifier, or the input stage of an oscilloscope to measure the response of a filter.

If we recognize that resistors and capacitors form a frequency-dependent voltage divider, we can intuitively analyze the filtering action of an RC low-pass topology.

Redraw the RC low pass filter to look like a voltage divider

When the frequency of the input signal is low, the impedance of the capacitor is high relative to the impedance of the resistor; therefore, most of the input voltage drops across the capacitor (and across the load, in parallel with the capacitor). When the input frequency is higher, the impedance of the capacitor is lower relative to the impedance of the resistor, which means the voltage across the resistor is reduced and less voltage is transferred to the load. Therefore, low frequencies are passed and high frequencies are blocked.

This qualitative interpretation of the RC low-pass function is an important first step, but it is not very useful when we need to actually design the circuit because the terms “high frequency” and “low frequency” are very vague. Engineers need to create circuits that pass and block specific frequencies. For example, in the audio system above, we want to keep the 5kHz signal and reject the 500kHz signal. This means we need a filter that transitions from pass to blocking between 5 kHz and 500 kHz.

5. Cutoff frequency

The range of frequencies where the filter does not cause significant attenuation is called the passband, and the range of frequencies where the filter does cause significant attenuation is called the stopband. Analog filters, such as RC low-pass filters, always have a gradual transition from the passband to the stopband. This means that there is no way to identify a frequency at which the filter stops passing the signal and starts blocking the signal. However, engineers need a convenient and succinct way of summarizing the frequency response of a filter, and this is where the concept of cutoff frequency comes into play.

When we look at the frequency response plot of an RC filter, we notice that the term “cutoff frequency” is not very accurate. The signal spectrum is “sliced” into two halves of the image, one of which is kept and one of which is discarded, not applicable because the attenuation gradually increases as the frequency moves from below the cutoff to above the cutoff.

The cutoff frequency of the RC low-pass filter is actually the frequency at which the input signal’s amplitude is reduced by 3dB (this value was chosen because a 3dB reduction in amplitude corresponds to a 50% reduction in power). Therefore, the cutoff frequency is also called the -3 dB frequency, which is actually a more accurate and informative name. The term bandwidth refers to the width of the filter passband, which in the case of a low-pass filter is equal to the -3 dB frequency (as shown in the figure below).

The figure represents the general characteristics of the frequency response of an RC low-pass filter.Bandwidth equals -3 dB frequency

As mentioned above, the low-pass behavior of an RC filter is caused by the interaction between the frequency-independent impedance of the resistor and the frequency-dependent impedance of the capacitor. To determine the details of the filter’s frequency response, we need to mathematically analyze the relationship between resistance (R) and capacitance (C), and we can also change these values ​​to design filters that meet precise specifications. The cutoff frequency (f C ) of the RC low-pass filter is calculated as:

Let’s look at a simple design example. Capacitance values ​​are more restrictive than resistor values, so we’ll start with a common capacitor value, such as 10 nF, and we’ll use this formula to determine the required resistor value. The goal is to design a filter that will preserve the 5 kHz audio waveform and reject the 500 kHz noise waveform. We will try a cutoff frequency of 100 kHz and later in the article we will analyze the effect of this filter on both frequency components more closely.

Therefore, a 160Ω resistor combined with a 10 nF capacitor will give us a filter that is very close to the desired frequency response.