RL and RC Low Pass Filter Circuit and Formula

RC low pass filter is one of the passive filter in electronic circuit. It is called RC low pass filter because it uses resistor and capacitor to make a low pass filter. Passive low pass filter can also be composed from resistor and inductor, called RL low pass filter.

Low Pass Filter Circuit

A common low pass filter can be made from a simple RC circuit with the capacitor as the output. Thus, a passive low pass filter is mentioned as a low pass filter RC circuit.  The example of low pass filter RC can be seen in Figure.(1).

fundamental low pass filter 1
Figure1

The transfer function will be

    \begin{align*}&\textbf{H}(\omega)=\frac{\textbf{V}_o}{\textbf{V}_i}=\frac{1/j\omega C}{R+1/j\omega C}\\&\textbf{H}(\omega)=\frac{1}{1+j\omega RC}\end{align*}

Keep in mind that H(0)=1, H(∞)=0. We can see the plot of |H(⍵)| in Figure.(2) along with its ideal characteristic.

fundamental low pass filter 2
Figure 2

The half-power frequency, which is equivalent to the corner frequency on the Bode plot but in context of filter is known as cutoff frequency, ⍵c. This value can be obtained by setting the magnitude of H(⍵) equal to 1/√2. Hence,

    \begin{align*}&H(\omega)=\frac{1}{\sqrt{1+\omega^{2}_{c}R^{2}C^{2}}}\\&\mbox{or}\\&\omega_{c}=\frac{1}{RC}\end{align*}

If the capacitor acts as an output:

telkom low pass filter

The transfer function in s domain:

    \begin{align*}H(s)=\frac{V_{out}(s)}{V_{in}(s)}=\frac{\frac{1}{sC}}{\frac{1}{sC}+R}=\frac{1}{1+sCR}\end{align*}

If s=j⍵, then its transfer function becomes:

    \begin{align*}H(j\omega)=\frac{1}{1+j\omega CR}\end{align*}

Thus frequency response:

    \begin{align*}&H|(j\omega)|=\frac{1}{\sqrt{1+(\omega CR)^{2}}}\\&\angle H(j\omega)=-\mbox{tan}^{-1}(\omega CR)\end{align*}

The magnitude of frequency response graph:

When:

    \begin{align*}&\omega=0\rightarrow |H(j\omega)|=1\\&\omega=\infty\rightarrow |H(j\omega)|=0\\&\omega=\frac{1}{CR}\rightarrow|H(j\omega)|=\frac{1}{\sqrt{2}}\rightarrow\mbox{cut-off frequency}\end{align*}

low pass filter magnitude of frequency response graph

Frequency response, phase domain graph

When:

    \begin{align*}&\omega=0\rightarrow\angle H(j\omega)=0^{o}\\&\omega=\infty\rightarrow\angle H(j\omega)=-90^{o}\\&\omega=\frac{1}{CR}\rightarrow\angle H(j\omega)=-45^{o}\rightarrow\mbox{cut-off frequency}\end{align*}

low pass filter Frequency response, phase domain graph

It is also common to call the cutoff frequency as rolloff frequency.

A low pass filter is designed to pass only frequencies from dc up to the cutoff frequency ⍵c.

A low pass filter can also be formed when the output of an RL circuit is taken off the resistor. There will be many other circuits to act as a low pass filter.

 

Keep in mind that:

The cutoff frequency is the frequency at which the transfer function H drops in magnitude to 70.71% of its maximum value. It is also regarded as the frequency at which the power dissipated in a circuit is half of its maximum value.

 

As the name implies, the low pass filter is mainly used to block or impede the high frequency signal while passing the low frequency signal. Say you want to limit the signal with frequency of 50Hz and below, then the signal with 50Hz or more will be blocked.

 

We can also say that low frequency signals have no difficulty in passing the filter while it is harder for high frequency signals to pass. Thus we get the low pass filter.

 

We can easily build a low pass filter using a resistor with a capacitor or inductor. A low pass filter using a resistor and capacitor is called RC low pass filter. A low pass filter using a resistor and inductor is called an RL low pass filter.

 

Don’t worry, we will learn those two here about how we build them, their formula and characteristics.

