A **passive low pass filter** is electrical circuit that design to filter only low frequency signal and block and impedes high frequency signal. The **passive low pass filter** is generally constricted using passive component like resistance and capacitor (RC) network.

While high pass filter use RLC (Resistor-Inductor-Capacitor) components, no amplifying elements (transistors, op-amps, etc) so have no signal gain. RC filter is a composed of a resistor and capacitor and the combination of resistor and inductor is called RL filter.

Both the RC and RL filter pass low frequency and block high frequency. In this tutorial we will go through both of these type of filter RC and RL. Filter are named as the range of frequencies, some are pass through them and some are blocked or attenuating by the filter. The mostly use of filter is designs are the:

**Low pass filter****–**The low pass filter can only pass low range of frequency signal from 0 HZ to its cut-off frequency point “ƒc”, above cut-off frequency are attenuated.

**High Pass Filter –**High pass filter can only pass above cut-off frequency while it block below cut-off frequency.

**Band Pass Filter –**Band pass filter only allow the certain range of frequency setup between two points.

**Passive Low Pass Filter**

Low pass is a circuit whose are made of resistance “R”, capacitor “C” and inductor “L”. When we use RL than it is called RL filter and when we use RC than it is called RC filter. The function of low pass filter is to reject all higher frequency and pass low frequency. The range of low frequency is 0-100 kHz and while above 100 kHz is range of higher frequency.

**Low Pass Filter Circuit**

The circuit of low pass filter is shown below. In this circuit diagram the circuit contains only resistance and capacitor element in series connection. The input signal appied across resistor and output taken obtain from capacitor.

The cut-off frequency of RC filter is occurs at resonance, the capacitive reactance “Xc” is Xc =1/2πfC, or 1/C, where = 2πf. Sometime the resistance is not an important element and we only are a simple capacitor element across a line to reference ground.

**First Order Low Pass Filter**

First order passive low filter are usually made series connection of single resistor and capacitor an input signal “V_{IN}” with the output of the filter “V_{OUT}” taken from the junction of these two components.

**Mathematical equation of first order low pass filter:**

Assume, Z** =**1/⍵ C_{1}

V_{out} = V_{IN} * / (R_{1} +) = V_{IN} (1/⍵ C_{1}) / R_{1}+(1 /⍵ C_{1})

V_{IN} 1/ C_{1} R_{1}+1

V_{IN} 1/ C_{1} R_{1}+1

Here s = j⍵

**Transfer function of Low pass filter transfer function is:**

V_{out}** /** V_{IN} =1 / C_{1} R_{1}+1

The output reduces because frequency is inversely proportional of system.

**Second Order Low Pass Filter**

The figure of second order RC filter is shown below.

Assume, Z_{1} = 1/⍵ C_{1}

V_{1} = V_{IN} Z_{1}/ R_{1}+ Z_{1}

V_{IN} *(1/⍵ C_{1})/ R_{1 }+ (1/⍵ C_{1})

V_{IN} 1/ C_{1} R_{1}+1

V_{IN} 1/ C_{1} R_{1}+1

Here s = j⍵

Transfer function of Low Pass Filter:

V_{1} / V_{IN} =1 / C_{1} R_{1}+1

Assume Z_{2} = 1/⍵ C_{1}

V_{1} = V_{IN} Z_{2}/ R_{2}+ Z_{2}

V_{IN} *(1/⍵ C_{2})/ R_{2}+(1/C_{2})

V_{IN} 1/ C_{2} R_{2}+1

V_{IN} 1/ C_{2} R_{2}+1

V_{IN} (1 / C_{1} R_{1}+1)* (1/ C_{2} R_{2}+1)

1 /( s_{2} R_{1} R_{2} C_{1} C_{2}+(R_{1} C_{1}+ R_{2} C_{2})+1)

Therefore transfer function is:

V_{IN} = 1 /( s_{2} R_{1} R_{2} C_{1} C_{2}+(R_{1} C_{1}+ R_{2} C_{2})+1)

**Filters are divided into two distinct types: **

**Active filter****Passive filter**

Active filter- the active filter contains the amplifying device like transistor and op-amp so that the strength of signal to be increase.

