# How does alternating current produce a sine wave over time?

I've only taken one introductory class to DC circuits where we learned Ohms law and series parallel combined circuits. Now I'm in an introductory class to AC circuits and last night's entire lecture (first day of the semester) was just over my head.

We didn't go over any fundamental concepts about HOW any of these things work, we just started talking about capacitance, and peak-to-peak voltage. I don't even understand how a voltage source that switches polarity produces a sine wave through time. Why are we even talking about "through time" suddenly? How does the voltage alternate in polarity?

We also talked about degrees on a circle and radians, I'm just confused :(
Any help is appreciated!

Annabell

dlgoff
Gold Member
... I don't even understand how a voltage source that switches polarity produces a sine wave through time. Why are we even talking about "through time" suddenly? How does the voltage alternate in polarity?

We also talked about degrees on a circle and radians, I'm just confused :(
Any help is appreciated!

Annabell

The fastest way to see how time enters the picture, in my opinion, is a visual. As you read through this Electric motors and generators page, play the animations. This should help in understanding where the sine wave comes from.

These and other animations can be found here:

http://www.animations.physics.unsw.edu.au/

Cool, thanks dlgoff for the references.

sophiecentaur
Gold Member
@Annabel
A normal AC generator doesn't switch + - + - + - bang bang. It actually produces a smoothly alternating voltage (sine wave) for a start. AC is usually generated in a rotating generator and the sine wave just comes from it naturally. It is possible to generate AC by 'quickly' reversing / alternating the connections to a battery (DC) and the result is a 'square wave'. That works fine in a lot of AC equipment, in fact, but they have to 'filter' that square wave to feed some equipment that really does need a sinewave. No real need to worry about that minority situation though.

Don't let Radians scare you. They come into AC theory and a lot of other places because, if you work in degrees, you find that 2∏ keeps turning up just when you don't want it to. Splitting a complete turn into 360 is a totally arbitrary choice but working in radians actually makes life easier in the end (you may not believe me now but it's true). They will also be using radians on the planet Zog, whose inhabitants have seven fingers and the planet circles their Sun in 413 of their days. (And their students are equally confused but they get over it! )

If you remember that there are 2∏ radii around the outside of a circle, you can also remember that there are 2∏ radians in a complete turn. 2∏ is the 'same as' 360° so one radian must be 360/2∏ degrees (about 57°).

Averagesupernova
Gold Member
The best way one could briefly describe why a sine wave is a sine wave in an alternator is because of the motion of the conductor cutting through the magnetic field. For obvious reasons there are certain times during one complete revolution of the rotor that NO lines are cut and NO voltage is produced. Go from there. I didn't look at the link that dlgoff posted but if it is what I think it is it will illustrate this very well.

jim hardy
Gold Member
Dearly Missed
We didn't go over any fundamental concepts about HOW any of these things work, we just started talking about capacitance, and peak-to-peak voltage. I don't even understand how a voltage source that switches polarity produces a sine wave through time. Why are we even talking about "through time" suddenly? How does the voltage alternate in polarity?

We also talked about degrees on a circle and radians, I'm just confused :(
Any help is appreciated!

Please don't take this simplistic post as talking down.
I have felt overwhelmed before too.

It is easy to forget "how hard it was" before you knew how to do something.
Do you have any memory of before you could walk?
I vividly remember my very first day of school, looking at that alphabet on a banner above the chalkboard and saying to myself "How will i ever learn all those letters!!!" .

Professors sometimes skip over the basics. Forgive them, they know a lot and may have forgot when they didnt.

In the beginning you may find it useful to "Freeze" time in you mind .
A movie is a sequence of stills, and AC is at any given instant DC.
So you can stop time in your mind while you figure things out.

As Mr Sophie said one could make AC by simply switching DC.
In practice it is better to have smoother electric power than a switched square wave and a sinewave was agreed on around a hundred years ago. It's fairly easy to make.

