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Transmission Line Worksheet Tutorial

Martin J. King 40 Dorsman Dr.

Clifton Park, NY 12065 [email protected]

Transmission Line Worksheet Tutorial By Martin J. King, 7/03/09

Copyright © 2009 by Martin J. King. All Rights Reserved.

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Introduction : My website has been on the Internet for about seven years and the number of visits has surpassed 500,000. This is well beyond any of my wildest expectations when I first uploaded the website on July 17, 2002. If I try and estimate how many people are actually using my MathCad transmission line worksheets, I can only make a guess based on a few simplifying assumptions. There were approximately 275,000 visits to my site when the MathCad worksheets were free. Assume that one in one hundred of these visits downloaded the free version of the MathCad worksheets. I assume this recognizing that some people take a quick look at the site while others return many times to read and digest what I have presented. But if we stick with this assumption and add the licensed users of the upgraded worksheets that have been available for the past few years, it means that approximately 3,000 people have tried the MathCad worksheets. This is a significant number of users and I believe that this estimate is very low. Based on the types of responses and questions I have received via e-mail, I know that the population using the worksheets is quite diverse in their technical/mathematical capabilities. I have heard from high school students, car mechanics, meteorologists, engineers, physicists, teachers, musicians, artists, chefs, college students, and at least one waitress to name just a few vocations. E-mail questions have come from all over the world including countries I had never heard of before and sometimes in valiant attempts at the English language. In most cases, after answering a question or two the user has been able to run the worksheets and design his, or her, own transmission line loudspeakers. A fair number of the questions have been asked more then once. There are certain aspects of the transmission line worksheets that are a little difficult to grasp, are poorly explained, or are only described deep in one of the many documents on my website. Using the MathCad program has been a new experience for non-engineers but surprisingly it has not been a huge issue for most people. The program takes some getting used to and thankfully the MathCad help system is very good. If a new user invests a few hours working with one of the simple transmission line worksheets, I believe they become proficient MathCad users very quickly. Unfortunately, a few users are never comfortable with MathCad and abandon the worksheets. In the next few pages, I am going to try and describe the required user inputs and the plotted outputs produced by the the latest version of the transmission line worksheets with an emphasis on the questions I have been asked most frequently. Hopefully this will answer some of the common questions, allow people to use the worksheets quickly with more confidence, and help produce better transmission line loudspeaker designs.



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Presentation Outline :

1. Pages 1 - 12 : Description of each page of the simulation worksheet "TL Sections Corner 7_03_09" including user inputs and model outputs.

2. Page 13 : Figure 1 - Focal TL Speaker Model Geometry Definition

3. Pages 14 - 30 : Attachment 1, "TL Sections Corner 7_03_09" Worksheet results

for Focal TL Model Geometry without Stuffing

4. Pages 31 - 47 : Attachment 2, "TL Sections Corner 7_03_09" Worksheet results for Focal TL Model Geometry with Stuffing

The “TL Sections Corner 7_03_09” Worksheet : I decided to review the “TL Sections Corner 7_03_09” worksheet in detail. This is the most general of the worksheets and almost all of the others were derived from this worksheet. The two attachments contain copies of the “TL Sections Corner 7_03_09” worksheet consistent with the Focal two-way transmission line that was my original project. This is the default geometry set up in the downloaded worksheet and it models the enclosure shown in the transmission line application note and also included as Figure 1. The only change I have made to the downloadable version of the worksheet is to remove the fiber stuffing in Attachment 1 so that all of the response peaks and nulls are shown clearly. The frequencies of these peaks and nulls will be identified and used when comparing different plots. Attachment 2 contains a copy of the same “TL Sections Corner 7_03_09” worksheet but with the stuffing distribution used in the Focal two way transmission line speaker system. Each page of the “TL Sections Corner 7_03_09” worksheet is labeled in both attachments. Page 1 of the “TL Sections Corner 7_03_09” Worksheet : The top half of the first page contains titles and defines some physical constants and units that will be used in the calculations. This region is locked to prevent tampering. The bottom half of this page is where the first user input is required. Input values are entered to the right of the “:=” sign that follows the appropriate variable label. The “:=” symbol is used by MathCad to assign a number to a variable. The “=” sign after a variable name will display the variable’s numerical value and associated units at that point in the MathCad worksheet. The first entry required on this page is a series resistance Radd that can be added to the simulation. This might be an estimate of the resistance of the speaker cables, the output impedance of a tube amp, or an intentional resistor placed in series with the driver to tame a rising SPL frequency response. Just below the series resistance the Thiele / Small parameters are entered for the driver. The series resistance is used to adjust the Thiele / Small parameters as shown in the two equations on the right side of this input region. The subscripts I have used to define the Thiele / Small parameters are different from what is commonly found in technical articles and on manufacture’s data sheets. I have replaced the “s” subscript, used by almost everybody else, with the “d” subscript. I



