Fourier of 50 Hz 100% AM of 20 KHz (PWM) ?

Guest
Hi,
an ex-colleage [who doesn't know how to use NewsGroups - he
won't even learn how to disable html on his emails !] writes:-

"The 50 Hz is chopped by the 20kHz.
The expected freq. spectrum of 20kHz +/- 50Hz;
perhaps 40kHz +/- 50Hz; 60kHz +/- 50Hz; ...
The fifth series for up to the 5th harmonic has to be established."

And his diagram shows 50 Hz 100% AM of 20 KHz.
Where the 20 KHz. is also pulse width modulated.
It's some kind of power supply.

I beleve the question is 'invalid' since the answer can only be
given for a specific PWM/duty-cycle.

Q1: what is the fourier series for 20 KHz 50% duty cycle
( T_positive=T_zero=T_negative) 100% amplitude modulated by 50Hz ?
Reduced to voltage (or power) ratios wrt. the fundamental, for the
first 5 greatest terms.
-----------

His lowest duty cycle shows 1microsec on (positive or negative) and
24 microSec off. Ie. 4% 'duty cycle'.

Q2: what is the fourier series for this case ?

Thanks for any answers,

== Chris Glur.
 
not@top-post wrote:

Hi,
an ex-colleage [who doesn't know how to use NewsGroups - he
won't even learn how to disable html on his emails !] writes:-

"The 50 Hz is chopped by the 20kHz.
The expected freq. spectrum of 20kHz +/- 50Hz;
perhaps 40kHz +/- 50Hz; 60kHz +/- 50Hz; ...
The fifth series for up to the 5th harmonic has to be established."

And his diagram shows 50 Hz 100% AM of 20 KHz.
Where the 20 KHz. is also pulse width modulated.
It's some kind of power supply.

I beleve the question is 'invalid' since the answer can only be
given for a specific PWM/duty-cycle.

Q1: what is the fourier series for 20 KHz 50% duty cycle
( T_positive=T_zero=T_negative) 100% amplitude modulated by 50Hz ?
Reduced to voltage (or power) ratios wrt. the fundamental, for the
first 5 greatest terms.
-----------

His lowest duty cycle shows 1microsec on (positive or negative) and
24 microSec off. Ie. 4% 'duty cycle'.

Q2: what is the fourier series for this case ?

Thanks for any answers,

== Chris Glur.
Click to expand...
The 20 KHz, 50% duty-cycle waveform is a square wave, with a DC term
added to make it never negative. If the peak is V volts, then its
unmodulated spectrum is

2V [ sin(3wt) sin(5wt) sin(7wt) sin(9wt) ]
-- * [1 + sin(wt) + -------- + -------- + -------- + -------- + ... ]
pi [ 3 5 7 9 ]


Each of those components will be modulated by 50 Hz.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
 
The spectrum of a PWM signal is complicated, because PWM is kind of like a
frequency or phase modulation.
The spectrum depends on exactly how the width of the pulse is determined
from the modulation waveform.
A width exactly proportional to the amplitude of the samples of the
modulation contains quite a bit of harmonic distortion. In this case, 50
Hz, as well as 100 Hz, 150 Hz, and so on will appear as sidebands on the odd
harmonics of the 20 Khz carrier. Sophisticated DSP is required to fix this
problem.
For more information, search with "PWM" and "Natural Sampling" on Google.

Regards,
Cliff Curry
 
"The 50 Hz is chopped by the 20kHz.
The expected freq. spectrum of 20kHz +/- 50Hz;
perhaps 40kHz +/- 50Hz; 60kHz +/- 50Hz; ...
The fifth series for up to the 5th harmonic has to be established."

And his diagram shows 50 Hz 100% AM of 20 KHz.
Where the 20 KHz. is also pulse width modulated.
It's some kind of power supply.

I beleve the question is 'invalid' since the answer can only be
given for a specific PWM/duty-cycle.

Q1: what is the fourier series for 20 KHz 50% duty cycle
( T_positive=T_zero=T_negative) 100% amplitude modulated by 50Hz ?
Reduced to voltage (or power) ratios wrt. the fundamental, for the
first 5 greatest terms.
-----------

His lowest duty cycle shows 1microsec on (positive or negative) and
24 microSec off. Ie. 4% 'duty cycle'.

Q2: what is the fourier series for this case ?
Click to expand...
Jerry wrote:-

The 20 KHz, 50% duty-cycle waveform is a square wave, with a DC term
added to make it never negative. If the peak is V volts, then its
unmodulated spectrum is

2V [ sin(3wt) sin(5wt) sin(7wt) sin(9wt) ]
-- * [1 + sin(wt) + -------- + -------- + -------- + -------- + ... ]
pi [ 3 5 7 9 ]


Each of those components will be modulated by 50 Hz.
Click to expand...
OK, but lets try to evolve/refine this to answer the QUESTIONS.
First ignore the complicating fact of PWM.

For a 50% duty-cycle the 'ratios' of 20 KHz and harmonics are: 1, 1/3, 1/5 ....
What are the ratios of the greatest powered harmonics if the 20 KHz,
square wave is 100% AM-ed at 50 Hz ?

Thanks for any answers,

== Chris Glur
 
Your Question needs more, clarification. What type of PWM is it? assume
random data? how many bits long?
The 50 and 20 kHz if independent form the PWM is easily solvable.


