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### Course: Electrical engineering > Unit 2

Lesson 5: AC circuit analysis- AC analysis intro 1
- AC analysis intro 2
- Trigonometry review
- Sine and cosine come from circles
- Sine of time
- Sine and cosine from rotating vector
- Lead Lag
- Complex numbers
- Multiplying by j is rotation
- Complex rotation
- Euler's formula
- Complex exponential magnitude
- Complex exponentials spin
- Euler's sine wave
- Euler's cosine wave
- Negative frequency
- AC analysis superposition
- Impedance
- Impedance vs frequency
- ELI the ICE man
- Impedance of simple networks
- KVL in the frequency domain

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# Sine of time

If we introduce time as the argument to the sine function, we get a sine wave. Animated. Created by Willy McAllister.

## Want to join the conversation?

- Just to confirm that I understand, are the frequency and the angular frequency the same parameter (just in different units)? For example, we could measure distance in miles or kilometers. Angular frequency just means that the frequency has different units?(12 votes)
- You are correct. Frequency has units of cycles per second (also known as hertz), while angular frequency has units of radians per second. You use whichever unit is best suited to the problem at hand.(8 votes)

- We know that the frequency is the reciprocal of the period (f = 1/T and T = 1/f). In your calculation you obtain the frequency from the reciprocal of the period (1/0.02). So it seems like that 50 derives naturally from the period. Why can't it be used directly as omega value ?

You said that frequency and angular frequency can be used interchangeably. However it seems like that the frequency in Hertz need to be always converted to rad/s in order to use it with sinusoids. So from a calculator point of view you end up using rad/s always. Assuming that a Hertz value is inserted, then the calculator need to convert it first to rad/s before performing the actual calculation. Did I get it right ?(2 votes) - what happens to the sine wave of the increasing frequency?(1 vote)
- The sine function has a period of 2π. That means the sin function completes one cycle when its entire argument goes from 0 to 2π.

ω represents the frequency of a sine wave when we write it this way: sin(ωt).

If ω=1 the sin completes one cycle in 2π seconds.

If ω=2π the sin completes one cycle sooner, every 1 second.

For f>1: If ω=2πf the sin completes one cycle even faster, every 1/f seconds. Or, equivalently, sin completes f cycles in 1 second. So the higher f is, the more cycles happen in 1 second. More and more cycles get crammed into that 1 second interval. That's what higher frequency is.

Vocabulary:

ω is called the*radian frequency*with units of radians per second. 2π radians is all the way around a circle (360°). 1 radian is about 57° of the way around a circle.

f is the*frequency*with units of cycles per second. Cycles per second has the honorary name*Hertz*. ω = 2πf (good to memorize)(2 votes)

- What is the difference between sin and cos(1 vote)
- Sine and Cosine have exactly the same shape, waving up and down between +1 and -1. The only difference is that sin(x) starts at 0 when x=0 and cos(x) starts at 1. They are "90 degrees out of phase."(1 vote)

- At about1:00you state that "omega is 1/time", and later you state that "omega is radians/sec". You really made this simple topic very confusing. Could you explain?(1 vote)
- The argument (the number inside) sin or cos is in radians, (a measure of the angle distance around a circle). Going around full circle is 2pi radians (of angle).

Radians are a unit of measure of circular angles. You might think that radians have some kind of dimension. However, oddly, radians are*dimensionless*. Recall the definition of radians: the ratio of the distance around the circumference of a circle divided by the circle's radius. That's the ratio of two distances, like meters/meters. The resulting dimension of a radian is: none.

If the argument of sin or cos is [\omega t] it has to be the case that the overall argument has dimension: none. That means \omega has dimension 1/t (1/time, 1/seconds, 1/minutes, whatever).

Since we know the argument to sin or cos is a radian value, it might be more precise to include "radian" in the dimension of \omega, like this: radian/sec. This reveals why \omega is called the*radian frequency*.

I perhaps made a small verbal slip by saying \omega is 1/t. It's quite common to forget the dimensionless "radians" when you are showing how \omega cancels out the seconds in t.(1 vote)

## 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.