Transmission Line Worksheet Tutorial - Quarter Wave

1 Transmission line Worksheet Tutorial Martin J. King 40 Dorsman Dr. Clifton Park, NY 12065 Transmission line Worksheet Tutorial By Martin J. King, 7/03/09 Copyright 2009 by Martin J. King. All Rights Reserved. Page 1 of 48 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.

2 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.

3 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.

4 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. Transmission line Worksheet Tutorial By Martin J. King, 7/03/09 Copyright 2009 by Martin J.

5 King. All Rights Reserved. Page 2 of 48 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.

6 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.

7 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.

8 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 Transmission line Worksheet Tutorial By Martin J. King, 7/03/09 Copyright 2009 by Martin J. King. All Rights Reserved. Page 3 of 48 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.

9 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.

10 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 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 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.