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In a filter network components is used

Filters as the name suggests, they filter the frequency components. That means, they allow certain frequency components and / or reject some other frequency components.

In this chapter, let us discuss about the passive filters. Those are the electric circuits or networks having passive elements like resistor, inductor and capacitor.

Types of Filters

Filters are mainly classified into four types based on the band of frequencies that are allowing and / or the band of frequencies that are rejecting. Following are the types of filters.

  • Low Pass Filter
  • High Pass Filter
  • Band Pass Filter
  • Band Stop Filter

Low Pass Filter

Low pass filter as the name suggests, it allows (passes) only low frequency components. That means, it rejects (blocks) all other high frequency components.

The s-domain circuit diagram (network) of Low Pass Filter is shown in the following figure.

Low Pass Filter

It consists of two passive elements resistor and capacitor, which are connected in series. Input voltage is applied across this entire combination and the output is considered as the voltage across capacitor.

Here, Vi(s)" id="MathJax-Element-1-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">Vi(s)Vi(s) and Vo(s)" id="MathJax-Element-2-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">Vo(s)Vo(s) are the Laplace transforms of input voltage, vi(t)" id="MathJax-Element-3-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">vi(t)vi(t) and output voltage, vo(t)" id="MathJax-Element-4-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">vo(t)vo(t) respectively.

The transfer function of the above network is

 

H(s)=Vo(s)Vi(s)=1sCR+1sC" id="MathJax-Element-5-Frame" role="presentation" style="display: inline; line-height: normal; font-size: 15px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">H(s)=Vo(s)Vi(s)=1sCR+1sCH(s)=Vo(s)Vi(s)=1sCR+1sC

 

 

⇒H(s)=11+sCR" id="MathJax-Element-6-Frame" role="presentation" style="display: inline; line-height: normal; font-size: 15px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">H(s)=11+sCR⇒H(s)=11+sCR

 

Substitute, s=jω" id="MathJax-Element-7-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">s=jωs=jω in the above equation.

 

H(jω)=11+jωCR" id="MathJax-Element-8-Frame" role="presentation" style="display: inline; line-height: normal; font-size: 15px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">H(jω)=11+jωCRH(jω)=11+jωCR

 

Magnitude of transfer function is

 

|H(jω)|=1(1+(ωCR)2" id="MathJax-Element-9-Frame" role="presentation" style="display: inline; line-height: normal; font-size: 15px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">|H(jω)|=1(1+(ωCR)2|H(jω)|=1(1+(ωCR)2

 

  • At ω = 0, the magnitude of transfer function is equal to 1.

  • At ω=1CR" id="MathJax-Element-10-Frame" role="presentation" style="display: inline; line-height: normal; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">