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A tank circuit in a radio transmitter is a series RCL circuit connected to an antenna. The antenna broadcasts radio signals at the resonant frequency of the tank circuit. Suppose that a certain tank circuit in a shortwave radio transmitter has a fixed capacitance of 1.6 x 10-11 F and a variable inductance. If the antenna is intended to broadcast radio signals ranging in frequency from 4.8 MHz to 9.0 MHz, find the (a) minimum and (b) maximum inductance of the tank circuit.

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a. The minimum inductance of the tank circuit is 56.85 H

b. The maximum inductance of the tank circuit is 199.86 H

Resonant frequency

The resonant frequency of a series RLC circuit is given by

f = 1/2π√LC where

  • L = inductance and
  • C = capacitance

Making L subject of the formula, we have

L = (2πf)²/C

a. Minimum inductance in the tank circuit

The minimum inductance of the tank circuit is 56.85 H

To find the minimum inductance, we substitute the value of

  • f = frequency = 4.8 MHz = 4.8 × 10⁻⁶ Hz and
  • C = capacitance = 1.6 × 10⁻¹¹ F into the equation, we have

L = (2πf)²/C

L = (2π × 4.8 × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = (9.6π × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = (30.159 × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = 909.583 × 10⁻¹² Hz²/1.6 × 10⁻¹¹ F

L = 568.489 × 10⁻¹ H

L = 56.8489 H

L ≅ 56.85 H

The minimum inductance of the tank circuit is 56.85 H

b. Maximum inductance in the tank circuit

The maximum inductance of the tank circuit is 199.86 H

To find the maximum inductance, we substitute the value of

  • f = frequency = 9.0 MHz = 9.0 × 10⁻⁶ Hz and
  • C = capacitance = 1.6 × 10⁻¹¹ F into the equation, we have

L = (2πf)²/C

L = (2π × 9.0 × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = (18π × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = (56.549 × 10⁻⁶ Hz)²/1.6 × 10⁻¹¹ F

L = 3197.75 × 10⁻¹² Hz²/1.6 × 10⁻¹¹ F

L = 1998.59 × 10⁻¹ H

L = 199.859 H

L ≅ 199.86 H

The maximum inductance of the tank circuit is 199.86 H

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