Easy-to-understand time and frequency domain characteristics of passive filters

I have read some information about passive filters recently. The article written by Robert Keim is easy to understand. Let us take a look at the most commonly used method for dealing with EMC problems - RC filtering.

This article introduces the concept of filtering and details the uses and characteristics 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 voltage over time. At any given moment, the signal has only one voltage value. What we see on an oscilloscope is a time domain representation of the signal.

A typical oscilloscope is intuitive, but it is also somewhat limited because 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 exist simultaneously.

Time domain representation of a sine wave (top) and a square wave (bottom)

Frequency domain representation of a sine wave (top) and a square wave (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 a signal into frequency components that will be passed and frequency components that will be blocked.

Let's assume we have an audio signal composed of a perfect 5 kHz sine wave. We know what a sine wave looks like in the time domain, and in the frequency domain we can only see a frequency "spike" at 5 kHz. Now let's assume we activate a 500 kHz oscillator, introducing high-frequency noise into the audio signal. The signal seen on the oscilloscope is still just a sequence of voltages, with one value at each moment in time, but the signal will look different because its time domain variations must now reflect both the 5 kHz sine wave and the high-frequency noise fluctuations. In the frequency domain, however, the sine wave and the noise are separate frequency components that exist simultaneously in that one signal. The sine wave and the noise occupy different parts of the frequency domain representation of the signal (as shown in the figure below), which means we can filter out the noise by directing the signal through a circuit that channels the low frequencies and blocks the high frequencies.

3

Types of filters

If the 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 is a high-pass filter. There are also band-pass filters, which only pass a relatively narrow range of frequencies, and band-stop filters, which only block a relatively narrow range of frequencies.

Filters can also be classified based on 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 operational amplifiers.

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 a resistive element with a reactive element. In other words, we need a circuit consisting of a resistor and either a capacitor or an inductor. In theory, the resistor-inductor (RL) low-pass topology is comparable in filtering capability to the resistor-capacitor (RC) low-pass topology. In practice, however, the resistor-capacitor version is much more common, so the remainder of this article will focus on RC low-pass filters.

RC Low Pass Filter

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

We can intuitively analyze the filtering action of the RC low-pass topology if we recognize that the resistor and capacitor form a frequency-dependent voltage divider.

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 is dropped across the capacitor (and across the load, which is in parallel with the capacitor). When the input frequency is higher, the impedance of the capacitor is low relative to the impedance of the resistor, which means that the voltage across the resistor is reduced, and less of it is transferred to the load. Therefore, low frequencies are passed and high frequencies are blocked.

This qualitative explanation of the function of an RC low-pass is an important first step, but it is not very useful when we need to actually design a 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 retain the 5kHz signal and reject the 500kHz signal. This means that we need a filter that transitions from passing to blocking between 5 kHz and 500 kHz.

5

Cut-off frequency

The frequency range where a filter does not cause significant attenuation is called the passband, and the frequency range where a filter does cause significant attenuation is called the stopband. Analog filters, such as an RC low-pass filter, always transition gradually from the passband to the stopband. This means that there is no way to identify a frequency where the filter stops passing signals and starts blocking them. However, engineers need a way to summarize the frequency response of a filter conveniently and concisely, and this is where the concept of cutoff frequency comes into play.

When we look at a frequency response graph of an RC filter, we notice that the term "cutoff frequency" is not quite accurate. The image of the signal spectrum being "cut" in half, with one being kept and one being discarded, does not apply because the attenuation gradually increases as the frequency moves from below the cutoff point to above the cutoff value.

The cutoff frequency of an RC low-pass filter is actually the frequency at which the input signal amplitude decreases by 3dB (this value is chosen because a 3dB decrease in amplitude corresponds to a 50% decrease in power). For this reason, 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's passband, which in the case of a low-pass filter is equal to the -3 dB frequency (as shown in the figure below).

This graph shows the general characteristics of the frequency response of an RC low pass filter. The bandwidth is equal to the -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 the resistor (R) and the capacitor (C). We can also vary these values ​​to design a filter that meets exact specifications. The cutoff frequency (f C) of an RC low-pass filter is calculated as follows:

Let's look at a simple design example. Capacitor values ​​are more restrictive than resistor values, so we will start with a common capacitor value (such as 10 nF), and then we will use the 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 more closely analyze the effects of this filter on the two frequency components.

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

6

Calculating filter responses

We can calculate the theoretical behavior of a low-pass filter by using a frequency-dependent version of the typical voltage divider calculation. The output of a resistor divider is expressed as follows:

The RC filter uses an equivalent structure, but we have a capacitor in place of R2. First, we replace R2 (in the numerator) with the reactance of the capacitor (XC). Next, we need to calculate the magnitude of the total impedance and put it in the denominator. Therefore, we have:

The reactance of a capacitor represents the amount of opposition to the flow of current, but unlike resistance, the amount of opposition depends on the frequency of the signal passing through the capacitor. Therefore, we must calculate the reactance for a specific frequency, and the equation we use for this is as follows:

In the design example above, R ≈ 160Ω and C = 10nF. We assume that the amplitude of V IN is 1 V, so we can simply remove V IN from the calculation. First let's calculate the amplitude of V OUT at the sine wave frequency:

The amplitude of the sine wave remains essentially unchanged. This is good, since our goal is to maintain the sine wave while suppressing the noise. This result is not surprising, since we chose a cutoff frequency (100 kHz) that is much higher than the sine wave frequency (5 kHz).

Now let us see how the filter successfully attenuates the noise component.

The noise amplitude is only about 20% of its original value.

7

Visualizing filter responses

The most convenient way to evaluate the effect of a filter on a signal is to examine a plot of the filter's frequency response. These graphs, often called Bode plots, have amplitude (in decibels) on the vertical axis and frequency on the horizontal axis; the horizontal axis usually has an exponential scale so that the physical distance between 1Hz and 10Hz is the same as the physical distance between 10Hz and 100Hz, and between 100Hz and 1kHz. This method of representation enables us to quickly and accurately evaluate the effect of a filter over a wide range of frequencies.