Passive filter – the passive filter only contains passive element like resistor, capacitor and inductor. It has no amplifying device. Output of LPF has smaller amplitude than its corresponding input signal. Therefore the gain of RC filter is unity or less than one.

The low pass filter contains only resistor, capacitor and inductor. It attenuates high frequency. The cut-off frequency are those frequency at witch transition occurs; it is also called corner frequency.

**Low pass filter cutoff frequency formula:**

Cut-off frequency of a low pass filter is that frequency at which transition occurs and also known a corner frequency.

Formula of cut-off frequency is:

**fc=1/2πRC**

** ****Π Filter**

The π low-pass filter is look like a Greek latter π so it is called π filter. It consist a shunt capacitor at input and it is followed by an L-section filter.

**Low Pass Filter**** Calculator**

Basically the calculation of low pass filter is the calculation of cut-off frequency, phase shift and voltage gain.

From the diagram of RC filter, “V_{I}” input voltage and “V_{O}” is output voltage.

The transfer function of RC filter is:

V₀(s)/ V_{I} (s) = (1/sC)/(R+(1/sC))

Since Vo(s) = 1/sC

Vi(s) = R + 1/s

**H(s) = 1 / (1+sCR)**

Let s= jω

Put the value of “s” then the above equation becomes

**H(jω) = 1 / (1+jωCR)**

The magnitude of transfer function is

**|H(jω)| =1 /√[1+(ωCR)^2]**

The magnitude of transfer is depend on the ω i.e, angular frequency

When

- ω = 0, the magnitude of transfer function is 0.
- ω = 1/ CR, the magnitude of transfer function is 0.707.
- ω = infinity, the magnitude of transfer function is 0.

Calculation of Gain and phase shift of low pass filter:

Assume ω = 1/RC and ω = ωc for the above equation

The new equation will be

**Gain |H(jω) = 1 / √[1+(ω/ωc)^2]**

Since the total voltage gain is,

**Av = 20log10 (Vout /Vin) in dB**

Consider f is operating frequency

fc is cut-off frequency

Phase shift of the LPF circuit is

**Φ = tan⁻¹ (ω/ωc)**

cut-off frequency of the LPF circuit is,

**fc=1/2πRC**

Hence, phase shift is,

**Φ ⁼ tan⁻¹(2πfRC)**

The equation of capacitive reactance

**Xc = 1/2πfC**

Where C = capacitance in Farads

f= operating frequency in Hz

The frequency response are drawn between gain (dB) and frequency (Hz).

At low frequencies, the gain is larger than passband gain of the filter

At high frequencies, the gain lower than passband gain and it falls to -20dB

Frequency increases from cut-off frequency, the output voltage falls 70.71% below the input voltage.

**Difference between Low pass filter and High pass filter**

**Low pass filter: **Low pass filter is the type of frequency domain filter that is used filter out high frequency or attenuate high frequency

**High pass filter: **High pass filter is the type of frequency domain filter that is used for sharpening the image or it pass high frequency and block low frequency

**Difference between Low pass filter and High pass filter in tabular form is given below:**

Low pass filter |
High pass filter |

LPF are use for smoothing the image. | LPF are used for sharpening the image. |

It blocks or attenuates high frequency. | It blocks or attenuates low frequency. |

Low frequency is preserved in it. | High frequency is preserved in it. |

LPF only pass below cut off frequency. | LPF only pass above cut off frequency. |

Resistor is in the input side capacitor at the output side. | Capacitor is the input side and resistor at output side. |

It is use to in remove of aliasing effect. | It is use to remove of noise. |

**Also read**: Sinusoidal wave, Digital to analog converter.