Simple calculation method for speaker cabinet

When buying a new speaker unit, enthusiasts often find that the speaker unit manufacturer recommends the best cabinet size. This aspect may include the volume of a closed box or an open box. Usually, this value is related to the equivalent air volume of the VAS or cone support compliance, which is composed of the mass of the cone and voice coil, as well as the rigidity of the surround and centering support called the speaker unit support
.

(I) Cabinet Proportions
When enthusiasts make speaker cabinets, there are various structural options, including cubes, tubes, or rectangles to many other shapes. Each shape has special characteristics,
advantages and disadvantages. However, the commonly used speakers, whether closed or reflex boxes, are mostly rectangular cabinets, so this article discusses the relationship between rectangular cabinet dimensions.

Assume that the cabinet volume Vb recommended in the speaker characteristics table is 0.09056 cubic meters. Enthusiasts can use this value to determine the ideal cabinet size for the actual speaker unit. If the volume is given, the required height, width, and thickness can be obtained by converting the required internal volume in cubic meters into cubic centimeters and then taking the cube root of the result. A square enclosure (i.e., an enclosure of equal height, width, and thickness) is very satisfactory for use with subwoofers because it increases the total output of the enclosure by enhancing internal standing waves. Many commercially available subwoofers are designed in this manner. However, this article is not intended for use with subwoofers, but rather with two-way or three-way speakers that cover the full audio range.

In practice, many speaker manufacturers have adopted an empirically derived "golden" ratio or "golden" ratio, which is related to the ratio of the enclosure dimensions determined by the ideal ratio of 0.618. For example, if integer dimensions are used, such as 6 units of depth, 10 units of width, and 16 units of height, the ratio of depth to width = 6:10 = 0.60, and the ratio of width to height = 10:16 = 0.625. The aspect ratios of these final dimensions are quite close to the ideal value of 0.618. Because this ratio allows the selected approximate dimensions to not have common normal frequencies that enhance internal resonances, this ratio has been confirmed to produce the best sound.

(II) Calculation of Internal Dimensions
Assuming that the required internal pure volume is 0.0864 cubic meters, the calculation process is as follows:
1. Convert 0.09056 cubic meters to 90,560 cubic centimeters.
2. Assuming the aspect ratio is 6:10:16, multiply these three numbers and get a product of 960.
3. Divide the total cubic centimeters 90,560 by 960, and the quotient is 94.3.
4. Now, find the cube root of 94.3, which is approximately 4.55.
5. Finally, multiply the three values ​​of the aspect ratio by 4.55, which are 6 x 4.55 = 27.3 (thickness), 10 x 4.55 = 45.5 (width), and 16 x 4.55 = 72.8 (height).
6. After these calculations, multiply the values ​​of the width, height, and thickness of the cabinet and compare them with the original required cabinet volume of 90620 cm3. Since the products need to be rounded to integers, they can be slightly different, and when there is a 1% error, it can be considered insignificant.

The above is the whole process of determining the optimal cabinet dimensions. As an example, the reader can also choose other aspect ratios of 7:11:17, or 34:55:89 and proceed in the same way as in the previous example. When the optimal value is different by about 5%, it will only have a small impact on the playback quality.

(III) About Errors
If the reader encounters a small-volume speaker, then the volume is related to the volume occupied by the speaker unit installed in the box. The reader can make the box volume slightly larger to compensate for the volume of the speaker unit. If the speaker unit displacement value is not given in the speaker unit characteristics, the approximate displacement value (or volume) can be calculated according to the following formula: V = πr 2h, where r is the radius of the magnet and h is the thickness or height of the magnet. Assuming the magnet diameter is 11.4cm (radius is 5.7cm), thickness is 2.5cm, volume is: 3.1416 × 5.7 2 × 2.5 = 255.2cm3 Now, calculate the cone volume using the following formula: V = πr2h ／3 Assuming the cone diameter is 22.9cm, and the height is 5.1cm, the cone volume is: 3.1416 × 11.52 × 5.1 ／3 = 706.3cm3 Adding the magnetic circuit volume (255.2cm3) to the cone volume (706.3cm3) gives the speaker unit volume of 961.5cm3. This value is only slightly larger than 1% of the required cabinet volume of 90560cm3. So the volume of the speaker unit is not important in this case. As long as the combined volume of the speaker unit does not exceed 5% of the total cabinet volume, it can be ignored in the calculation.

No matter what ratio the reader uses, the dimensions of depth, width and height should not be integer multiples of any number. For example, 8, 16 and 24 should not be used because these numbers are integer multiples of 8, so harmful resonance will appear in the box.

For subwoofers, because this box needs resonance, it is often made into a square. Moreover, this speaker only covers a narrow frequency band, so the resonance of the box enhances the output. Of course, the bass can also be further enhanced by using the open box form.

(IV) Mathematical Golden Cut Ratio
The number representing the golden cut ratio (also called the golden mean, golden ratio and golden section) is obtained from dividing the line segment. At this time, the ratio of the shorter part to the longer part is equal to the ratio of the longer part to the total length of the line segment (Figure 1). Let the total length of the line be 1, and let the longer part be x, then the shorter part is 1-x, and the ratio derived is:
[(1-x)/x] = (x/1) or x2 = 1-x (1) A little rearrangement gives the quadratic equation: x2 + x-1 = 0 (2)
Comparing this with the basic quadratic form, we get ax2 + bx + c = 0, and applying this formula, x = (-b)/2a The positive value of x (the longer line segment) gives 0.61803..., which is rounded to 0.618 for practical purposes. By subtraction, the length of the shorter part is 0.382, which is the square of the longer line segment, as equation (1) directly shows.

The reader can also (in theory) find a correct split point obtained by geometric splitting. In Figure 2, ABC is a right triangle. For convenience, AB is chosen to be 2 units, and BC (perpendicular to AB) is chosen to be 1. According to the Pythagorean theorem, AC =. Draw an arc with C as the center and radius = BC = 1, intersecting the hypotenuse at point D, and we get AD = -1. Draw another arc with A as the center and AD as the radius, intersecting AB at point G, which is the golden ratio that divides AB. The longer part AG = -1, and the shorter part GB = 2-(-1) = 3-. Applying these values, we can see that
GB/AG = AG/AB is the same.

The golden ratio can also be obtained from other mathematical operations. For example, there is a FIBONACCI SERIES, in which each number is equal to the sum of the previous two numbers): 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, and so on. With a little calculation, it becomes clear how the number sequence is established. Take the ratio of consecutive pairs of numbers and look at the results:
1:1=1; 1:2=.5; 2:3=.67…; 3:5=.6; 5:8=.625; 8:13=.61538…; 13:21=.61904…21:34=.61764…; 34:55=.61818…; and so on.

黄金比率在许多方面都有出现，例如，正五边形对角线的线段，在测量五个正几何立体金字塔的一定比率，而最显著的是在自然界中，假如读者能获得一个大的成熟的向日葵，请注意花簇头部的顺时针和逆时针方向的螺旋纹，仔细数出两个方向的螺纹数，取较小的数和较大的数的比率，再和Fibonacci数序的比率比较。

显然，这是一个值得注意的比率，而且当引入到扬声器箱体尺寸后，扬声器箱放音非常优良就没有什么奇怪了。