A Radio Tuning Circuit?

Question

A radio can be tuned into a particular station frequency by adjusting the capacitance in an L-C circuit. Suppose that the minimum capacitance of a variable capacitor in a radio is 4.11 picoFerrads (pF).

What is the inductance of a coil connected to this capacitor if the oscillation frequency of the L-C circuit is 1.67MHz, corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance?

L=?

The frequency at the other end of the broadcast band is 0.533 MHz. What is the maximum capacitance of the capacitor if the oscillation frequency is adjustable over the range of the broadcast band?

C=?

Answer 1

The natrural frequency of an LC circuit is

f = 1/( 2·π ·√(L·C) )

So:

f_max = 1/( 2·π ·√(L·C_min) )

<=>

L = 1/( (2·π·f_max)² ·C_min)

= 1/( (2·π · 1.67×10⁶Hz)² · 4.11×10⁻¹² F)

= 2.21×10⁻³H

= 2.21×10mH

and

f_min = 1/( 2·π ·√(L·C_max) )

<=>

C_max = 1/( (2·π·f_max)² ·L)

= 1/( (2·π · 0.533×10⁶Hz)² · 2.21×10⁻³H)

= 4.03×10⁻¹¹F

= 40.3pF