RC Low Pass Filter

As mentioned before, an RC low pass filter is a circuit built from a resistor and a capacitor which only passes low frequency signal and blocks high frequency signal. It is very easy to build an RC low pass filter, we just need to put a resistor in series with the source and a capacitor in parallel with the source. You can see an RC low pass filter below:

rc low pass filter 1

Looking from the circuit above, we need to analyze a few things about how the circuit works exactly. The capacitor is a reactive component and provides very high resistance to low frequency signals, especially to DC signal. If you ask why, then you can read about capacitors for dc circuits. The simplest explanation can be seen from its material, a pair of dielectric plates with a small gap between them. A low frequency signal or DC signal can’t pass an open-circuit path.

 

But it is different for high frequency. The capacitor provides small resistance to high frequency signals. Thus from these two characteristics we can conclude that:

The capacitor will block low frequency signals from entering and make it flow to the next path of the circuit, while the high frequency signal will be passing the capacitor and not able to move to the next part of the circuit.

Just remember that the current will always flow through the smallest resistance path. Thus, high frequency signals prefer to flow through the capacitor not to the next part of the circuit, while the low frequency will flow to the next part of the circuit because of the high resistance from the capacitor.

How to Make an RC Low Pass Filter

After learning how it works, we will step on how to make one. The circuit will be the same, a resistor connected parallel with a capacitor. For an example, we will use a 10nF capacitor and 1k ohm capacitor. The circuit will be shown below:

rc low pass filter 2

The formula for cutoff frequency is very simple and straightforward. For an RC low pass filter the low pass filter cutoff frequency can be calculated from:

    \begin{align*}f_{c}=\frac{1}{2\pi RC}\end{align*}

Using the value above, the low pass filter cutoff frequency will be:

    \begin{align*}f_{c}&=\frac{1}{2\pi RC}\\&=\frac{1}{2(3.14)(1k\Omega)(10nF)}\\&=15,923\mbox{Hz}\end{align*}

approximately 15.9KHz.

 

It means that the RC low pass filter above will block the signal with 15.9KHz or more. Signals with frequency 15,9KHz and below will pass easily through the filter.

RL Low Pass Filter

An RL low pass filter is not different from an RC low pass filter. It is composed from a resistor and an inductor which blocks high frequency signals and allows low frequency signals to pass.

 

A bit different from an RC low pass filter, we will connect an inductor in series with the source and a resistor connected in parallel. The example can be seen below:

rl low pass filter 1

The RL low pass filter above works under the inductive reactance principle. Just like impedance and capacitance, the inductance will cause phase shift and differing frequency signals. Unlike the capacitor, the inductor provides very low resistance to low frequency signals and very high resistance to high frequency signals. Thus, we place the inductor in series with input while the resistor is connected in parallel. Don’t worry it will work as perfectly as an RC low pass filter.

How to Make RL Low Pass Filter

Now after we learn how an RL low pass filter works, we can move on to how to make one. For example we will use a 470mH inductor and a 10k ohm resistor. The example circuit will be:

rl low pass filter 2

The formula to find the low pass filter cutoff frequency will be:

    \begin{align*}f_{c}=\frac{R}{2\pi L}\end{align*}

Then we put the value for the inductor and resistor to get the low pass filter cutoff frequency as shown below:

    \begin{align*}f_{c}&=\frac{R}{2\pi L}\\&=\frac{10k\Omega}{2(3.14)(470mH)}\\&=3,388\mbox{Hz}\end{align*}

approximately 3.39KHz.

 

It means that this RL low pass filter will block the signal with frequency 3.39KHz above and allow signals with less than 3.39KHz to pass.

 

An active low pass filter can be built from a passive RC low pass filter operated with an operational amplifier. This way the passive low pass filter can have amplification.

 

Even without any calculation we know that the output voltage of the passive filter is always lower than the input. If we observe carefully, the passive filter can be seen as a voltage divider then the input voltage is divided into resistor and reactive component (inductor or capacitor).

 

The ‘loss’ produced by a passive low pass filter is called “attenuation” which can give small impact or great impact. But we can solve it with amplification from an Active Filter.

