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Re: [Amps] PEP power..

To: "Gary Schafer" <garyschafer@comcast.net>,"'Tom W8JI'" <w8ji@w8ji.com>,"'Ian White GM3SEK'" <gm3sek@ifwtech.co.uk>, <amps@contesting.com>
Subject: Re: [Amps] PEP power..
From: Bill Fuqua <wlfuqu00@uky.edu>
Date: Mon, 16 Oct 2006 16:24:41 -0400
List-post: <mailto:amps@contesting.com>
Every couple of years I have to drag out the following example to 
illustrate what RMS means and send it to AMPS. It think it is a good 
example and goes to show that you can't just arbitrarily call average power 
RMS power just because so many people do so. It is like calling all 
amplifiers "linears" even if they are not linear at all but class C 
amplifiers. Technical correctness is much different than political 
correctness.

Just like taking the average of a number of data points. The discrete 
(non-calculus) version of RMS is a definite procedure but just a little 
more complicated. Below are examples of both calculating RMS voltage and 
average power by using a 8 data points in a sine wave. And the applying the 
RMS "proceedure" to power you will see that you get a erroneous result that 
is meaningless.  My example only uses 8 data points which works well for 
sine waves but for arbitrary waveforms (ones that contain many frequencies) 
you must use many more data points or an infinite number in some cases. 
This is a non-calculus example after all.

R   take square Root
         of
M   the Mean (average)
         of
S    the Squares of the values of all the points

  It may look complicated  as an equation or an integral expression (calculus)
but it is SIMPLE!!!!

For example...


Take one cycle of a 1 volt sine wave and divide into at least 8 equally 
spaced points starting at zero degrees.
Calculate instantaneous voltage, voltage squared and power into 50 ohms 
load at each point.
Do not use 360 degrees because it is the same point as 0 degrees. You may 
do this for any number
of full cycles and you may want to do this on a spread sheet  for each 
degree (ie 360 points) or what ever.

Phase     Voltage      Voltage Squared      Power into 50 ohms
                                                              Watts
0                  0                0                       0
45              .7071           .5                      .01
90                1                1                       .02
135             .7071           .5                      .01
180                0               0                       0
225             -.7071          .5                      .01
270              -1               1                       .02
335             -.7071          .5                      .01

totals             0                4                      .08
average of totals               .5                     .01watts Average Power
(divided by 8)
Square Root of average   .7071 volts RMS


The RMS value of the signal  is .7071 times the peak (1 Volt)= .7071 
volts  GEE look at above bold and underlined.

What is the power of this signal into a 50 ohm load?
.7071 volts squared divided by 50 ohms=  .01 watts   Look again  above at 
the average power bold.

It is all very consistent with calculations using RMS voltage or 
Current  and resistance and power.

Now what happens when we take RMS of the Power??


Phase   Voltage         Voltage Squared      Power into 50 ohms      Power 
Squared
                                                               Watts
0                 0                 0                       0 
                    0
45              .7071           .5                      .01 
               .0001
90               1                 1                      .02 
                  .0004
135             .7071           .5                     .01 
               .0001
180             0                  0                      0 
                   0
225             -.7071          .5                     .01 
               .0001
270             -1                 1                     .02 
                 .0004
335             -.7071          .5                     .01 
               .0001
totals           0                 4                     .08 
                  .0012
average of totals             .5                      .01 Average 
power           .0015
Square Root of 
average   .7071                                                     0.01225 
  RMS power ?????

OK above is your RMS power  for  what ever good it is....
And it does not relate to any of the other numbers you may use such as 
voltage, current.  resistance or power. But does give you 22.5% bigger 
number in this example.
73
Bill wa4lav
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