This is the "MAXNR.DOC" file distributed with Z-Track. No attempt has been made to beautify it for presentation on the web... it is almost exactly in its original form, which was intended for viewing on an 80x25 text screen or printing on one of the old "text only" printers that some of us were still using at the time. See also Polarization of EME Signals: How To Succeed More Often


                             MM Software Systems
                                 April, 1993

        I've been asked about the "Max NR" figure generated in Z-TRACK by
     several users of the program.  Most of the questions were just plain
     old  "Huh ???" , indicating that I really goofed.  Max NR, as far as
     I know,  had never been used in any  moon-tracking software prior to
     the first release of Z-TRACK.   I developed it as a new indicator to
     supplement the  "Spatial Polarization Offset"  so often seen in such
     software.   I firmly believe it is a much more meaningful and easily
     interpreted indicator of polarization conditions.   So, here goes my
     best shot at removing the shroud of mystery from the Max NR index !!
     I've had a go (or two, or...)  at explaining this  verbally, without
     the kind of success I was hoping for.  MAYBE I can do better writing
     it down (???).

                               THE PRE-BASICS

        I suspect few people will argue these days that the polarization
     of an  incoming  signal  with  respect  to the  polarization of the
     receiving  antenna  makes a tremendous  difference in  how well the
     signal can be heard.   A signal  perfectly  aligned in polarization
     with the  receiving  antenna system will suffer no loss of strength
     due to polarization mis-alignment.  A signal arriving perpendicular
     to the  polarization plane of the antenna  (90 degrees to it)  will
     suffer  the greatest loss - How much  depends on many  factors, but
     for a  typical modern yagi it is  generally taken as between 20 and
     30 dB.   In between these extremes  the loss varies logarithmically
     with  the  cosine of the  angle  (by angle I mean the difference in
     degrees  between the  polarization plane of the incoming signal and
     that of the antenna).  A formula for this is:

             Loss (dB) =  20 log(cos )

             where  = the angle in degrees.

     This formula assumes a  perfect yagi  with no  unwanted response to
     signals of the opposite polarization.  It is VERY close up to about
     the 18 or 20 dB point,  but  beyond  that needs a correction factor
     to keep the result down to a realistic level.

        That's all well and good  for two  antennas of like polarization
     pointing toward each other along the Earth's surface.   But what of
     two antennas of like  polarization  pointing  off into space?   The
     situation  becomes a little more complicated.   In my own crude way
     let me try to help you  visualize  what I mean.   Suppose you had a
     large round ball  sitting in  front of you.   If you took two samll
     model yagis and  planted one on the  very top of the ball  (be sure
     to align the  elements so they are  parallel to the  surface of the
     ball  directly below the antenna)  so that it is  pointing straight
     away  from you  toward a  distant  wall,  and the other on the left
     side of the ball (elements  parallel to the  surface  of the  ball)
     pointing to the same distant wall, You will have the  basis of this
     visualization.    Both  antennas are  horizontally  polarized  with
     respect to the surface of the ball (simulated Earth),  right?   But
     stand back and sight along the boom of each yagi, toward that wall.
     If you can  visualize a signal  leaving each antenna and travelling
     to the the wall (simulated moon),you will see that the polarization
     planes of the two signals  hitting the wall are actually 90 degrees
     out  of  sync with  each  other.   This  difference is what we call
     "Spatial" Polarization Offset!  Gee Whiz!

        The geometry involved on an  EME path is a little more  complex,
     what with elevation angles and so on.   The point of reference used
     for such  calculations is  generally the  Earth's polar axis, but a
     complete  discussion of the mechanics involved is beyond the intent
     of this discourse (and is a fantastic way to get a headache).  Here
     is the basic formula used to calculate a spatial polarizatoion  for
     an antenna with respect to the Earth's polar axis:

                      (sin L * cos E - cos L * cos A * sin E)
             P = ATN  ---------------------------------------
                                  (cos L * sin A)

             where L = Latitude of station           {footnote 1}
                   A = Azimuth of antenna
                   E = Elavation of antenna
                   P = Polarization angle

     The spatial polarization offset (read 'difference') between any two
     stations is simply P1 - P2, but keeping it within a range of -90 to
     +90 takes additional steps.

                          MOVING ON TO THE REAL WORLD
        For some time we have been accustomed to seeing a  "Polarization"
     or  "Spatial Polarization Offset"  figure in  moon-tracking programs
     that  track the moon  for two stations.   This figure is in degrees,
     with  some programs  keeping it within a range of -90 to +90 degrees
     while others allow it to approach 180.  Along with this, many of the
     programs have also tried to include a  "Polarization Loss" figure in
     dB, with 0 dB indicating a polarization offset of 0 (or 180) degrees
     and 25 dB or more indicating a polarization offset of 90 derees.

