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02 - Sinusoidal AC Voltage Sources in Circuits, Part 1

hello I'm Jason welcome to this lesson

of the AC analysis tutor what we're

going to do now is pick it up from the

last section we had a general overview

of what alternating current is we said

that alternating current literally comes

out and in and then out the other way

back in and it literally switches the

direction so many times per second for a

wall socket at 60 Hertz or 60 times per

second but we're gonna study in our

circuits many many different frequencies

and so all of the techniques we're going

to use is applicable to any problem that

you have but before you go any further I

want to give a little bit of review of

trigonometry not a long review I think

you really need to have some skills from

back then but I do want to take a few

seconds to show you something because it

is going to tie in so let's look at the

plot of sine of theta right theta

remember it's just an angle so you can

represent angles in degrees 360 degrees

in a circle right

you can also express an angle in radians

there are two pi radians in a circle

right so it's a conversion factor 360

degrees in a circle two pi radians also

in a circle so here is a sine of theta

plotted against theta now I'm not going

to be exact with this but the point of

this is not to be exact here but what I

want to do is just kind of show you what

it generally looks like because you kind

of need to have this in your head a

little bit I think it's helpful sine

functions just plain sines sine waves

however you want to look at it pretty

much look like this they start at the

origin that's the word start sitting it

goes up and then down and then up and

then down and then up and then down

notice my little marks I'm trying to

evenly space them out that's pretty

decent looking sign there and the thing

that you need to remember a couple

things I want you to remember is that it

starts at the origin so the starting

point starts at the origin and it goes

up to plus one and this goes to minus

one so that's what the sign looks like

now what are we plotting is we're

plotting against angles

all you're doing is you're putting an

angle in here and in in circuit analysis

believe it or not you deal with degrees

sometimes and you deal with radians

other times so it's important that you

have some familiarity with degrees and

radians I already talked about that a

little bit

basically you're plotting the angle in

here and you're giving a the sine

function as a result of that angle so it

goes up it goes through zero goes to

negative goes through zero and you can

see it's cyclic it just keeps coming

doing the same thing over and over and

over again forever basically now if we

wanted to compare that with cosine of

theta

it would look similar but there's an

important difference and the reason I

want to show you this even though you

probably already know this a lot of you

guys I want to show you this because it

is gonna be helpful for you later so I'm

gonna put a little marks here as best I

can I'm not trying to be too terribly

exact cosines do not start at the origin

cosine start right here so if you'll had

to do that cosine would come down here

like this and it would be up like that

reach a peak like that okay something

like that all right so when you look at

this here it's a maximum that the cosine

gets to is +1 and the minimum that it

gets to over here is -1 so I want you to

stare at these and remember that I'm not

an artist so so they don't look exactly

perfectly right but they get the point

across for what I'm trying to show first

thing is I want you to know that both of

these functions sine and cosine have a

maximum value of 1 and a minimum value

of -1 so the extent of the the height of

the function is always plus or minus 1

for sine and cosine the other thing I

want you to remember is that the shape

of these guys look identical if I took

away the black axis right and just had

these red things on the board they would

look the same the shape of the waves

look identical to one another sines and

cosines so what's the difference between

them well the difference really is the

starting point right there here we start

at the origin here we start at the

maximum so if you use your imagination

if I could like grab this sine if I

could just grab the red part leave the

axis alone if I could just grab the red

part and

pull it this way so that the maximum

here lines up here then this would

exactly equal a cosine so if I could let

me say that again if I could grab this

sine wave here and just literally pull

it over a little bit then they would

match up and overlay exactly with this

or you can think of it another way I

could grab this cosine because don't

forget the cosine even though I've

stopped it here it continues on on the

other side of the origin as well if I

could grab this and pull it this way

like this so that this ends up here if

you can use your imagination then this

again would look exactly like that so

why am i going through all this trouble

to explain the sine and cosine the thing

is they have the same amplitude same

height they have the same shape the only

difference between a sine and a cosine

is that they're shifted versions of one

another

I'm gonna say that again because you

know a lot of people when they study

trig they just learned that sine and

cosine is a button on the calculator and

it is a button on the calculator but

fundamentally they are the same thing

