Question

If the variable capacitor in an FM receiver ranges from 14.9 pF to 22.4 pF , what inductor should be used to make an LC circuit whose resonant frequency spans the FM band

Answer 1

Complete Question

The FM radio band covers the frequency range 88-108 MHz

If the variable capacitor in an FM receiver ranges from 14.9 pF to 22.4 pF , what inductor should be used to make an LC circuit whose resonant frequency spans the FM band

Answer:

The  inductor that should be made use of is and inductor with an inductance of $L = 1.46 *10^{-7} \ H$

Explanation:

From the question we are told that

   The  frequency range  for the FM band is  88-108 MHz

   The range of the capacitance is  $C = 14.9 \ pF \ \ to \ 22.4 pF = 14.9 *10^{-12} \ F \ \ to \ \ 22.4 *10^{-12}\ F$

  The resonant frequency is  $f_r = 108 MHz. = 108 *10^6 \ Hz$ this is in the question we are told that the resonant frequency spans the FM band

 Generally the at resonance the reactance of the capacitor is equal to that of the inductor i.e

       $R_c = R_l$

Here  $R_c$  is the reactance of the capacitor which is mathematically represented as

       $R_c = \frac{1}{2 \pi f C}$

while the reactance of the inductor is  $R_i = 2 \pi f L$

So

     $\frac{1}{2 \pi f C} = 2 \pi f L$

=>  $L = \frac{1}{4 \pi^2 * f ^2 * C }$

When  C is  $C = 14.9 \ pF = 14.9 *10^{-12} \F$

=>  $L = \frac{1}{4 (3.142 )^2 * (108*10^6) ^2 * 14.9 *10^{-12} }$

=>  $L = 1.46 *10^{-7} \ H$

When  C is  $C = 22.4 \ pF = 14.9 *10^{-12} \F$

=>  $L = \frac{1}{4 (3.142 )^2 * (108*10^6) ^2 * 22.4 *10^{-12} }$

=>  $L = 1.46 *10^{-7} \ H$