>> There are lots of different things in the physical world
around us that we model sinusoidally.
It could be the voltage or current in an electrical circuit.
It could the vibrations of a mechanical system.
We could be referring to the propagation of
light energy or electromagnetic energy.
So in this video here, I'd just like to review with you
the various parameters or
the various markings of a sinusoidally varying signal.
By sinusoidally varying, I mean anything that could either
be modeled as a sine or a cosine.
So for example, let's just talk in terms of voltage.
As I'm an electrical engineer,
let's talk in terms of voltage.
That's the time-varying voltage that we would model as having
some amplitude V sub m times this cosine of Omega t,
plus some phase angle Phi.
All right. What is V sub m refer to?
Well, first of all, what is cosine?
Well, you know that the cosine function varies
from negative one to positive one.
So if you multiply that plus or minus one by some constant V sub m,
then we have a waveform is varying between
negative V sub m and positive V sub m. In this picture right here,
we've got down here at negative five and on up here to a positive five,
we refer to this amplitude or this distance here
from where it crosses the horizontal axis on up,
we refer to that as the amplitude.
Amplitude of the wave,
and for our model,
that amplitude is V sub m. Now,
this waveform varies periodically.
That means that it repeats itself every so often.
The period is the length of time that it takes to go
through one of these cycles or until it repeats itself.
So we can say then that we've
got a distance from
its peak on the top to the peak on the top would represent one period,
and we use the letter T,
capital T to represent period.
So T is the period of the signal,
and it has units of seconds or time to be more accurate.
So from peak to peak is T from negative going zero crossing,
to negative going zero crossing.
That's another interval of T seconds.
Or we could talk about from positive going,
zero cost crossing to positive going, zero crossing.
Again, that would represent one period.
Now, if instead of talking in terms of time
we talk about how rapidly the signal is changing,
we refer to its frequency and we use the letter lowercase f for frequency.
It has the units of cycles per second.
If you stop and think about it,
if the period is the length of time that it takes to go through one cycle,
it will be so many seconds per cycle.
Then frequency, the cycles per second is just equal
to one over T.
That gives us a frequency in cycles per second,
common in electrical engineering we might talk about 1,000 cycles per second.
Another term for cycles per second is Hertz.
A 100 cycles per second or a 100 hertz,
1,000 cycles per second or a kilohertz,
or a million cycles per second,
one million or one megahertz.
If you think about the FM radio dial,
say you're listening to some radio stations broadcasting its FM 96.3,
that's referring to a carrier signal is oscillating at
96.3 megacycles per second or 96.3 megahertz.
Now, the argument of
a cosine or a sine wave is frequently measured in terms of radians per second.
So we introduce a another frequency
or type of frequency that we're going to call the radio frequency.
The radio frequency is in radians per second,
and that is equal to,
because there are two Pi radians per cycle,
we can calculate Omega by
two Pi times f. Dimensionally, let's just look at that.
If f is in cycles per
second and radio frequency is in radians per second,
we need to convert from cycles to radians.
Well, we simply multiply by two Pi radians per cycle.
Cycles cancel, and we're left with radians per second.
So we have amplitude,
we have the period T,
we have the frequency f which is the frequency in cycles per second,
and we have the frequency omega which is in radians per second.
There's one final parameter we want to introduce,
and that is the concept of phase.
We mentioned that we model this in terms of
this V sub m cosine of omega T plus some Phi.
Phi we refer to as the phase term.
If you go back to your college algebra,
you remember that if you have some function f of t,
then f of t plus some constant was simply
the function f of t shifted by c units.
Now, if c is a positive number,
then f of t plus c represented a shift of
the original function f of t to the left.
If c was negative,
it represented a shift of f of t to the right.
So here we have our original function,
V sub m cosine omega t in the blue,
and we've got this red function here which if you'll notice,
it's basically the blue function shifted to the left,
in this case by 90 degrees.
So the red function could be written in terms of the blue function shifted to
the left or we'd say that the red function
is V sub m. It has the same amplitude,
cosine of Omega t. It has the same radio frequency,
but it's been shifted ahead by 90 degrees.
Now, we electrical engineers get a little bit schizophrenic here,
because we do mix our units.
Sorry, that's just the way it is.
You going to have to get used to it I guess,
Omega is in terms of radians per second,
and phase frequently is given in terms of degrees.
So how do you convert from degrees to radians?
Well, once again, let's just take the dimensional analysis approach.
So if we have degrees and we want to convert to radians,
we multiply it by something that's going to have
degrees in the denominator and radians in the numerator.
So then we have to say, all right well how many radians are there per degree?
Or we know that there are two Pi radians per 360 degrees.
So to convert from degrees to radians,
we multiply by two Pi over 360 or by Pi over 180.
In other words, degrees times Pi over 180 equals radians.