Truncated match.
PICList
Thread
'PIC as controller for sinewave inverter  magic si'
1997\02\03@071245
by
Mike

Hi gals and guys,
I'm looking (again) at feasability of using HC11 architecture as controller
for sinewave inverters 48V DC in for 240V AC out 50Hz and 5000VA.
Now after being exposed to PICs  I'm considering using a PIC with built
in A to D...
Although I have most experience with 8051 architecture, it seems that the
HC11 with OTP would be a more viable solution in terms of usability
of programming model and level of integration  thats an option but, can
anyone care to suggest a suitable PIC for such a task ?
There has been some discussion in various hobby magazines about 'magic
sinewaves' in 384 bit sequences  anybody know about these or the
background maths etc  I was going to look at Taylor's series etc to
produce the HC11 table constants but there might be a more efficient way ?
I'm interested in any +ve/ve comments/observations on this...
Rgds
Mike
spam_OUTerazmusTakeThisOuTwantree.com.au
There is no a'priori reason that the ultimate truth will be interesting
or even useful, those moments of frustration during philosophical debate
would be replaced by the sheer terror which accompanies true knowledge.
1997\02\03@191707
by
Dave Mullenix

Mike wrote:
>There has been some discussion in various hobby magazines about 'magic
>sinewaves' in 384 bit sequences  anybody know about these or the
>background maths etc  I was going to look at Taylor's series etc to
>produce the HC11 table constants but there might be a more efficient way ?
>
>I'm interested in any +ve/ve comments/observations on this...
Don Lancaster has written about "magic sinewaves". Don has a homepage at
http://www.tinaja.com/
Enter the Reference Library, then the Magic Sinewaves Shelf and find this blurb:
Don Lancaster's
Magic Sinewave Library
"Magic sinewaves" are repeating long sequences of ones and
zeros. Digital sinewaves with precisely controlled amplitudes
and amazingly low distortions.
Compared to traditional PWM, magic sinewaves can offer far
higher efficiencies and lower distortions. With circuitry that is
elegantly simple and microcontroller friendly. For use in such
applications as induction motor speed controls, electric autos,
solar panels, power factor correction, inverters, home energy
efficiency improvers, and specialized utility services.
There are more magic sinewaves than there are particles in the
universe. Only recently have specialized analysis tools become
available. Don Lancaster, Synergetics, and several Synergetics
Partners are in the forefront of this billion dollar opportunity.
Extensive seminars, consulting, sourcecode, tutorials, and a new
Partners program are now being offered.
These files are available for downloading:
Magic Sinewave Development Proposal.
Formal proposal outlining seminars, partners, and services.
A wide range of opportuinities and possibilities are listed,
along with full references.
Blatant Opportunist #35 MAGSINE.PDF (*)
Magic sinewaves are a brand new method of driving ac induction
motors, solar panels, inverters, or electric cars. Offer ing outstanding
simplicity and efficiency. Introductory Don Lancaster tutorial.
Blatant Opportunist #37 EMERGOP4.PDF (*)
Fourth annual release of emerging technical opportunities.
Push me pull you. The TeraHertz mystery band. PIC microcontollers.
Magic Sinewaves. Solitons. DNA computing. Binary chain codes.
Bionomial Goes Binary ONESWORD.HTML *
A PostScript utility to generate all possible binary words of length
"n" having "k" ones in them. Use for magic sinewaves and exploration
of the binomial theorm. Requires two way recordable com.
An analysis of Magic Sinewave Higher Harmonics.
Just where do those square corners come from on a low harmonic
waveform? Jim Fitzsimons analyzes the first 999 (!) harmonics
of five typical 384 bit nondelta magic sinewaves.
Demo sampler of 288Bit Magic Sinewaves.
A detailed analysis pf a few of the 288bit "delta Friendly"
magic sinewaves. There are 49 amplitudes in the complete
288 sequence, having distortions as low as 0.01 percent.
Tech Musings #101 MUSE101.PDF ( June 1996 )
Using DeltaWye transforms, Professional audio resources,
Acrobat amber beta software, Building your own Tesla coil,
Three phase magic sinewaves, more.
Tech Musings #93 MUSE93.PDF ( October 1995 )*
More on magic sinewaves, stepper motor driver chips, yet
another patent horror story, mid range Mench computer, a
collection of PCMCIA plugin card resources.
Tech Musings #90 MUSE90.PDF ( July 1995)*
Fourier Series analysis, alternatives to plated through holes,
classic computer resources, a $200 GPS receiver, Newtek's
new video toaster for Windows.
Hardware Hacker #87 HACK87.PDF (April 1995 )*
AC motor drive fundamentals. New Basic Stamps from Parallax.
PIC Microcomputer resources. Additional details on forced zero
magic sinewaves. Dealing with "too good to be true" results.
Hardware Hacker #86 HACK86.PDF (March 1995 )*
The DNA computer language. Some wavelet book resources.
Magic digital sinewave codes. Engineering economics review.
Hot new fringe FM RBDS tuner.
1997\02\03@194411
by
Steve Hardy

