Question

The figure shows a simple FM antenna circuit in which \(L=8.22 \mu \mathrm{H}\) and \(C\) is variable (the capacitor can be tuned to receive a specific station). The radio signal from your favorite FM station produces a sinusoidal time-varying emf with an amplitude of \(12.9 \mu \mathrm{V}\) and a frequency of \(88.7 \mathrm{MHz}\) in the antenna. a) To what value, \(C_{0}\), should you tune the capacitor in order to best receive this station? b) Another radio station's signal produces a sinusoidal time-varying emf with the same amplitude, \(12.9 \mu \mathrm{V}\), but with a frequency of \(88.5 \mathrm{MHz}\) in the antenna. With the circuit tuned to optimize reception at \(88.7 \mathrm{MHz}\), what should the value, \(R_{0}\), of the resistance be in order to reduce by a factor of 2 (compared to the current if the circuit were optimized for \(88.5 \mathrm{MHz}\) ) the current produced by the signal from this station?

Short answer

$$R_{0} = \sqrt{3Z^2}$$ After plugging the calculated value of the second station's angular frequency impedance into the equation and solving for \(R_{0}\), we find the optimal resistance value needed to reduce the current produced by the second station's signal by a factor of 2.

Step-by-step solution

Calculate the resonant frequency of the tuned capacitor and inductor

To best receive the station at the frequency 88.7 MHz, the circuit needs to be tuned to this frequency. The resonant frequency of an LC circuit can be found using the formula: $$f_{0} = \frac{1}{2 \pi \sqrt{LC}}$$ We are given the inductor value \(L = 8.22\,\mu \mathrm{H}\), the frequency \(f_{0} = 88.7\,\mathrm{MHz}\), and we need to find the value of the capacitor \(C_{0}\). Rearrange this equation to find \(C_{0}\): $$C_{0} = \frac{1}{(2 \pi f_{0})^2 L}$$

Find the value of the capacitor \(C_{0}\)

Plug the given values for \(L\) and \(f_{0}\) into the equation to find the value of the capacitor \(C_{0}\): $$C_{0} = \frac{1}{(2 \pi \times 88.7 \times 10^6)^2 \times (8.22 \times 10^{-6})}$$ Calculate \(C_{0}\): $$C_{0} = 2.61 \times 10^{-12}\,\mathrm{F}$$ So, the value of the capacitor \(C_{0}\) should be \(2.61\,p\mathrm{F}\) to best receive the station at 88.7 MHz.

Determine the impedance at the frequency of the second station

To decrease the current produced by the signal from the second radio station, we need to find the optimal value of the resistance \(R_{0}\). First, we'll find the impedance \(Z\) of the circuit at the frequency of the second station, \(f = 88.5\,\mathrm{MHz}\): $$Z = \sqrt{R_{0}^2 + (\omega L - \frac{1}{\omega C_{0}})^2}$$ where \(\omega = 2 \pi f\) is the angular frequency. Let's substitute the given values: $$Z = \sqrt{R_{0}^2 + (2 \pi \times 88.5 \times 10^6 \times 8.22 \times 10^{-6} - \frac{1}{2 \pi \times 88.5 \times 10^6 \times 2.61 \times 10^{-12}})^2}$$

Calculate the current with the optimized capacitor

We want to reduce the current produced by the second station's signal by a factor of 2 compared to the current if the circuit were optimized for the second station's frequency. To do that, we need to first find the current \(I_{1}\) with the optimized capacitor for the second station's frequency: $$I_{1} = \frac{V}{Z} = \frac{12.9 \times 10^{-6}}{Z}$$

Determine the desired current and solve for \(R_{0}\)

We want the current in the circuit for the second radio station to be half of \(I_{1}\): $$I_{2} = \frac{I_{1}}{2} = \frac{12.9 \times 10^{-6}}{2Z}$$ Now, using Ohm's law, we can write: $$R_{0} = \frac{V}{I_{2}} = \frac{12.9 \times 10^{-6}}{\frac{12.9 \times 10^{-6}}{2Z}} = 2Z$$ Rearrange the equation for the impedance \(Z\): $$2Z = \sqrt{R_{0}^2 + (2 \pi \times 88.5 \times 10^6 \times 8.22 \times 10^{-6} - \frac{1}{2 \pi \times 88.5 \times 10^6 \times 2.61 \times 10^{-12}})^2}$$ Now we can solve the equation for \(R_{0}\): $$R_{0}^2 = 3Z^2$$ Plug the calculated value of second station angular frequency's impedance in the equation and find \(R_{0}\).

Key concepts

Resonant Frequency

When optimizing an FM antenna circuit, understanding the resonant frequency is crucial. This frequency refers to a specific value at which an LC (inductor-capacitor) circuit naturally oscillates with the greatest amplitude. For receiving radio signals, an antenna's circuit must be tuned to the resonant frequency of the desired station.

For an LC circuit, the resonant frequency, denoted as \(f_0\), is given by the equation \(f_{0} = \frac{1}{2 \pi \sqrt{LC}}\). Tuning the circuit involves adjusting the capacitor so that \(f_0\) matches the frequency of the broadcast signal. If the circuit is correctly tuned, it will maximize the signal reception by resonating at the same frequency as the transmitted signal from the FM station.

LC Circuit

An LC circuit forms the basic building block of many electronic devices, including radio receivers. It consists of an inductor, represented by the letter 'L', and a capacitor, represented by the letter 'C'. These components are usually connected either in series or in parallel. Their collaborative interaction creates an oscillatory system, capable of storing and releasing energy alternately between the electric field of the capacitor and the magnetic field of the inductor.

When the LC circuit is not connected to an external power supply, it can oscillate freely at its resonant frequency, which depends on the values of the inductance and capacitance. Engineers leverage this property to create filters and frequency-selective circuits, essential for tuning a specific radio frequency.

Impedance

The concept of impedance is a fundamental aspect of any AC circuit, including an FM antenna. Impedance, denoted as 'Z', is a measure of opposition that a circuit presents to the flow of alternating current (AC). It is a complex quantity that combines resistance, which impedes direct current (DC), and reactance, which impedes AC at certain frequencies.

For an LC circuit, impedance is affected by the inductance, capacitance, and resistance within the circuit as well as the frequency of the AC signal. A key formula for finding the impedance in an LC circuit is \(Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}\), where \(\omega\) is the angular frequency of the signal. By changing resistance, either by adding a resistor or optimizing the existing circuit elements, you can manipulate the circuit's impedance to control the current flow accordingly.

Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics and electrical engineering, proving essential for understanding current flow in circuits. It states the relationship between voltage (V), current (I), and resistance (R) as \(V = IR\), making it a powerful tool for circuit analysis.

This law underscores that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. In the context of the FM antenna circuit, Ohm's Law allows us to calculate the necessary resistance to alter the current when the circuit receives a signal from a frequency it wasn't optimized for. This is pivotal in ensuring that you can still get adequate reception, even if the frequency isn't an exact match with the resonant frequency of the LC circuit.