# Low Pass Filter: LPF using Op-Amp & Applications

A filter can be defined as; it is one kind of circuit used for reshaping, modifying, and otherwise rejecting all unwanted frequencies of a signal. An ideal RC filter will divide & allows passing input signals (sinusoidal) depending on the frequency. Generally, in low-frequency (<100 kHz) applications, passive filters are constructed using resistor and capacitor components. So it is known as a **passive RC filter**. Similarly, for high-frequency (> 100 kHz) signals passive filters can be designed with resistor-inductor-capacitor components. So these circuits are named as passive **RLC circuits**. These filters are so called based on the range of frequency of the signal which they let to pass them. There are commonly three filter designs are used such as **low pass filter, high pass filter, and bandpass filter**. This article discusses an overview of the low pass filter.

## What is a Low Pass Filter?

The **definition of low pass filter** or LPF is one kind of filter used to pass signals with low frequency as well as attenuate with high frequency than a preferred cut-off frequency. The **low pass filter frequency response** mainly depends on the **Low pass filter design**. These filters exist in several forms and give the smoother type of a signal. The designers will frequently use these filter like a prototype filter with impedance as well as unity bandwidth.

The preferred filter is acquired from the sample by balancing the preferred impedance, and bandwidth, & changes into the preferred band type like low-pass (LPF), high-pass(HPF), band-pass (BPSF) or band-stop(BSF).

### First Order Low Pass Filter

A first-order LPF is shown in the figure. What is this circuit? A simple integrator. Note that integrator is the basic building block for LPFs.

Assume **Z1 =1/𝑗⍵𝐶1**

**V1 = Vi *𝑍1/𝑅1+𝑍1 = Vi (1/𝑗⍵𝐶1) / 𝑅1+(1 /𝑗⍵𝐶1)**

**= Vi 1/ 𝑗𝜔𝐶1𝑅1+1**

**= Vi 1/𝑠𝐶1𝑅1+1**

Here **s = j⍵**

**low pass filter transfer function** is

**𝑉1/𝑉𝑖 =1 / 𝑠𝐶1𝑅1+1**

The output reduces (attenuates) inversely as the frequency. If frequency doubles output is half (-6 dB for every doubling of frequency otherwise – 6 dB per octave). This is an LPF of the first order and the roll-off is at -6 dB per octave.

### Second Order Low Pass Filter

The **second order low pass filter** is shown in the figure.

Assume **Z1 = 1/𝑗⍵𝐶1**

**V1 = Vi 𝑍1/𝑅1+𝑍1**

**Vi*(1/𝑗⍵𝐶1)/𝑅1+(1/𝑗⍵𝐶1)**

**Vi 1/ 𝑗𝜔𝐶1𝑅1+1**

**= Vi 1/𝑠𝐶1𝑅1+1**

Here **s = j⍵**

Low Pass Filter Transfer Function

**𝑉1/𝑉𝑖 =1 / 𝑠𝐶1𝑅1+1**

Assume **Z2 = 1/𝑗⍵𝐶1**

**V1 = Vi 𝑍2/𝑅2+𝑍2**

**Vi*(1/𝑗⍵𝐶2)/𝑅2+(1/𝑗⍵𝐶2)**

**Vi 1/ 𝑗𝜔𝐶2𝑅2+1**

**= Vi 1/𝑠𝐶2𝑅2+1**

**Vi (1 / 𝑠𝐶1𝑅1+1)* (1/ 𝑠𝐶2𝑅2+1)**

**= 1 /(𝑠2𝑅1𝑅2𝐶1𝐶2+𝑠(𝑅1𝐶1+𝑅2𝐶2)+1)**

Therefore transfer function is a second order equation.

**𝑉𝑜/𝑉𝑖 = 1 /(𝑠2𝑅1𝑅2𝐶1𝐶2+𝑠(𝑅1𝐶1+𝑅2𝐶2)+1)**

Output reduces (attenuates) inversely as the square of the frequency. If frequency doubles output isc1/4th.(- 12 dB for every doubling of frequency or – 12 dB per octave). This is a low pass filter of second order and the roll of is at -12 dB per octave.