{Quote hidden}> From: Mike <
.....erazmusKILLspam@spam@WANTREE.COM.AU>
> Hi gals and guys,
>
> I'm looking (again) at feasability of using HC11 architecture as controller
> for sinewave inverters 48V DC in for 240V AC out 50Hz and 5000VA.
>
> Now after being exposed to PICs  I'm considering using a PIC with built
> in A to D...
>
> Although I have most experience with 8051 architecture, it seems that the
> HC11 with OTP would be a more viable solution in terms of usability
> of programming model and level of integration  thats an option but, can
> anyone care to suggest a suitable PIC for such a task ?
>
> There has been some discussion in various hobby magazines about 'magic
> sinewaves' in 384 bit sequences  anybody know about these or the
> background maths etc  I was going to look at Taylor's series etc to
> produce the HC11 table constants but there might be a more efficient way ?
>
> I'm interested in any +ve/ve comments/observations on this...
Socalled 'magic sinewaves' seem to be a pet subject of a bloke
called Don Lancaster. The actual mathematics to generate the
sequences is not in the public domain; however there are a few
published sequences of length 384 and above  look at his web
page. I'm not sure about this, but I think it is possible to
patent algorithms in the US. However, you can't in Australia so
if you can discover his algorithm for yourself there would be
nothing to stop you commercially develping products in Oz.
Basically, there are an infinite number of PWM sequences which will
generate 'sine waves' with a greater or lesser harmonic content.
The aim of 'magic sinewaves' is to minimise (or even zero) the first
n harmonics, such that the first nonzero harmonics will be at a
very high frequency and thus easy to filter using an LC lowpass
power filter. The complete cycle (20ms) is divided into fixed
time slots (20ms/384). Each slot is then assigned an output voltage
of +V, 0 or V. Naturally, because of the 4fold symmetry of a sine
wave only one quadrant of the sine wave needs to have table entries.
The power of the fundamental (which is what you want) is proportional
to the total number of +V slots (= the number of V).
The mathematical technique is based on Fourier analysis (not
Taylor series, AFAIK).
If you need to regulate the output, you can presumably narrow the
utilisation of the +/V slots. Thus you would aim for 50% duty cycle
(of the slot) for noload output, allowing this to rise to 100%
for full load.
As it so happens, I am currently designing a 6KW inverter, except
that I am using rectified 3phase as the power source (giving about
580V avg). The DC (with 4.2% ripple) is switched by a full bridge
of MOSFETs using optoisolated control. Control will be provided
by a 16C74  primarily because I have a few of these floating around.
The '74 will be used to monitor voltage and current (current being
crudely measured from the voltage drop across the MOSFETs Rds(on)).
It will also determine frequency of operation and PWM sequencing.
Might also add temperature sense.
One of the most difficult things about controlling 600 odd volts at
50KHz is the tremendous dV/dt (those FETs switch fast!) which will
couple through the tiniest capacitance to potentially disrupt ones
delicate control circuitry. I am having to build up my own
optoisolators and isolated power supplies. Total capacitance has to
be limited to 0.2pF, otherwise currents of over 1mA will be induced
(6000V/us).
I would be interested in your overall converter topology. Please
correspond privately since this is wildly offtopic.
Regards,
SJH
Canberra, Australia
1997\02\04@111945
by
Brian Boles

