# Types of Low Pass Filters – RL and RC Passive Filters – Examples

**What is Low Pass Filter? Passive Low Pass Filter & Its Types With Examples.**

A **passive low pass filter** is a type of low pass filter that is made up of passive electronic components such as resistor, capacitor & inductor. The gain of a passive low pass filter is always less than or equal to 1. So its output signal’s amplitude is always less than it’s input signal’s amplitude. However, they are simple & easy to design. In this article, we will discuss the passive low pass filter & its types with examples.

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Table of Contents

**Low Pass Filter**

Low pass filter or **LPF** is a type of filter that allows low-frequency signals and blocks high-frequency signals. The frequencies lower than a selected frequency known as the cut-off frequency are passed while any frequency higher than cut-off frequency is blocked by the filter.

**Low pass filters are of two types:**

- The passive low pass filter
- The active low pass filter

We will only discuss the passive low pass filter in this article as active low pass filters are already explained in another post. Passive low pass filters are classified according to the order of the filter. we will discuss **1 ^{st}** &

**2**order low pass filter.

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**First Order Low Pass Filter**

First order low pass filter is the simplest form of low pass filters that are made of only one reactive component i.e Capacitor or Inductor. A resistor is used with the Capacitor or Inductors to form RC or RL passive low pass filter respectively. RC & RL low pass filters are briefly discussed below with examples.

- RC Low Pass Filter
- RL Low Pass Filter

**RC Low Pass Filter**

The most simple passive low pass filter is made of a resistor connected in series with a capacitor & the output is taken across the capacitor as shown in the figure below.

As we know that the capacitor allows a high-frequency signal (operate as short wire) & block low-frequency signal (operate as open wire). So, when a low frequency is applied to the circuit, the capacitor will become open & the signal will appear across its terminal, which will eventually flow out as output. However, when the high-frequency signal reaches the capacitor it becomes a short circuit & the output becomes zero.

The reason the capacitor blocks & allows frequency is because of its reactance which is given by.

**X _{c} = 1/C = 1/(2πfC)**

Where **X _{c}** is the reactance of the capacitor

**f** is the applied signal’s frequency

**C** is the capacitance of the capacitor

From the above equation, we can say that the capacitive reactance **X _{c}** is inversely proportional to the applied frequency

**f**. if the applied frequency is too low, the reactance

**X**will be greater than the resistance of the resistor & the input signal will be established across the capacitor. But when the frequency f goes higher, the reactance

_{c}**X**becomes lower than the resistor’s resistance. This results in a low voltage drop (almost negligible) across the capacitor as compared to the resistor.

_{c}**Frequency Response:**

The frequency response, also known as bode plot of a circuit shows the output to input ratio for a specified frequency range. The frequency response of Low Pass Filter is given below:

It contains some key points such as the **cutoff frequency** **f _{c}**,

**passband**,

**stopband**,

**bandwidth**&

**roll-off**etc.

**Cutoff Frequency:**

Cutoff frequency, also known as **corner frequency** denoted by **f _{c}** is the selected frequency point where the output signal’s power becomes

**-3db**or

**70.7%**of the input signal. At this frequency, the capacitive reactance

**X**& resistor’s resistance R become equal.

_{c}The low pass filter allows frequency below the cutoff frequency and blocks any frequency higher than the cutoff frequency.

Where the cutoff frequency is calculated by :

**R = X _{c}**

**R = 1/(2πfC)**

**f _{c} = 1/(2πRC)**

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**PassBand**:

The passband is the range of frequency that gets passed through the filter. In the low pass filter, the passband frequency is lower than the cutoff frequency **f _{c}**.

**StopBand:**

The stopband is the range of frequency which is blocked by the filter. In low pass filter, the range of frequency higher than the cutoff frequency **f _{c}** is referred to as

**stopband**.

**Bandwidth:**

The bandwidth of the filter is a range of frequency that gets passed without any attenuation. The low pass filter allows frequency from **0 Hz** to **f _{c} Hz**. So its bandwidth is

**f**.

_{c}-0 = fc Hz**Roll-off:**

When the frequency increases and reaches the cutoff frequency the gain of the filter starts to decrease. The rate of decrement of the gain is known as roll off.

The roll off of a **1 ^{st}** order low pass filter is

**-20db per decade**.

**Output Voltage:**

To calculate the output voltage of a passive low pass filter at any frequency, the voltage divider rule is applied between the resistor and capacitor. So, the output voltage v_{out }is given by.

**V _{out} = V_{in} * (X_{c}/Z)**

Where

**X _{c}** =capacitive reactance

**Z** = total Impedance of circuit

**Z = R + jX _{c}**

**Z = √(R ^{2} + X_{c}^{2})**

So

**V _{out} = V_{in} * (X_{c}/√(R^{2} + X_{c}^{2}))**

Where **X _{c} = 1/(2πfC)**

**Gain:**

Gain in the ratio of output voltage to input voltage.

