Technical Discussions: Re: k factor

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Re: k factor
Posted by: Neil Burleigh , E-mail: Address, on December 15, 2008 at 23:49 :
In Reply to: Re: k factor posted by : James Barshinger , E-mail: Address, on December 15, 2008 at 23:02 :
----------- Start Original Message -----------
: Ed,
: To derive the values for "k" in the beam divergence equation for circular oscillators, 1st start with the equation:
: R(g)=2*J1(x)/x where x=pi*D/l*sin(g)
: - These equations are in krautramer though I have used g instead of "gamma" and l instead of "lambda"
: To find a "k" for a certain dB drop, you start with the J1(x)/x term, equating that to the desired amplitude reduction. For example: J1(x)/x=.5 for a 6dB drop of the "free field", or for the "echo field", (J1(x)/x)^2=.5. You need to then use a numerical solver to obtain the x that satisfies the equation for the particular dB drop you are interested in.
: Now, using the second equation, x=pi*D/l*sin(g). You can rewrite this equation as: sin(g)=(x/pi)*(l/D). "k" is simply the quantity (x/pi), thus you get the equation in the Krautkramer book.
: For rectangular oscillators, the process is the same with the exception of using Sin(x)/x instead of J1(x)/x
: -Jim
: : I have been asked how the values for "k" used in the beam divergence equations have been derived. My favourite references of Krautkramer and Ermolov touch on the subject and allude to the use of the Bessel J1(x) function. However, I cannot find the link to the k values for dB drop using this function described in either text.
: : Does anyone have a more complete formulation of the equations to derive k?
------------ End Original Message ------------
Morning Ed,
Some years ago I had a similar situation, we were using a K factor of 1.08 for the 20dB edge and the theoretical determination of the beam was always larger than the practical determination. The 2 K factors we were told to use (from the training and examination boards) were 1.22 for the infinite edge and 1.08 for the 20 dB edge.
In the little pocket book, the "Krautkramer Blue Book",
listed K factor for the 20dB edge for rectangular crystals as 0.87. When we used this constant the practical and the theoretical determination of the edge of the beam were very similar. I asked the author of the "Blue Book" Udo Schlengermann and he explained that originally that these constants came from astronomy and light transmission from celestial bodies. Which meant that the K factor of 1.08 is for through transmission whereas the K factor of 0.87 is for pulse echo.
Hope this helps in the history.
Regards
Neil Burleigh

Follow Ups:

Re: k factor Udo Schlengermann 12:06 Dec-16-2008 (4)

Re: k factor Ed Ginzel 22:59 Dec-17-2008 (0)

Re: k factor S.V.Swamy 07:45 Dec-17-2008 (0)

Re: KK factor Joe Buckley 20:29 Dec-16-2008 (1)

Re: KK factor Neil Burleigh 23:44 Dec-16-2008 (0)

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----------- Start Original Message ----------- : : Ed, : : To derive the values for "k" in the beam divergence equation for circular oscillators, 1st start with the equation: : : R(g)=2*J1(x)/x where x=pi*D/l*sin(g) : : - These equations are in krautramer though I have used g instead of "gamma" and l instead of "lambda" : : To find a "k" for a certain dB drop, you start with the J1(x)/x term, equating that to the desired amplitude reduction. For example: J1(x)/x=.5 for a 6dB drop of the "free field", or for the "echo field", (J1(x)/x)^2=.5. You need to then use a numerical solver to obtain the x that satisfies the equation for the particular dB drop you are interested in. : : Now, using the second equation, x=pi*D/l*sin(g). You can rewrite this equation as: sin(g)=(x/pi)*(l/D). "k" is simply the quantity (x/pi), thus you get the equation in the Krautkramer book. : : For rectangular oscillators, the process is the same with the exception of using Sin(x)/x instead of J1(x)/x : : -Jim : : : I have been asked how the values for "k" used in the beam divergence equations have been derived. My favourite references of Krautkramer and Ermolov touch on the subject and allude to the use of the Bessel J1(x) function. However, I cannot find the link to the k values for dB drop using this function described in either text. : : : Does anyone have a more complete formulation of the equations to derive k? : Morning Ed, : Some years ago I had a similar situation, we were using a K factor of 1.08 for the 20dB edge and the theoretical determination of the beam was always larger than the practical determination. The 2 K factors we were told to use (from the training and examination boards) were 1.22 for the infinite edge and 1.08 for the 20 dB edge. : In the little pocket book, the "Krautkramer Blue Book", : listed K factor for the 20dB edge for rectangular crystals as 0.87. When we used this constant the practical and the theoretical determination of the edge of the beam were very similar. I asked the author of the "Blue Book" Udo Schlengermann and he explained that originally that these constants came from astronomy and light transmission from celestial bodies. Which meant that the K factor of 1.08 is for through transmission whereas the K factor of 0.87 is for pulse echo. : Hope this helps in the history. : Regards : Neil Burleigh ------------ End Original Message ------------

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