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made this switch because the “d” subscript to me denotes driver while the “s” subscript to me signifies speaker system. This notation extends back over 20 years in my personal notes and design calculations. It has caused some confusion for which I apologize. I have also over specified the Thiele / Small parameters required as input. In reality, if Vad is known then Bl can be calculated using the other parameters. The reverse is also true; if Bl is known then Vad can be calculated from the other parameters. When originally setting up the worksheets I could not decide which variable, Vad or Bl, would be more common and convenient for use in the worksheets. Since I measure the Thiele / Small parameter values for my own drivers, all the parameters are readily available so I included both in the input section. I have added two short worksheets to my site that calculate Bl given Vad or Vad given Bl. This should help users who have an incomplete set of Thiele / Small parameters or would just like to double check a manufacturer’s data sheet to see if the information provided is consistent. I strongly recommend running one of these two worksheets before doing any simulations to assess the quality of the driver’s parameters being used as inputs. If the driver parameters are not consistent, then the results of any subsequent simulation are questionable. Just below the Thiele / Small parameters, the input power can be specified. The applied voltage is calculated from this input power and Rref the voice coil resistance. For a 1 watt input, the default applied voltage to the driver is 2.828 volts since Rref is equal to 8 ohms. This applied voltage is consistent with the way a lot of manufacturer’s test speakers and derive the specifications shown on the data sheets.. The input power, or the Rref voice coil resistance, can be altered by the user to produce any applied voltage. The very last section at the bottom of page 1 provides a reference for the geometric dimensions used in the downloaded worksheet. Reading this reference article will provide the user with some background information about the Focal two-way transmission line being analyzed. Page 2 and 3 of the “TL Sections Corner 7_03_09” Worksheet : Page 2 has probably causes the most confusion. On this page all of the transmission line geometry is entered, the driver position specified, and the placement and amount of fiber stuffing prescribed. The first two variables defined are n_closed and n_open.

n_closed = the maximum section number used to model the closed end n_open = the maximum section number used to model the open end

Referring to Figure 1, the section numbers for the closed and open ends of the Focal transmission line are labeled. There are 5 sections used to model the closed end of the transmission line and 10 sections used to model the open end of the transmission line. Also recognize that the counter starts at 0. So for the closed end there are five sections labeled 0, 1, 2, 3, and 4. A similar section labeling is used for the open end. The counter variables n_open and n_closed are the maximum section subscripts used in the transmission line model. Looking at the next two regions, the detailed inputs for the closed and open ends of the transmission line are defined as rows and columns. Both sets of input definitions start at the driver and move towards the open or closed end of the transmission line.