The **low pass filter bode plot** is shown below. Generally, the frequency response of a low pass filter is signified with the help of a Bode plot, & this filter is distinguished with its cut-off frequency as well as the rate of frequency roll off

### Low Pass Filter Using Op Amp

Op-Amps or **operational amplifiers** supply very efficient low pass filters without using inductors. The feedback loop of an op-amp can be incorporated with the basic elements of a filter, so the high-performance LPFs are easily formed by using the required components except for inductors. The **applications of op-amp** LPFs are used in different areas of power supplies to the outputs of **DAC (Digital to Analog Converters)** for eliminating alias signals as well as other applications.

#### First Order Active LPF Circuit using Op-Amp

The** circuit diagram** of the single pole or first order **active low pass filter** is shown below. The circuit of the **low pass filter using op-amp** uses a capacitor across the feedback resistor. This circuit has an effect when the frequency increases for enhancing the feedback level then the capacitor’s reactive impedance falls.

The calculation of this filter can be done by working on the frequency at which the capacitor reactance can equal the resistance of the resistor. This can be obtained by using the following formula.

** Xc = 1/ π f C**

Where ‘Xc’ is the capacitive reactance in ohms

‘π’ is the standard letter and the value of this is 3.412

‘f’ is the frequency (Units-Hz)

‘C’ is the capacitance (Units-Farads)

The in-band gain of these circuits can be calculated in a simple way by eliminating the capacitor’s effect.

As these types of circuits are helpful to give a reduction within gain at high frequencies, as well as offers an ultimate speed for roll-off of 6 dB for each octave, which means the o/p voltage divides for each repetition in frequency. So, this kind of filter is named as first order or single pole low pass filter.

#### Second Order Active LPF Circuit using Op-Amp

By using an operational amplifier, it is possible for designing filters in a wide range with dissimilar gain levels as well as roll-off models. This filter offers a bandwidth response as well as unity gain.

The circuit values calculations are uncomplicated for the response of **Butterworth low pass filter** & unity gain. Significant damping is necessary for these circuits & the ratio values of the capacitor and resistor conclude this.

**R1 = R2**

**C1 = C2**

**f = 1 – √4 π R C2**

While selecting the values, make sure that the values of the resistor will drop in the region among 10 kilos ohm to 100 kilo-ohms. This is worthwhile as the circuit’s o/p impedance increases by the frequency & outside values of this section may change the act.

### Low Pass Filter Calculator

For an RC **low pass filter circuit**, the **low pass filter calculator** calculates the crossover frequency and plots the **Low pass filter graph** which is known as a bode plot.

**For example:**

The low pass filter transfer function can be calculated by using the following formula if we know the values of the resistor and capacitor in the circuit.

** Vout(s) /Vin(s) + 1/CR/s + 1/CR**

Calculate the frequency value for the given resistor as well as capacitor values

fc** = 1/2 πRC**

### Low Pass Filter Applications

The applications of low pass filter include the following.

- Low pass filters are used in telephone systems for converting the frequencies of audio in the speaker to a band-limited voice band signal.
- LPFs are used to filter high-frequency signal which is known as ‘noise’ from a circuit, as the signal is passed through this filter, then the most of the high-frequency signal is eliminated as well as an obvious noise can be produced.
**Low pass filter in image processing**for enhancing the image- Sometimes these filters are known as a treble cut or high cut due to the applications in audio.
- A low pass filter is used in an RC circuit which is known as an
**RC low pass filter**. - LPF is used as an integrator like an RC circuit
- In multi-rate DSP, while executing an Interpolator, LPF is used as an Anti – Imaging Filter. Similarly, when executing a decimator this filter is used as an anti-aliasing filter.
- Low pass filters are used in receivers like super heterodyne for an efficient response of the baseband signals.
- Low pass filter is used in the signals of medical devices coming from the human body while testing using the electrodes are less in frequency. So these signals can flow through the LPF for removing some unwanted ambient sound.
- These filters are used in the conversion of duty cycle amplitude as well as phase detection in the phase locked loop.
- LPF is used in AM radio for the diode detector to change the AM modulated intermediate frequency signal to the audio signal.

Thus, this is all about a low pass filter. The designing of op-amp based LPF is simple to design, as well as more complicated designs using different types of filters. For more applications, the LPF provides an outstanding performance. Here is a question for you, what is the main function of the low pass filter?