In article ,
(John Byrns) wrote:

In article ,
"Frank Dresser" wrote:

"John Byrns" wrote in message
...

The whole analysis on this web page is too simplistic and is irrelevant to
the subject at hand.

He came up with actual numbers, which is more than most do. Anyway, I also
noticed that there was no mention of the actual voltages the detector was
being driven at.

As far as the square law stuff goes, Terman says a the distortion of a true
square law detector will be m/4. So 80% modulaton will result in 20%
distortion. He might have derived that number, I don't remember.

Yes, that's exactly what I thought, given the analysis methodology he
seemed to be using on the web page, the distortion seemed way too low to
me. You have inspired me to take a closer look and see exactly what he
did, and where he went wrong, or if I have just misinterpreted his
methodology. I will report back in a few days time.

OK, I have taken a closer look at the analysis on the web page at this URL:
http://www.amwindow.org/tech/htm/diodedistortion.htm
and it is more screwed up than I thought.

The analysis starts with the Shockley diode equation, and then the
exponential power series equivalent to the Shockley equation is stated as
equation #2. At this point the author "examines" the second power
component of the equation, not quite making it clear that is all he is
going to examine, and will base the entire analysis on only the second
power component of the diode characteristic. An equation for the output
of a square law diode is given as equation #4, which is derived by
squaring the equation representing a carrier AM modulated by a single
tone. Equation #4 actually represents the V/I characteristic of a square
law diode, and does not necessarily represent the output of such a diode,
but we will accept it as such for the purposes of this analysis.

After considerable mathematical manipulation and six more equations the
author comes to the final diode output in equation #9, and after low pass
filtering to eliminate the carrier and carrier terms in the output he
comes to equation #10 which represents the demodulated signal output from
the detector. The author's equation #10 is:

(10) m(t) = (m**2)/4 + m*cos wmt + (m**2)/8[cos 2wmt]

The authors derivation of equation #9 from equation #4 was too convoluted
for me to easily follow, so I did my own derivation which required only
two intermediate steps rather than the 5 steps the author required, my
result for equation #10 was:

(10) m(t) = 1/2 + (m**2)/4 + m*sin wmt - (m**2)/4[cos 2wmt]

Neglecting the sin in place of cos for the main modulation term, and the
sign on the second harmonic term, we notice that the author lost the DC
term somewhere, and his second harmonic term is half of mine with an 8 in
the denominator rather than the 4 I derived. These differences could be
due to errors in my derivation, which often happen on the first pass, but
considering that my distortion result, discussed next, is the same as
Terman's, it seems likely that 4 is the correct value for the denominator
of the second harmonic term. I plan to eventually try to plow through the
authors derivation of equation #9 to see where he made his errors.

The error in the denominator would only account for a factor of two in the
distortion percentage, but he compounds the error when he calculates
distortion as power ratio rather than a voltage ratio which I believe is
conventional.

Taking the ratio between the amplitude of the fundamental, m*cos wmt, and
the amplitude of the second harmonic, (m**2)/8[cos 2wmt] and squaring the
author comes up with his equation #12 for percent distortion:

(12) THD (%) = (((m**2)/8)**2)/(m**2)) * 100, or ((m**2)/64) *100

which yields 1% distortion at 80% modulation and 1.5625% distortion at
100% modulation.

My version of equation #12, based on the ratio of the fundamental, m*sin
wmt, and the amplitude of the second harmonic, (m**2)/4[cos 2wmt] becomes:

(12) THD (%) = ((m**2)/4)/m) * 100, or (m/4) *100

which yields 20% distortion at 80% modulation and 25% distortion at 100%
modulation. These results show distortion more than an order of magnitude
greater than the distortion figures calculated on the web page. Note that
my result is in agreement with Terman's result at 80% modulation, as
quoted above, which leads me to suspect that I didn't make any serious
mathematical errors in my derivation. The errors in the web page author's
analysis stem from two sources, first the amplitude of the second harmonic
term is too small by a factor of two due to an error or some sort in the
derivation of the equation. The second cause of the error is due to the
fact that the web page author expresses distortion as a power ration
rather than the conventional voltage ratio.

Now of course all this is for a perfect square law detector, which does
not apply to what we have been talking about, which is a peak envelope
detector. The peak envelope detector which is considerably more difficult
to analyze, which probably explains why the author of the web page didn't
even try, and some authorities have gone so far as to say the problem is
so complex that it is basically intractable to rigorous mathematical
analysis.

I hope I didn't make too many typos in this, please let me know if I did
so I can correct them when I post the results of my analysis of exactly
where the author's derivation went wrong.


Regards,

John Byrns


Surf my web pages at, http://users.rcn.com/jbyrns/