Question

3) In a radio receiver the antenna consists of a coil (see lecture notes). The signal Vin(t)-Voeis filtered by the combination of the coil inductance L, the coil resistance R and a capacitor C (see Fig. below). Equating input (signal) voltage to the sum of the voltages across the inductance, resistance and capacitance find, the voltage azross the capacitor Vc(t). Show that it is maximal when o-00-1/LC. Thus by changing the capacitor one can tune in to a given emission at frequency co V. in

Answer 1

using the above figure

known equation

X_L = 2 $\pi$ f L

but $\omega$ = 2 $\pi$ f

so X_L = $\omega$ L

and X_C = 1 / 2 $\pi$ f C

X_C = 1 /$\omega$ C

the total resistance of the circuit is

X_T = X_L - X_C

Z = ( R² + ( X_L - X_C )² )^1/2

at X_L = X_C

$\omega$ L = 1 / $\omega$ C

$\omega$ = 1 / ( LC )^1/2

at this $\omega$ value the circuit act as a shunt circuit and voltage is maximum across C.