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Each row defines one section. If for example, you increase n_closed from 4 to 5 then an additional row of input is required corresponding to a subscript of 5. This row would be added below the last row used to define the closed end of the transmission line. The easiest way to add another section is to copy and paste the preceding row and then just change the subscript number for each entry in the new row. The section length, initial area, final area, and stuffing density are input moving from left to right across any row. When defining the area values, I tend to use a multiple of the cone area Sd. Exact geometry dimensions could also be used. For example, Sc0,0 = 3 x Sd could have been defined as Sc0,0 = 13.5 inches x 7.875 inches as shown in Figure 18 of the original Focal transmission line article. You can use any number of sections as long as n_closed and n_open are greater than or equal to 1 (two lines of input). You can input step changes in area by setting the initial area of a section larger or smaller than the final area of the previous section. You can accurately model corners, as was done in this example, but at low frequencies this is really not critical. In Figure 1, the dotted line traces the path and the dashed lines show the initial and final areas of each section. Notice that in this part of the worksheets only the internal air volume is being modeled, all dimensions are internal measurements not external cabinet dimensions. You should understand everything in Figure 1 and the attachments before you attempt to modify “TL Sections Corner 7_03_09” or any of the other MathCad worksheets. Almost all of the other worksheets are derived from the “TL Sections Corner 7_03_09” worksheet. For example, if you look at one of the back loaded horn worksheets, you can see another creative application of “TL Sections Corner 7_03_09”. Two last points about the information input on this page should be highlighted. The maximum stuffing density that can be used is 1 lb/ft3. Inputting a density greater then this value will produce suspect results. The curve fits to the measured and correlated test line data are only accurate up to a density of 1 lb/ft3. If a greater stuffing density is required, to suppress peaks and nulls in a transmission line’s SPL response, then the enclosure geometry should probably be redesigned.

The SPL calculated is at 1 m for the power level input on page 1 and assumes that the speaker system is radiating into 2π space. As described above, the input power can be changed but remember that the applied voltage is calculated based on an 8 ohm resistor. Parts 2 and 3 of the worksheet calculate the response at different listening positions considering the enclosure external geometry and some of the room boundaries. On page 3, the total length of the transmission line is calculated. Double check this summed length to make sure it matches the length shown in your enclosure design sketch (hopefully you made a sketch before starting to run simulations). And finally the total mass of stuffing is calculated, this determines how much stuffing is inserted into the finished enclosure. With the total stuffing weight it is easy to measure a fixed amount rather then guessing at a density in each section. If a TL design has a constant stuffing density, measure out the total amount and do your best to make sure it is placed equally along the line length. It is not critical to place the stuffing exactly, small local areas of density variation will not cause the design to perform poorly. Also on page 3 of the “TL Sections Corner 7_03_09” worksheet is a plot depicting the cross-sectional area as a function of length. After entering all of the



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detailed lines of input on page 2, this plot serves as a visual double check that the model is correct. Compare this plot with your sketch or drawing of the design, the only difference is that the model is represented as one long straight path and the folds are represented as local changes in cross-sectional areas. Once the information on these pages is entered, just scroll down and the calculations will automatically update the plotted output. Some of the regions on the pages that follow are locked to prevent tampering with the calculation algorithm. Page 4 of the “TL Sections Corner 7_03_09” Worksheet : This is the first page of calculated results. Two variables are plotted as functions of frequency, the acoustic impedance seen by the rear of driver and the ratio of air velocities. There are a number of pieces of useful information in these plots that can help the user understand what is going on in the quarter wavelength enclosure. In Attachment 1, the top two plots show the magnitude and phase of the acoustic impedance. Looking at the magnitude plot, the 1/4 wavelength tuning frequency is seen as the first peak at 41 Hz. The 3/4, 5/4, and 7/4 wavelength frequencies are defined by the next three peaks at 124 Hz, 208 Hz, and 289 Hz respectively. Since this is approximately a straight constant area transmission line these values are consistent with closed form solution results.

