Homework Project 3: Building a Tuner
FROM: Steven R. Dunbar
Department of Mathematics and Statistics
University of Nebraska-Lincoln
Lincoln, NE 68508-0323, USA
Well, you did it! You finally graduated and got a job! Too bad it's
working for minimum as the technician for Fried Electronics (Company motto:
If your electronics are Fried, they must be from us!) One day, your boss
comes to you with a terrible problem. The store's best customer, the mysterious
`` Mr. D.'' (rumored to have been a mathematician before he got rich in
shadowy circumstances) needs a tuning circuit for the ship-to-shore receiver
on his yacht built immediately. It's a good thing that you studied
Sections 3.8 and 3.9, pages 179-198 of Boyce and DiPrima, your differential
equations text. You remember, the sections that explain analyzing circuits
with differential equations and forced vibrations.
The only parts in the store (you knew it was a crummy store) are
1. a capacitor of strength C = 1,
2. resistors of sizes R = 1, R = 0.2, and R = 0.04.
and
3. a variable inductor capable of achieving various sizes of
L by the turning of a knob.
The Coast Guard sends a navigational signal out to sea by radio waves
of the form F0 sin(g t). An antenna on the receiver collects
the signals as a forcing input. Since radio waves of all frequencies are
collected by the antenna, the tuning circuit is supposed to amplify one
particular frequency to a great extent, without significant amplification
of the frequencies of other stations.
When you examine the parts however, you discover that the inductor is
damaged (you knew it was a really crummy store!). You can still change
the inductance L with a screwdriver; however once the receiver has been
assembled, no one will be able to change the tuning (Geez!). You must therefore
decide what value of L should be used. After a hurried conference with
the boss, you learn that the Coast Guard navigational signal has (circular)
frequency g = 1, but there are alternative music stations (which Mr. D.
detests) with frequencies g = 0.9 and g = 1.2 that might interfere.
Your task is to determine the best setting for the inductor, and best
choice of resistor for the circuit. You must also report to the boss explaining
what you did, so that he can file it in company records, should there ever
be a lawsuit. Be sure to include plenty of appropriate graphs, which make
it easier for the boss and the lawyers to understand the report
1. Write down the differential equation for I(t), the current in an
RLC circuit with C=1 and F(t) = sin(g t). Take the amplitude of the forcing
function arbitrarily to be 1; the amplitude of the steady-state current
I(t) is then the ratio of the strength of the response to the strength
of the stimulus.
2. Determine the steady-state solution to your equation. Find the amplitude
of the steady-state solution. The gain G is defined to be the ratio of
current amplitude to signal amplitude. Find the gain. (Hint: Your result
for G should have the property that G2 is a ratio of polynomials
in g depending on the parameters R and L. The numerator should be second
degree in g and the denominator a fourth degree polynomial in g.)
3. Determine what value is to be preferred for L. The answer should
depend on R. (Hint: As often occurs in maximization problems, whatever
value of L maximizes G will also, of course, maximize G2 and
conversely. You will find it easier to work with G2).
4. Given your choice for L, determine which of the three resistors will do the best job of amplifying the signal having the desired frequency relative to the signals having the undesired frequencies.