        It doesn't  take a rocket scientist to  figure out that this just
     cannot be the case.  We've all made some fine EME QSOs when software
     told us to  expect 20+ dB loss of signal based on polarization!   So
     what's the deal here?   The problem is that programs like that don't
     consider the affects of Faraday  rotation.   The fact is that due to
     Faraday rotation in the ionosphere signals can arrive at the receive
     antenna at ANY polarization, reguardless of the spatial polarization
     relationship  between the two stations.    Given that consideration,
     the next thing  we're  confronted  with is the  argument that ALL of
     this spatial polarization business should be  disreguarded since the
     unpredictable  Faraday is going to modify  the  polarization anyway.
     In other words we "make our skeds blindly and take our chances".

        BUT - and this is a B-I-G but - the ionosphere has another unique
     little  property that should make us re-think the situation.  Simply
     put, a signal  polarization which rotates in a given direction  when
     passing  through  the  ionosphere on its way to the moon will rotate
     the same amount in the same direction (from the observer's viewpoint)
     on the return  trip.   You can  visualize it this way:   Suppose you
     were standing  behind the reflector on your EME antenna and sighting
     down the  boom to the moon.   Further  suppose for a moment that you
     could see the  polarization plane of the signal leaving your antenna
     and  traveling to the  moon and back  again to the antenna.   If you
     sent  out a  burst of RF and  observed  that the  polarization plane
     shifted  45  degrees to the right  (clockwise) on the way out to the
     moon, your first  assumption  would probably be that the signal will
     rotate in the  opposite direction (from your perspective) on its way
     back, thus  "un-twisting" itself and arriving perfectly in line with
     the elements on your antenna.   But the ionosphere doesn't see it in
     that way.   In fact the  polarization  plane will rotate by the same
     amount in the same direction  (clockwise) from your vantage point on
     the  return trip - thus making it now  90 degrees  out of  alignment
     with your antenna.

        So what?   Consider this:  If you add a second  station whose QTH
     is far enough  away  from  yours so  that the  spatial  polarization
     offset between  the two  of you is 45 degrees, something interesting
     begins to take shape.   Suppose you  transmit and the Faraday causes
     the polarization plane of your signal to rotate 45 degrees clockwise
     by  the  time  it  reaches  the  other  station.   It started out at
     0 degrees (the spatial polarization of your antenna) and was twisted
     45 degrees clockwise, so it arrives at his antenna perfectly aligned
     with it (his spatial  polarization is 45  degrees greater than yours,
     don't forget).   Sounds  like the  Faraday is  doing us a  big favor,
     right?  WRONG!   Now suppose the  other  station  transmits a string
     of O's back to you  (having heard your signal very well by virtue of
     the perfect polarization alignment).   The polarization plane of the
     signal starts out at 45 degrees  (his spatial polarization) and then
     gets  rotated 45 degrees  clockwise  on its way to you,  making  the
     polarization of the signal  90 degrees in  "spatial" terms.   Yikes!
     Your  antenna is  still at 0 degrees "spatial" polarization, and his
     signal  is coming in at  90 degrees!   Unless you're a  BIG GUN, you
     just won't be  able to  hear him.   Good grief, one-way propagation!

        Fortunately,  there  is a way  to  figure out what times are most
     likely to yield  frustrating one-way  conditions in advance.   Going
     back to the example above...   If the spatial offset between the two
     stations were 0 deegrees,  then  Station A  transmitting to  station
     B would  result in a clockwise twisting of  45 degrees,  arriving 45
     degrees  out of alignment with his antenna.  On the return trip, the
     exact  same  relationship  holds.   If  the  Spatial  offset were 90
     degrees, then  station A  transmitting to  station B would result in
     a signal at 45 degrees  "spatial polarization"  being received by an
     antenna with a  polarization of 90 degrees,  for a  difference of 45
     degrees.   On the return trip,  90 degrees starting from his antenna
     gets twisted  clockwise to  135 degrees  (135 degrees is  equivalent
     to -45 degrees.   It  should  always be  kept in a  range of  +/- 90
     degrees by adding or  subtracting  180  degrees  as  necessary.  180
     degrees is  physically the same as 0 degrees), which is an offset of
     45 degrees with reguard to your antenna.   In  either of these cases
     the signal will be  mis-aligned by  the same  amount (45 degrees) at
     both stations.  There will be a resulting loss of 3 dB, but at least
     it's reciprocal (and 3 dB isn't as bad as 20+ dB).