<not@top-post> wrote in message news:37KdndnJeJXJdjvdRVn-hg@is.co.za...
"The 50 Hz is chopped by the 20kHz.
The expected freq. spectrum of 20kHz +/- 50Hz;
perhaps 40kHz +/- 50Hz; 60kHz +/- 50Hz; ...
The fifth series for up to the 5th harmonic has to be established."

And his diagram shows 50 Hz 100% AM of 20 KHz.
Where the 20 KHz. is also pulse width modulated.
It's some kind of power supply.

I beleve the question is 'invalid' since the answer can only be
given for a specific PWM/duty-cycle.

Q1: what is the fourier series for 20 KHz 50% duty cycle
( T_positive=T_zero=T_negative) 100% amplitude modulated by 50Hz ?
Reduced to voltage (or power) ratios wrt. the fundamental, for the
first 5 greatest terms.
-----------

His lowest duty cycle shows 1microsec on (positive or negative) and
24 microSec off. Ie. 4% 'duty cycle'.

Q2: what is the fourier series for this case ?

Jerry wrote:-

The 20 KHz, 50% duty-cycle waveform is a square wave, with a DC term
added to make it never negative. If the peak is V volts, then its
unmodulated spectrum is

2V [ sin(3wt) sin(5wt) sin(7wt) sin(9wt) ]
-- * [1 + sin(wt) + -------- + -------- + -------- + -------- + ... ]
pi [ 3 5 7 9 ]


Each of those components will be modulated by 50 Hz.

OK, but lets try to evolve/refine this to answer the QUESTIONS.
First ignore the complicating fact of PWM.

For a 50% duty-cycle the 'ratios' of 20 KHz and harmonics are: 1, 1/3, 1/5
.....
What are the ratios of the greatest powered harmonics if the 20 KHz,
square wave is 100% AM-ed at 50 Hz ?

Thanks for any answers,

== Chris Glur
Click to expand...
 
not@top-post wrote:

(snip)


For a 50% duty-cycle the 'ratios' of 20 KHz and harmonics are: 1, 1/3, 1/5 ....
What are the ratios of the greatest powered harmonics if the 20 KHz,
square wave is 100% AM-ed at 50 Hz ?
Click to expand...
I think superposition and/or separability works here.

The 20kHz square wave is the sum of its harmonics at the
appropriate amplitude. That, along with either the associative,
commutative, or distributive laws should give you the answer.

-- glen
 
not@top-post wrote:

(snip)

For a 50% duty-cycle the 'ratios' of 20 KHz and harmonics are: 1, 1/3, 1/5 ....
What are the ratios of the greatest powered harmonics if the 20 KHz,
square wave is 100% AM-ed at 50 Hz ?

glen herrmannsfeldt <gah@ugcs.caltech.edu> wrote:
I think superposition and/or separability works here.

The 20kHz square wave is the sum of its harmonics at the
appropriate amplitude. That, along with either the associative,
commutative, or distributive laws should give you the answer.

Perhaps I'm wrong, but:
Click to expand...
* superposition assumes linearity [addition], whereas,
* modulation is based on non-linearity [multiplication] ?

== Chris Glur.
 
<snip>

?
Click to expand...
Yes it is multiplication, and original post is OK if no PWM.

50 Hz sin wave chopped by 20 kHz square wave will give multiples of 20 kHz
spectral lines each with two sidebands +/-
50 Hz away. With DC term reflecting the average DC content of the 20 kHz
chop. If not 50 Hz sin wave but 50 Hz square wave, then you have more
harmonic terms of multiples of 50 Hz about each of the muliptles of the 20
kHz spectral lines.

If any PWM at all, the spectrum is not the same as the 20 kHz spectral
lines can go away, depending upon type/frequency of PWM. Components do not
add up in phase anymore.
 
not@top-post wrote:

...

Jerry wrote:-


The 20 KHz, 50% duty-cycle waveform is a square wave, with a DC term
added to make it never negative. If the peak is V volts, then its
unmodulated spectrum is

2V [ sin(3wt) sin(5wt) sin(7wt) sin(9wt) ]
-- * [1 + sin(wt) + -------- + -------- + -------- + -------- + ... ]
pi [ 3 5 7 9 ]


Each of those components will be modulated by 50 Hz.


OK, but lets try to evolve/refine this to answer the QUESTIONS.
First ignore the complicating fact of PWM.

For a 50% duty-cycle the 'ratios' of 20 KHz and harmonics are: 1, 1/3, 1/5 ....
What are the ratios of the greatest powered harmonics if the 20 KHz,
square wave is 100% AM-ed at 50 Hz ?

Thanks for any answers,

== Chris Glur
Click to expand...
The spectrum due to chopping DC at any duty cycle is calculable and
available in tables. (It's in "ITT Reference Data for Radio Engineers",
but I won't type it out in Adobe Courier New.) The 50% case was intended
as a model. Higher harmonics diminish in amplitude vs. frequency most
rapidly with that case. Very small and very large duty cycles approach a
train if delta functions. In those limiting cases, all harmonic
amplitudes are the same.

Whatever the spectrum due to the chopping, if the 20 KHz is 100% AM
modulated, then each of the component frequencies will be 100% modulated
as well. Each term will be multiplied by (1 + sin(2*pi*50*t), leaving
each of them intact (the additive 1 in the expression) and adding half-
amplitude sidebands 50 Hz away. I thought that was, ahem, obvious.

Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
 
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