**Gain = V _{out}/ V_{in}**

This gain is usually described is** dB**, which is the logarithmic form of the gain & it is given by:

**Gain _{dB} = 20 log (V_{out}/V_{in})**

Notice the gain of the passive low pass filter in the frequency response graph given above. The maximum gain remains **0dB**. In an active filter, this gain can be modified according to the requirement.

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**Example :**

Let’s take an example of an **RC** low pass filter with the resistor **R** of **1K ** & capacitor **C** of **47 nF**.

The cutoff frequency **f _{c}** is calculated as:

**f _{c} = 1/(2πRC)**

**f _{c} = 1/(2π*10^{3}*47*10^{-9 })**

**f _{c} = 3.38 KHz**

Frequency response simulation using Proteus for the given **RC** low pass filter is

It clearly shows **-3dB** gain at **3.38KHz**, which is the cutoff frequency of this filter.

This filter will allow any frequency below **3.38KHz** and block any frequency greater than **3.38KHz**. Thus its bandwidth is **3.38KHz**.

**RL Low Pass Filter:**

This type of low pass filter is made from a resistor & Inductor in series where the output is taken across the resistor as shown in the figure below.

The inductor allows the low-frequency signal to pass which eventually establishes across the resistor and it blocks high-frequency signal which does not make it to the load resistor.

Inductor provides low reactance for low-frequency signal & high reactance for a high-frequency signal. Its reactance is given by:

**X _{L} = L = 2πfL**

According to the equation above, the inductor reactance is directly proportional to the input signal’s frequency.

So when the input signal’s frequency is low, it reactance will be lower than the resistance of the resistor. Thus, the resultant input signal’s voltage drop will be maximum at the load resistor.

When the input signal’s frequency increases, the inductive reactance **X _{L}** becomes greater than the resistor’s resistance

**R**. Due to which the result is a negligible voltage drop at the load resistor. thus the input signal gets blocked.

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**Frequency Response:**

The frequency response of RL low pass filter is similar to RC low pass filter.

The cutoff frequency of the RL low pass filter is given by:

At cutoff frequency, the reactance is equal to the resistance

**X _{L} = R**

**2πfl = R**

**f _{c} = R/(2πL)**

Where

**L** = inductance of the inductor

**R** = Resistor’s resistance

**Output Voltage:**

The output voltage at any frequency can be calculated by voltage divider rule.

**V _{out} = V_{in} * R/Z**

Where **Z** = total impedance of the circuit.

**Z = R +JX _{L}**

**Z = √(R ^{2} + X_{L}^{2})**

So

**V _{out} = V_{in * }R/√(R^{2} + X_{L}^{2})**

Where **X _{L} = 2πfL**

**Example:**

In this example we will take **1K** resistor & **3mH** inductor.

We will calculate the cutoff frequency for this circuit.

**f _{c} = R/(2πL)**

**f _{c} = 10^{3}/(2π*3*10^{-3})**

**f _{c} = 53kHz**

Let’s simulate this circuit using Proteus. The frequency response using proteus is given below:

It clearly shows the **-3dB** (cutoff frequency) point where the frequency is **52.9 KHz** which is almost equal to **53KHz**.

**Second Order Passive Low Pass Filter**

Second order low pass filter is made by cascading two first order low pass filters. The first order low pass filter can be **RC** or **RL** circuit. We will discuss both with examples.

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**RC Low Pass Filter**

Two **1 ^{st}** order RC low pass filter are cascaded together to form

**2**order low pass filter. The schematic of

^{nd}**2**order RC low pass filter is given below;

^{nd}The first order Low pass filter stage is made of **R _{1}C_{1 }**& second stage is made of

**R**

_{2}C_{2.}

The output can be taken from either of the two stages. The first stage will provide **1 ^{st} **order low pass filter output with a roll off of

**-20db/decade**. The second stage will provide

**2**order low pass filter output with a steeper roll-off of

^{nd}**-40db/decade**. Its frequency response is shown below.

The green signal shows the frequency response of **1 ^{st}** order low pass filter with

**-20db/decade**roll-off. The red signal shows the frequency response of

**2**order low pass filter with

^{nd}**-40db/decade**roll-off.

**Cutoff Frequency:**

The cutoff frequency or corner frequency **f _{c}** of

**2**order low pass filter is given by

^{nd}**f _{c} = 1/{2π√(R_{1}R_{2}C_{1}C_{2})}**

If the resistor **R=R _{1}=R_{2 }**& Capacitor

**C=C**, then

_{1}=C_{2}**f _{c} = 1/(2πRC)**

**Gain at Corner Frequency:**

The gain of general **n** order low pass filter at corner frequency is given by:

**Gain = (1/√2) ^{n}**

So the gain of **2 ^{nd}** order low pass filter at corner frequency

**f**is

_{c}**Gain = (1/√2) ^{2} = 0.5**

The gain in **dB** will be:

**Gain _{(dB) }= 20 log (0.5) = -6dB**

So the gain of **2 ^{nd}** order LPF at cutoff frequency is

**-6db**.

**-3dB Frequency:**

The calculated corner frequency **f _{c}** provides

**-6dB**gain whereas, the filter passband frequency lies at a –

**3dB**gain which is calculated as:

**f _{ (-3db)} = f_{c}√(2^{(1/n)}-1)**

Where **n** is the order of the filter & **f _{c}** is the calculated corner frequency. The

**f**decrease with the increase in the order of the filter. This implies that the increasing order of the filter provides a steeper or fast roll off.