Before you go for that HC11, check out the 17C756 that will be
sampling soon. It will "out motor" a HC11!
Rgds, Brian.
______________________________ Reply Separator _________________________________
Subject: PIC as controller for sinewave inverter  magic sinewaves
Author: Mike <erazmusKILLspamWANTREE.COM.AU> at Internet_Exchange
Date: 2/3/97 8:10 PM
Hi gals and guys,
I'm looking (again) at feasability of using HC11 architecture as controller
for sinewave inverters 48V DC in for 240V AC out 50Hz and 5000VA.
Now after being exposed to PICs  I'm considering using a PIC with built
in A to D...
Although I have most experience with 8051 architecture, it seems that the
HC11 with OTP would be a more viable solution in terms of usability
of programming model and level of integration  thats an option but, can
anyone care to suggest a suitable PIC for such a task ?
There has been some discussion in various hobby magazines about 'magic
sinewaves' in 384 bit sequences  anybody know about these or the
background maths etc  I was going to look at Taylor's series etc to
produce the HC11 table constants but there might be a more efficient way ?
I'm interested in any +ve/ve comments/observations on this...
Rgds
Mike
.....erazmusKILLspam.....wantree.com.au
There is no a'priori reason that the ultimate truth will be interesting
or even useful, those moments of frustration during philosophical debate
would be replaced by the sheer terror which accompanies true knowledge.
1997\02\04@113613
by
Mike
At 09:04 AM 4/02/97 0700, you wrote:
> Before you go for that HC11, check out the 17C756 that will be
> sampling soon. It will "out motor" a HC11!
>
> Rgds, Brian.
Sounds OK, I checked it out on microchip's site, nice chip.
Might have a bit more horsepower than I need, can I get it for A$9 each
in tube lots and the assembler for free ?
Rgds
Mike
1997\02\05@205411
by
sdattalo
1997\02\09@131012
by
Sten Dahlgren
For all of you interested in PIC's and power inverter, check out there web
site
for the RD001 paper, "Uninterupptible Power Supply Reference Design".

Sten Dahlgren CelsiusTech Systems ! "I'd rather have 39 Hp under my arm
S175 88 Jaerfaella Sweden ! than one under my backside"
EraseMEsedaspam_OUTTakeThisOuTcelsiustech.se +468 58084430 ! join your nearest karting club now !!
1997\02\09@151012
by
peter
Sten Dahlgren wrote:
>
> For all of you interested in PIC's and power inverter, check out there web
> site
> for the RD001 paper, "Uninterupptible Power Supply Reference Design".
> 
> Sten Dahlgren CelsiusTech Systems ! "I'd rather have 39 Hp under my arm
> S175 88 Jaerfaella Sweden ! than one under my backside"
> sedaspam_OUTcelsiustech.se +468 58084430 ! join your nearest karting club now !!
Which web site are you refering to ??

Peter Cousens
email: @spam@peterKILLspamcousens.her.forthnet.gr
snailmail: Peter Cousens, karteros, Heraklion, Crete, 75100, Greece,
phone: + 3081 380534, +3081 324450 voice/fax
After Bill Gates announced to the world that he was Microsoft,
his wife was asked to comment. She said that as his wife, she
had been the first to notice this problem
1997\02\10@000947
by
Mike
At 07:09 PM 9/02/97 +0100, you wrote:
>For all of you interested in PIC's and power inverter, check out there web
>site
>for the RD001 paper, "Uninterupptible Power Supply Reference Design".
>
>Sten Dahlgren CelsiusTech Systems ! "I'd rather have 39 Hp under my arm
>S175 88 Jaerfaella Sweden ! than one under my backside"
>KILLspamsedaKILLspamcelsiustech.se +468 58084430 ! join your nearest karting club now !!
Which web site are you referring to  here ?
Rgds
Mike
1997\02\11@121200
by
Zack Cilliers
Hi there!
Can some one send me some more information as how the sinewave can improve
on a squarewave.
thanks

