An inductor is connected across a capacitor whose...

An inductor is connected across a capacitor whose capacitance can be varied by turning a knob. We wish to make the frequency of oscillation of this $L C$ circuit vary linearly with the angle of rotation of the knob, going from $2 \times 10^{5}$ to $4 \times 10^{5} \mathrm{~Hz}$ as the $\mathrm{knob}$ turns through $180^{\circ}$. If $L=2.0 \mathrm{mH}$, plot the required capacitance $C$ as a function of the angle of rotation of the knob.

Key Concepts

Graphical Representation of Relationships

Plotting a function to represent the relationship between capacitance and a control variable involves understanding the mathematical relationship between the two quantities and then visualizing how one affects the other. This process is essential for designing systems where a mechanical adjustment, like turning a knob, results in a predictable and controlled electrical response.

Function Transformation for Linear Control

Achieving a linear response in the frequency by controlling a nonlinear parameter like capacitance involves rearranging and transforming the resonant frequency equation. By expressing capacitance in terms of frequency, designers can determine the required capacitance values that produce a linear variation in frequency in response to a linear change in another physical parameter, such as a control angle.

Resonant Frequency Formula

The resonant frequency of an LC circuit is given by the formula f = 1/(2??(LC)), where f is the frequency, L is the inductance, and C is the capacitance. This relationship shows that the frequency is inversely proportional to the square root of the capacitance, meaning any change in C will nonlinearly impact the frequency of oscillation.

LC Circuit

An LC circuit consists of an inductor and a capacitor connected together and can oscillate at a natural frequency. The energy stored in the magnetic field of the inductor and the electric field of the capacitor continuously transfers from one form to the other, creating oscillations. This concept is fundamental in electrical circuits used for tuning and filtering signals.

Nonlinear Relationship between Frequency and Capacitance

Because the frequency in an LC circuit is determined by the inverse square root of the capacitance, there is a nonlinear relationship between the two. When attempting to control one parameter linearly (for example, through a mechanical adjustment), additional mathematical transformations are needed to compensate for the nonlinearity of the frequency response due to changes in capacitance.

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