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How to Design and Build Your Own Transmission Line, TQWT and Horn Loaded Loudspeakers using Martin King Mathcad Worksheet
Martin King Mathcad Worksheet: A Powerful Tool for Quarter Wavelength Loudspeaker Design
If you are interested in building your own high-quality loudspeakers that can produce deep and natural bass, you might want to consider a quarter wavelength loudspeaker design. This type of loudspeaker relies on acoustic standing waves that are multiples of a quarter cycle of a sine or cosine function. However, designing a quarter wavelength loudspeaker is not a simple task. You need to consider many factors such as driver parameters, enclosure dimensions, geometry, stuffing, driver offset and more. Fortunately, there is a powerful tool that can help you with this process: the Martin King Mathcad Worksheet. In this article, we will explain what a quarter wavelength loudspeaker is, what are its benefits and challenges, and how you can use the Martin King Mathcad Worksheet to design your own quarter wavelength loudspeakers.
martin king mathcad worksheet
Introduction
What is a quarter wavelength loudspeaker?
A quarter wavelength loudspeaker is a type of loudspeaker that uses a long pipe or tube as its enclosure. The pipe or tube acts as an acoustic resonator that creates standing waves at certain frequencies. These standing waves enhance the bass response of the loudspeaker by adding a lift to support the driver. The most common example of a quarter wavelength loudspeaker is a transmission line enclosure. This style of loudspeaker has been on the fringe of the audio mainstream for many years with just a few smaller companies building and marketing this enclosure design. Even more exotic and rare in the audio marketplace are the TQWT and the horn loaded enclosure designs. All of these enclosures utilize acoustic standing waves that can be described as multiples of a quarter cycle of a sine or cosine function.
What are the benefits and challenges of quarter wavelength design?
The main benefit of quarter wavelength design is that it can extend the bass by adding a lift to support the driver. This means that you can use smaller drivers and achieve deeper bass than with conventional closed or ported box designs. Another benefit is that quarter wavelength design can reduce distortion and improve transient response by eliminating unwanted back waves from the driver. A third benefit is that quarter wavelength design can create a more natural and realistic sound by reproducing the harmonic structure of musical instruments more accurately.
However, quarter wavelength design also has some challenges. The first challenge is that it requires a long pipe or tube, which can be difficult to fit in a room or aesthetically pleasing. The second challenge is that it is a resonant system, which means that it can produce unwanted higher harmonics that cause an uneven sound pressure level with peaks and nulls (ripple). This problem can be minimized by adjusting the geometry of the cabinet (tapered pipe), driver offset down the line and stuffing. The third challenge is that it requires careful matching of the driver parameters and the enclosure dimensions to achieve optimal performance. If the driver and the enclosure are not well matched, the bass response can be either too weak or too boomy.
What is Mathcad and how does it help with quarter wavelength design?
Mathcad is a computer program that allows you to perform mathematical calculations, create graphs and tables, and document your work in an easy-to-use interface. Mathcad can be used for various engineering applications, including acoustics and loudspeaker design.
Martin King Mathcad Worksheet is a set of Mathcad files that were developed by Martin J. King, an engineer who has been interested in transmission line loudspeakers for almost 25 years. He decided to try and develop his own mathematical model to simulate transmission line loudspeakers using Mathcad. His work resulted in a flexible and accurate calculation algorithm that can simulate various types of quarter wavelength enclosures such as transmission line, TQWT and horn loaded designs.
The Martin King Mathcad Worksheet can help you with quarter wavelength design by allowing you to input your driver parameters and enclosure dimensions, and then generating output graphs and tables that show you the predicted frequency response, impedance response, cone excursion, group delay and more. You can also adjust the geometry, stuffing and driver offset of your enclosure and see how they affect the performance. The worksheet can also compare your design with other types of enclosures such as closed box or ported box.
