The Fresnel Zone explained

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  • Опубліковано 6 вер 2024
  • What is the Fresnel Zone (or zones) and what do they mean for those who fly RC planes, drones and racing mini-quadcopters, especially when using FPV systems? In this video I attempt to explain the science and theory behind this interesting phenomenon without going too heavily into the math.

КОМЕНТАРІ • 272

  • @alaanoor3679
    @alaanoor3679 Рік тому +1

    its never too late ....
    working in radios for 20 years , and only after watching this video i understood what the heck is fresnel zones
    thank you man !!!

  • @ElmerFuddGun
    @ElmerFuddGun 8 років тому +31

    Bruce... nice explanation and video but my only critiques would be that it is "Hz" not "hz" and capital "G" and "M" in GHz or MHz. Lower case "m" is for millihertz or 1/1000 of a Hertz. And a full cycle would be 360° so the peak high point of your wave is at 90°, back at zero is 180°, negative peak is at 270°, and then back to zero at 360°. Enjoyed the video, keep it up!

  • @bigshotdadz
    @bigshotdadz 8 років тому +1

    The d1d2/d1+d2 portion of your equation looks eerily familiar... High school Physics class, Cornwall College circa 1985. Parabolic curves... And we all thought this stuff would NEVER, EVER, EVER be useful in real life!!
    Thanks for the flashback Bruce!!
    Now, if only I can remember where exactly I put the house keys!!

  • @sandfordstebbings125
    @sandfordstebbings125 Рік тому +1

    Amazing, I sat in a masters level course and listened to the professor talk technical nonsense for 2 days on this, I didn't have a clue what he was talking about. Now I know... Thank you - amazing video.

  • @gjalovec
    @gjalovec 4 роки тому +1

    I've decided I only want middle-aged Australian men to give educational lectures. This guy and EEV have such a pleasant tone and cadence that I feel like I could learn anything from these guys.

  • @daemn42
    @daemn42 8 років тому +5

    Bruce, you've got the effects of your odd/even zones backwards.
    The F1 (innermost) zone ellipse is specifically calculated to account for 180 degree phase reversal due to the reflection + another 180 degree reversal due to increased path length. Reflections near F1 zone ellipse produce a strengthened signal.
    The F2 zone ellipse is includes the same 180 degree phase reversal due to the reflection, pus 360 degree phase from path length, which means the signal arrives 180 out of phase, and weakens the primary signal. F3 is like F1 but path length produces 540 degree phase shift, and so forth. Reflections on odd numbered F zones strengthen the signal, and even numbered zones weaken the signal.
    Also, all the zones should be drawn so that they extend in a roughly parabolic curve behind both the Tx and Rx. F zones are all about reflections, and the signal bouncing off an object nearby behind either antenna is just as important as one in front of it. People often fly with their back to a structure or their vehicle which violates the F zones. This is a situation where a directional antenna can help by heavily attenuating reflected signals from behind.

  • @navigator902
    @navigator902 5 років тому

    I will miss your instruction ... until times and circumstances change, I will stay subscribed, I don't let teachers go so easily..... and your instruction has taught me much.... Thank you Bruce.

  • @robertogalvez3865
    @robertogalvez3865 5 років тому

    Thanks a billion for explain how the F zones work. This video helps me undertand how OTA television antennas are affected by their surroundings. It is not just LOS what you need.

  • @wb8ujb
    @wb8ujb 8 років тому +7

    Thank YOU! Man, I love your teaching techniques. That cleared up a lot of questions I have had about this subject. Great video.

  • @world_traveller61
    @world_traveller61 3 роки тому +1

    Greetings from Japan, I was checking some of pathloss tool, Canadian tool, videos. Then, yours was recommended for me today. Nice crash course for Fresnel zones. You reminded me of a Swedish instructor from Ericsson.

  • @Rcschim
    @Rcschim 8 років тому +2

    thanks for that lesson Bruce! very understandable and aligning with the standard rule of thumb that you should fly at 10% alt of your distance (100m @ 1km...)Also one could move his groundstation on a "lamp holder" tripod thingie instead of a normal tripod. I got one of these and it extends easily to 3m of height. should give a great experience to fly miniquads low and far with the RX so high... greez, Mario

    • @daemn42
      @daemn42 8 років тому +2

      +RCSchim The 1/10 rule isn't based on any real violation of the Fresnel zones. It's just something someone randomly made up. I and many others regularly fly out at 1/100 or lower with no visible multipathing.

