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Fostex Rear-Horn SimulationHORNSIMULATION with the program AJ-Horn by MARVIN (revised 09-24-2000) The Thiele-Small parameters of the driver were taken from the original manufacturer data sheet . Putting them into the program "Speaker Unit" proves them to be correct and consistent . 
Next we put the drivers parameters into the program "BoxCalc" and see whether a reflex loading makes sense or not . By playing with box size and reflex port tuning frequency we decided to go for two box arrangements with slightly different goals in which one has a more precisely impulse response and the other one gives more output around the lower cut-off frequency , with an acceptable increase of overshoot/ringing compared to the previous reflex design with this driver . Two frequency responses now , see actual box volume and tuning frequency in the lower right corner of every plot : 
The box size of the latter reflex loading example will be used as a "reference" to see how much acoustic gain provides the Horn loading of this driver and is the resulting cabinet size acceptable for common tastes , in the next two plots this *reference* is represented by the red curve : 
We see above the response of an undamped T-Line which is 172cm long and having the same cross-sectional area as the driver itself . The line length of 172cm was chosen because this length is equal to one-quarter of the wavelength at the drivers basic resonance of 50Hz . If we increase the terminus area of the T-Line by a factor of 5 ( 5 x 132cm2 = 660cm2 ) to create a Hornspeaker ( backloaded ) , raises only the efficiency but lacks in getting deeper bass from the current version : 
The only thing to do to get deeper bass from a Horn is to make it longer !! So we increased the horn-length in three steps from 200cm over 220cm to 240cm but we kept the mouth size at 660cm2 , BTW - the actual Horn contour is exponential : 
This seems to be not much of an improvement , but let us compare the most different lengths ( 172 vs. 240cm ) to make things clear : 
( we have now raised the lower plot limit to 20Hz , this stretches the image a little bit and gives better optical resolution ) You see a difference of 7.8dB at 50Hz by making the Horn-Length longer , not that bad !! The price to pay for this improvement is an increase of the cabinet size , from 23Ltrs of the TL - version ( 172cm long ) to the three Horn - Variations : 66Ltrs ( 200cm Length ) , 72Ltrs ( 220cm Length ) and 79Ltrs ( 240cm Length ). But the overall response is very ragged due to the time delay dips and the typical high-Q resonances that are unavoidable by using an undersized Hornmouth . One way to reduce the irregularities is to smooth them a little by making the throat area somewhat smaller and inserting a pressure chamber at the beginning of the backloaded Hornspeaker : 
But you can see , making the size of the pressure chamber too big ( red curve ) is not good , the response will get deeper and wider nulls/dips and the peaks are higher than before , resulting in a more uneven overall-response . A general rule to size the pressure chamber that we can recommend is to make it between 2 and 4 Liters big . ( the "4.0Ltr" response is excluded in the plot above , because an overlay of more than 3 curves generates often some confusion , especially if the curves are too close together or can't be separated enough from another ) . So you maybe asking : Why is this a general rule ?? As far as we can judge the situation by now , it is related to the ratio between the upper and lower cut-off frequency , for most designs this will be between 40-60Hz and 300-400Hz . A different view of this aspect can be done by transforming the actual backloaded Hornspeaker into a frontloaded-horn of the same dimensions , but without the part that normally would come from the back of the driver . The following simulation show what happens to the response when three different pressure chamber sizes ( 2 - 4 - 8Ltrs ) are used : 
The Q-factor of the resonances goes up ( especially for the first three dominant spikes in the red curve ) and the shape of the lower cut-off changes , for most designs we investigated there is a significant loss in output , expect several dB's !! And what happens by reducing the throat area you would probably like to ask also ?? First you lose additional output at the lower cut-off but you can reduce the ripple somewhat ( of course distortion rises also , but that's another story ) : 
The same situation as above but now the back side of the driver radiates into the Horn , transforming the previous front-loading into a backloading condition : 
See that reducing the throat area is not a good thing , we would like to recommend to reduce the throat area not under 66% of the drivers effective area , it is better to use a value of 80% to get a good compromise between box size and bass extension respectively overall linearity . These bunch of simulations should only show some basic effects when the dimensions of a Hornspeaker changes . Designing an optimum Hornspeaker is difficult , it is very hard to determine design rules , everything is "floating" … Therefore any speaker constructor who is simulating a Hornspeaker should make his decisions carefully regarding what is an improvement and what tends to be critical , one very nice tool that can make these decisions easier and squeezes the most out of AJ-Horn so far , is the freeware-program ResGraph 2.6 we have used here to make the overlay of curves in different colours , my congratulations to the one and only Dan Wiggins (used to be www.avataraudio.com ) , the man who offers such a wonderful goodie like ResGraph for FREE !!! At last we introduce a modified version of the Hornspeaker that we've treated here so much , we added a little damping at the first half of the Horn-Length ( see factor Beta=100 , a value of 1000 is the maximum ) , this brings the response partially down by one or two dB ( to account for some losses in real-world designs ) , but isn't there enough bass in the response ?? Sure it is … 
Second screenshot of the current AJ-Horn simulation with the cone excursion instead of the impedance curve : 
Time to say goodbye for now , I hope you have liked my simulations , so all of you out there take care of yourself and keep on building the best speakers you can do !! GreetingZ from Germany - MARVIN Remarks :For those of you who would like to have the contour file for a Hornspeaker with a constant width along to your specification , I can quickly re-simulate the Hornspeaker and send you the contour file via e-Mail in accordance to the constant width you want to have . ( Editor's note: you can also get Horn contours from my Exponential/Hyperbolic and Tractrix contour calculators . ) This contour file , actually with an inner cabinet width of 20cm , is a text-file like the one you can see below : ( the number format is a bit humble , sorry for that , not my fault … )
----------------------------------- First row is the horn length , second is the cross-sectional area of the Horn in square centimeters , third is the height of the cabinet's inner constant width. AJ-Horn cuts the horn length into 50 pieces , therefore the somewhat fuzzy values , again , sorry … Top
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