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**Example**

In this example we have taken resistor **R _{1 }= 1K**,

**R**& capacitor

_{2}= 10K**C**,

_{1 }= 47nF**C**

_{2}= 4.7nF_{2}>10R

_{1}& C

_{2}<10C

_{1}is because of the loading effect otherwise the output signal will contain errors. To reduce the loading effect the impedance of the second stage need to be much greater than the first stage.

The cutoff frequency of this circuit will be

**f _{c} = 1/{2π√(R_{1}R_{2}C_{1}C_{2})}**

**f _{c} = 1/{2π√(10^{3}*10*10^{3}*47*10^{-9}*4.7*10^{-9})}**

**f _{c} = 3.38 KHz**

This frequency lies at **-6dB** gain. Now we will calculate **f _{ (-3db)}**

**f _{ (-3db) }= fc √(2^{(1/2)}-1)**

**f _{ (-3db) }= 3.38*10^{3 }√(2^{(1/2)}-1)**

**f _{ (-3db) }= 2.17 KHz**

So this low pass filter will allow frequency less than **2.17 KHz**, thus its bandwidth is **2.17 KHz**.

Let’s simulate this example using **Proteus**. The following is the frequency response of the said example.

This graph clearly shows the **-3dB** frequency at **2.17KHz** which we calculated earlier.

**RL Low Pass Filter:**

Two stages of **RL** low pass filter are cascaded together to form **2 ^{nd}** order low pass filter. The first stage consists of

**L**& the second stage consist of

_{1}R_{1 }**L**. It schematic is given below.

_{2}R_{2}The **1 ^{st}** stage is a

**1**order low pass filter whose output provides a roll off of

^{st}**-20db/decade**. The 2

^{nd}stage provides

**2**order low pass filter with a roll-off of

^{nd}**-40db/decade**.

**Corner Frequency**

The corner frequency or cutoff frequency f_{c} of **2 ^{nd}** order low pass filter is given by.

**f _{c} = {√(R_{1}R_{2})}/{2π√(L_{1}L_{2})}**

If the resistor **R=R _{1}=R_{2 }**& Inductor

**L = L**, then

_{1 }= L_{2}**f _{c} = R/(2πL)**

**Gain at Corner Frequency:**

The gain at corner frequency is given by:

**Gain = (1/√2) ^{n}**

Where **n** is the order of the filter. So the gain of **2 ^{nd}** order Low pass filter is:

**Gain = (1/√2) ^{2} = 0.5**

The gain in **dB** is

**Gain _{(dB) }= 20 log (0.5) = -6dB**

**-3dB Frequency**

The actual cutoff frequency of a filter lies at **–3dB** gain. To calculate **-3dB** frequency we will use:

**f _{(-3dB)} = f_{c}√(2^{(1/n)}-1)**

Where **n** is the order of the filter.

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**Example:**

Lets take resistor **R _{1 }= 1K**,

**R**& Inductor

_{2 }= 10K**L**,

_{1 }=3mH**L**.

_{2 }= 30mHWe took **R _{2}>10R_{1 }**&

**L**

_{2}>10L_{1 }because of the loading effect explained earlier. The impedance of the succeeding stage needs to be at least

**10**times higher than the previous stage.

The cutoff frequency of this filter is

**f _{c} = {√(R_{1}R_{2})}/{2π√(L_{1}L_{2})} **

**f _{c} = {√(10^{3}*10*10^{3})}/{2π√(3*10^{-3}*30*10^{-3})} **

**f _{c} = 53 KHz**

The frequency lies at **-6db** so we will calculate **-3db** frequency, which is given by:

**f _{(-3dB)} = f_{c}√(2^{(1/2)}-1) **

**f _{(-3dB)} = 53*10^{3}√(2^{(1/2)}-1) **

**f _{(-3dB)} = 34.11 KHz**

This is the **-3dB** frequency of this filter. The Proteus simulation of this filter is given below.

The graph shows the **-3dB** gain where the frequency is **34.11 KHz**. and you can clearly see the difference between the roll off of **1 ^{st}** &

**2**order low pass filters.

^{nd}**Limitations of Passive Low Pass Filter:**

- There is no amplification of the input signal in a passive low pass filter.
- Its gain remains less than or equal to 1.
- Cascading stages for higher order passive filters results in the loss of amplitude of the signal.
- The load impedance affects the characteristics of the filter.

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