For a communication receiver an engineer wishes to know how sensitive the resonant…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

For a communication receiver an engineer wishes to know how sensitive the resonant frequency of its tuning circuit is to small adjustments of the variable capacitor. The circuit uses a fixed inductance

L

and a variable capacitance

C

. (a) Starting from the expression

f=\dfrac{1}{2\pi\sqrt{LC}}

, derive the relation

\dfrac{\Delta f}{f}=-\dfrac{1}{2}\,\dfrac{\Delta C}{C}

for small changes in

C

while

L

remains constant. (b) In one particular set

L=180\,\mu\text{H}

and the capacitor varies with knob angle

\theta

according to

C=C_0(1+\alpha\theta)

, where

C_0=100\,\text{pF}

and

\alpha=0.020\,\text{rad}^{-1}

. Find

\dfrac{df}{d\theta}

\theta=0\,\text{rad}

. (c) Based on your answer in part (b), comment quantitatively on whether fine tuning of this receiver would be easy or difficult.

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