Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 15

Radio Receiver. A simplified version of the front end of an AM radio receiver is shown in Figure 2.13. This receiver consists of an \(R L C\) tuned circuit containing a resistor, capacitor, and an inductor connected in series. The RLC circuit is connected to an external antenna and ground as shown in the figure. The tuned circuit allows the radio to select a specific station out of all the stations transmitting on the AM band. At the resonant frequency of the circuit, essentially all of the signal \(V_{0}\) appearing at the antenna appears across the resistor, which represents the rest of the radio. In other words, the radio receives its strongest signal at the resonant frequency. The resonant frequency of the LC circuit is given by the equation $$ f_{0}=\frac{1}{2 \pi \sqrt{L C}} $$ where \(L\) is inductance in henrys (H) and \(C\) is capacitance in farads (F). Write a program that calculates the resonant frequency of this radio set given specific values of \(L\) and \(C\). Test your program by calculating the frequency of the radio when \(L=0.1 \mathrm{mH}\) and \(C=0.25 \mathrm{nF}\).

Short Answer

Expert verified
The resonant frequency of the AM radio receiver with \(L = 0.1\,\text{mH}\) and \(C = 0.25\,\text{nF}\) is approximately 1,006,230 Hz or 1.0062 MHz.

Step by step solution

01

Understand the given formula

In this problem, we are given a formula to calculate the resonant frequency of an RLC circuit: $$f_{0} = \frac{1}{2\pi\sqrt{LC}}$$ f₀ represents the resonant frequency we are looking to determine. L represents inductance in henrys (H) and C represents capacitance in farads (F).
02

Converting given values

We are given the values of L and C: \(L = 0.1\,\text{mH}\) and \(C = 0.25\,\text{nF}\). We need to convert these values to henrys (H) and farads (F) respectively: 1 milli-Henry (mH) = 0.001 Henry (H), so \(L = 0.1\,\text{mH} = 0.0001\, \text{H}\). 1 nano-Farad (nF) = 0.000000001 Farad (F), so \(C = 0.25\,\text{nF} = 0.00000000025\,\text{F}\).
03

Calculate the resonant frequency

Now, we have the values of L and C in henrys and farads, respectively. We can use the given formula to calculate the resonant frequency: $$f_{0} = \frac{1}{2\pi\sqrt{LC}}$$ Substitute the values of L and C: $$f_{0} = \frac{1}{2\pi\sqrt{(0.0001)(0.00000000025)}}$$
04

Solve and find the resonant frequency

Now, we can solve the equation for f₀: $$f_{0} = \frac{1}{2\pi\sqrt{0.000000000025}} \approx 1,006,230.086\,\text{Hz}$$ So, the resonant frequency of the receiver is approximately 1,006,230 Hz or 1.0062 MHz.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Radio Receiver. The voltage across the resistive load in Figure \(2.13\) varies as a function of frequency according to Equation (2.18). $$ V_{R}=\frac{R}{\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}} V_{\theta} $$ where \(\omega=2 \pi f\) and \(f\) is the frequency in hertz. Assume that \(L=0.1 \mathrm{mH}\), \(C=0.25 \mathrm{nF}, R=50 \Omega\), and \(V_{O}=10 \mathrm{mV}\). a. Plot the voltage on the resistive load as a function of frequency. At what frequency does the voltage on the resistive load peak? What is the voltage on the load at this frequency? This frequency is called the resonant frequency \(f_{0}\) of the circuit. b. If the frequency is changed to \(10 \%\) greater than the resonant frequency, what is the voltage on the load? How selective is this radio receiver? c. At what frequencies will the voltage on the load drop to half of the voltage at the resonant frequency?

Are the following MATLAB variable names legal or illegal? Why? a. dog1 b. \(1 \mathrm{dog}\) c. Do you_know_the_way_to_san_jose d. help e. What's_up?

Decibels. Engineers often measure the ratio of two power measurements in decibels, or \(\mathrm{dB}\). The equation for the ratio of two power measurements in decibels is $$ \mathrm{dB}=10 \log _{10} \frac{P_{2}}{P_{1}} $$ where \(P_{2}\) is the power level being measured, and \(P_{1}\) is some reference power level. a. Assume that the reference power level \(P_{1}\) is I milliwatt, and write a program that accepts an input power \(P_{2}\) and converts it into \(\mathrm{dB}\) with respect to the \(1 \mathrm{~mW}\) reference level. (Engineers have a special unit for dB power levels with respect to a \(1 \mathrm{~mW}\) reference: \(\mathrm{dBm}\).) Use good programming practices in your program. b. Write a program that creates a plot of power in watts versus power in \(\mathrm{dBm}\) with respect to a \(1 \mathrm{~mW}\) reference level. Create both a linear \(x y\) plot and a log-linear \(x y\) plot.

Solve the following system of simultaneous equations for \(x\). \(\begin{aligned}-2.0 x_{1}+5.0 x_{2}+1.0 x_{3}+3.0 x_{4}+4.0 x_{5}-1.0 x_{6}=0.0 \\ 2.0 x_{1}-1.0 x_{2}-5.0 x_{3}-2.0 x_{4}+6.0 x_{5}+4.0 x_{6}=& 1.0 \\\\-1.0 x_{1}+6.0 x_{2}-4.0 x_{3}-5.0 x_{4}+3.0 x_{5}-1.0 x_{6}=&-6.0 \\\ 4.0 x_{1}+3.0 x_{2}-6.0 x_{3}-5.0 x_{4}-2.0 x_{5}-2.0 x_{6}=10.0 \\\\-3.0 x_{1}+6.0 x_{2}+4.0 x_{3}+2.0 x_{4}-6.0 x_{5}+4.0 x_{6}=&-6.0 \\ 2.0 x_{1}+4.0 x_{2}+4.0 x_{3}+4.0 x_{4}+5.0 x_{5}-4.0 x_{6}=&-2.0 \end{aligned}\)

Assume that array arrayl is defined as shown, and determine the contents of the following sub-arrays. \(\operatorname{array} 1=\left[\begin{array}{rrrrr}1.1 & 0.0 & 2.1 & -3.5 & 6.0 \\\ 0.0 & 1.1 & -6.6 & 2.8 & 3.4 \\ 2.1 & 0.1 & 0.3 & -0.4 & 1.3 \\ -1.4 & 5.2 & 0.0 & 1.1 & 0.0\end{array}\right]\) a. array \(1(3, t)\) b. array \(1(:, 3)\) c. array \(1\left(1: 2: 3,\left[\begin{array}{lll}3 & 3 & 4\end{array}\right]\right)\) d. array \(\left.1\left(\begin{array}{ll}1 & 1\end{array}\right], 4\right)\)
See all solutions

Recommended explanations on Psychology Textbooks

Issues and Debates in Psychology

Read Explanation

Research Methods in Psychology

Read Explanation

Psychological Treatment

Read Explanation

Cognition and Development

Read Explanation

Addiction

Read Explanation

Relationships

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free