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The Mathematics of FM Radio

The Mathematics of FM Radio. 6 . 1 . 00. Randy Franks. Jon Nakashima. Matt Snider. Radio Basics. AM = Amplitude Modulation. FM = Frequency Modulation. FM waves require line-of-sight between transmitter and receiver.

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The Mathematics of FM Radio

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  1. The Mathematics of FM Radio 6 . 1 . 00 Randy Franks Jon Nakashima Matt Snider

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

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

  4. Our FM Receiver : a Schematic

  5. 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)

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

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

  8. Parallel Plate Capacitor d = separation of plates A = area of Gaussian surface 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. E and dA are parallel, and dA is a constant, in this case. So…. So….

  9. Cylindrical Capacitor (cross section) a = inside radius b = outside radius r = radius of Gaussian surface From the last derivation: 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….

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

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

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