Physics

Problem

For a tuned radio receiver, it is desirable that the average power transferred to the circuit falls to half its resonant value at frequencies

f_0 \pm \Delta f

. Consider a series

LCR

circuit with resistance

R

, inductance

L

and capacitance

C

whose current amplitude is

I_m(f)

. a. (i) Starting from the condition that the average power is proportional to

I_m^2 R

, derive an expression that relates the quality factor

Q

of the circuit to the fractional bandwidth

\dfrac{\Delta f}{f_0}

. b. (ii) A receiver is required to distinguish two stations broadcast at

1.005\,\text{MHz}

and

1.010\,\text{MHz}

. Estimate the minimum value of

Q

necessary for the tuning circuit so that the two stations can be resolved. c. (iii) If the circuit uses

L=150\,\mu\text{H}

and achieves the minimum

Q

found in part (ii), calculate the resistance that must be employed, assuming an ideal capacitor of suitable value is available.