3 x 41 Hz = 123 Hz 5 x 41 Hz = 205 Hz 7 x 41 Hz = 287 Hz

These relationships only hold for a straight constant area transmission line, a tapered or expanding line will not produce a predictable pattern of peaks. This pattern would continue if the driver had been placed at the closed end and not offset six inches. The next interesting features of the acoustic impedance plot are the deep nulls that occur at even increments of frequency. Looking at the first three nulls which occur at 88 Hz, 176 Hz, and 264 Hz it is often stated that these are the half wavelength modes of the transmission line. This is not correct; these nulls are created by the phase shift associated with the quarter wavelength standing wave just above and below the null. Looking at the phase plot, at each quarter wavelength resonance the phase angle rapidly shifts from a maximum positive (essentially +90 deg) to a maximum negative phase (essentially -90 degrees) passing through 0 degrees at the frequency of the peak impedance magnitude. For the standing wave mode below a null, the magnitude of the acoustic impedance is decreasing and the phase is approaching -90 degrees as frequency increases. For the standing wave mode above the null, the magnitude of the acoustic impedance is increasing and the phase is approaching +90 degrees as frequency decreases. At the frequency midway between the two peaks, the magnitudes of the two modes are almost equal and the phases are 180 degrees apart resulting in a deep null and a zero phase angle. Reading the “Anatomy of a Transmission Line Loudspeaker” article, located elsewhere on this website, details of this peak and null phenomenon are presented. There is one more null that should be recognized. At 562 Hz there is a very deep null generated by the driver offset. In this design the driver is offset six inches from the



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closed end of the transmission line. The quarter wavelength frequency associated with this offset is calculated as follows.

f = c / (4 L) f = (342 m/sec) / (4 x 6 inches) f = (342 m/sec) / (4 x 0.152 m) f = 562 Hz

At this frequency, the volume velocity of the driver is essentially equal to the volume velocity into the closed end of the transmission line. In this situation the quarter wavelength resonator, formed by the closed end of the line, is being excited at the point where the pressure is almost zero. The sound waves are being absorbed by the closed end just like a tube trap would at a prescribed frequency in a listening room. The acoustic impedance is defined as the pressure divided by the volume velocity. If the pressure is almost zero, the acoustic impedance is also zero and a deep null will result as seen in the magnitude plot at 562 Hz.

The second pair of plots on this page, the ratio of the air velocities, is not nearly

as useful. The only information that is really important is seen in the magnitude plot. At the 1 Hz frequency, the ratio of the velocities should equal the ratio of the cross-sectional area at the driver position to the cross-sectional area at the open end. This is required at low frequencies for mass continuity. At this frequency, the volume velocity of air moving into the open end at the driver location must be equal to the volume velocity leaving the open end at the terminus. Therefore the ratio of velocities must equal the ratio of areas. This is a nice double check that the sections have been entered correctly.

Also notice in the second set of plots that the peaks occur at different frequencies

when compared to the acoustic impedance plotted above. The first peak is at 44 Hz while the second and third peaks are at 133 Hz and 222 Hz respectively. The shape of this plot is determined by the open end length defined for the transmission line and not the entire transmission line length. The only time that the peaks in the acoustic impedance plot and the velocity ratio plot will be at the same frequencies is when the driver is mounted exactly at the closed end.

Adding stuffing to the transmission line, as seen in Attachment 2, will cause the

peaks in the acoustic impedance to become less pointed and smear out over a wider frequency band. The deep sharp nulls between the peaks will become shallow and rounded valleys. The acoustic impedance phase shifts will also become more gradual, not as steep or sudden.

One final thought, the plots on this page are derived for the enclosure geometry

and stuffing arrangement. They are completely independent of the driver or the driver/transmission line combined system response. Therefore the plots are characteristics only of the air and stuffing in the transmission line enclosure.

