As discussed in the course policies, all pre-laboratory
assignments are individual assignments; they are
NOT to be submitted by a group. You may discuss the
assignment with other classmates, but the work you turn in to me
should be your own and reflect your personal
comprehension of the laboratory material. Do NOT
split pre-lab work up among several people; each student should
complete the pre-laboratory assignment in its entirety and should
provide a unique submission to me at the beginning of the
class.
INDIVIDUAL pre-laboratory assignment (out of 100
points): (due from each student at the beginning of
class)
-
(50 points) Use Equation (2) from the
second exercise of the laboratory experience to
verify the RMS values of
the square, sinusoidal ("sine"), triangle, and sawtooth ("ramp")
waves. The expected values are shown as Vrms
on each of the graphs shown in the lab book’s exercise. In
particular,
-
Vrms = A/sqrt(2)
for a (symmetric) sine wave.
- Vrms = A for a
(symmetric) square wave.
-
Vrms = A/sqrt(3)
for a (symmetric) triangle wave.
-
Vrms = A/sqrt(3)
for a (symmetric) ramp (or "sawtooth") wave.
Some hints:
- In Equation (2), the variable T is the
period of the waveform.
- Remember to square the waveform. You are
integrating f2(t) and not
f(t).
- Remember to scale the integral by 1/T as is shown
in Equation (2).
- For the sinusoidal wave, use frequency 1/T so that
f(t) = A sin(2π t/T).
- Integrate the other three waveforms in
pieces so that f(t) looks like a
line on each piece. That way
f2(t) is a simple quadratic on each piece.
- Make sure each piece has the correct
slope and correct vertical
intercept!
- In every case, the integral of f(t) from
0 to T should be zero!
If your function has some nonzero average value, then you
constructed it incorrectly.
- Remember that
sqrt(A+B+C) ≠ sqrt(A) + sqrt(B) + sqrt(C).
We will discuss the physical significance of RMS signals in
class.
-
(50 points) Use Equation (2) to verify
Equation (3) from the third exercise of the
laboratory experience. In particular, for the square, sinusoidal,
triangle, and sawtooth waves, replace f(t) with f(t)
+ C where C is a constant offset that represents a
non-zero average (i.e., "DC") component of f(t). You are
verifying that the square of the new RMS value
is C2 greater than the old RMS value. Some
hints:
- Remember that (f(t)+C)2 = f2(t) +
2 f(t) C + C2.
- Remember to scale the integral by 1/T as is shown
in Equation (2).
- Complete this exercise for each of the
four waveforms.
-
BONUS (10 points): Give a strong
quantitative mathematical argument (i.e., a
rigorous proof) that
Equation (3) holds for any integrable
periodic function f(t) with period T. Some
hints:
- Let f(t) = g(t) + C where C is the
average value of f(t) and g(t) is a version
of f(t) with zero average.
- What is the integral of g(t) over one period?
(remember that g(t) is periodic and has zero
average)
-
BONUS (5 points): As with laboratory reports, a
pre-laboratory assignment submission generated with any flavor of
TeX (e.g., LaTeX)
will earn the author 5% extra credit on that
assignment.
We will discuss the solutions in class.
