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6 . 1 . 00 Randy Franks Jon Nakashima Matt Snider The Mathematics of FM Radio
11

The Mathematics of FM Radio

Jan 14, 2016

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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. - PowerPoint PPT Presentation
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Page 1: The Mathematics of FM Radio

6 . 1 . 00

Randy Franks

Jon Nakashima

Matt Snider

The Mathematics of FM Radio

Page 2: The Mathematics of FM Radio

AM = Amplitude AM = Amplitude ModulationModulation

FM = Frequency FM = Frequency ModulationModulation

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

Page 3: The Mathematics of FM Radio

transistransistortorresistresistororcapacicapacitortor

battebatteryry

inductinductoror

An arrow across a component indicates its level is variable.

Schematic of a basic transmitter

Basic Schematic Symbols

Page 4: The Mathematics of FM Radio

Our FM Receiver :

a Schematic

Page 5: The Mathematics of FM Radio

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)

Page 6: The Mathematics of FM Radio

Internal Capacitor Design

A parallel-A parallel-plateplate

A A cylindriccylindric

al al capacitorcapacitor

(cross-section from above)

capacicapacitortor

A A variable variable capacitocapacitorr

(really, it’s just a stack of parallel plates)

Page 7: The Mathematics of FM Radio

Basic Strategy: use equations for q and V separately , solve for the situation, divide and simplify results

electrostatic charge

voltage

Calculating the Capacitance

Page 8: The Mathematics of FM Radio

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….

Page 9: The Mathematics of FM Radio

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….

Page 10: The Mathematics of FM Radio

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:

Page 11: The Mathematics of FM Radio

And that’s it!

Well, almost….

Matt will now perform Matt will now perform the ritual dance the ritual dance

necessary to achieve necessary to achieve decent radio reception decent radio reception

on this campus.on this campus.