{Quote hidden}> From: Scott Dattalo <
RemoveMEsdattaloTakeThisOuTunix.sri.com>
> To: Multiple recipients of list PICLIST <
spamBeGonePICLISTspamBeGoneMITVMA.MIT.EDU>
> Subject: Re: PIC as controller for sinewave inverter  magic sinewaves
> Date: Thursday, February 06, 1997 3:51 AM
>
> Mike wrote:
> >
> > There has been some discussion in various hobby magazines about 'magic
> > sinewaves' in 384 bit sequences  anybody know about these or the
> > background maths etc  I was going to look at Taylor's series etc to
> > produce the HC11 table constants but there might be a more efficient
way ?
{Quote hidden}
1997\02\12@180550
by
sdattalo

Zack Cilliers wrote:
>
> Hi there!
>
> Can some one send me some more information as how the sinewave can improve
> on a squarewave.
>
> thanks
Zack,
Are you asking how the sinewave can be generated from the square wave?
Well, if you aren't then I'm answering the wrong question...
A couple of weeks ago John Payson and I were discussing some details
on how it is possible to suppress the 2nd and 3rd harmonics in a
sampled square wave. This led me on an analytical rabbit chase that
turns out to be the inverse of the so called "magic sine wave".
Skipping over the details, let me just pull a formula out of
my magic hat. Suppose you have a very narrow duty cycle pulse train:
^
 tau
 <>

 ++ ++
    
++ ++ +>
t
phi
<>
< T >
In other words, if you have a pulse stream that has
a frequency of 1/T, pulse widthes of tau, and an
initial phase of phi, then one (of an infinite many)
series expansion is:
f(t) = d +
inf

\ sin(n*pi*d) / n*2*pi tau \
2* /  * cos  *( t    phi) 
 n*pi \ T 2 /
n=1
where,
d = tau/T is the duty cycle
pi = 3.141592653...
t = time
For 50% duty cycles, you may recall that there are no
even harmonics present in the fourier series. So as one
simple check, you can substitute d=0.5 and see that the
sin(n*pi/2) kills the even harmonics in this parameterized
expansion.
The goal of the magic sinewaves is to suppress many if not
all of the harmonics beyond the fundamental. This can be
accomplished by chopping the period into many fine pieces;
in other words, let tau<<T. Mike made a reference to there
being 384 divisions. Now if you strategically place these
narrow pulses among the 384 possible slots, it's possible
to suppress higher harmonics.
The technique of finding the optimum placement is far
from trivial. However, here are few helpful observations.
The fundamental frequency is 1/T.
If the number of subdivisions is N (i.e. tau = T/N) then
1) Harmonics that evenly divide N/2 can be totally suppressed.
2) If the n'th harmonic is suppressed then the Nn'th is
also suppressed.
3) To suppress the 2nd through the n'th harmonic at least
2^(n1) pulses are needed.
As an example, if you wanted to suppress the 2'nd, 3'rd and
4'th harmonics then the number of needed subdivisions is
N = 2*3*4 = 24
Out of the 24 possible pulses, 2^(41) = 8 pulses are needed.
(And if I had my notes, I would tell you where these optimum
pulse positions are located). In addition to suppressing
these harmonics, the 20'th, 21'st and 22'nd are also
suppressed. The fifth harmonic is not completely suppressed.
There is also a 1/n factor that will attenuate those harmonics
that are not suppressed. However for small n, the sin(n*pi*d)/n*pi
is approximately n*pi*d/(n*pi) = d. So the 1/n factor only
becomes important for larger values of n.
There are many, many more details that I'm ignoring...
Scott
1997\02\14@131742
by
sdattalo

