Download Example 21-6 Tuning an FM Radio

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.