Page 5 of the “TL Sections Corner 7_03_09” Worksheet : The top pair of plots on this page is really the bottom line when it comes to the optimization of the driver/transmission line design. The red curves represent the magnitude and phase response of the driver/transmission line system. The dashed blue curves represent the magnitude and phase of the same driver mounted on an infinite



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baffle. The difference between these two curves is the contribution of the transmission line terminus output. The transmission line enclosure should augment the natural rolled off response of the driver to extend the useful SPL response lower in frequency. The transmission line system is assumed to be radiating only into 2π space so any baffle step response or room influences are not included in this part of the worksheet. The SPL plots on this page can be compared directly with the plots produced by most other freeware speaker design/simulation programs. The assumption of 2π space means that in effect the driver and terminus are mounted coincident on an infinite baffle and the measurement microphone is exactly 1 m away along the driver axis. This is the way most speaker design simulations calculate and present the SPL frequency response plots; clearly this is a very limited result

The combined baffle step and room response will lower the bass output, typically 3 - 4 dB, below a frequency determined by the size and shape of the front baffle. If you don’t use some form of compensation, either room placement or electrical filtering, a transmission line speaker system that is predicted to have a flat extended SPL response in MathCad will probably exhibit weak bass performance. This observation is also true of most other speaker enclosure designs calculated by other freeware speaker design programs. The bottom two plots on this page separate the contributions of the driver and the terminus to the system SPL response that was shown in the top two plots. The red curves represent the magnitude and phase response of the driver. The dashed blue curves represent the magnitude and phase of the terminus. In Attachment 1, the magnitude plot shows the peaks at the transmission line quarter wavelength frequencies and the influence on the driver output. The driver’s output goes through a sharp null at each of these frequencies and most of the system response is generated by the terminus. This behavior is similar to a bass reflex design around the ported box tuning frequency except for a transmission line enclosure it occurs at multiple additional frequencies above the fundamental tuning frequency. Comparing the results shown in Attachment 1 with those shown in Attachment 2, the effectiveness of the fiber stuffing is obvious. The fiber stuffing greatly attenuates the sharp narrow resonant peaks and nulls to produce a response that contains a few gentle ripples. Adding more stuffing reduces the amount of ripple but at the expense of bass extension. The exact density and placement of the fiber stuffing in the transmission line needs to be optimized so resulting ripples are minimized and bass extension is maintained. The MathCad worksheets can be used to provide a good initial distribution of fiber stuffing but in the end listening is the preferred method for getting the best stuffing results. Page 6 of the “TL Sections Corner 7_03_09” Worksheet : The pair of plots on this page display the electrical impedance. The red curves represent the magnitude and phase of the driver in the transmission line. The dashed blue curves represent the magnitude and phase of the same driver mounted on an infinite baffle. In Attachment 1, the impedance curve for the empty transmission line exhibits the double humped shape typical of a bass reflex design. The only difference is a third peak in the impedance curve at 128 Hz due to the 3/4 wavelength resonance. As



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stuffing is added, the double humped impedance curve approaches a single humped curve as seen in the electrical impedance plot in Attachment 2. Probably the best method for using the impedance curves is as a double check of a design. Measuring the unstuffed transmission line’s electrical impedance and comparing the results against the calculated impedance will help establish the accuracy of the MathCad simulation. Below 100 Hz, the impedance is predominately a reflection of the driver’s motion. If the measured and calculate impedance curves match then the driver is behaving as predicted and therefore the transmission line system calculated SPL response should be accurate. This is a good way of double checking a design if you don’t have the capabilities to perform near field and far field SPL measurements. If you can measure Thiele / Small parameters, then you can check the transmission line design using the impedance curve. Page 7 of the “TL Sections Corner 7_03_09” Worksheet : In the upper plot the driver’s RMS displacement is plotted for the voltage determined from the user specified input power referenced to Rref. If you read a deflection off this plot and multiply it by 1.414, the result can be compared to Xmax. I don’t usually worry too much about the results shown in this plot. It would be more important if you spent a lot of time listening to test tones, for music I am not sure it is a critical result. Many speaker designers really emphasize the driver’s 1 watt displacement relative to Xmax. I tend to review the curve and as long as the deflection is not grossly different from the infinite baffle result, I do not worry about the impact on performance. In the bottom plot the pressure impulse response is provided. In Attachment 1, the initial peak is created by the driver. This is followed by a second out of phase peak from the terminus that is delayed in time by the length of the transmission line. Then comes a series of reflections as a portion of the initial pressure pulse reaching the terminus is reflected back into the transmission line traveling the full length to the closed end and reflecting again to arrive at the terminus later in time. This partial reflection process is repeated resulting in a string of pulses of decreasing magnitude. In Attachment 2, the stuffing does an excellent job of attenuating the reflected pulses and cleaning up the impulse response. Page 8 of the “TL Sections Corner 7_03_09” Worksheet : The group delay and the air velocity at the terminus are plotted on this page. Group delay is a plot that was requested several times a year so I added it to the worksheets. Some people are very interested in group delay and use it as an indication of the type and quality of bass produced by a speaker system. But to be honest, it is not a result that I spend much time worrying about.