This is my second time to send this to the list. The first time
apparently didn't make it. (At least it was not in this month's
log.)
Zack Cilliers wrote:
>
> Hi there!
>
> Can some one send me some more information as how the sinewave can improve
> on a squarewave.
I believe the context of your question is WRT power inverters? In which
case the sine wave drive is lethal to the switching devices. The so
called "magic sine waves" is a glitzy synonym for pulse width
modulation.
(O.k. there's a little more to it than PWM, but the underlying theory is
identical.) And PWM is the proper technique for driving many these
devices.
A couple of weeks ago John Payson and I were discussing some details
on how it is possible to suppress the 2nd and 3rd harmonics in a
sampled square wave. This led me on an analytical rabbit chase that
turns out to be the inverse of the so called "magic sine wave".
Skipping over the details, let me just pull a formula out of
my magic hat. Suppose you have a very narrow duty cycle pulse train:
^
 tau
 <>

 ++ ++
    
++ ++ +>
t
phi
<>
< T >
In other words, if you have a pulse stream that has
a frequency of 1/T, pulse widthes of tau, and an
initial phase of phi, then one (of an infinite many)
series expansion is:
f(t) = d +
inf

\ sin(n*pi*d) / n*2*pi tau \
2* /  * cos  *( t    phi) 
 n*pi \ T 2 /
n=1
where,
d = tau/T is the duty cycle
pi = 3.141592653...
t = time
For 50% duty cycles, you may recall that there are no
even harmonics present in the fourier series. So as one
simple check, you can substitute d=0.5 and see that the
sin(n*pi/2) kills the even harmonics in this parameterized
expansion.
The goal of the magic sinewaves is to suppress many
of the harmonics beyond the fundamental. This can be
accomplished by chopping the period into many fine pieces;
in other words, let tau<<T. Mike made a reference to there
being 384 divisions. Now if you strategically place these
narrow pulses among the 384 possible slots, it's possible
to suppress higher harmonics.
The technique of finding the optimum placement is far
from trivial. However, here are few helpful observations.
The fundamental frequency is 1/T.
If the number of subdivisions is N (i.e. tau = T/N) then
1) Harmonics that evenly divide N/2 can be totally suppressed.
2) If the n'th harmonic is suppressed then the Nn'th is
also suppressed.
3) To suppress the 2nd through the n'th harmonic at least
2^(n1) pulses are needed.
4) It is possible to suppress all of the even harmonics.
This is accomplished by an pulse train that is followed
by it inverse. For example, if you had a traing of pulses
like 011000 the even harmonics are suppressed by the
doubled chain: 011000100111.
To illustrate, suppose you wanted to suppress the 2'nd and
the 3'rd harmonics. The number of needed subdivisions is
N/2 = 2*3 = 6
or N = 12
Out of the 12 possible pulses, 2^(31) = 4 pulses are needed.
And their locations are:
101101000000
(actually any cyclic rotation of this stream works).
In addition to suppressing the 2'nd and 3'rd harmonics, the
10'th and 11'th are also suppressed. The fourth harmonic is
not completely suppressed. If you wanted to suppress the
4'th or all of the even harmonics for that matter, then
the following pulse stream would do the trick:
original
<>
101101000000010010111111
<>
inverted from the original
There is also a 1/n factor that will attenuate those harmonics
that are not suppressed. However for small n, the sin(n*pi*d)/n*pi
is approximately n*pi*d/(n*pi) = d. So the 1/n factor only
becomes important for larger values of n.
There are many, many more details that I'm ignoring...
Scott
1997\02\15@160058
by
Sten Dahlgren

