Problem 11 - Fundamentals of Physics

Question

A variable capacitor with a range from 10 to \(365 \mathrm{pF}\) 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 \(0.54 \mathrm{MHz}\) 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?

Step-by-Step Solution

Answer

(a) Frequency ratio is 6.04. (b) Add 35.75 pF capacitance. (c) Coil inductance should be 2.515 μH.

Step 1: Identify Key Parameters

The variable capacitor has a capacitance range from 10 pf to 365 pf. We need to calculate the frequency range based on these capacitance values. The radio requires frequency scanning between 0.54 MHz and 1.60 MHz. Let's denote the maximum capacitance as \(C_{max} = 365 \text{ pF}\) and the minimum capacitance as \(C_{min} = 10 \text{ pF}\).

Step 2: Calculate Frequency Range Based on Capacitance

The frequency in an LC circuit is given by \(f = \frac{1}{2\pi \sqrt{LC}}\). Therefore, the minimum frequency \(f_{min}\) occurs at maximum capacitance, and the maximum frequency \(f_{max}\) occurs at minimum capacitance. Using the formula, the ratio \(\frac{f_{max}}{f_{min}} = \sqrt{\frac{C_{max}}{C_{min}}}\).

Step 3: Determine Frequency Ratio Calculation

Substitute the capacitance values into the frequency ratio expression: \[\frac{f_{max}}{f_{min}} = \sqrt{\frac{365}{10}} = \sqrt{36.5} \approx 6.04\]

Step 4: Address Frequency Constraint

The calculated ratio is approximately 6.04. However, the desired frequency ratio from the problem is \(\frac{1.60}{0.54} \approx 2.96\). This discrepancy indicates the range is too broad, and we need to adjust the capacitive components.

Step 5: Determine Required Capacitance Addition for Desired Frequencies

To adjust the frequency range by switching effective minimum capacitance, add fixed capacitance \(C_p\) parallel to the variable capacitor, transforming the minimum capacitance from 10 pf to 10 pf + \(C_p\), and maximizing accordingly, we calculate: \[2.96 = \sqrt{\frac{365+C_p}{10+C_p}}.\] Solving this equation yields: \[\frac{(365+C_p)}{(10+C_p)} = (2.96)^2 = 8.76 \] making \[365 + C_p = 87.6 + 8.76C_p\] and solving \[277.4 = 7.76C_p\] resulting in \(C_p \approx 35.75 \text{ pF}\).

Step 6: Find Required Inductance for Maximum Frequency

To achieve the maximum desired frequency (1.6 MHz) with the new total capacitance configuration (\(C_{min} = 10 \text{ pF} + C_p\)): \[ 1.60\times10^6 = \frac{1}{2\pi \sqrt{L (10 + 35.75) \times 10^{-12}}}\] Solving for \(L\), \[L = \frac{1}{(2\pi\times1.60\times10^6)^2 (45.75\times10^{-12})} \approx 2.515\underline{\phantom{xxx}} \mu H\]

Key Concepts

Variable Capacitor

In an LC circuit, a variable capacitor plays a vital role in tuning. It's a component whose capacitance can be adjusted. This adjustment allows it to store and release a precise amount of electrical energy. It's like adjusting the volume on a radio. By turning a knob, you control how much charge flows into and out of the capacitor.

This is crucial for tuning because the variable capacitor can change its capacitance from a minimum of 10 picofarads (pF) to a maximum of 365 pF. This wide range is what gives the LC circuit its flexibility. Each change in capacitance alters the resonant frequency of the circuit, allowing it to tune into different radio frequencies. The versatility of a variable capacitor is essential for devices that need to pick up a broad range of frequencies, like radios.

Frequency Tuning

Frequency tuning is the process of adjusting the frequency at which an LC circuit resonates. This is primarily achieved by adjusting the components of the circuit: the inductance and capacitance.

• Frequency is highest when capacitance is at its lowest.
• Conversely, frequency is lowest when capacitance is at its highest.

Given that the frequency of an LC circuit, \( f \), is defined by the formula \( f = \frac{1}{2\pi \sqrt{LC}} \), we see that frequency tuning involves manipulating the capacitance \( C \) and the inductance \( L \).
Frequency tuning in radios allows users to hop across radio stations, moving from one signal to another smoothly. Here, the adjustment is continuous due to the variable capacitor, which gradually changes the capacitance, thus changing the tuned frequency.

Inductance

Inductance is a property of an electrical component that determines its tendency to oppose changes in current. It is represented by the symbol \( L \). In an LC circuit, the coil or inductor provides this inductance. Think of it as a storehouse of magnetic energy that impacts how quickly the circuit can "tune" to a new frequency.

In practical terms, when the inductance is higher, the time the circuit takes to adjust is longer. For frequency tuning in a radio using the above circuit, the inductance needs to be precisely calculated to allow tuning over the desired frequency range. In this case, the calculated inductance for the desired frequency range of 0.54 MHz to 1.60 MHz was determined to be approximately 2.515 microhenries (\( \mu H \)). This precise calculation ensures the circuit performs optimally across the desired frequencies.

Capacitance

Capacitance is a measure of how much electric charge a capacitor can hold at a given voltage level. In an LC circuit, capacitance is provided by the capacitor, which is often variable in radio applications.

• Capacitance affects the resonant frequency of the circuit.
• It is measured in farads, typically microfarads (\(\mu F\)) or picofarads (pF) for small circuits.

A variable capacitor's ability to change its capacitance allows for adjusting the frequency of the circuit, which is crucial for radio tuning.

The variance between minimum and maximum capacitance values directly affects the tuning range. If needed, additional capacitance can be added parallel to the variable capacitor, as seen in the solution, to achieve the desired frequency tuning range. This ensures the LC circuit operates efficiently within the set frequency parameters.

Radio Frequency

Radio frequency refers to the rate of oscillation within the range of about 3 kHz to 300 GHz. It is a crucial aspect of wireless communications, including radio broadcasting.
In the context of an LC circuit, radio frequency is the specific rate at which the circuit can naturally oscillate, allowing it to pick up specific signals broadcasted at that frequency.

• The design of the LC circuit is often tailored to the specific radio frequency range it needs to cover.
• Factors like capacitance and inductance are manipulated to ensure resonance at these frequencies.

Achieving precise tuning to these frequencies is vital for clear radio communication. The circuit discussed in the original exercise was specifically designed to range between 0.54 MHz and 1.60 MHz, allowing for the reception of typical AM radio signals.