The passive analog filter is the simplest and compact way to filter analog signals. They are very reliable and require no additional power supply. All of these can be designed with a simple configuration of resistors and capacitors. The basic filter types are:
- Low pass
- High pass
- Band pass
- Band reject / Notch
Critical Equations and Order
The main equation you deal with in basic passive filter design is the cutoff frequency equation as seen below where fc is the desired cut-off frequency. The cut-off frequency is at what point we want our filter to have an impact on our signal. The R and C components in the denominator represent the values of the resistor and capacitor used.
The "Order" of a filter is how many stages it has. The benefit of a multiple stage filter is that you end up achieving a sharper transition at your cut-off frequency determined in the formula above. This is illustrated in the graph below. Blue is a single-order, red is a second-, green is a third-, and black is a tenth-. As the order increases the transition becomes sharper and more defined.
Filter Configurations
Let’s start with the low-pass and high-pass filters. The schematic examples can be seen in the figure above. You may notice that these both resemble a simple voltage divider, and you’re right! One of the interesting things about capacitors is that they behave differently based on frequency of the signal that they are exposed to. The higher the frequency of the signal the less resistance (reactance) the capacitor would have on the current flow. Likewise, at lower frequencies the capacitor exhibits greater resistance (reactance) in regards to current. To help visualize, let’s consider the formula for a voltage divider.
As we can see in the formula above, as the value of R2 becomes much greater than R1 the fraction becomes close to 1 and the input equals the output. in the example of the low pass filter the capacitor begins to have a much greater reactance and thus the output signal begins to look more and more like the input signal. The opposite is true for the high pass filter circuit.
Just as I explained earlier about cascading a low pass filter to achieve a sharper transition, different types of filters can be chained together to create different overall behaviors. This is where the band-pass and band-reject/notch filters come from. If we use a high-pass filter with a low cut-off frequency and then follow it with a low pass filter with a high cut-off frequency then we isolate the frequencies between them. This produces a band-pass filter. A similar approach can be used by paralleling the two filters to create a band-stop/notch filter to eliminate a specific frequency range.
Disadvantages of Passive Filters
Passive filters are simple, require no additional power source but as can be seen in the bode plot at the top, are not especially great in performance. You can create basic filters but if you need a really steep cut-off and precise filtering you are going to need to create some really high order filters as opposed to using operational amplifiers to create an active filtering solution.