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## AC circuit analysis

# Sine of time

## Video transcript

- [Voiceover] Now I wanna
introduce a new idea, and that is the idea of voltage or current, some electrical signal being a function of time, cosine of omega t. So here what we're doing
is we're introducing time as the argument to a cosine. And time is that stuff
that always goes up, this is a number that increases forever. And we have another variable
in here called omega, this is the Greek lowercase omega, and omega has an important job in this. The argument to cosine,
whatever is inside the cosine, this has to be dimensionless,
this has to have no units. And so if we put in a unit
of seconds, that means omega is something that has the
units of one over seconds, or one over time. So omega is one over time. And when we multiply those
two numbers together, we get something that has no
units, and then we can take the cosine of it. So, this is referred to as a frequency. Something that has units of
one over time is a frequency. This is a constant number,
this is some number, this is a number, time is
a number that increases all the time, and so when we have that cosine, we now have something we call
a cosine wave or a sine wave, or a sinusoid. And that sine wave goes on. As time increases, it keeps
going and going and going. So now we've turned our
trigonometric cosine function, which is right here, which
is something that was well defined between
zero and two pi radians. Notice that I've changed the
axis, the axis is now in time over here, and now we're
counting off time in seconds, there's two seconds, three, four, five, and that dot there, that's at pi seconds, and this is at two pi seconds, right at that dot right there. And you can see that that is
the full cycle of one cosine before it starts repeating again, so that's 6.28 seconds. So for this image here,
omega has the value of one. So when time t reaches two pi seconds, we've gone through one full cycle. So this idea of this continuously changing cosine or sine wave going on forever, that gives us the term sine waves, and sine waves are a good
model for a lot of things that happen in nature. If you ever hear a pure
tone or a pure note, a bell being rang or a whistle, or if you sing a note,
the shape of those tones looks like a sine wave or a cosine wave. And these are often the
things that come into our electronic systems and
we wanna do things with them. So now I wanna talk a little
bit more about the details of this kind of a sinusoid wave. We're gonna learn some new words for this. So one important concept is the idea of any repeating waveform,
any repeating signal, is the idea of a period. Let's just do the zero crossings here. If I take the time change
from there to there, this is the repeating
interval of this function, and I'm gonna call that,
that distance right there, this is the period of this sine function. This is actually a cosine wave. The period of this sine is this
distance in time right here, and the symbol we use
is typically a capital T to indicate the period. So let's look at this
cosine wave, this sinusoid, and identify what its period is. I can do it easily if I go right here. It looks like it repeats on
this interval right here. Every time we hit one of those points, so this would be, if
this is time zero here, this is time big T, this is time two T, on and on like that. And I read off this graph,
and what I see right here is that the time is T equals 0.02 seconds. So that's how you find out what
the period of something is. You can take any two points. We could actually go right here and then go through one cycle, and go to right here and I can read off that
period there, there's T, and that's the same value
as that T right over there. So the time T, we can
also call that a cycle. That's the time it takes
to go through one period, that's one cycle. So one of the questions I can
ask about this waveform is, how many cycles fit in one second? How many cycles per second,
is another way to say that. So we can say that one cycle happens every T seconds, and in our particular case, it's one cycle per 0.02 seconds, and if take the reciprocal of 0.02, we get the answer to be, that's 50 cycles per second. That's the speed, that's
the repetition rate of this sinusoid, 50 cycles per second. And this has another name,
it's named in honor of a German scientist, and
this is called a Hertz. Heinrich Hertz is the first
person to send a radio wave and receive it on purpose,
he knew what he was doing. We named the unit cycles
per second in his honor, and that's called the Hertz. So now we have two ways
to measure frequency. One is f, which is frequency,
which is measured in Hertz, and that's cycles per second. And one cycle equals two pi radians per second. So the two measures are cycles per second and radians per second, and
we'll flip back and forth between those. Okay, and radians per second,
or the variable is omega, and that's called angular frequency, or radian frequency, and you'll sometimes see the word rad used to indicate that
we're talking about angular or radian frequency, and
the variable is omega. So let's work out what,
the relationship between f and omega. It's actually sitting right here, okay. So if I give you an f, given f, what is omega? So I write down a number f,
and it's in cycles per second, and I'm gonna multiply
that by a conversion factor that I'm gonna make up, so
we're gonna multiply that by two pi radians per second is the same as one cycle per second. And that equals, cycles per second cancels
with cycles per second, so that equals two pi f radians per second. So the conversion factor is omega equals two pi f, and that's worth remembering. So if I have a sine wave, a
voltage sine wave, for instance, v of t equals cosine omega t, I can write that equivalently as v of t equals cosine two pi f times t. So one of the frequencies, this
one is in cycles per second, and this one is in radians per second, and we can interchange 'em that way using this conversion factor. So if we take the example
from earlier in the video, we had a signal that was 50
Hertz, or 50 cycles per second, so we would write that here like this. We'd say v of t equals cosine two pi f, and f is 50, times t, and that's the same as cosine of 100 pi t. So this number right here,
100 pi, that's omega, and this number right here is f. So that does it for our
review of trigonometry, and we've introduced
the idea of a sine wave where t is the argument
to the trig function.