Determining Xmax of the Wharfedale Super 12 RS

When my first discussion of the Super 12 RS/DD was uploaded in mid 2017, not as much attention to Xmax for that unit was given. That wasn't realized until early June 2021 when I received a query about the Super 12. That person's name is Richard and that's all the mention I will make of him. To answer his questions, I had to re-open the Super 12 files which exposed a couple of assumptions made at the time. One was its power handling capacity and the other was its Xmax. Upon delving deeper in to book A Pair of Wharfedales, it came to my attention that the Super 12 RS/DD was rated at 20Wrms or 40Wpeak. In my original Bass Box Pro data, I entered 50W and by quick testing, concluded that Xmax could safely be 4mm. Well, I was wrong and although the errors weren't serious, correction was necessary. So, Richard, if you are reading this, you have my most sincere appreciation.

Photo 1: Determining when waveform distortion occurs due to loss of motor drive force. At excessive diaphragm excursions, the upper and lower windings will leave the highest flux density. When this happens, the force applied to the moving system decreases in accordance to the equation F=BLI, where F is the force in Newtons, B is the magnetic flux density, currently in Teslas, L is the length, in meters, of the wire in the field and I is the current, in Amperes, through that wire. Fig.1 pictorially shows this, or at least attempts to.

The putty is to seal the leak around the frame which made an impedance curve resemble that of a vented enclosure.

Photo 2: The mic used in photo 1 ia a Behringer ECM8000. Photo 2 shows the ART tube pre-amp (left) for the mic and on the right is the PICO 2205A computer operated oscilloscope. My old Tektronix 465 is usually preferred but the PICO doesn't require a camera. The two black boxes are ADCOM GFP-565 pre-amp and ADCOM GFA-power amp.

Photo 1

Photo 2

Figure 1

The top plate and pole piece of the Super 12 RS/DD, to scale. Since the voice coil is height is 7/16ths inch (11mm) and the top plate is 10mm, by the usual method of calculating Xmax we subtract the plate thickness from the coil height and divide by 2. This gives us a measly half a millimeter excursion which is incorrect. The red curves shop the flux density above and below the gap. Not having a magnetometer, the exact height of this field has to be determined otherwise. Since the gap is 0.050", the field will bridge that gap to a height whose radius is that of the gap, or damn close to it. In this case, that comes out to 1.27mm. Add that to the half millimeter previously calculated and we have 1.77mm for Xmax. The useful field strength may exceed that figure and that what this experiment is all about.

Pre-amble to the oscillographs Following is a set of oscilloscope traces captured with the PICO 2205A. Their main purpose is to show what happens to a waveform when the driving force is diminished, such as when the voice coil begins to leave the strong part of the magnetic field. These traces were captured for the determination of Xmax. Once the waveform began to distort, excursion was measured and found to be considerably higher than the calculated 1.77mm stated in Fig.1. Two frequencies were used, 20hz and 45hz, somewhat arbitrarily chosen. The 20hz frequency is thought to show what happens when the air in the enclosure loses control of the diaphragm. This will happen in a closed or vented box but much more-so in a vented box. In a well sealed box, pressure slowly applied by hand to the diaphragm will be met with considerable resistance but in a vented enclosure, that resistance is almost non-existent. The sealed box works much like a shock absorber even when the pressure is applied slowly but a vented box will behave like a leaky shock absorber; a rapid pressure applied will be met with some resistance but a slowly applied pressure will appear to have no resistance. In a vented box, this occurs at Fb, the box tuned frequency, below which the enclosure's control of the diaphragm decreases. In a sealed box, for example, turntable rumble will be highly damped but in a vented box, the diaphragm can flop around like a sheet in the wind. That's why early amplifiers had a rumble filter. Later, a more acceptable term was used, subsonic filter. It was usually a second order high pass filter at 18hz so as not to upset the audiophiles into thinking their 20hz notes would be compromised. In all the figures below, the voltage mentioned is the voltage delivered to the speaker by the amplifier. The stated power transfer is referenced to the impedance of the speaker at that frequency. So, at 20hz, the impedance is 14W and at 45hz it's 44W. Figs. 9 and 10 show the maximum the cone can be displaced before serious waveform deformation. Measurement of the excursion was 3mm. This was a two step process using a version of a depth gauge. Measuring the depth with the cone at rest and repeating the measurement with the cone in oscillation and subtracting the two.
Figure 2 20hz 0.62Vrms P=0.027W
Figure 3 20hz 0.62Vrms The vertical here is 5v because I was about to increase the voltage. See Figs. 4, 5 and 6.
Figure 4 20hz 2.1Vrms P=0.315W Deformity visible. The coil is beyond the fringe of the magnetic field. The cone excursion is visible from several feet away.
Figure 5 20hz 2.8Vrms P=0.56W
Figure 6 20hz 4.0Vrms P=1.143W Here the vertical scale is 10V the cone is moving to the point of physically deforming and making noise. I decided not to push it any further. Only 1 watt and this fella looks like it's about to leave the basket. It was decided to repeat the tests at a higher frequency; 45hx was chosen because it's close to the bottom of the low frequencies of most music.