I suggest you take a few minutes and a pocket calculator and sheet of graph paper, and plot a graph of volts versus time
VOLTS = 1.414*sin(2 * pi * 60 * time) for time increment of 1.388889 milliseconds (that's 1/720 second)
from time = zero to 16.66667 miliseconds

I suppose you could do it in excel or something, but myself i took one look at xcel and said "How will i ever learn all those commands?" and stuck with my slide rule.
If you do it by computer, make a column for what's inside those parentheses following 'sin'.

now you have drawn one cycle of a one volt RMS 60 cycle sine wave voltage.
And you have shown your brain how to freeze time.
And if you paid attention to the argument of sin, what's inside the parens, you notice it went from zero to 360 which is # of degrees in a circle.
So 'talking in circles' is a handy way to deal with sinewaves no pun intended (well not much of one)
Note with time increment selected we hit every thirty degrees.
It'll be handy in AC circuits to know sin of every thirty degrees, and 45 too..

Now, you also asked "Why are we even talking about "through time" suddenly?"

Because Tesla and Steinmetz realized around 1900 that if electic industry stayed with direct current there could never be long distance transmission of power.
So they came up with the transformer which requires alternating current,
and the "alternator" to produce it.
AC varies with time, DC does not.
Had we stuck with DC time would be not be so much involved.

But you are exactly where i was that first day of school.
It's not hard to learn this stuff, it's just hard to believe you can. At first.

Admit these ideas. (Admit in sense of open and allow to enter.)

You'll do fine.

again, no offense intended , just an encouraging word.

old jim

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vk6kro
Like this?

http://dl.dropbox.com/u/4222062/sine%20wave.PNG [Broken]

Congratulations on the Science Advisor award Jim.

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jim hardy
Gold Member
Dearly Missed
Thanks Vk6 -

yes ! the dots are instants in time...
how'd you draw that?
you guys amaze me.

I'd bet an awful lot that Sophie remembers working from sliderules and five place log tables-
i still remember from high school math class(1963) that log pi = 0.49715
and log degrees in a radian= 1.75812

but where did i leave the car keys?

old jim

vk6kro
They reckon that if you lose your car keys, that is normal. If you forget you own a car, that is Alzheimers.

That graph was a 5 minute quicky in Excel.

I couldn't think of 3 commands I use all the time.

Just put "=" if you want to enter a formula. That's it, I think. Otherwise just enter a number or text.
You can use it without using Macros or a lot of its other fancy tricks.

I use it to plot the downward spiral of my financial empire.

I have a nice calculator now that takes care of the log tables for me, but I used it to check your recollections and they were right.

davenn
Gold Member
2021 Award
The fastest way to see how time enters the picture, in my opinion, is a visual. As you read through this Electric motors and generators page, play the animations. This should help in understanding where the sine wave comes from.

These and other animations can be found here:

http://www.animations.physics.unsw.edu.au/

Thanks Don,

excellent links, right from my home town university and I didnt know the pages existed

bookmarked for future reference for helping other out

Dave
cheers

sophiecentaur
Gold Member
Thanks Vk6 -

yes ! the dots are instants in time...
how'd you draw that?
you guys amaze me.

I'd bet an awful lot that Sophie remembers working from sliderules and five place log tables-
i still remember from high school math class(1963) that log pi = 0.49715
and log degrees in a radian= 1.75812

but where did i leave the car keys?

old jim

I sure do. Logarithm calculation with 'Bar Numbers', for values less than 1. Then we had two new mechanical calculators at School (turn the handle) in the sixth form. At University we had Mc Murdo (?) motorised calculators kerchunk kerchunk. I bought a Sinclair Cambridge 4 function LED calculator with about one week's wages and, in the lab, we bought the first HP Reverse Polish Calculator. I've seen 'em all arrive and become old fashioned. Remember those quaint old ipads? Oh no - that's next year.

I miss the bright red LEDs on the scientific calculators. I know it's all very irrational but numbers on that crisp red display seemed so much more alive than when we switched to LCDs.

vk6kro
I miss the bright red LEDs on the scientific calculators. I know it's all very irrational but numbers on that crisp red display seemed so much more alive than when we switched to LCDs.