the exact same thing they're just

shifted versions of each other okay I'm

not going to talk much more about that

here but I just need you to believe me

they're that they're shifted versions of

one another the reason I'm telling you

this is because we said that when we're

doing alternating current we're going to

be exclusively really studying

sinusoidal stuff sinusoids meaning I'm

gonna have a voltage source that's gonna

look like a sinusoid they want to see

what's the current and the voltage in

the circuit I'm going to have a

sinusoidal current and I'm gonna may

want to calculate something else in the

circuit and what we're gonna find is if

the circuit is driven by a sinusoid then

all the other stuff in the circuit all

the voltages and the currents everywhere

else they're also behaving like

sinusoids they may be shifted with

respect to one another but they're all

going to look like sinusoids they may

have different heights and stuff but

they're gonna have the same overall

basic shape but whenever we're trying to

figure out how to write these things

down mathematically we got to figure out

do we want to represent our voltages as

sines or as cosines because they really

are the same thing and in almost every

electrical engineering text we're gonna

use the cosine function to represent and

write down all of our voltages currents

and everything else so keep that in mind

we're going to

the cosine function to represent

voltages and currents throughout our

circuit but it's really no different

than using sine the bath would look a

little different because they are very

similar except for the shift between

them just kind of keep that in mind so

if you ever wonder why are we using

cosines as those signs that there's no

real reason it's just we do that by

convention all right so we choose a

cosine function if we had chosen a sine

function to represent our voltage it

would look very similar we'd be a

shifted version but let me show you what

a voltage would look like in terms of

mathematical weeks we've drawn it you

know that's a cosine over there but

let's write it down so you might have a

voltage that's V of T notice that

voltage is now a function of time

because it's changing with time it might

look something like this capital V sub M

cosine Omega I'll talk about that in a

minute T plus Phi all right this is what

a typical voltage is gonna look like in

electrical engineering it carries all

the information to describe what the

sinusoid looks like so think about the

things you need to know you when you're

building a circuit or you're analyzing a

circuit that's AC you're gonna want to

know how tall is that sine wave if

you're talking about the source I'm

talking about the voltage source of

driving the circuit you might want to

know how tall that thing is the

amplitude we'll talk about that in a

minute and that's represented up here

you might want to know the frequency how

fast is it also oscillating back and

forth we'd say that the wall sockets you

know 60 Hertz but our circuit may be 500

Hertz or thousand Hertz or something

like this and we also have something

called a phase angle we'll talk about a

little bit later so let's analyze this

and just see what the parts of this

really looks like see if we can make

some sense of it the number is called V

M V maximum that sits out in front of

the cosine that is called the amplitude

right that's the amplitude this is the

height of the signal the height of the

of the cosine this guy is course cosine

and this varies between plus and minus

one because the cosine only goes plus

and minus one remember that

this guy is called the angular frequency

I'll explain why in a minute but

basically it's it's you know we talked

about sixty Hertz that sixty times per

second this is basically very very

closely related to Hertz and I'll

explain what it is in just a second but

basically it's the frequency of your

cosine it's telling you how fast is it

oscillating how many times per second

you can kind of gather by looking at

this and converting it to regular old

frequency don't worry too much about

what it says angular frequency I'll

explain that a little bit later this

last part is called the phase angle and

the phase angle are kind of actually

introduced you to you a minute ago

without you realizing it you can take

these functions and shift them around

left and right and so the phase angle is

telling you where you start measuring

from basically and I'll draw you a

picture and show you that in just a

minute but every part of this is equally

important to characterize what our

voltage source looks like the cosine it

gives you the shape it's sinusoid right

it varies between plus or minus one so

this entire thing inside of the cosine

just basically governs how the cosine is

going to look the frequency here is how

fast it's oscillating the phase angle is

telling us how we're dragging it left

and right right and the VM on the

outside because remember the cosine only

goes between plus or minus 1 whatever

number is out here is going to govern

the actual amplitude you know maybe it's

a 50 volts or so that means 50 would be

out in the front and the cosine is

making it dive up and down between plus

or minus 50 because this goes between

plus or minus 1 if this were 50 that

would be driving the overall amplitude

but enough of that a picture is worth a

thousand words and so we're going to

draw a picture here to try to illustrate

what we're talking about here so here is

