WEBVTT

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>> There are lots of different things
in the physical world

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around us that we model sinusoidally.

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It could be the voltage or
current in an electrical circuit.

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It could the vibrations
of a mechanical system.

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We could be referring to the propagation of

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light energy or electromagnetic energy.

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So in this video here, I'd
just like to review with you

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the various parameters or

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the various markings of
a sinusoidally varying signal.

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By sinusoidally varying, I mean
anything that could either

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be modeled as a sine or a cosine.

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So for example, let's just
talk in terms of voltage.

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As I'm an electrical engineer,

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let's talk in terms of voltage.

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That's the time-varying voltage
that we would model as having

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some amplitude V sub m times
this cosine of Omega t,

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plus some phase angle Phi.

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All right. What is V sub m refer to?

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Well, first of all, what is cosine?

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Well, you know that
the cosine function varies

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from negative one to positive one.

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So if you multiply that plus or
minus one by some constant V sub m,

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then we have a waveform is varying between

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negative V sub m and positive
V sub m. In this picture right here,

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we've got down here at negative five
and on up here to a positive five,

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we refer to this amplitude
or this distance here

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from where it crosses
the horizontal axis on up,

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we refer to that as the amplitude.

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Amplitude of the wave,

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and for our model,

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that amplitude is V sub m. Now,

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this waveform varies periodically.

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That means that it repeats
itself every so often.

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The period is the length of
time that it takes to go

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through one of these cycles
or until it repeats itself.

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So we can say then that we've

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got a distance from

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its peak on the top to the peak on
the top would represent one period,

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and we use the letter T,

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capital T to represent period.

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So T is the period of the signal,

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and it has units of seconds
or time to be more accurate.

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So from peak to peak is T from
negative going zero crossing,

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to negative going zero crossing.

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That's another interval of T seconds.

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Or we could talk about from positive going,

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zero cost crossing to
positive going, zero crossing.

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Again, that would represent one period.

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Now, if instead of talking in terms of time

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we talk about how rapidly
the signal is changing,

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we refer to its frequency and we use
the letter lowercase f for frequency.

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It has the units of cycles per second.

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If you stop and think about it,

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if the period is the length of time
that it takes to go through one cycle,

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it will be so many seconds per cycle.

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Then frequency, the cycles
per second is just equal

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to one over T.

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That gives us a frequency
in cycles per second,

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common in electrical engineering we might
talk about 1,000 cycles per second.

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Another term for cycles
per second is Hertz.

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A 100 cycles per second or a 100 hertz,

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1,000 cycles per second or a kilohertz,

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or a million cycles per second,

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one million or one megahertz.

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If you think about the FM radio dial,

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say you're listening to some radio
stations broadcasting its FM 96.3,

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that's referring to
a carrier signal is oscillating at

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96.3 megacycles per
second or 96.3 megahertz.

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Now, the argument of

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a cosine or a sine wave is frequently
measured in terms of radians per second.

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So we introduce a another frequency

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or type of frequency that we're
going to call the radio frequency.

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The radio frequency is
in radians per second,

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and that is equal to,

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because there are two Pi radians per cycle,

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we can calculate Omega by

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two Pi times f. Dimensionally,
let's just look at that.

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If f is in cycles per

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second and radio frequency
is in radians per second,

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we need to convert from cycles to radians.

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Well, we simply multiply by
two Pi radians per cycle.

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Cycles cancel, and we're left
with radians per second.

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So we have amplitude,

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we have the period T,

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we have the frequency f which is
the frequency in cycles per second,

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and we have the frequency omega
which is in radians per second.

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There's one final parameter
we want to introduce,

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and that is the concept of phase.

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We mentioned that we model this in terms of

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this V sub m cosine of
omega T plus some Phi.

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Phi we refer to as the phase term.

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If you go back to your college algebra,

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you remember that if you
have some function f of t,

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then f of t plus some constant was simply

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the function f of t shifted by c units.

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Now, if c is a positive number,

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then f of t plus c represented a shift of

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the original function f of t to the left.

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If c was negative,

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it represented a shift
of f of t to the right.

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So here we have our original function,

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V sub m cosine omega t in the blue,

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and we've got this red function
here which if you'll notice,

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it's basically the blue function
shifted to the left,

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in this case by 90 degrees.

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So the red function could be written in
terms of the blue function shifted to

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the left or we'd say that the red function

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is V sub m. It has the same amplitude,

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cosine of Omega t. It has
the same radio frequency,

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but it's been shifted ahead by 90 degrees.

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Now, we electrical engineers get
a little bit schizophrenic here,

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because we do mix our units.

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Sorry, that's just the way it is.

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You going to have to
get used to it I guess,

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Omega is in terms of radians per second,

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and phase frequently is
given in terms of degrees.

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So how do you convert
from degrees to radians?

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Well, once again, let's just take
the dimensional analysis approach.

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So if we have degrees and we
want to convert to radians,

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we multiply it by something
that's going to have

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degrees in the denominator
and radians in the numerator.

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So then we have to say, all right well
how many radians are there per degree?

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Or we know that there are two
Pi radians per 360 degrees.

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So to convert from degrees to radians,

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we multiply by two Pi over
360 or by Pi over 180.

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In other words, degrees times
Pi over 180 equals radians.