L7 1 6Characteristics of Sinusoid Waveforms

>> 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.