    • @daemn42
      @daemn42 8 років тому

      +RCSchim For example. Check out vimeo.com/27963240
      I started the return run only about 4m up, 3200m out with clear video. That's a 1:800 ratio. Most of the railroad track deep within the F1 zone, but no multipathing issues.
      What's more important than the height to distance ratio, is the distance from terrain of the Tx and Rx antennas themselves. Most of the reflections of interest occur near the endpoints, not in the middle. Also, I'm sure you've problem seen this in your own cliff diving flights, flying in front of a rock face with a specific angle can produce horrible multipathing, even when it's not inside the F1-F2 zones. I tend to visualize large rock slabs as big Rf mirrors. I've gotten mulipathing off them from 300m above.

  • @markusritter3636
    @markusritter3636 8 років тому

    So going to the flying field with a 10ft folding ladder has now 2 advantages! 1. you can get your plane easier out of the trees, and 2. when sitting on the ladder you have better FPV reception.
    Good Video Bruce, always looking forward for the next. Keep them coming....
    It.s cold here in Canada, so building time

  • @TheKetsa
    @TheKetsa 8 років тому +10

    As a French native speaker I can tell you : Silent S on Fresnel.
    Great video and awesome chanel btw, thanks.

  • @sethralphs2815
    @sethralphs2815 8 років тому

    Good to see this type of education out there. As a ham it's disappointing to see so many people ignore this entire side of the tech that people use on a day to day basis. Keep it up 😃!

  • @jimmac
    @jimmac 8 років тому +3

    Thanks for the simplified outline. I'd never make it through an accurate lecture :)

  • @fulano31231
    @fulano31231 7 років тому +1

    Just one thing, you should have added that this is for vertical polarized antenna. For horizontal polarization it will be the opposite, the 1st zone is constructive intereference and the 2nd destructive.

    • @mikeellicott6977
      @mikeellicott6977 6 років тому

      What does this mean for average joe with his cloverleaf type omnidirectional fpv antenna? and helical receiver antenna, does it behave different again ?

  • @mckeznak
    @mckeznak 7 років тому +2

    Man I could listen to bearded Picard teach me about technology all day! This guy is great.

  • @RandomChaos851
    @RandomChaos851 8 років тому

    For me, you kept the knowledge flow just out of zone 1 :). Not too much at once, but enough to understand the basics of it! You sir, are a good teacher. Thanks for this video.

  • @jayrobinson9016
    @jayrobinson9016 8 років тому

    being an engineer in training, i am very glad you are giving the math. thank you! i do love it somewhat simple as i am new to fpv and drones. hopefully i will be able to start to afford these things and experiment on my own after my degree. keep up the videos, i love them.

  • @allthingsrandom8137
    @allthingsrandom8137 8 років тому

    This actually explained a lot about the slight issues with my 1.3 ghz fpv. Very minor but I wondered by it was in and out with clarity. Thank you!

  • @GNARGNARHEAD
    @GNARGNARHEAD 8 років тому +1

    I come for the information, I stay for the sound effects.

  • @RobWhittlestone
    @RobWhittlestone 8 років тому

    Hi Bruce - you saved me making a video about the F zone. I have handheld European "PMR" UHF walkie-talkies (446 MHz) and did a range test video with different locations. The one surprising result was good reception with 500mW over about 5km just because nothing was in the F zone (across a plain from hill to hill). All other attempts had the ground and buildings/trees in the way. I first had to take the F zone into account when engineering microwave links some years ago. I love your channel, keep up the good work. All the best, Rob

  • @ssam7195
    @ssam7195 8 років тому

    Nice and quick understanding of fresnel zone..

  • @blackmennewstyle
    @blackmennewstyle 8 років тому

    Since mister Augustin-Jean Fresnel was a french folk, you should pronounce his family name [ Fray-nel ]
    Anyway, thanks for this beautiful and very efficient explanation about this phenomenon!
    Happy chinews new year :)

  • @tabeshforoughi1380
    @tabeshforoughi1380 3 роки тому

    A Real Great Teacher, Made For It.

  • @dhenzgofigure6795
    @dhenzgofigure6795 3 роки тому +1

    Please correct your waveform...thank you. One complete cycle equals 360 degrees if I am not mistaken. Anyway...loved your video.