        I could  (and probably should)  sit here  and go  through various
     possible combiantions of "spatial offset" and  Faraday  rotation for
     a L-O-N-G time.  I'm not going to, but I would  like to bring up one
     other point before going on.  If one works away at this problem long
     enough  (I've been  fretting  away at it for 5 years now),  one then
     begins  to think  in terms of "windows of opportunity".  Considering
     the original  45 degree spatial  offset again...  The  absolute best
     that can be achieved (under normal circumstances)  in this case is a
     Faraday of 0 degrees or 90 degrees.   This would  indeed result in a
     reciprocal path with 3 dB loss  of signal at both ends.  BUT, if the
     Faraday changes by only  15 degrees in either  direction, the signal
     misalingment  at one  station will become  30 degrees (1.24 dB loss)
     and at  the other station  60 degrees  (6 dB).   Notice  how quickly
     the signal  falls off at one end of the path for very small  changes
     in prevailing  Faraday - thus a  small  "window of opportunity"  for
     reciprocal  conditions,  what with  Faraday being such a  changeable
     entity.   On the other  hand, consider either of the perfect cases -
     that is  either  spatial 0, Faraday 0    or  spatial 90, Faraday 90.
     Under  these  circumstances the path is reciprocal with 0 dB loss at
     either end.   If the Faraday  changes 15 degrees in either direction
     the reciprocity will remain, and the signal will drop by only 0.3 dB
     at  both  stations.    Indeed,  the  Faraday  must  shift 45 degrees
     one  way or the other  before the  signal  drops to  -3 dB  and even
     then  it is still reciprocal.   Hence a greater "window of opportun-

        The bottom line is  that one-way  conditions are much more likely
     to occur when the  spatial  offset is in the  vicinity of 45 degrees
     and drops to  nearly 0 when  the  offset is 0 or 90  degrees.   Data
     collected over the past few years  clearly shows that  statistically
     the chances of  completing  an  EME QSO drop  off sharply  when  the
     spatial offset  approaches  45 degrees,  especially for the  smaller
     stations who just don't have the signal to spare. Many have  clearly
     benefitted in  the past few  years by careful consideration of these
     polarization issues.

                               A BETTER WAY ???
        Considering  the spatial  offset in degrees is all  right up to a
     point, but it can become a nuiscance.  I couldn't help thinking that
     what we really  wanted to  know was how  non-reciprocal a given path
     could get at a particular time.   The spatial offset information was
     nice, but I would find myself wanting to know more.  When working up
     sked times,  I often wondered  about the actual non-reciprocity that
     might come into play. Hmm, the offset is 38.2 degrees, the cosine of
     38.2 is ahhh...  and the log of that is ummm.....    Blast and drat!
     After a few false starts (not to mention sleepless nights) I came up
     with the "Max NR" figure.   This  gives a good  approximation of the
     non-reciprocity that is  likely to be  encountered  based on a given
     spatial offset.   The name  is somewhat  of a misnomer, since Max NR
     does not always  give the  absolute  maximum non-reciprocity that is
     possible for a given time.   It gives  a precise  maximum  only  for
     offsets of 0, 45,  or 90 degrees.   At other offsets it gives a very
     good  approximation  of  typical  non-reciprocity  values  that  are
     actually observed on EME.  Spatial offsets near 45 degrees will show
     the highest Max NR figures, as offsets in that range are the most
     likely to cause severe one-way conditions.

        One must bear in  mind that  this is  neither a  worst-case nor a
     best-case indication.  However, it is  a very good  indicator of the
     typical non-reciprocities seen on EME and correlates quite well with
     EME logs  analyzed  to  date.   What the  Max NR  does is  provide a
     convenient and user-friendly way of weeding out times that have been
     statistically  proven  to have the lowest chance for success.

                              A VERY GOOD "FIT"
        When I devised the new Max NR index, I needed  to see how well it
     would  fit with  observed EME  results,  compared to  other  methods.
     I analyzed several EME logs for percentage of QSOs vs spatial offset
     and  Max NR.   The  graphs  of  QSOs vs. spatial  offset  showed the
     anticipated  dip around 45 degrees (+/- a fair bit), but it was less
     than dramatic, all  things  considered.   I next plotted the QSOs vs.
     Max NR and  found an  amazing  correlation.   I was not prepared for
     the results.   All of the  logs checked showed 90 percent or more of
     the QSOs  occurred when the  Max NR  was  less than 3 dB!   It was a
     smooth  curve with a very sharp  drop for low but  increasing values
     of Max NR,  leveling out as it  approached 25 dB.  There was usually
     a small "anomaly" at  25 dB,  caused by the method used to round off
     (or limit)  Max NR  indices that  soar above  that point.   Remember
     my  previous  comment  about  the  equation  for  dB  loss needing a
     correction  factor above  18 or 20 dB?   I am very pleased  with the
     results obtained.

        Here is an attempt at reproducing the graphs in text form  (for a
     typical log):    [These  are  lacking in  resolution - it's tough to
     reproduce graphics on an 80x25 text screen!!!].