The air velocity at the terminus is typically not an issue with a TL since the terminus area is so large compared to the port in an equivalent bass reflex or ML TL system. However in some of the other worksheets, such as the ML TL and DBR worksheets, it is something to consider when sizing the port diameter and length.



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Pages 9 and 10 of the “TL Sections Corner 7_03_09” Worksheet : All of the inputs for Part 2 of the worksheet are entered on these two pages. This is where the external dimensions of the enclosure and a room’s corner are considered in the calculation of the SPL response. Using the dimensions of the baffle, the depth of the enclosure, the positions of the driver and terminus on the baffle, and the enclosure’s location relative to the nearest room boundaries, the SPL response is calculated at a prescribed distance, height, and angle relative to the driver’s axis. Starting at the top of page 9, some basic definitions and assumptions are provided for the user. Directly below, six key variables are input defining the distance from the rear and side walls, the angle the enclosure is rotated relative to the corner, the ceiling height, the distance that a stand raises the enclosure above the floor, and the number of baffle edge sources per unit length. The origin of the coordinate system for the X0 and Y0 distances is the room corner at the floor. The first two distances define the position of the outside front corner of the enclosure relative to the origin. The angle can be used to rotate the enclosure toward or away from the room center. The stand height raises the bottom of the speaker enclosure above the floor. Finally, the variable num_r controls the number of sources to be used per unit length of baffle edge. Increasing num_r increases both the accuracy of the calculations and unfortunately the length of time to perform the calculations. The default value of 10 used in the worksheets seems to work well for most speaker baffle sizes.

Directly below these entries, the geometry of the front baffle is described using four corner points. The local origin for defining these points is the bottom left front corner of the enclosure. A non rectangular baffle can be input by specifying the corner locations of the baffle, at this time the baffle shape is limited to four sides but it could be easily increased. Looking at a ML TQWT design for example, the truncated pyramid speaker enclosure that was designed, built, and described elsewhere on the website can be accurately modeled using four corner points.

Below the baffle geometry section, at the top of page 10, the locations of the

driver and the terminus are input along with the number of simple sources to be used across each of the horizontal dimensions. The local origin of the coordinate system is still the lower left front corner of the front baffle. One other option for the user is to locate the terminus (port or horn mouth) on the front or rear baffle, it is easy to try both to see which provides the best performance at the user defined listening position. Increasing the number of simple sources used to represent the driver or terminus increases both the accuracy of the calculations and unfortunately the length of time to perform the calculations.

The next item to be input on page 10 is the listening position. There are two

options for specifying the listening position, relative to the driver axis or relative to the room corner. The first possible listening position is defined as a radius and angle along the driver’s axis and a height above the bottom of the front baffle, typically the driver height. The second possible listening position is a set of fixed coordinates from the room corner. The variable “n_listen” is used to select which listening position is to be used in the SPL calculations. For example, selecting the first option, “n_listen := 0”, allows the speaker and listening position to be moved around together relative to the room boundaries. Selecting the second option, “n_listen := 1”, allows the response at a fixed position to be calculated as you move the speaker around relative to the room corner.



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The floor reflective boundary condition can be modified from a hard reflection typical of a hardwood, tile, or concrete floor to a frequency dependent softer reflection typical of a carpeted floor. The wall and ceiling reflections are all assumed to be typical of a drywall and wood frame construction.