  • @F2Dcombat
    @F2Dcombat 8 років тому +3

    Nice video, thanks :)
    BTW, Tape some cheap magnets on your whiteboard pens, then they can stick to the whiteboard ;)

  • @Vass881234
    @Vass881234 8 років тому

    are we to assume the signal strength is what we are talking about and not the speed the inut signal from the Tx travels down the wave?

  • @akujiwar
    @akujiwar 8 років тому

    Interesting. But how does the f zone differ from other multipathing? Also, while lower frequencies got a larger f zone, arent lower frequenzies less prone to multipathing as penetration is better? On top of this, how does circular polarization affect this?
    And to add another twist. Say you got a vertical obstruction like a spiky mountain entering the f zone. There wouldnt be much, or any surface at all to reflect the signal. What would happen then?

  • @lasersbee
    @lasersbee 8 років тому

    Good one Bruce... It explains in simple laymen's terms the Snow (F) Zones when flying FPV.

  • @michaeldemetriou1399
    @michaeldemetriou1399 8 років тому

    Thank you for all the very good videos you are a brilliant teacher and extremely funny the way you bounce off of Ron. He is a good sport and you are a scholar and a gentleman.

  • @hankus253
    @hankus253 8 років тому

    Hey Bruce you and I have the same problem but I know that 1 cycle is a positive excursion followed by a negative excursion... or 360 degrees. Peaks are 90 and 270 with a mid point of 180 degrees. But more important I had never heard of Fresnel outside of the lens type, so you taught me something again sir. Thank you!

  • @toddhowitt54
    @toddhowitt54 8 років тому

    Thanks Bruce, there is no snot running out of my nose because of confusion. So placing my Rx on a pole will help .

  • @richgixxer6
    @richgixxer6 8 років тому

    I got a little confused at the start of the video but as it went on I understood what you were explaining really enjoyed that well explained 👍🏻

  • @mattwilson5185
    @mattwilson5185 8 років тому

    Is this why on some clear nights you can pick up radio stations from other citys? (zone 2 bouncing off atmosphere )?

  • @oriannarangel7455
    @oriannarangel7455 4 роки тому

    Hi, I have a question, why don't we try to increase the frequency rather than decreasing it? this way "r" would be smaller, as well as the chances to lose the signal

  • @cbluebeard
    @cbluebeard 8 років тому

    Is there a general rule of thumb for distance/altitude? 2.4ghz control and 5.8 video, for example?

  • @warptmanbmx
    @warptmanbmx 8 років тому

    Thank you needing this info and love how it was not boring and made since a+ video

  • @thefpvlife7785
    @thefpvlife7785 8 років тому

    How does water, flying over lakes or Oceans, change the reflection of the signals? I fly much further and remain low with 1.2ghz over water.

  • @JD_1
    @JD_1 8 років тому

    Great illustration Bruce

  • @rwogh
    @rwogh 6 років тому

    This seems to be the calculation for F1 zone. Does the formula for F2, F3, ... simply change the constant in the formula?

  • @chrisclay9203
    @chrisclay9203 8 років тому

    How does this affect circularly polarized antennas? Reflections change the direction of polarization from what I understand, so would the signals not combine?

  • @jamesceastep8874
    @jamesceastep8874 8 років тому

    Bruce:
    A very animated presentation=good attention & better retention. grade=4+.
    I wonder how circular polarization affects F zone behavior on signals.....
    Jim

  • @bryanfpv
    @bryanfpv 8 років тому

    Thanks alot Bruce... will sleep smarter tonight. Your an excellent teacher ;)

  • @AlanBathurst
    @AlanBathurst 8 років тому

    excellent video.......didn't hurt my brain at all....

  • @haliseful
    @haliseful Рік тому

    🤣I LOVE your energy! Thanks for the fun and informative video :)

  • @jameslamb4573
    @jameslamb4573 8 років тому

    How does FM v AM effect the Fresnel zones, same goes for inversion layers or are the frequencies too high to bother considering them? What, too, about the signals that bounce off Gordon's head, possibly not only a direct phase shift but an orthogonal shift to boot?
    On a completely different topic, what's happened to Roddy's flying meat pack?

  • @chuckcrunch1
    @chuckcrunch1 8 років тому

    very cool. works like phase cancelling with audio waves

    • @xjet
      @xjet 8 років тому +1

      +chuckcrunch1 Yep, pretty much exactly the same.

