RC filters

In electronics, a filter is an electrical circuit that can pass certain frequencies while attenuating other frequencies of the input signal. These circuits filter out unwanted components from the input signal - look at "Electric filter simulation".

Filters pass sinusoidal input signals based on their frequency. There are different types of filters known, we can choose such criteria:
• Low pass filters,
• High pass filters,
• Bandpass filters (very often an appropriate combination of the previous two).
All types of filters can be constructed using simple RC circuits (Resistor-Capacitor).
The simplest RC filter can be made by connecting together a single resistor and a single capacitor in series. Below you can see two variants of the use of this circuit.

RC filter

We will be interested in the transition of time-varying sinusoidal signals UIN(t) = U0sin(ωt) through such systems. The sinusoidal angular frequency ω is expressed in radians per second. On the basis of Kirchhoff's second law, we have for any given moment of time t:
     UIN(t) = UR(t) + UC(t) .
Appropriate considerations show that a current I that flows in such the circuit is equal:

The same considerations also allow us to write:

UIN(t) = U0sin(ωt),     UR(t) = URosin(ωt - φ),     UC(t) = -UCosin(ωt - φ + π/2)

where:    URo = RI0,    UCo = I0/ωC    and    tg φ = -1/RωC
I described these circuits, avoiding solving differential equations, I took ready-made solutions. Further analysis can now be performed using phasor diagrams.
Note, that:      U20 = U2Ro + U2Co
The phasor diagram for the considered RC series circuits is shown below.

phase diagram

As you can see, all relations described by the above formulas UIN(t), UR(t) and UC(t) can be presented graphically. On the left side we have the phase diagram for any moment of time t. The projection of vectors U0, URo and UCo onto the vertical axis expresses the instantaneous voltage values of the generator output UIN(t) and across the resistor URo(t) and capacitor UCo(t).

The most important conclusions:
• As the sinusoidal angular frequency ω increases, the voltage amplitude UCo across capacitor C decreases. Thus, the voltage amplitude URo across the resistor R increases (maximum value URo → U0). It should also be noted that as the value of ω increases, the phase shift angle φ decreases.
• For small values of ω, the amplitude of the voltage UCo across the capacitor increases (maximum value UCo → U0).
Thus, the same RC series circuit can be a high or low pass filter.
We have a low-pass filter (LPF) when the capacitor is the filter output. If the filter output is a resistor, we will get a high-pass filter (HPF).

Very often the properties of filters are expressed by a transfer ratio TR. The transfer ratio is a gain factor for the sinusoidal input signal with given frequency. This gain factor, and therefore the transfer ratio is a function of the frequency - TR(ω).
We define the transfer ratio as follows:

TR(ω) = UOUT(ω)/UIN(ω)

In the case of the considered series RC circuit, this function looks like this:

Transfer ratios

Below you can find two graphs of the transfer ratio TR versus angular frequency ω for the series RC circuit with the following parameters: R = 1.5kΩ, C = 100nF. Note, that ω = 2πf.

Transfer ratio diagram

There is one more parameter that is used to describe filters - it is called the filter cut-off frequency. The cut-off frequency fC of an filter is the frequency at which the amplitude of the input signal is reduced by 3 dB on output. This practically means that this is the frequency for which TR is equal to 1/√2  ≅ 0.71.
For both considered filters, the cut-off frequency fC is expressed by the formula:

fC = 1/2πRC    or      ωC = 1/RC

In the example used here, the theoretical value is 1,060 Hz. There are many calculators on the internet dedicated to counting fC, e.g. look here. We interpret the obtained result as follows. The low pass filter only pass low frequency signals from 0Hz to 1060 Hz and the high pass filter high frequency signals from 1060 Hz to infinity.

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You already have a basic knowledge of simple RC filters, so it's time to experiment on your own. Such the experiment is easy to perform in in the electronics lab. However, if you do not have an oscilloscope and capacitors and resistors, you can use my online laboratory, it is waiting for you.
If you want to learn more about the discussed issue, I recommend you to take a look at the internet (e.g. here or there).