What Low Pass Filter Is and How It works

A filter is a type of circuit used for reshaping, modifying, and rejecting unwanted frequencies of a signal. An ideal RC filter divides and allows passing input signals (sinusoidal) based on their frequency. Typically, in low-frequency (<100 kHz) applications, passive filters are made using resistor and capacitor components, hence called passive RC filters. For high-frequency (>100 kHz) signals, passive filters use resistor-inductor-capacitor components, known as passive RLC circuits. These filters are categorized based on the frequency range they allow to pass. The three common filter designs are the low pass filter, high pass filter, and bandpass filter. This article provides an overview of the low pass filter.

 

What is a Low Pass Filter?

 

A low pass filter (LPF) is a type of filter that allows signals with low frequencies to pass and attenuates higher frequencies beyond a specified cut-off frequency. The low pass filter frequency response primarily depends on its design. These filters come in various forms and provide a smoother signal. Designers often use these filters as prototype filters with impedance and unity bandwidth.

 

The desired filter is obtained by balancing the preferred impedance and bandwidth, then converting it into the desired band type such as low-pass (LPF), high-pass (HPF), band-pass (BPF), or band-stop (BSF).

 

First Order Low Pass Filter

 

A first-order LPF is shown in the figure. What is this circuit? It is a simple integrator. Note that an 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

 

The low pass filter transfer function is

 

𝑉1/𝑉𝑖 =1 / 𝑠𝐶1𝑅1+1

 

The output reduces (attenuates) inversely with increasing frequency. If the frequency doubles, the output is halved (-6 dB for every doubling of frequency, or -6 dB per octave). This is a first-order LPF, 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, the transfer function is a second order equation.

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

The output reduces (attenuates) inversely as the square of the frequency. If the frequency doubles, the output is one-fourth (-12 dB for every doubling of frequency or -12 dB per octave). This is a second order low pass filter, and the roll-off 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 depicted using a Bode plot, and this filter is characterized by its cut-off frequency and the rate of frequency roll-off.

 

Low Pass Filter Using Op Amp

 

Operational amplifiers (Op-Amps) provide very efficient low pass filters without the need for inductors. The feedback loop of an op-amp can be integrated with the basic elements of a filter, allowing for the creation of high-performance LPFs using the necessary components, except inductors. Applications of op-amp LPFs include power supplies and the outputs of Digital to Analog Converters (DACs) to eliminate alias signals, among other uses.

 

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 low pass filter circuit using an op-amp incorporates a capacitor across the feedback resistor. This design is effective as the frequency increases, enhancing the feedback level while the capacitor’s reactive impedance decreases.

 

The calculation of this filter can be done by determining the frequency at which the capacitive reactance equals the resistance of the resistor. This can be obtained using the following formula:

Xc = 1/ π f C

Where:

 

• Xc is the capacitive reactance in ohms
• π is the constant pi, approximately 3.412
• f is the frequency in Hz
• C is the capacitance in Farads

 

The in-band gain of these circuits can be calculated simply by ignoring the effect of the capacitor.

 

These circuits are useful for reducing gain at high frequencies and provide an ultimate roll-off rate of 6 dB per octave, meaning the output voltage halves with each doubling of frequency. Therefore, this type of filter is called a first order or single pole low pass filter.

 

Second Order Active LPF Circuit using Op-Amp

 

Using an operational amplifier, it is possible to design filters with a wide range of gain levels and roll-off models. This filter provides a bandwidth response as well as unity gain.

The calculation of circuit values for the response of a Butterworth low pass filter with unity gain is straightforward. Significant damping is necessary for these circuits, and the ratio values of the capacitor and resistor determine this.

R1 = R2

 

C1 = C2

 

f = 1 – √4 π R C2

When selecting values, ensure that the resistor values fall within the range of 10 kilo-ohms to 100 kilo-ohms. This is important because the circuit’s output impedance increases with frequency, and values outside this range may affect performance.

 

Low Pass Filter Calculator

 

For an RC low pass filter circuit, the low pass filter calculator determines the crossover frequency and generates the low pass filter graph, known as a Bode plot.

 

For example:

 

The low pass filter transfer function can be calculated using the following formula if the resistor and capacitor values are known:

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

To calculate the frequency value for given resistor and capacitor values:

fc = 1/2 πRC

Low Pass Filter Applications

 

The applications of low pass filters include the following:

 

• Low pass filters are used in telephone systems to convert audio frequencies in the speaker to a band-limited voice band signal.
• LPFs filter high-frequency signals, known as ‘noise,’ from a circuit. As the signal passes through this filter, most of the high-frequency noise is eliminated, reducing obvious noise.
• Low pass filters are used in image processing to enhance images.
• Sometimes, these filters are referred to as treble cut or high cut due to their applications in audio.
• A low pass filter is used in an RC circuit, commonly known as an RC low pass filter.
• LPF functions as an integrator in an RC circuit.
• In multi-rate DSP, LPFs are used as anti-imaging filters during interpolation and as anti-aliasing filters during decimation.
• Low pass filters are used in receivers, such as superheterodyne receivers, to efficiently process baseband signals.
• In medical devices, low pass filters are used to process signals from the human body during testing with electrodes, allowing low-frequency signals to pass through while filtering out unwanted ambient noise.
• These filters are utilized in the conversion of duty cycle amplitude and phase detection in phase-locked loops.
• LPFs are used in AM radios with diode detectors to convert the AM modulated intermediate frequency signal to an audio signal.

 

Thus, this covers everything about a low pass filter. Designing an op-amp based LPF is straightforward, while more complex designs involve various types of filters. For a wide range of applications, the LPF delivers excellent performance. Here is a question for you: What is the main function of a low pass filter?

 

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