Finally, options are provided for removing one or all reflective surfaces from the

SPL calculations. By setting any of the reflective surface selectors to zero, the influence of the boundary condition can be removed from the simulation. Setting them all to zero represents an anechoic condition where sound radiates from the speaker to infinity without being reflected back.

On the following pages the SPL calculated will have more peaks and nulls then were seen in the plots generated in Part 1. These are caused by the relative positions of the driver and terminus, the room reflections, and the baffle step phenomenon. By changing some of the inputs on this page you can investigate the cause of particular anomalies in the calculated SPL response. For example you could remove the side wall to eliminate these reflection contributions. Another trick is to make the driver and terminus coincident to push the simulation closer to the Part 1 calculated results. By changing some of the entries and watching the resulting effect on the SPL response, you can get a feel for what is controllable and what is just part of the response the speaker design produces in a typical listening room corner.

One cautionary note, there is nothing in the MathCad worksheets that double

checks to make sure what you are simulating is realistic. For example, if you are designing a ML TL enclosure that has an internal air volume height of 42 inches in Part 1 and then in the Part 2 you place the port 48 inches away from the driver, the MathCad worksheet will calculate a solution. Now all that is needed is some new laws of nonlinear physics to make this enclosure design physically possible. It is up to the User to enter a realistic physically achievable speaker enclosure design and placement relative to a room corner. Pages 11 and 12 of the “TL Sections Corner 7_03_09” Worksheet : A picture is worth a thousand words. First on page11, the dimensions in meters of the enclosure and the positions of the driver and the terminus are shown. The Y axis represents the bottom horizontal edge of the front baffle. The location of the driver is always on the front baffle while the terminus (port or mouth) can be located on the front or the rear baffle. The density of the simple sources used to model the driver and the terminus are shown as the little red and blue circles respectively. If this picture does not look like what you envisioned then you should double check all of the inputs in Parts 1 and 2. Then on page 12, a set of plots are shown that define the position and rotation of the enclosure relative to the room’s corner. Once again the same red and blue simple sources are shown for the driver and terminus (port or mouth). In addition, a magneta square is shown that represents the listening position for the SPL calculations. To maintain the scale of the plots, and not distort the pictures, a single entry “axis” at the top of page 12 is used to set the lengths of all the axes in all three plots. If this picture does not look like what you envisioned then you should double check all of the inputs in Parts 1 and 2 of the MathCad worksheet.



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Pages 13, 14, and 15 of the “TL Sections Corner 7_03_09” Worksheet : These three plots should be compared to the plots presented on page 5. The differences between the corresponding plots are attributable to the relative position of the driver and terminus, the room reflections, and the baffle step response. Comparing these same plots with and without stuffing, shows that the resonance peaks and nulls are attenuated when the stuffing is added to the transmission line but there are still a few nulls that are not significantly impacted. The potential for achieving a nice flat SPL frequency response as more external influences are accounted for in the analysis of the speaker system, typically a goal when using simple lumped parameter modeling tools, becomes much more difficult if not impossible to achieve. By using actual geometric dimensions the impact of driver and terminus placement, the baffle size, and the distance from the room boundaries can be studied and optimized to minimize the anomalies in the summed SPL plot on page 15. One point not covered here is the distance and off axis angle used for the calculations, if you listen to your speakers at a distance of 10 feet and 15 degrees off the axis of the driver, then those are probably the best entries for the listening position on page 10. I have concluded that while the potential for achieving a perfectly flat SPL response is minimal, the potential for improving the SPL response is large if good design decisions are made with respect to the external geometry, room placement, and listening position. Just keep in mind that only a few external influences are included in the calculation, hopefully a future upgrade will extend these capabilities to represent all of the room boundaries. People using simple lumped parameter freeware simulation programs to generate nice flat SPL responses are really misleading themselves. Page 16 of the “TL Sections Corner 7_03_09” Worksheet : All of the inputs for Part 3 of the worksheet are entered on this page. After reviewing the summed SPL response on page 15, it should be clear that a mismatch in SPL level exists as frequency increases due to the baffle step influence on the SPL response. One option to rectify this problem is the use of a correction circuit. My experience with full range driver speaker systems is that without a correction circuit the bass response will sound weak and thin compared to the midrange. This type of response is often described as “shouty”, it is a piercing and fatiguing listening experience. The baffle step correction circuit in this simulation includes the effect of any added series resistor Radd and Re the driver voice coil resistance. One other point to keep in mind concerns the use of a Zobel circuit across the driver terminals to offset the rising voice coil inductance. Notice that there are no entries made in the MathCad worksheet to simulate a Zobel circuit. If such a circuit is present in the design, make sure that the voice coil inductance entered as part of the Thiele / Small parameters is set to zero. If a Zobel is not being used in the design, then the actual voice coil inductance value should be entered into the worksheet for inclusion in the calculations.