  • @miguelh.m.m.6911
    @miguelh.m.m.6911 8 років тому +2

    So.... if the ground is not plane, or if it has trees or buildings, Will it change the direction or the timing of the wave ?

    • @xjet
      @xjet 8 років тому

      +MIGUEL H. M.M. Any reflective object that intersects the Fresnel zone will affect the signal so yes, trees, buildings, cars and such will have an effect. It's also worth remembering that the Fresnel zone is 3D -- not just up and down but sideways so even flying past a tree or building that intrudes into the side of the Fresnel zone can have an effect on your signal.

    • @ElmerFuddGun
      @ElmerFuddGun 8 років тому

      +MIGUEL H. M.M. - you bet. Everything can have an effect and it depends on how well they each reflect (or refract) the signal. Metal objects are going to have more of an effect than the dry ground for example. If you are flying in downtown NY you are likely to have more affects on your signal than if you are flying even over a larger distance out in a rural area.

    • @malelonewolf80
      @malelonewolf80 8 років тому

      +xjet my head hurt from trying to picture it in my head how a 3D fresnel zone would look in an enviroment with obstacles. But at least it made sense from a 2D perspective. 😀

  • @DutchRC
    @DutchRC 8 років тому +1

    This left with more questions then were answered BUT that's how it works huh? :)
    Very interesting & well explained in relative layman's terms :) Thanks!

  • @Djhg2000
    @Djhg2000 8 років тому

    But according to the sketch the phase offset also decreases (since the phase offset is a function of sin^2) as you go towards the ground. Wouldn't that theoretically mean you could go down to a height that exactly matches the amplitude of your signal to minimize the offset?

  • @vladmirputin7139
    @vladmirputin7139 8 років тому +6

    Is it .433 MHz or .433 GHz? That's gonna make a big difference.

    • @xjet
      @xjet 8 років тому +4

      +Vladmir Putin Yeah... .I meant 0.433GHz. I hate it when that happens :-)

    • @MrDanoman812
      @MrDanoman812 8 років тому

      +xjet
      Lol...

    • @jameslamb4573
      @jameslamb4573 8 років тому

      +xjet Bit like hz instead of Hz. Nice though that Vladimir (son of Vladimir) Putin is paying close attention to your channel Bruce, next thing you know Barack (son of Barack) Obama will chime in with a comment.

  • @TheTrippicm
    @TheTrippicm 8 років тому

    Thxs for this!Gunna make the team watch this!!!

  • @sum1else83
    @sum1else83 4 роки тому

    Thanks for these videos, very informative, i made my first cloverleaf and have a 5 turn helicoil in the plans..my fpv has never looked soo good with as much range.

  • @filipnordgren5134
    @filipnordgren5134 8 років тому +12

    Thanks Bruce for all the awesome videos you have posted the latest years!! Newer leavingyour videos without hitting the like button! =)

  • @drewhodge3820
    @drewhodge3820 8 років тому

    Cheers for putting in the time and effort into this video and all the other fantastic videos.
    Plus hello from England.

  • @TonyRule
    @TonyRule 6 років тому

    Good informative video.
    A couple of things though - and bear in mind I'm no RF guru by any stretch of the imagination - but surely as the reflected signal isn't 100% strength (in the case of a directional antenna such as a wireless network bridge the 'stray signals' don't even start at 100% of transmission power) so it wouldn't actually cancel out the direct signal wave - it'll just reduce its strength (amplitude).
    Also, the phase shift wouldn't necessarily be 180° for all reflected signals - it would depend on the total distance of the reflected path and the frequency. In any case, realistically, who even flys 3.6m off the ground at 1km or 5.0m off the ground 2km away anyway?

  • @bageldrone
    @bageldrone 8 років тому

    So with this info what is the height needed for a ground station for low flying miniquads on 5.8Ghz?

  • @allant9432
    @allant9432 8 років тому

    thanks Bruce,very informative, I never thought about the ground affecting the signal, I sit close to the ground

  • @wmdayman
    @wmdayman 8 років тому

    Fantastic. Now I understand it. Thanks.

  • @vipero00
    @vipero00 8 років тому +1

    I have to say Bruce that this lecture was better than 90% of EE professors I've seen. Go Purdue BSEE, MSEE 1983.

