Friday 7-26-96
The relevant section in the book is 20.5
** Note that the abbreviation sqrt in some of the equations below stands for square root. **
Now we're going to learn how a radio works (or at least learn about the tuning system!). Consider the impedance equation:
Z = sqrt (R^2+(XL-Xc)^2)
For a fixed resistance, the minimum impedance is obtained when the inductive reactance XL and the capacitive reactance Xc are equal. Both XL and Xc depend on frequency, so setting XL and Xc equal to each other will give the frequency, known as the resonance frequency, needed to minimize the impedance:
w = 1/sqrt(LC)
or, equivalently,
f = 1/(2 pi sqrt (LC))
Minimizing the impedance does two things. First, it maximizes the current in the circuit, and secondly it gives a phase angle of zero degrees, maximizing the power transferred to the circuit.
AM/FM radios use a resonant circuit in which the current increases sharply at resonance. When the radio is tuned to one radio station, which will be broadcasting at a particular frequency, it means that the resonant circuit has a resonance frequency which is the same as the station's broadcast frequency. This amplifies that frequency selectively, picking that radio station out from the many signals (at all sorts of frequencies) which are being picked up by the radio's antenna.To tune the radio to another station, you turn a knob which changes the value of the capacitor (and/or inductor) in the resonant circuit, changing the resonance frequency to the frequency of the other radio station, amplifying that signal selectively over signals of other frequencies.