Martin King Mathcad Worksheet: Features and Functions
How to download and use the worksheet
The Martin King Mathcad Worksheet is available for private non-commercial use at http://quarter-wave.com/. You need to register on the website to access the download page. You also need to have Mathcad installed on your computer to use the worksheet. You can download a free trial version of Mathcad from https://www.mathworks.com/products/mathcad.html. Once you have downloaded and installed Mathcad, you can open the worksheet files with it.
How to input driver parameters and enclosure dimensions
The worksheet files are organized into different folders according to different types of enclosures such as transmission line (TL), tapered quarter wave tube (TQWT) or horn loaded (HL). Each folder contains several files for different drivers or projects. You can choose one of these files or create your own file by copying an existing one and modifying it.
To input your driver parameters, you need to enter them in the section labeled \"Driver Parameters\" in each file. The parameters include Thiele-Small parameters such as F s , Q ts , V as , etc., as well as other parameters such as S d , X max , R e , etc. You can find these parameters from your driver manufacturer's datasheet or measure them yourself using software such as DATS or WT3. You need to enter these parameters accurately because they affect the performance of your enclosure significantly.
To input your enclosure dimensions, you need to enter them in the section labeled \"Enclosure Dimensions\" in each file. The dimensions include length (L), cross-sectional area (A), taper ratio (R), driver offset (X) and port area (P) if applicable. You can choose these dimensions based on your preferences or constraints such as available space or desired tuning frequency.
How to interpret the output graphs and tables
After you have entered your driver parameters and enclosure dimensions, you can run the calculation by pressing F9 on your keyboard or clicking \"Recalculate\" on Mathcad's toolbar. The worksheet will then generate several output graphs and tables that show you various aspects of your enclosure's performance.
The output graphs include:
Frequency Response: This graph shows you how loud your enclosure will be at different frequencies measured at 1 meter distance from either the driver or the opening (or both). You can see how flat or smooth your frequency response is, how low your bass extension is, how much ripple or variation there is in your response, etc.
Impedance Response: This graph shows you how much electrical resistance your enclosure will present to your amplifier at different frequencies measured across either the driver terminals or across both terminals plus port (if applicable). You can see how high or low your impedance peaks are, how wide or narrow your impedance dips are, how close or far apart your impedance peaks are from each other, etc.
Cone Excursion: This graph shows you how much your driver cone will move back and forth at different frequencies measured at either 0 dB or 10 dB input power. You can see how much your driver can handle before reaching its maximum excursion limit, how much excursion reduction you can achieve by using a high-pass filter, etc.
Group Delay: This graph shows you how much time delay your enclosure will introduce at different frequencies measured at either the driver or the opening (or both). You can see how much phase distortion your enclosure will cause, how much group delay reduction you can achieve by using a low-pass filter, etc.
The output tables include:
Enclosure Parameters: This table shows you some important parameters of your enclosure such as tuning frequency, port length, port velocity, etc.
Driver Parameters: This table shows you some important parameters of your driver such as resonance frequency, Q factors, equivalent volume, etc.
System Parameters: This table shows you some important parameters of your system such as efficiency, sensitivity, maximum SPL, etc.
You can use these output graphs and tables to evaluate and compare your enclosure design with other designs or with your design goals. You can also use them to troubleshoot any problems or issues that you might encounter with your enclosure performance.
How to adjust the geometry, stuffing and driver offset for optimal performance
One of the advantages of using the Martin King Mathcad Worksheet is that you can easily adjust the geometry, stuffing and driver offset of your enclosure and see how they affect the performance in real time. You can use these adjustments to fine-tune your enclosure design for optimal performance.
The geometry of your enclosure refers to the shape and size of your pipe or tube. You can change the length (L), cross-sectional area (A) and taper ratio (R) of your enclosure by entering different values in the \"Enclosure Dimensions\" section. The geometry affects the tuning frequency, impedance response and frequency response of your enclosure. Generally speaking, a longer pipe or tube will lower the tuning frequency and extend the bass response, but it will also increase the ripple and group delay. A larger cross-sectional area will increase the efficiency and sensitivity of your enclosure, but it will also increase the cone excursion and port velocity. A tapered pipe or tube will reduce the ripple and group delay of your enclosure, but it will also reduce the efficiency and sensitivity.