    • @xjet
      @xjet 8 років тому +1

      +vipero00 LOL... I guess I've sat through enough crap lectures myself that I at least know what to avoid :-)

  • @TheDrunkenMug
    @TheDrunkenMug 8 років тому

    another verry informative and FUN video Bruce, thanks so much !! :)

  • @behnamjamali284
    @behnamjamali284 8 років тому

    Very Instructive video. I love it.

  • @vsyncci
    @vsyncci 8 років тому

    the markings for the wave phases are actually shifted 180 degrees to the right...

  • @kadaj2k6
    @kadaj2k6 8 років тому

    u are an excellent teacher

  • @billbixby4033
    @billbixby4033 8 років тому +2

    Another informative video Bruce. You were saying about Fresnel zone 2 (at aprox 9:15) that because this signal is in phase but one wave behind the straight line signal and you had a net gain in signal strength. How is this late arriving signal used by the Rx if the picture information it is carrying is "late" when compared to the straight line signal? It seems the zone 2 signal would be trying to show images that have already been broadcast by the earlier arrival of the straight line signal. Hope this question makes sense.

    • @xjet
      @xjet 8 років тому +6

      +Bill Bixby No, because the frequency of the radio wave is so very high in comparison to the frequency of the video or control signal it is carrying, consecutive waves are all but identical. The change (created by the modulated information) is so very gradual that it takes thousands of waves to carry just a single bit of information.

    • @BillRoberts5
      @BillRoberts5 8 років тому

      Wow, so that is why lower frequency waves are bad for transmitting lots of information, right?

    • @xjet
      @xjet 8 років тому +2

      +Bill Roberts Yep, a higher frequency carrier can generally carry more data than a lower-frequency one.

    • @Nightsd01
      @Nightsd01 7 років тому

      Bill Roberts if you think of it, a higher frequency wave can fit more data (1's and 0's).

  • @thebridgeninja
    @thebridgeninja 8 років тому

    This is the best way to learn science!

  • @nikhilchalla6658
    @nikhilchalla6658 2 роки тому

    Thank you very much for the videos. Would be great if you could enable the subtitle settings for this video for me to turn on CC.
    Also, have a few questions...
    1. Assuming the ground surface structure is such that it just enters zone1 but not a lot, and the cumulative ground surface area in zone2 is greater than zone1. I assume the cumulative effect is signal amplification?
    2. Does the impact of subsequent zones reduce (zone3

  • @madmarty7103
    @madmarty7103 8 років тому +1

    great video. What about the different antenna types? I read that helical antennas reduce mulitpathing?

  • @julianwaters3946
    @julianwaters3946 8 років тому

    So is zone 2 communication why meteor scatter is so effective?

  • @navigator902
    @navigator902 5 років тому

    Brilliant... Now I understand F zone, and my fuzziness...and now its raining ....

  • @thexpto51
    @thexpto51 8 років тому

    Has always another great video! I liked it a lot! I think and I have said it earlier you could do some videos explaining what ever you want! You are a very good speaker! Good Work!

  • @mickthebass6086
    @mickthebass6086 8 років тому

    Explained very well. Thank you Bruce.

  • @Ruben-fk7zz
    @Ruben-fk7zz 7 років тому

    Thanks for the video,
    I have a one question for understanding:
    Does the multipating within the FZone, free of objects, come from dust particles in the air, that distribute the rf power over the space of the rotated ellipsoid?
    Would there also be a 1.2. and nth fresnel zone in a vacuum?
    I am wondering why the rf power would not just take the shortest way from transmitter to receiver.
    Please tell me your thoughts.

  • @agentbertram4769
    @agentbertram4769 6 років тому

    Very interesting and informative. Thanks.

  • @colcol83ful
    @colcol83ful 8 років тому

    Thanks Bruce, this was extremely enlightening.

  • @kev.j.goodwin8612
    @kev.j.goodwin8612 8 років тому

    Many thanks for the explanation Bruce, certainly food for further study. Just wondering what level of attenuation might be expected to the indirect signal?. Presumably reflecting surfaces are not perfect, so to what degree might different materials have an effect - e.g wet or dry grass, concrete, tarmac etc and how might this affect RX performance?

  • @djsomers100
    @djsomers100 8 років тому

    Outstanding. Time Gentlemen Please!