Ah yes. And calculators which had flat batteries half way through an exam.

Wait until you see OLED displays. I just got one which is a 2 row 16 digit OLED display and it is really nice.
As bright as a LED but with current consumption more like a LCD.

On the other hand I put a AA alkaline cell in a LCD scientific calculator in 1989 and it is still working just fine.

FOIWATER
Gold Member
relative motion between a conductor and a magnetic field induces a voltage in said conductor.

Whatever voltage is induced, is DC. DC is the only naturally occurring voltage.... you can think of an AC waveform as DC over time, over time.

As the coil travels 2pi radians around the interior of the alternator, the electromagnets in your field induce a voltage in the coil, each instant the coil moves, a new magnitude of voltage gets induced - as the coil approaches a electromagnetic more closely.

As the coil travels away from the electromagnet, the value decreases to a point where the voltage induced is zero (the coil is directly between a north and south pole)

the speed of the alternator coil turning inside the field determines the frequency in hertz.

any corrections welcomed.!

sophiecentaur
Gold Member
relative motion between a conductor and a magnetic field induces a voltage in said conductor.

Whatever voltage is induced, is DC. DC is the only naturally occurring voltage.... you can think of an AC waveform as DC over time, over time.

As the coil travels 2pi radians around the interior of the alternator, the electromagnets in your field induce a voltage in the coil, each instant the coil moves, a new magnitude of voltage gets induced - as the coil approaches a electromagnetic more closely.

As the coil travels away from the electromagnet, the value decreases to a point where the voltage induced is zero (the coil is directly between a north and south pole)

the speed of the alternator coil turning inside the field determines the frequency in hertz.

any corrections welcomed.!

Ac vs DC? There are plenty of examples of Alternating Currents in Nature. All the natural sources of Radio Waves involve alternating current - just at a higher frequency than the mains.
It is best to try to avoid categorising these things because it will always fail at some point and it really doesn't help with understanding.

There is a general principle (a result of Maxwells Equations and discovered by Michael Faraday) is that whenever the magnetic field around a conductor changes, there will be an induced emf. The size of the emf is due to the rate of change of the so-called Magnetic Flux. In a rotating generator, the flux will change cyclically and, at some points it will change faster than at others, due to the geometry, giving you alternating directions of emf with smooth transitions between the maximum values. In simple generators, the variation follows a sine wave in time.

NascentOxygen
Staff Emeritus
I sure do. Logarithm calculation with 'Bar Numbers', for values less than 1. Then we had two new mechanical calculators at School (turn the handle) in the sixth form. At University we had Mc Murdo (?) motorised calculators kerchunk kerchunk. I bought a Sinclair Cambridge 4 function LED calculator with about one week's wages and, in the lab, we bought the first HP Reverse Polish Calculator. I've seen 'em all arrive and become old fashioned. Remember those quaint old ipads? Oh no - that's next year.
The orange NIXIE tubes hold a special place in my heart. As the ghostly digit floated near and far in its tube, they gave the appearance of being alive.

The orange NIXIE tubes hold a special place in my heart. As the ghostly digit floated near and far in its tube, they gave the appearance of being alive.

Prepare to be enchanted:

http://www.cathodecorner.com/

dlgoff
Gold Member
The orange NIXIE tubes hold a special place in my heart. As the ghostly digit floated near and far in its tube, they gave the appearance of being alive.

Now we're talking special.

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NascentOxygen
Staff Emeritus

Mesmerizing.

By the speed it's whirling, I think it says my computer screen is highly radioactive?

psparky
Gold Member
ω
Thanks Vk6 -

yes ! the dots are instants in time...
how'd you draw that?
you guys amaze me.

I'd bet an awful lot that Sophie remembers working from sliderules and five place log tables-
i still remember from high school math class(1963) that log pi = 0.49715
and log degrees in a radian= 1.75812

but where did i leave the car keys?

old jim

Hey Jim....congrats on Science advisor. All your hard work finally pays off with recognition:)

Now my two cents on AC.

All other points are good....in additon....