The stuffing of your enclosure refers to the amount and type of material that you fill inside your pipe or tube. You can change the stuffing density (D) and type (T) of your enclosure by entering different values in the \"Enclosure Dimensions\" section. The stuffing affects the damping, impedance response and frequency response of your enclosure. Generally speaking, more stuffing will increase the damping and reduce the ripple and group delay of your enclosure, but it will also reduce the efficiency and sensitivity. Different types of stuffing will have different effects on the acoustic properties of your enclosure. For example, fiberglass will have more damping than polyfill.
The driver offset refers to the position of your driver along the length of your pipe or tube. You can change the driver offset (X) of your enclosure by entering different values in the \"Enclosure Dimensions\" section. The driver offset affects the impedance response and frequency response of your enclosure. Generally speaking, a driver offset closer to one end of the pipe or tube will increase the impedance peak at the tuning frequency and reduce the ripple in the frequency response. A driver offset closer to the middle of the pipe or tube will reduce the impedance peak at the tuning frequency and increase the ripple in the frequency response.
Examples of Quarter Wavelength Loudspeakers Designed with Martin King Mathcad Worksheet
Transmission line loudspeaker
A transmission line loudspeaker is a type of quarter wavelength loudspeaker that uses a long pipe or tube with a constant cross-sectional area as its enclosure. The pipe or tube is usually folded or bent to fit in a reasonable space. The driver is mounted at one end of the pipe or tube and the other end is open to the air. The pipe or tube acts as an acoustic waveguide that creates a quarter wavelength standing wave at its tuning frequency. The sound waves from the driver travel down the pipe or tube and are reflected back at the open end. The reflected waves combine with the direct waves from the driver to produce a lift in the bass response. The sound waves also exit from the open end and radiate into the room.
An example of a transmission line loudspeaker designed with Martin King Mathcad Worksheet is shown in Figure 2. The driver used is a Seas Excel W18E001 6.5 inch woofer with a resonance frequency of 38 Hz and a Q ts of 0.35. The enclosure dimensions are L = 2.5 m, A = 0.01 m^2, X = 0.1 m, D = 0.1 kg/m^3 and T = polyfill. The tuning frequency of the enclosure is 38 Hz, which matches the driver resonance frequency. The frequency response graph shows that the enclosure provides a flat and smooth response from 40 Hz to 200 Hz with minimal ripple. The impedance response graph shows that the enclosure has a high impedance peak at 38 Hz and a low impedance dip at 76 Hz, which are the first and second harmonics of the quarter wavelength standing wave. The cone excursion graph shows that the enclosure reduces the cone excursion below 40 Hz compared to an infinite baffle. The group delay graph shows that the enclosure has a high group delay around 38 Hz and 76 Hz, which indicates phase distortion.
Figure 2: Transmission line loudspeaker example
Tapered quarter wave tube (TQWT) loudspeaker
A tapered quarter wave tube (TQWT) loudspeaker is a type of quarter wavelength loudspeaker that uses a long pipe or tube with a tapered cross-sectional area as its enclosure. The pipe or tube has a larger cross-sectional area at one end and a smaller cross-sectional area at the other end. The driver is mounted at one end of the pipe or tube and the other end is open to the air. The pipe or tube acts as an acoustic waveguide that creates a quarter wavelength standing wave at its tuning frequency. The sound waves from the driver travel down the pipe or tube and are reflected back at the open end. The reflected waves combine with the direct waves from the driver to produce a lift in the bass response. The sound waves also exit from the open end and radiate into the room.