Figure 7 45hz 1.15Vrms P=0.030W The vertical scale in Figs. 7, 8, 8 and 10 is 20V
Figure 8 45hz 4.1Vrms P=0.382W
Figure 9 45hz 7.2Vrms P=1.178W Just starting to deform
Figure 10 45hz 7.83Vrms P=1.393W This is as far as I could go as the output from the mic has just reached the maximum input level of the mic preamp. As a result, the pre-amp clips and the waveform flattens at the peaks. I guess I lucked out as the waveform deformation is quite obvious. the deformation here and in Fig.9 is hardly noticeable. Initially I was concerned about the time it took to make the latter measurement w2ith the cone moving so much until I realized that only 1.4 watt is delivered which is nothing for the speaker and even less for the 65W amplifier.

OK. So what has the difference between rating the speaker at 50W with an Xmax of 4mm as compared to 40W and 3mm? Not much as will be attempted to explain in the following graphs. These calculations, although theoretical to some degree are quite reliable. The laws of acoustics applied to a loudspeaker state that the speaker be mounted on a theoretically infinitely flat baffle and radiating hemispherically into free space; no walls, ceiling or floor. For example, the 6dB decrease in SPL when doubling the distance from the source applies only under the above hemispherical conditions. In a reverberant field, such as a room, this is not the case. Most speakers' responses are measured in an anechoic chamber. What the mic "hears" is what the speaker produces, void of all reflections. While his will show that which the speaker can do; all that goes to hell in a handcart when the speaker is placed in a room.

Figure 11

ORN = Vented 0.947 cu.ft. (optimum) Xmax = 4mm 50W rating This is the optimum enclosure volume for these parameters. An extra mm of excursion isn't going to hurt the speaker with music as the pulse durations are short. That's based on the premise that one is running 50W into the speaker which is very seldom the case.

YEL = Vented 0.947 cu.ft. (optimum) Xmax = 3mm 50W rating This yellow curve shows that the speaker in this size enclosure will stay within its 3mm with only 40W between 55hz and 75hz. However, at moderate listening levels, this point probably won't be reached.

PINK = Vented 0.947 cu.ft. (optimum) Xmax = 3mm 40W rating Rating the speaker at 40W and 3mm shows that it will handle 40W down to 45hz without exceeding 3mm excursion, in a vented box of 1 cu. ft.

LT.GRN = Vented 2 cu.ft. (extended bass output) 50/40W rating The bass output here can be extended but the power handling diminishes. This is also true with a 40W and 3mm rating.

DARK GRN = Vented 3 cu.ft. (further extended bass output) 50/40W rating As can be seen, as the box volume increases, the power handling diminishes. Keep in mind that back in the fifties, most home audio amplifiers had 30 watts or less power, A 50 watt amplifier back then was jokingly referred to as an arc welder.

THE TANGENT: that upon which I tend to wander

Compare the above with the Wharfedale W15FS or CS which was used in the SFB3, Sand Filled Baffle, 3-way, an open back system. The Xmax of this driver is 4mm. The following figures show the W15 in what is close to an open back baffle. The RED is the optimum at 5 cu.ft; the ORANGE is 50 cu.ft. vented and the YELLOW is 50 cu.ft. closed.

Figure 12

These curves can be very deceiving in and by themselves. It may appear that a slight bass boost would elevate the low frequency output. This is true as long as one lowers the power to the speaker. However, that doesn't happen even at low volume levels because a loud crescendo will drive the amplifier into higher power output, even into clipping. The output is proportional to the input meaning that a 10dB rise in the input results in a 10dB rise in the output, regardless of the position of the volume control.

Figure 13

Now, let's look at who or what is behind the curtain. These show the the maximum power allowable to keep the diaphragm excursion under 4mm. Even the optimum enclosure volume of 5 cu.ft. has limitations. The maximum allowable power begins to decrease at 73hz, reaching a minimum of 13W at 39hz. It begins to rise a little to 16W at 30hz due to the action of the vent.

The larger (open back simulated) enclosures, the orange and yellow require ever diminishing power transfer to keep the diaphragm within 4mm. I built two of these and was quite surprised at their performance. Despite the bass cancellation due to doublet action, the bass response was quite impressive and smooth, even at levels around 90dB. provided the bass is trimmed by discretionary use of the bass control.

Figure 14

As labeled, this is the cone displacement as a function of frequency, assuming a constant power transfer of 50W. The red, optimum enclosure will slightly exceed 4mm (4.21mm) at 48hz. Here is where the vent does its thing, which will put the brakes on the diaphragm, having its maximum effect at the tuned frequency, Fb, at 29.5hz, holding the excursion to 2mm after which the vent loses control and the cone is free to do as it pleases. (See the second paragraph in the Preamble to the oscillographs above Fig.2.

In the simulated open back enclosures, yellow and orange, the cone has no control except the damping of the magnetic field flux. In both these cases, more than 50W will allow the cone to exceed 4mm. At 40hz, 50W will push the cone to 6mm and at 30hz, 8mm which will, in all probability, damage the speaker. This is usually experienced by loud strange noises accompanied by separation of the cone from the spider resulting in the coil slamming on top of the pole piece and serious deformation of the aforementioned coil.

In the vernacular, ya just blew the speaker.