 

When we get the active filter then it will have an active element in the circuit. The active element can be an operational amplifier, FET, MOSFET, IGBT, or even simple transistor. These elements are used to amplify the output signal using external power.

 

Amplification of a filter can fix the waveform, frequency response, adjust the bandwidth, or adjust the output signal. After all the examples we can conclude that “amplification” is the main business here.

Active Low Pass Filter

We can build an active low pass filter from a passive low pass filter and an operational amplifier on the output side. The principle of this active low pass filter will be the same as before in passive low pass filter. We only add an op-amp for output amplification and control the gain. For a simple example, we can add a non-inverting amplifier or inverting amplifier.

First Order Active Low Pass Filter Without Amplification

Check the circuit below:

first order active low pass filter without amplification

This active low pass filter type is made from a passive RC low pass filter and a non-inverting amplifier acts as a voltage follower. The gain of the voltage follower is Av=1 or unity gain. The previous passive RC low pass filter has a gain less than unity or less than one.

 

What is the good thing for this configuration? The high impedance on the input side limits the excessive loading on the output side. The low impedance on the output side prevents the change of cut-off frequency caused by the change of load impedance.

 

What is the bad thing for this configuration? The main issue is its gain is never more than one. Even the maximum voltage gain is only one, the power gain is very high because the output impedance is lower than input impedance. If we need the gain more than unity then we can use the circuit below.

Amplification First Order Active Low Pass Filter

Check the circuit below:

first order active low pass filter with amplification

This circuit will provide both RC low pass filter frequency response along with its increased output amplitude by the gain Av. For a non-inverting op-amp, the voltage gain will be:

    \begin{align*}\mbox{Gain}=(1+\frac{R_2}{R_1})\end{align*}

Hence the gain formula for an active low pass filter in frequency domain will be:

    \begin{align*}\mbox{Voltage gain}&=\frac{V_{out}}{V_{in}}\\&=\frac{A_{V}}{\sqrt{1+\frac{f}{(f_c})^2}}\end{align*}

Where:

Av = gain of the filter

f = input signal frequency (Hz)

fc = cut-off frequency (Hz)

There will be three operations of active low pass filter from the gain formula above:

    \begin{align*}\mbox{Low frequency},f<f_c&=\frac{V_{out}}{V_{in}}\cong A_V \\\mbox{Cut-off frequency},f=f_c&=\frac{V_{out}}{V_{in}}=\frac{A_V}{\sqrt{2}}=0.707A_V \\\mbox{High frequency},f>f_c&=\frac{V_{out}}{V_{in}}<A_V \\\end{align*}

From the operations above we can conclude that the Active Low Pass Filter will have the gain Av when the input frequency is between 0 Hz to fc. We will get 0.707Av gain if the input frequency is the same as fc. And lastly we will get the gain less than Av if the input frequency is higher than fc.

 

We can change the equation above into dB value:

    \begin{align*}\mbox{Av(dB)}&=20\mbox{log}_{10}(\frac{V_{out}}{V_{in}})\\-3\mbox{dB}&=20\mbox{log}_{10}(0.707\frac{V_{out}}{V_{in}})\end{align*}

Second Order Low Pass Filter

Up until now we have learned about a first order low pass filter, a low pass filter with one resistor and one capacitor or inductor.

 

The term “order” in passive filters represents how many reactive components we use to make a passive filter. When a low pass filter only has one reactive element (capacitor or inductor) then we call it a first order low pass filter. When a low pass filter has two reactive elements then we call it a second order low pass filter.

second order low pass filter 1

We can put a simple example just like below:

second order low pass filter 2

Then we can use the Gain equation below:

    \begin{align*}A=(\frac{1}{\sqrt{2}})^{n}\end{align*}

Where:

n = number of order or stages

 

The cut-off frequency for second order low pass filter is given by

    \begin{align*}f_{C}=\frac{1}{2\pi \sqrt{R_{1}C_{1}R_{2}C_{2}}}\end{align*}

Frequency equation for second order low pass filter is given by

    \begin{align*}f_{(-3dB)}=f_{c}\sqrt{(2^{(1/n)}-1)}\end{align*}

Where:

fc = cut-off frequency

n = number of order or stages

f(-3dB) = pass band frequency

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