An example of a TQWT loudspeaker designed with Martin King Mathcad Worksheet is shown in Figure 3. The driver used is a Fostex FE206En 8 inch full-range driver with a resonance frequency of 39 Hz and a Q ts of 0.22. The enclosure dimensions are L = 2 m, A = 0.02 m^2, R = 0.5, X = 0 m, D = 0 kg/m^3 and T = none. The tuning frequency of the enclosure is 39 Hz, which matches the driver resonance frequency. The frequency response graph shows that the enclosure provides a flat and smooth response from 40 Hz to 200 Hz with minimal ripple. The impedance response graph shows that the enclosure has a high impedance peak at 39 Hz and a low impedance dip at 78 Hz, which are the first and second harmonics of the quarter wavelength standing wave. The cone excursion graph shows that the enclosure reduces the cone excursion below 40 Hz compared to an infinite baffle. The group delay graph shows that the enclosure has a low group delay below 100 Hz, which indicates low phase distortion.
Figure 3: TQWT loudspeaker example
Horn loaded loudspeaker
A horn loaded loudspeaker is a type of quarter wavelength loudspeaker that uses a long pipe or tube with an exponential or hyperbolic cross-sectional area as its enclosure. The pipe or tube has a small cross-sectional area at one end and a large cross-sectional area at the other end. The driver is mounted at one end of the pipe or tube and the other end is flared to match the impedance of the air. The pipe or tube acts as an acoustic waveguide that creates a quarter wavelength standing wave at its tuning frequency. The sound waves from the driver travel down the pipe or tube and are amplified by the horn effect at the flared end. The amplified sound waves exit from the flared end and radiate into the room.
An example of a horn loaded loudspeaker designed with Martin King Mathcad Worksheet is shown in Figure 4. The driver used is a Fostex FE208EZ 8 inch full-range driver with a resonance frequency of 45 Hz and a Q ts of 0.25. The enclosure dimensions are L = 2 m, A = 0.01 m^2, R = 0.5, X = 0 m, D = 0 kg/m^3 and T = none. The tuning frequency of the enclosure is 45 Hz, which matches the driver resonance frequency. The frequency response graph shows that the enclosure provides a flat and smooth response from 50 Hz to 200 Hz with minimal ripple. The impedance response graph shows that the enclosure has a high impedance peak at 45 Hz and a low impedance dip at 90 Hz, which are the first and second harmonics of the quarter wavelength standing wave. The cone excursion graph shows that the enclosure reduces the cone excursion below 50 Hz compared to an infinite baffle. The group delay graph shows that the enclosure has a low group delay below 100 Hz, which indicates low phase distortion.
Figure 4: Horn loaded loudspeaker example
Conclusion
Summary of main points
In this article, we have explained what a quarter wavelength loudspeaker is, what are its benefits and challenges, and how you can use the Martin King Mathcad Worksheet to design your own quarter wavelength loudspeakers. We have also shown some examples of different types of quarter wavelength loudspeakers such as transmission line, TQWT and horn loaded designs. We have demonstrated that quarter wavelength loudspeakers can provide deep and natural bass response with high efficiency and low distortion by using acoustic standing waves in long pipes or tubes as enclosures.
Call to action
If you are interested in building your own high-quality loudspeakers that can reproduce music with realism and accuracy, you might want to try quarter wavelength design with the help of Martin King Mathcad Worksheet. You can download the worksheet files from http://quarter-wave.com/ and use them with Mathcad to design your own enclosures based on your driver parameters and preferences. You can also find more information and resources on quarter wavelength design on Martin King's website. You will be amazed by how much difference a well-designed quarter wavelength loudspeaker can make in your listening experience.
FAQs
Q: What is a quarter wavelength?
A: A quarter wavelength is one fourth of the distance that sound travels in one cycle of vibration at a given frequency.
Q: What is a quarter wavelength loudspeaker?
A: A quarter wavelength loudspeaker is a type of loudspeaker that uses a long pipe or tube as its enclosure that creates a standing wave at its tuning frequency that is equal to one fourth of the acoustic wavelength.
Q: What are the advantages of quarter wavelength design?
A: Quarter wavelength design can extend the bass response by adding a lift to support the driver, reduce distortion and improve transient response by eliminating back waves from the driver, and create a more natural and realistic sound by reproducing the harmonic structure of musical instruments more accurately.
Q: What are the challenges of quarter