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Download Example 21-6 Tuning an FM Radio
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Transcript
Example 21-6 Tuning an FM Radio The tuner knob on an FM radio moves the plates of an adjustable capacitor. This capacitor is in series with a 0.130-mH inductor and a net resistance of 755 . The peak current induced in this circuit by a radio wave becomes large when the natural frequency of the circuit matches the carrier frequency of the radio wave. (a) What is the frequency of the FM station that is tuned in when the capacitor in the radio is adjusted to 19.6 pF? (b) If the peak operating voltage in the tuning circuit is 9.00 V, what is the peak current? Set Up The tuning circuit resonates when driven by a radio wave with an angular frequency v that equals the natural angular frequency v0 of the circuit. We’ll use Equation 21-38 to determine the value of v0 for this circuit, and convert it to an ordinary frequency in Hz. We’ll find the peak current, or current amplitude, using Equations 21-37. Natural angular frequency of the tuning circuit: v0 = 1 A LC (21-38) R = 755 Ω V(t) from radio station: f=? Relationship between frequency and angular frequency: C = 19.6 pF L = 0.130 µH v f = 2p Amplitude of the oscillating current: i 0, max = Z = Solve (a) Use the given values of capacitance and inductance to determine the natural frequency of the circuit. V0 Z 2 1 - vL b + R2 B vC a (21-37) We are given L = 0.130 mH = 0.130 * 1026 H and C = 19.6 pF = 19.6 * 10212 F. From Equation 21-38 the natural angular frequency is v0 = 11>(LC ), and the natural frequency is v0 divided by 2p : f0 = v0 1 1 1 1 = = -6 2p 2p A LC 2p A 10.130 * 10 H2 119.6 * 10-12 F2 = 99.7 * 106 Hz = 99.7 MHz (b) Find the amplitude of the oscillating current. At resonance the term 11>(vC )2 - vL in the expression for impedance is equal to zero. Then Equation 21-37 tells us that Z = R = 755 . If the voltage amplitude is 9.00 V, the current amplitude is i 0, max = Reflect V0 9.00 V = = 0.0119 A = 11.9 mA Z 755 The carrier frequencies of FM radio stations lie in the megahertz (MHz) range. So the frequency we found, 99.7 * 106 Hz, is 99.7 MHz on your FM radio dial. The peak current, about 10 mA, is typical for a portable FM radio. Note that you tune a television set in the same way. When you use the remote control to change the channel (Figure 21-1b), you’re commanding a circuit in the television to change its capacitance. This adjusts the natural frequency of the circuit to match the carrier frequency of the channel you want to watch.