> From: Peter Cousens <TakeThisOuTpeterEraseMEspam_OUTcousens.her.forthnet.gr>
> To: RemoveMEPICLISTTakeThisOuTMITVMA.MIT.EDU
> Subject: Re: PIC as controller for sinewave inverter  magic sinewaves
> Date: den 10 februari 1997 07:09
>
> Sten Dahlgren wrote:
> >
> > For all of you interested in PIC's and power inverter, check out there
web
> > site
> > for the RD001 paper, "Uninterupptible Power Supply Reference Design".
> > 
> > Sten Dahlgren CelsiusTech Systems ! "I'd rather have 39 Hp under my
arm
> > S175 88 Jaerfaella Sweden ! than one under my
backside"
> > sedaEraseME.....celsiustech.se +468 58084430 ! join your nearest karting club
now !!
>
> Which web site are you refering to ??
Sorry, I was a little bit faster than i should be. It is http://www.microchip.com
Note also that it is already in it's b version as i discovered thursday.
1997\02\15@191005
by
Dmitry Kiryashov

Scott Dattalo wrote:
[some tech details skipped]
{Quote hidden}>
> The goal of the magic sinewaves is to suppress many
> of the harmonics beyond the fundamental. This can be
> accomplished by chopping the period into many fine pieces;
> in other words, let tau<<T. Mike made a reference to there
> being 384 divisions. Now if you strategically place these
> narrow pulses among the 384 possible slots, it's possible
> to suppress higher harmonics.
>
> The technique of finding the optimum placement is far
> from trivial. However, here are few helpful observations.
>
> The fundamental frequency is 1/T.
>
> If the number of subdivisions is N (i.e. tau = T/N) then
> 1) Harmonics that evenly divide N/2 can be totally suppressed.
> 2) If the n'th harmonic is suppressed then the Nn'th is
> also suppressed.
> 3) To suppress the 2nd through the n'th harmonic at least
> 2^(n1) pulses are needed.
> 4) It is possible to suppress all of the even harmonics.
> This is accomplished by an pulse train that is followed
> by it inverse. For example, if you had a traing of pulses
> like 011000 the even harmonics are suppressed by the
> doubled chain: 011000100111.
>
> To illustrate, suppose you wanted to suppress the 2'nd and
> the 3'rd harmonics. The number of needed subdivisions is
> N/2 = 2*3 = 6
> or N = 12
> Out of the 12 possible pulses, 2^(31) = 4 pulses are needed.
> And their locations are:
>
> 101101000000
> (actually any cyclic rotation of this stream works).
>
> In addition to suppressing the 2'nd and 3'rd harmonics, the
> 10'th and 11'th are also suppressed. The fourth harmonic is
> not completely suppressed. If you wanted to suppress the
> 4'th or all of the even harmonics for that matter, then
> the following pulse stream would do the trick:
>
> original
> <>
> 101101000000010010111111
> <>
> inverted from the original
>
> There is also a 1/n factor that will attenuate those harmonics
> that are not suppressed. However for small n, the sin(n*pi*d)/n*pi
> is approximately n*pi*d/(n*pi) = d. So the 1/n factor only
> becomes important for larger values of n.
>
> There are many, many more details that I'm ignoring...
> Scott
While i've read this i've thinking  "May be exist reverse technique ?"
For example we have an input digitized signal ( only logical "1" and "0"
sequence).
And we want to detect some fundamental frequencies in this stream while
harmonics
should not have influence on result of detection. Are there some way to
apply
that magic sinewaves technique reversly i.e. not for synthesis but for
detection ?
I think there are ability in use simultaneosly magic waves and
correlation techniques
to achieve cutting harmonics in each correlation sum (in DTMF detection
for example).
I hope that i have not speak some nonsense ;)
And second question  what happen in case when we take into account N =
odd value,
i.e. not 384 but for example 385 or 383 pieces ?
WBR Dmitry.
1997\02\17@145839
by
sdattalo