There are three distinct steps used to size the correction circuit on this page of the MathCad worksheet. First, review the summed SPL plot on page 13 and determine the center frequency of the rising response and the number of dB of correction to be applied. In theory 6 dB of baffle step correction is required but in a real listening room environment this value is probably closer to 3 or 4 dB. Enter the center frequency and



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amount of dB attenuation at the top of page 16. Second, just below these entries a theoretical value for the parallel resistor is calculated. To the right of this calculated resistance, enter the actual value to be used; this usually corresponds to an even value that can be easily purchased. Third, after entering the actual resistor value, an ideal inductor value will be calculated. To the right of this calculation, and just below the actual resistor entry, enter an even value for the actual inductor to be used. I programmed this somewhat convoluted method of entry so that the user could see what ideally would be required and then enter what is readily available. By iterating on the actual component values of the baffle step correction circuit, different compromises can be tried. At the bottom of this page, the corrected SPL response is plotted. Page 17 of the “TL Sections Corner 7_03_09” Worksheet : The next page of the worksheet shows the revised impedance curve with the baffle step correction circuit in place. You should see slightly rising impedance at higher frequencies as the baffle step circuit transitions to an additional resistance in series with the driver. This curve can be used to check the impedance of you finished speaker system and assess the accuracy of the modeling. One word of caution, when adding a baffle step correction circuit it is best to double check the terminal DC resistance with an ohm meter before connecting an amplifier. I did not do this simple check once with a pair of speakers constructed by a friend, one circuit had been wired incorrectly resulting in a short circuit and my test amp did not survive after being turned on. Unfortunately, a lesson learned the hard way. Pre-Formatted Transmission Line Worksheets : As stated earlier, the “TL Sections Corner 7_03_09” worksheet is the most general worksheet from which most of the other worksheets were derived. In most of the other worksheets, only simplified input is required for Part 1 on the first page. Nothing more is needed to run the simulation. The section definitions on the second page are automatically populated so the user is not required to edit any of these input values. The only time the user should edit data on the second page of a simplified worksheet is if an adjustment to the amount of stuffing, location of stuffing, or a small tweak to the geometry definition is desired. Conclusions : I have tried to provide a brief tutorial that covers most of the common questions that new users ask frequently. This is truly a dynamic document and I will add to it as people ask the next level of detailed questions or point out other uses for the plotted results. If you have a tip or an observation, please e-mail me directly so it can be included in the next revision of this tutorial.



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Figure 1 : Focal TL Model Geometry Definition

Closed End

OpenEnd

01

2

34

0

1

2 3 4

5

6

7

8 9



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Attachment 1 : “TL Sections Corner 7_03_09” MathCad Worksheet (w/o Fiber Stuffing)

Page 1 of the “TL Sections Corner 7_03_09” Worksheet



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Attachment 2 : “TL Sections Corner 7_03_09” MathCad Worksheet (w/ Fiber Stuffing)




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