If you rotate a magnetic field around a wire, you get a current (generator). If you run a current through a wire, you get a rotating magnetic field (motor). AC is the natural state of generators and motors. DC can be forced with diodes and so forth. Batteries are DC as well.

V=IR in DC.

Same goes for AC....except now frequency dependent V(ω)=I(ω)*R(ω)

If you have an AC circuit with strictly resistive elements....and are trying to find the current with a resistance of same 10 ohms....simply divide the source...say for example 170sin377t volts by 10. Giving a current of 17sin377t amps. Or in RMS terms...you could say 120/10 = 12 amps rms. Two ways of saying the same thing.

As soon as you add in a capacitor (1/jωc) or inductor (jωl).....now you get a shift in the current in relation to the voltage. Affecting power factor or producing filters for example.
But, the same rule applies V(ω)=I(ω)*R(ω).

So if you know the voltage and resistance (reactance) in an AC circuit....simply divide the voltage by the resistance (reactance) to get the current. Although at this point....you will have to be somewhat of a pro with vector math and so forth. That little "j" is equivalent to 1<90. This will complicate things....but we like complicated things. Also, keep in mind that since inductors and capacitors are frequency dependent....their resistance (reactance) changes with frequency.

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start revising complex numbers ;P thats how you will understand this the best.

NascentOxygen
Staff Emeritus
start revising complex numbers ;P thats how you will understand this the best.
Somehow I think a course in complex algebra may not be an appropriate recommendation here. Let's just take a look at where the OP stands:
I've only taken one introductory class to DC circuits where we learned Ohms law and series parallel combined circuits.
Okay.
Now I'm in an introductory class to AC circuits
Right.
we just started talking about capacitance, and peak-to-peak voltage.
Fine.
We also talked about degrees on a circle and radians, I'm just confused
Uh, huh.

To recap, the question was:
How does alternating current produce a sine wave over time?
I reckon that can be answered and well understood without the need to take a course in complex algebra!

psparky
Gold Member
We also talked about degrees on a circle and radians, I'm just confused :(
Any help is appreciated!

Annabell

Frequencies are described with Sin waves and vectors....and other things as well.

But I'll just talk about Sin vs Vectors.

One complete cylce of a Sin wave is 360 degrees. You probably know that. Now take a vector sitting at zero degrees. Spin the vector 360 degrees and you just described the same thing. The magnitude of the Sin wave changes throughout....the vector magnitude does not. The vector is the RMS value of the sin wave. Important to keep that in mind.

Now lets talk radians. One complete cycle of Sin wave is 2∏. Take the same vector sitting at 0....and rotate all the way around and you just went 2∏ radians.

ω=2∏f

In USA....the sin wave produced by a receptacle in your house is 170sin(377t) Volts

You also know that the frequency in USA is 60 hertz....or 60 Sin waves per second.

If you multiply 60 hertz times 2∏....you get 377....the exact term in 170sin(377t)

170 is the magnitude....take the RMS (root mean square)....divide by square root of 2 (1.41)
170/1.41 = 120. That's how we get our 120 volts AC out of the outlets.

Hope this helps a bit and spurs further conversation.

sophiecentaur
Gold Member
In USA....the sin wave produced by a receptacle in your house is 170sin(377t) Volts

You also know that the frequency in USA is 60 hertz....or 60 Sin waves per second.

If you multiply 60 hertz times 2∏....you get 377....the exact term in 170sin(377t)

170 is the magnitude....take the RMS (root mean square)....divide by square root of 2 (1.41)
170/1.41 = 120. That's how we get our 120 volts AC out of the outlets.

Hope this helps a bit and spurs further conversation.

That's not 'exact'. It's an approximation. ∏ is not a whole number. It may be useful for working with but ∏ is not something that you should get rid of until the last possible minute and radians are something to get into and not to avoid at all costs.
Also, everyone is saying that √2 is the magic number. It's only right for a sine wave and many AC waveforms (even from Power sources) are not sinusoidal. RMS for a square wave will be half the peak to peak voltage.