a plot of time and seconds like this

seconds and then we're going to have

volts up here we want to basically plot

the function that we have put on the

board here and see if we can get some

information out of it first of all this

is a cosine

right cosines start at the top and I'm

not gonna draw it too perfectly but

basically cosines look like this and

they just go on and on and on forever

equally spaced crossings equally spaced

peaks and troughs and so on now when

we're we're looking at just the raw

cosine function we said it varies

between plus or minus 1 but we have a

coefficient out in front of it and so

that means that this varies between the

M and over here minus V M so positive

and negative BM so what we have here

this is called the amplitude amplitude

when you think of ant the word amplitude

it means maybe the strength of something

or the height of something that's what

amplitude means high amplitude means

something that's tall maybe something

like there's something that's loud

well here it means the height of your of

your source and now I'm talking about

voltammetry ting this as if it were a

voltage source you know and all the DC

analysis is just a steady 5 volt that's

boring now everything's gonna be

changing with a cosine right the height

of that guy whether it's 5 volts or 10

volts or ninety five volts or a thousand

volts is going to be governed by this

number out in front called the amplitude

that's what that is all right and the

other thing here when you look here you

can see where the signal here I should

say the source you can see where it

starts to repeat if you look here

between this point and this point right

here notice everything starts over here

you go from here to the bottom to the

bottom everything starts over so this is

called one complete cycle or one

complete period so what we have is T is

equal to the period and that means you

repeat right they repeat alright so I

want to write a few things down

underneath here that we're going to be

drawing upon as we learn about this

stuff and the first one is so let me

kind of draw like a little divider line

here first thing we want to talk about

is the frequency

all right when you look at the word

frequency it means how much does

something oscillate that's what the word

frequency means in electrical

engineering the frequency is one over

the period so when you when you talk

about the period of this guy the period

is going to be in seconds so you may

plot this and I may say hey the period

of this wave is half a second that

basically means that it starts to repeat

itself after half a second that's called

the period if I take one over the period

then I'm going to end up with another

unit called Hertz which means cycles per

second how many times does does the

thing could do complete oscillation in

one single second so this F here this is

the 60 Hertz that we're talking about in

the wall socket that's the F that's the

the frequency or the amount of cycles

per second there so the unit here for

frequency is Hertz which is cycles per

second how many cycles per second is

this way of doing now notice that we did

not use this frequency up here we use

something different called Omega it's

like a little W here and I called it an

angular frequency and I told you yeah

it's kind of a frequency but it's a

little different I'll explain later well

now I'm going to explain it to you what

we have here is I'll write it down here

angular frequency right Omega is

directly related to the regular

frequency by two times pi times F in

fact this I want you to remember I don't

tell you to memorize too many things in

any of my classes but one of them is

this Omega is 2 pi F Omega is 2 pi hat

Omega is 2 pi F Omega is 2 pi F I want

you to say it because you're going to

need to convert back and forth alive you

need to remember that Omega is 2 pi F

right now what would the units of this

be since this is cycles per second and

here we have kind of an angle and

radians 2 pi radians in a circle what

you get here I'll go ahead and use the

the red here the units is radians per

second that is the unit of angular

frequency

all right so I want to take just a

second to kind of make sure everybody

understands that okay I want to make

sure everybody understands that when we

talk about things that repeat ok there's

kind of two ways especially in terms of

sinusoids there's two ways to look at

one is to go figure out the frequency

that's how many cycles per second the

thing is changing and the other is to

talk about the angular frequency which

is how many radians per second the thing

is changing right they both kind of mean

the same thing if you have a higher

frequency it's changing faster if you

have a higher angular frequency the

thing is changing faster in fact they're

actually related directly by just a

constant here so really F and Omega are

really the same thing

they're just related by a constant so

why do we even do that why don't we just

deal with frequency because that's the

one that makes sense to me when you

first study this stuff you're like

frequency that's something I can wrap my

brain around why are we talking about

radians and angular frequency and the

reason is because if you look inside of

the cosine function we have the

frequency times time plus some phase

angle here all right this is the angular

frequency times time

don't forget whenever we take the cosine

of something on your calculator let's

say you're always taking the cosine of

an angle okay you're not taking a cosine

