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AC Sine Waves

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MikeT1982

Fri Aug 24 2007, 08:46AM

Registered Member #621 Joined: Sun Apr 01 2007, 12:37AM
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I was thinking about this, If you took a DC source and just switched its polarity back and forth 60 times per second, wouldn't you just get a 60hz square wave but still considered 60hz ac? So where does the smooth rise and fall of the sine wave in typical AC current come from? Does it have to do with peaks in magnetism of the rotational assembly of the generator as it actually physically passes the magents or something? Thats all I can think of that would cause it. Hmm, these things always make me think LOL.

Steve Conner

Fri Aug 24 2007, 10:52AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

The smooth sine wave from a generator is a natural consequence of the physics and geometry of spinning things. If you try to cast your mind back to high school math, the sine and cosine waves are just projections of the end of a spinning vector of constant length.

Likewise in electromagnetism, if you spin a magnet near a stationary coil, you will see a sine wave induced.

If you pulled the cylinder head off your car engine, you would see that turning the crankshaft at a constant speed makes each piston go up and down in a sine wave. From a mathematical point of view, the explanation is exactly the same as the explanation for the electric generator's sine wave output.

BTW: Pedants will no doubt point out that the engine piston would only describe a perfect sine wave if the connecting rod were infinitely long, and for all finite lengths the wave is distorted. To which I say get a life.

Bjørn

Fri Aug 24 2007, 11:12AM

Registered Member #27 Joined: Fri Feb 03 2006, 02:20AM
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A sine wave is the most natural way to go through a cycle, it requires the least energy and contains only one frequency component. Imagine a pendulum swinging back and forth.

Your square wave is a 60 Hz square wave but it also contains higher frequency components often in the MHz range. Imagine a pendulum swinging abruptly like that, you need to apply huge forces and you will get vibrations (ringing) that will radiate away energy and generate losses.

So many things tend to generate sinewaves naturally and designers will also make an effort to get perfect sinewaves since that will give the least amount of problems.

Simon

Sat Aug 25 2007, 01:29AM

Registered Member #32 Joined: Sat Feb 04 2006, 08:58AM
Location: Australia
Posts: 549

As Steve said, sine waves are inherent in anything rotating. Sine waves are even more pervasive than that.

Sine waves are produced whenever the force pulling something inwards is proportional to how far away the thing is. (E.g. a spring. The force pulling a spring back to equilibrium gets bigger at a constant rate as the spring gets pulled or squashed away from equilibrium.)

On top of that, whenever you have a smooth energy well in a system (like with a pendulum, where potential energy is lowest when the bob is in the middle and rises smoothly each side), you'll get oscillations that are approximately sinusoidal, at least when small.

A cylindrical buoy bobs sinusoidally in water. A spherical one approximately so.

Even non-sinusoidal things can be described using sine waves, thanks to Fourier Analysis. If you have some way of graphing functions, try plotting sin(x), then sin(x) + sin(3x)/3, then sin(x) + sin(3x)/3 + sin(5x)/5... Or else, have a look at the pictures here.

What that means is that a square wave is identical to whole bunch of sine waves put together. You can bring out each component with filters if you want.

Hazmatt_(The Underdog)

Sat Aug 25 2007, 03:00AM

Registered Member #135 Joined: Sat Feb 11 2006, 12:06AM
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Posts: 1735

I was going to post the convergence of sin into square waves but I wasn't sure if that was going off topic.

What is of interest of the sum of the Fourier sieries is that its only the ODD sine components that produce the square wave. The even components cancel, so we are left with odd harmonics.

In order to produce a nearly clean square wave you need ~ 49 iterations of the sinc function, you could go as far as 100 iterations but its not necessary.

Steve Conner

Sat Aug 25 2007, 10:53AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

Simon: I like to think that the pervasiveness of sine waves is because the cos and sin waves are each other's derivatives, so they make handy solutions to differential equations.

Better still, e^x is its own derivative and can transform into any combination of growing, decaying, or constant cos and sine waves you fancy.

Simon

Mon Aug 27 2007, 06:47AM

Registered Member #32 Joined: Sat Feb 04 2006, 08:58AM
Location: Australia
Posts: 549

Steve: you're quite right. That's the fundamental property of sine waves that's the mathematical basis of it all.

Hazmatt_(The Underdog) wrote ...

In order to produce a nearly clean square wave you need ~ 49 iterations of the sinc function, you could go as far as 100 iterations but its not necessary.

"Nearly" clean depends purely on your application. If your system has a bandwidth that only reaches up to 1kHz, there is no point using 8kHz harmonics in your analysis, even if you can tell the difference by eye.

More economical use of harmonics can be possible using weighted "windows" in terminating the series, rather than just jamming on the brakes at n=50. This stops ringing in the approximation, with tradeoffs in sharpness. Read up on the Gibbs Phenomemon if you're interested.

Dr. Slack

Mon Aug 27 2007, 11:40AM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

Seeing as a square wave requires an infinite bandwidth, you can never have a square wave in reality. Any approximation will be just that, with a rise-time related to the generator bandwidth. The quality of the approximation required depends on your application. Some approximations can be very fast indeed, with mercury-wetted relays or step recovery diodes, and the output edge of an ACMOS gate is pretty damned fast when viewed on a slow scope.

Hazmatt_(The Underdog)

Mon Aug 27 2007, 06:18PM

Registered Member #135 Joined: Sat Feb 11 2006, 12:06AM
Location: Anywhere is fine
Posts: 1735

Matlab project we did for DSP EE 386

The MatLab code:

%This program generates a sine approximation to a square wave
%Through Fourier Series Decomposition.

%Written by Matt Keyser

%T = Period
T = 1;

% T_0 = Period of one wave
T_0 = 2*T;

%f_0 = Frequency of the particular wave
f_0 = 1/T_0;

%t = time interval
t= -T:.01:3*T;

%--------------------------- Third Harmonic --------------%
%m = harmonic mode content
m = 49;

s=0;
for n=-m:2:m; %odds only
Cn = 2./(j*n*pi); %Coefficient for each sinusoid
s=Cn.*exp(j*2*pi*n*f_0*t)+s; %Generate the harmonic waveform
q=square(2*pi*f_0*t); %Generate the square waveform
end
%------------------------------------ ---------------------%

plot(t,s,'b',t,q,'r') ,title([' Harmoinc = ',num2str(m)]),xlabel('Time'),ylabel('Amplitude')

sine converges pretty well into a square wave even though its still ringing, but I think that's pretty darn close.

Simon

Tue Aug 28 2007, 01:14AM

Registered Member #32 Joined: Sat Feb 04 2006, 08:58AM
Location: Australia
Posts: 549

Hazmatt_(The Underdog) wrote ...

sine converges pretty well into a square wave even though its still ringing, but I think that's pretty darn close.

It will always ring ~10%, no matter how many harmonics you add, if you terminate the series suddenly. (Kind of like how a sound system is going to pop if you turn it off suddenly. You'll only stop this by fading out.)

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