Randy Franks Jon Nakashima Matt Snider The Mathematics of FM Radio

AM = Amplitude Modulation FM = Frequency Modulation AM waves use line-of-sight, but can also reflect off the ionosphere and then to a receiver. FM waves require line-of-sight between transmitter and receiver. Radio Basics

transistor resistor capacitor battery inductor An arrow across a component indicates its level is variable. Schematic of a basic transmitter Basic Schematic Symbols

Our FM Receiver : a Schematic

Capacitors - like a battery - stores charge - when the device needs extra power, say for a particularly loud sound, the capacitor discharges “dielectric”- a material placed between the plates of the capacitor to increase capacitance “electrolytic” – type of capacitor which has strict polarity (charge may only flow in one direction through this type of capacitor)

Internal Capacitor Design A parallel-plate A cylindrical capacitor (cross-section from above) capacitor A variable capacitor (really, it’s just a stack of parallel plates)

Basic Strategy: use equations for q and V separately, solve for the situation, divide and simplify results  electrostatic charge voltage  Calculating the Capacitance

Parallel Plate Capacitor d = separation of plates A = area of Gaussian surface E and dA are parallel, and dA is a constant, in this case. So…. Again, E and ds are in the same direction. Also the integral of ds from one plate to the other is equal to the total separation. So….

Cylindrical Capacitor From the last derivation: a = inside radius b = outside radius r = radius of Gaussian surface (cross section) A = area of Gaussian surface, which is now cylindrical. A = 2πrl, So…. Solving the above for E will prove useful in the determination of V, So….

Cylindrical Capacitor (cont.) a = inside radius b = outside radius r = radius of Gaussian surface First, replace ds with dr, since the Gaussian surface is circular. Second, substitute for E (using result of last page). Third, change the integral limits to a and b (from the inside radius to the outside radius. After all that…. Pull out constants, and…. Evaluating the integral leaves: Finally, it’s time to divide the two results. A lot of Algebra later:

And that’s it! Well, almost…. Matt will now perform the ritual dance necessary to achieve decent radio reception on this campus.