of cycles per second you're not taking

the cosine of inches you're not taking

the cosine of cubic centimeters you're

always for it to make sense taking the

cosine of an angle which is let's say

radians because Radian measure and math

is what we typically use in engineering

radians right so at the end of the day

whatever is inside of this guy has got

to be an angle and it typically needs to

be in Radian measure right so if we just

put F here for frequency then that

cycles per second and then we'd be

multiplying times time which is seconds

so then what we end up getting there is

cycles so we'd be taking the cosine of

cycles that makes no physical sense we

have to take the cosine of radians right

so we introduced this thing called the

angular frequency we say it's directly

related to the regular frequency so

instead of talking about how many cycles

we're gonna talk about how many radians

per second the thing makes you know in

one single second so if you take radians

per second which is this unit

apply by seconds then you just get

radians here so this product gives you

radiance this is just an angle so you

just have radians in there and then you

can take the cosine of it that's one way

to look at it in terms of in terms of

units basically you have to have

radiance somewhere in there to take

cosine of it and so we're gonna be

dealing with Omega all the time which is

radians per second the other way to look

at it is is just another way of

representing cycles per second here if

we want to know how many cycles per

second we just look at the graph look at

one second and count how many cycles we

have and so on but when we multiply by

2pi what we're doing is think about the

unit circle from trigonometry that's why

I said probably good idea to review some

trig think about the unit circle there

are two pi radians in one full

revolution so here we're talking about

kind of like one full cycle of this wave

but instead of thinking about the way

think about it going through the unit

circle one time that's one full

revolution that's 2pi radians every time

we go around two more pi two more pi

radians two more pi radians so i can

express anything cyclic especially in

terms of cosines i can express how many

times is it going to go around the unit

circle in one second how many integer

multiples of 2pi is it going to go all

the way around per second and that ends

up boiling down to being radians per

second so when you take the frequency

multiplied by 2pi what you're getting is

you're basically figuring now how many

times does this thing take a full trip

around the unit circle 2pi which would

be radians per second so there's a

couple different ways to look at it I'm

trying to give you a little bit of a

solid foundation there because sometimes

you look at it you're like why we using

Omega why I'm multiplying by two pi it's

just another way of representing a

cyclic term right so instead of saying

we're changing looking at the sinusoid

as it goes around and we're counting how

many cycles per second we multiply by 2

pi that's how many radians how many 2 pi

times we go around the unit circle per

second all right now the next thing we

want to do I think we beat angular

frequency into the ground is amplitude

and amplitude we already talked about

fundamentally that means the height the

height which is V sub M right that's the

height of this guy this were 36 then the

amplitude would be 36 volts and by the

way amplitude is measured from the top

to the axis here it's not measured from

peak to peak like this so if this were

36 from here to here would be 36 volts

if this were 150 and be 150 volts

basically whatever number sits in front

of the cosine is the amplitude which

tells you the height and then we've

already talked about it but I'm going to

write it down again

the period the period is called T which

is the length of time for one

oscillation so I have some more stuff to

talk about here related to this but I

want to stop this lesson here let's you

absorb this in the next lesson we're

going to talk about that what we call

this phase angle here I want to draw

some more pictures and show you what

that means but it's a really really

quick recap just kind of keep in mind

that every single voltage or current

that we talked about in this class is

basically going to look like what we've

written written up here so you have to

have something to go in each one of

these spots whatever number you put out

here is going to be the amplitude so if

it's voltage then this would be the

amplitude of the voltage that we're

talking about if this is a current

source or something or a measuring

current the number out here is the

amplitude of the current it's the height

of this wave it's cosine that basically

tells you the shape of the thing inside

here we have frequency but it's not

expressed as just how many times per

second it's expressed as angular

frequency how many radians per second is

it going around keep in mind that 2pi

radians is one time around the unit

circle so this is radians per second

when you multiply by time you end up

with a unit of radiance which matches

with what we need and then we have the

phase angle here which is going to shift

it left and right so what I like to do

is close it down now make sure you

understand this

and we'll do some examples and solidify

it a little bit later

follow me on to the next section we'll

talk about the phase angle in terms of

alternating current alternating current

circuits