  • @paraschauhan1214
    @paraschauhan1214 7 років тому

    yeah we like the voices lol... you teach way better than the existing teachers tbh.....pleas keep on making vdos I love physics and liked your style of teaching but please add some more syllabus or something like that...😜 thank you

  • @iflylilplanes
    @iflylilplanes 8 років тому

    Bruce, I recently had a brown out with my FrSky gear, first time ever in 2 years of use. Reconnection was fast, but, the aircraft was close to the ground when the signal was lost, hit the only rock close to the field, it will be a long rebuild. Does 2.4 GHz suffer the same Fresnel effect? and, is this effect a possible cause of the brown outs suffered by DSM2 gear?
    Keep up the info vids, great job. Thanks

  • @kip383
    @kip383 8 років тому

    Hi Bruce. Can I use a 2.4 cloverleaf on my tx an the stock antenna on the rx. Will I have more range?

  • @mattkerle81
    @mattkerle81 8 років тому

    thanks Bruce, I really appreciate your educational videos, i always learn something!

  • @rvmorgan47
    @rvmorgan47 7 років тому

    I used to have to know this for the microwave comms I was using to figure out how high we had to be. Whoever did your freq. study and approval would always tell you the fresnel zones for freq. and distance.

  • @snaprollinpitts
    @snaprollinpitts 7 років тому

    great lesson Bruce, thanks couldn't we use a cable from the goggles to a pole with the antenna on top?

  • @377Pro
    @377Pro 8 років тому

    Could you also explain how to calculate the 2nd Fresnel zone?

  • @SaccoBelmonte
    @SaccoBelmonte 3 роки тому

    lol, I like the funny bits :D Now we have two fun geeky Aussies to watch.

  • @Piggish450
    @Piggish450 8 років тому

    man I love it. you are my favorite teacher and your lighting is very good too with your video. but I do have one question from it is deflecting on water different then grass? I figured it will still cancelling zone.

  • @ME-zu5kg
    @ME-zu5kg 8 років тому

    Well a master class. Clean clear and thank you for it.

  • @beatnik1855
    @beatnik1855 8 років тому

    so is this a non issue if you are using circular polarized antenna. with these antenna the rotation of the signal changes when reflecting off something making the reflected wave very difficult to receive by matched circular polarized receiver antenna

    • @billybbob18
      @billybbob18 6 років тому

      Conrad Brisebois
      True, but only if both antennas have an exceptional axial ratio. The lumieer AXii is known for this.

  • @nguyentruonggiang9654
    @nguyentruonggiang9654 4 роки тому

    Thank you. I have never known it before watching this video.

  • @crashgordon9684
    @crashgordon9684 8 років тому

    Excellent great lecture. I learned a lot from that. Thanx

  • @LA6UOA
    @LA6UOA 8 років тому

    This is a great explenation! Thanks Bruce!

  • @parajared
    @parajared 8 років тому

    Why don't they make 433 circular polarized antennas for fpv?

    • @blackmennewstyle
      @blackmennewstyle 8 років тому

      +parajared fpv Maybe because of the size of the antennas they will have to design, i think they will be insaleny huge, and not sure if you will be happy to carry one and even put one in your aircraft...
      But i am pretty sure there is some patch circularly polarized antennas already in the market, have a look and you will see...

  • @shawntoomath9328
    @shawntoomath9328 6 років тому

    Those pink dots are so ++ Feynman 😎

  • @camaro92026
    @camaro92026 7 років тому

    if im flying with an other pilot and im using left hand cp antennas and hes using right hand am i going to be picking up his F zone and vis versa ? granted hes using 1258 and im on 1280 here. and is that enough separation in channels? it seems those are the only two legal channels we can use on the 1.2G band.

  • @MrVinchenza
    @MrVinchenza 8 років тому

    Fantastic explanation thank you.

  • @jonathanchen5470
    @jonathanchen5470 7 років тому

    Please make a video on how the calculations work. I need it for my regional science fair

  • @sadistix82
    @sadistix82 8 років тому

    Thanks Bruce! Love your explanations.

  • @qvisionstudios
    @qvisionstudios 8 років тому

    Oooooh. My head hurts from all that math! 🤕. Thanks for the explanation and making things clear though.

  • @hansleatherby1032
    @hansleatherby1032 8 років тому

    good stuff Bruce
    It gives us all more insight into the hobby