A variable capacitor with a range from 10 to 365 pF is used...

A variable capacitor with a range from 10 to 365 pF is used with a coil to form a variable-frequency $L C$ circuit to tune the input to a radio. (a) What is the ratio of maximum frequency to minimum frequency that can be obtained with such a capacitor? If this circuit is to obtain frequencies from 0.54 MHz to 1.60 $\mathrm{MHz}$ , the ratio computed in (a) is too large. By adding a capacitor in parallel to the variable capacitor, this range can be adjusted. To obtain the desired frequency range, (b) what capacitance should be added and (c) what inductance should the coil have?

Key Concepts

Parallel Capacitance Effect

Adding a fixed capacitor in parallel with a variable capacitor changes the effective total capacitance, shifting both the minimum and maximum values. This modification allows designers to tailor the resonant frequency range of an LC circuit so that it matches specific operational needs, such as narrowing or extending the tuning range.

LC Resonance

The LC resonance concept is based on the equation f = 1/(2??(LC)), which describes the natural frequency of oscillation for an inductor-capacitor circuit. This principle is crucial in circuits such as radio tuners, where setting a specific resonant frequency allows for the selective reception of desired signals.

Variable Capacitance

A variable capacitor is designed to have an adjustable range of capacitance values, which in turn allows for the tuning of the resonant frequency in an LC circuit. By altering the capacitance, the circuit can be fine?tuned across a range of frequencies, making it essential for applications that require frequency agility, such as radio receivers.

Frequency Tuning Range

The frequency tuning range pertains to the span between the maximum and minimum frequencies that an LC circuit can achieve. This range is influenced by the variability in capacitance according to the f = 1/(2??(LC)) formula. Understanding the frequency ratio helps determine whether the native range of a variable capacitor meets the design requirements or needs additional modification.

Inductance Determination

Once the desired capacitance range and frequency range are set, the appropriate coil (inductor) value can be calculated using the resonant frequency formula. Determining the correct inductance ensures that the LC circuit will resonate within the target frequency band, making this step critical for achieving optimal circuit performance in applications like radio tuning.

Transcript

00:03 In this problem, we have an lc circuit with a variable capacitance.

00:13 The minimum capacitance of this capacitor is going to be 10 pico -pharats, and the maximum capacitance of the capacitor is going to be 365 picofarhertz.

00:26 What we have to have, we need to find in part a, we need to find what is the ratio between the maximum frequency, of that oscillating circuit over the minimum frequency.

00:40 Now, how do we find this ratio? now, the frequency of an lc circuit is given by 1 over 2 pi times 1 over the square root of the inductance times capacitance, and that means that the frequency, like if the only variable is the capacitance, the frequency is proportional to 1 over the square root of the capacitance.

01:02 So the bigger the capacitance, the smaller the frequency.

01:06 And that means that the maximum frequency correspond to the minimum capacitance, and the minimum frequency correspond to the maximum capacitance.

01:20 Then having said that, we find that the ratio of the maximum frequency over the minimum frequency is going to be equal to the square root of the ratio between the maximum capacitance over the minimum capacitance.

01:40 And that is the square root of 365 picofarats over 10 picofarad.

01:48 And so this ratio is equal to 6.

02:00 Now in part b, we are told that that ratio is actually too big for our purposes.

02:06 And what we actually want is we actually want for the minimum frequency to be equal to 0 .54 megahertz and the maximum frequency equal to 1 .60 megahertz.

02:26 And in order to achieve these frequencies, we're going to add a capacitor in parallel.

02:40 Sorry, we have to add a capacitor in series.

02:49 And we need to find how, what is the, what is the capacitance of this capacitor that is going to make our frequencies take these values.

03:01 Okay, so first we need to find what is the ratio of frequencies that we actually need.

03:07 So we have that f max over f mean is going to be equal to 1 .60 over 0 .54.

03:16 So our ratio of frequencies is 2 .96...

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