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Assuming Expression for Impedance in a Parallel Resonant Circuit, State the Conditions for Parallel Resonance. Define Resonant Frequency and Obtain an Expression for It. - Physics

Sum

Assuming expression for impedance in a parallel resonant circuit, state the conditions for parallel resonance. Define resonant frequency and obtain an expression for it.

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Solution

`i=i_0 sin  ωt`

`"i" ="i"_"c" - "i"_"L"`

`"e"/"z"="e"/"X"_"c"-"e"/"X"_"L"`

`1/"z"=1/"X"_"c"-1/"X"_"L"`

For parallel resonance

`"X"_"L"="X"_"C"`

`1/"z"=0`

`"z"= alpha`

Resonant frequency (fr) :

The frequency at which impedance ( L )  because infinity and current becomes zero is called resonant frequency.

`ω "L"= 1/ (ω "L") ` 

`ω ^2 "LC" =1`

`ω^2=1/("LC")`

`ω =1/sqrt("LC")`

`2pi "f"_"r" = 1/sqrt("LC")`

`"f"_"r" = 1/(2pi) 1/sqrt("LC")`

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