                          EME QSOs VS SPATIAL OFFSET

     #   100-|*
          90-|**  *
     O    80-|*****
     F    70-|******* **
          60-|*************   **                                 * *    *  **
     E    50-|********************                       *   ****************
     M    40-|**********************                 ************************
     E    30-|*********************** **           **************************
          20-|****************************   **  ****************************
     Q    10-|***************************************************************
     S     0-|***************************************************************
     O    -------------------------------------------------------------------
     s       0     10     20     30     40     50     60     70     80     90
                              SPATIAL OFFSET (DEGREES)

                              EME QSOs VS MAX NR

     #   1000-|*
     O    800-|**
     F    700-|**
     E    500-|****
     M    400-|*****                                       round-off anomaly
     E    300-|********                                                    
     Q    100-|**************                                              **
     S      0-|**************************************************************
     O    -------------------------------------------------------------------
     s        0    2    4    6    8   10   12   14   16   18   20   22   24
                                       Max NR (dB)

                            RECOMMENDATIONS FOR USE
        Having  introduced  the  concept  of  "Max NR" , here is a brief
     comment on what I have used it for.  When making skeds I start with
     the smallest stations I want to run with,  scheduling them when the
     Max NR is at its lowest (typically less than 1 dB).  Then I move up
     to progressively larger  stations,  accepting somewhat poorer times
     as the size of the station increases.   If you make  many skeds you
     will find that it is not  possible to schedule  them all at "ideal"
     times!   Generally speaking,  I try to keep the Max NR on all skeds
     under 3 dB.  I rarely suffer the "one-way blues" any more, although
     it does  happen on  rare  occasions.   (For every rule, there is an

        Spatial Offset (and hence Max NR) varies drastically from day to
     day as the  declination of the  moon  changes.   Hint: if you can't
     find a time with low Max NR on a given day, try a few days later or
     earlier.   Sometimes, you have to give up...  During the recent EME
     expedition to KC6, I ran a statistical analysis of the path from my
     QTH to KC6 and found that the  lowest  Max NR for the entire common
     moon window during that month was 12.6 dB!  Arghhh!!!

                    EXCEPTIONS - QSOs NEAR 25 dB MAX NR
        Any discussion on the  subject of  one-way EME conditions  would
     not be complete (or fair) without some comments about why some good
     QSOs are indeed possible at 45 degrees spatial offset (or 25 dB Max
     NR).   You will have noticed that while the graphs above show a dip
     in the  number of  EME QSOs in this  region,  it certainly does not
     drop to zero.  There are at least 3 reasons for this.

        Perhaps the most obvious exception is when the two stations have
     enough power and combined antenna gain to overcome very substantial
     losses.  It's certainly possible to give up many dB and  still work
     some of the super-stations.

        As mentioned before, a situation exists when the  spatial offset
     is 45 degrees and the  Faraday is 0 or 90 degrees (and very stable)
     causing reciprocal conditions but 3 dB loss of  signal at both ends
     of the QSO.   Sometimes  the  Faraday is  indeed  stable  enough to
     permit complete QSOs in this case  (assuming both stations can deal
     with the 3 dB loss).

        The third exception is that it is possible for the  polarization
     of a signal  passing  through the  ionosphere to become diffused or
     elliptical.  In this case weak signals may be heard at a variety of
     receive-antenna  polarizations,  but not  without some sacrifice in
     signal strength.  Depending on the size of the two stations and the
     exact nature of the  polarization  diffusion,  some QSOs are indeed
     possible  in this  case  reguardless  of the  spatial  polarization
     offset.  From observations made over a period of time it seems that
     such  conditions  occur  more  frequently  at times  of high  solar

        Again, the whole point is that it is less likely to complete EME
     QSOs  when  the  spatial offset  is near 45 degrees  (and Max NR is
     high) than when the offset is  closer to 0 or 90 degrees  (in which
     case Max NR is low).    Therefore, a careful  consideration  of the
     polarization issue when scheduling EME activities can significantly
     improve the chances for success (by weeding out times that have the
     lowest  chance  for  success),  but  will  never  offer an absolute
     guarantee of anything.  While working on the Max NR concept for use
     in Z-TRACK, many controlled tests were run with stations at various
     spatial polarization offsets.  It was found that QSOs could be made
     more frequently  and with less effort  when  the  offset  was  near
     either end  of the scale  than when  it was  approaching 45 degrees.
     Some QSOs were made in the  worst-case 45 degree  condition, but in
     almost every case involved  significantly  weaker signals at one or
     both ends of the QSO than when the offset was more favorable.

     1.  This equation was developed by C.H Hustig and was taken from an
         article "Spatial  Polarization  and  Faraday  Rotation"  by Tim
         Pettis, KL7WE.    This  article  appeared in PROCEEDINGS of the
         22nd CONFERENCE of the CSVHF SOCIETY (Lincoln, NE  1988).

[....just a fancy line....]

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