Dmitry Kiryashov wrote:
>
>
> While i've read this i've thinking  "May be exist reverse technique ?"
> For example we have an input digitized signal ( only logical "1" and "0"
> sequence).
> And we want to detect some fundamental frequencies in this stream while
> harmonics
> should not have influence on result of detection. Are there some way to
> apply
> that magic sinewaves technique reversly i.e. not for synthesis but for
> detection ?
Yes. But unfortunately there are some practical problems...
> And second question  what happen in case when we take into account N =
> odd value,
> i.e. not 384 but for example 385 or 383 pieces ?
Well for starters, it's not theoretically possible to suppress all of
the even harmonics if N is an odd value (recall N is the number of
subdivisions of the fundamental frequency; ie. if the fundamental
frequency is f=1/T, then the time resolution for pulse transitions
are limited to the time quanta T/N.) Pushing theory aside however,
it is possible to come fairly close to suppressing the even harmonics.
And with a little magic, it's possible to suppress a large subset
of the lower harmonics.
Think about how you would generate a pulse stream that could be
lowpass filtered to generate a sine wave. Intuitively, you might
expect to have some sort of frequency modulated square wave. When the
the frequency is "low" the filter has a large output and when the
frequency is "high" the filter has a small output. Your intuition
may further lead you to this equation:
g(t) = (1 + sign(A1*cos(2*pi*f1*t) + A2*cos(2*pi*f2*t)) )/2
This dual tone signal (which for the context of our current
discussion has nothing to do with DTMF) consist of two harmonically
related sinusoids:
f2 = m*f1
f1 is the frequency of the fundamental waveform that we wish
to synthesize (e.g. DTMF generation) or detect (DTMF decoding).
f2 is a much high frequency that can be easily filtered with
our low pass filter.
The sign() function returns the sign of its arguement. Loosely
speaking, it's the mathematical function of a comparator. The
rest of the equation for g(t) shifts the DC offset so we have
a stream of 1's and 0's.
There's a simple way to see what this function is doing. Imagine
that the lower frequency sinusoid's amplitude, A1, is twice the higher
one. When the low frequency cosine is passing through zero, there
are many zero crossings in the higher frequency cosine. For every
zero crossing, there is a corresponding pulse transition. Now as
the lower frequency cosine approaches it's peak, the higher frequency
cosine at some point will no longer be able to cause the sum (of
the sinuoids) to change signs.
Here's another way to see what's happening. If the amplitudes A1
and A2 are equal then the sum can be expressed as a product:
sum = A1*cos(2*pi*f1*t) + A2*cos(2*pi*f2*t)
= A1*(cos(2*pi*f1*t) + cos(2*pi*f2*t))
= 2 * A1 * cos(pi*(f1+f2)*t) * cos(pi*(f1f2)*t))
Which is seen to be one cosine wave being modulated by another.
Now to address Dmitry's question. So far we've made no assumption
about the sampling rate. But it's sort of obvious that the faster
we can generate the pulse stream the easier it will be to filter
the high frequency components. If it's not obvious, then please
take my word for it.
I haven't investigated the case for there being an odd number of
samples per T1 (=1/f1). I can only speculate that it might induce
a subharmonic oscillation. There are two reasons. First, the phases
of the cosine waves could be such that every other cycle differs
by one extra pulse. The frequency of this extra pulse is 1/2 the
frequency of the fundamental (f1). Thus the introduction of the
subharmonic. Secondly, if you take the odd number of samples and
multiply it by two you get an even number of samples that is
twice as long. Using the same theory as above, it's then possible
to generate a sine wave of half the frequency of f1. And as we
all know from O.J., if it's remotely possible then we can assume
it as true.
One more point. Think about what happens when the relative amplitudes
of the cosines are varied. For example looking at just the extremes:
if A1>>A2 then g(t) is a low frequency square wave and the low pass
filter's output will have a large output.
if A1<<A2 then g(t) is a high frequency square wave and the low pass
filter's output will have a small output.
The actual output will be somewhere between these two.
One more point. Amplitude control of the synthesized sine wave can
be traded off for harmonic supression.
Scott
More... (looser matching)
 Last day of these posts
 In 1997
, 1998 only
 Today
 New search...