Computer-Aided Vented-Box Loudspeaker Design with SPICE
by Isaac MCN
loudspeakers with SPICE is shown.
Builders often purchase modelling software in an attempt to design and build cheaper, and hopefully better, loudspeaker systems in comparison to commercially-available units. Accurate loudspeaker simulation software are often very expensive, often the cost of such a package is greater than a commercially-available loudspeaker system. Of course, there are cheap software packages that are quite accurate, but such applications are almost always limited in what features were integrated by the software programmer. With such a limited software at hand, the user is left with the need to buy a separate program that will fill whatever design requirement arrises.
Loudspeakers systems, especially low-frequency systems, can be accurately simulated using lumped-paramter modelling. Parts of the loudspeaker is represented with electronic components such as resistors, capacitors and inductors. Information about electroacoustic modelling can be found in articles from the Journal of the Audio Engineering Society. The more accurate simulations we want, the bigger the required analogous model. As an example, the model used in this paper used a minimum of 24 components to simulate a vented box loudspeaker. Such a big circuit is certainly possible to analyze by hand, but it would take a great deal of calculations. An easier and significantly faster way would be to use a free circuit simulator such as PSPICE. One can make fast analyses to make appropriate system changes which in turn help to attain the required system performance.
The vented-box model in this paper was derived by W. Marshall Leach, Jr. As with the original paper by Leach, a design example is described and the corresponding frequency response plot and impedance magnitude graphs are given. In addition to these two sets of graphs, other screenshots of group delay, cone excursion, cone velocity, port velocity, acoustic phase, impedance phase and impulse response. "This engineering report presents an introduction to the use of SPICE as an aid to electroacoustic system design. SPICE models are given for the basic electrical, mechanical and acoustical elements of electroacoustic modeling and for the transducer elements that couple electrical to mechanical systems and mechanical to acoustical systems. One of the features of the modeling is that controlled sources are used in place of the usual transformer. This makes it possible to use either impedance or mobility models in the analogous circuits. Only the impedance analogous circuits are covered here." [after Leach].
Figure 1 shows a typical vented-box configuration. The distance between the centers of the port and woofer cone, d, is used to determine a particular design's mutual coupling coefficients.
The analogous model in this paper simulates a woofer in a vented-box with a single cylindrical port. Figure 2 shows the slightly modified circuit diagram of the analogous model. PSPICE's Schematic function was used to construct the said schematic diagram.
Fig.2. Controlled-source analogous circuit for vented-box system.
Because no permissions have been granted yet to reproduce significant
parts of the original paper from Leach, no equations and component
descriptions for the analogous model could be included in this paper.)
As can be seen from the circuit diagram, several dependent sources are used to transform electrical energy into acoustical energy. Because of the size and complexity of the analogous model, it will take a considerable amount of time and calculation if one were to analyze the circuit by hand. It would be best to analyze the circuit with a circuit simulator like SPICE. Unlike commercial software that also use lumped-parameter modeling, the model in this paper is very flexible. Various factors in the amplifier side can be included in the simulation. Non-idealities such as ampflier output resistance can be included. The analogous model can also accomodate the effects of voice-coil inductance, which is known to have significant negative effects on passive crossover networks. Both the driver unit's diaphragm and port are modelled as rigid pistons and mutual coupling between both is also modelled. Enclosure leakage losses can also be simulated as well as port losses and box losses. Other lumped-parameter models neglect or approximate such losses, but it is evident that the same losses have a significant effect on actual vented-box performance. The following sections give examples of several data that can be derived from the analogous model.
To give an example, a Vifa P17WJ-00-08 woofer was used. The following is a list of the woofer's Thiele-Small parameters.
The box ratios (d = 1, w = 0.618 and h = 1.618) were chosen to minimize internal resonances. Although such resonances cannot be simulated with the electroacoustic model, they were still used to derived the internal box dimensions, which in turn affect mass loading on the back of the driver unit's cone. Furthermore, the net box volume is 18.84L after taking into account volume displacement in the box such as braces.
Av was chosen to be 1.0in. as the recommended absolute minimum Av was 0.98 in. The absolute minimum Av can be calculated from the following equation [after R.H. Small].
Anything smaller than Av = 1.0in. will result in noisy and turbulent air flow through the port; whereas anything significantly bigger will require a longer Lv. Lv can be calculated from the following equation.
The SBB4, or Super Butterworth Boom-Box, "flat-frequency-response" alignment was chosen for this particular example and the enclosure dimensions are
Figure 3 shows some of the graphs combined into one picture.
Fig.3. Response plots in one picture
Without suspension break-in, the measured sensitivity of the Vifa P17WJ-00-08 was 88dB at 1m with an input of 2.83V. The calculated reference efficiency, no, is 0.00376 or 0.376% and it can be calculated from the following equation.
Therefore the calculated SPL is equal to 87.85dB and it can be calculated from the following equation.
Figure 4 is a closer look at the frequency response.
Fig.4. Overall low frequency response (green).
Port response (red). Woofer response (blue)
The dip in the output of the cone occurs at 36.8Hz or 37Hz, which is the calculated resonant frequency, Fb, of the vented-box. Consequently, the peak port output is centered around 37.5Hz. At Fb, most of the volume velocity comes from the port and there is little, to no, cone movement. The simulated -3dB frequency is 44.5Hz at 85dB.
Figure 5 shows the impedance magnitude and phase plot.
Fig.5. Impedance magnitude (green). Impedance
One can see the characteristic impedance curve of a woofer in a vented-box, which has 2 peaks and a rising impedance, with frequency, due to the voice coil reactance. Minimum impedance magnitude occurs at (400Hz, 5.89ohms). Impedance phase was calculated with respect to the input signal's phase.
From the green curve,
Fl = 19.2 Hz
Assuming for a moment that these are real-world impedance measurements, the total box losses, Qb, can be calculated from the following equations
which yield the following values.
Qb = 6.63
Qtsb = 0.37
Qesb = 0.47
Qmsb = 1.63
Since enclosure leakage losses are accepted to be dominant in vented-box enclosures, then Ql = Qb, and since our target Ql = 7, there's no need to adjust the box volume. One can see that Ql was simulated quite accurately.
Figure 6 shows the cone excursion graph.
Fig.6. Cone Excursion
Naturally the cone will be moving the greatest at the lowest frequency because the cone will be "unloaded." At 10Hz, the cone excursion is equal to 3.82mm, which is almost equal to the Xmax of the P17WJ-00-08. At 37.2Hz, or Fb, cone excursion is equal to 272 micrometers, which is quite small and this is because most of the volume velocity comes from the port. Just above Fb, the cone excursion peaks at 0.815mm (at 57.2Hz) before going down again to near-zero excursion at higher frequencies.
Figure 7 shows the group delay plot.
Fig.7. Group delay (red). Total SPL (green).
Peak group delay is 14.3ms at 20.2Hz. As with any vented-box design, the group delay is worse than closed-box systems. In fact, the best vented-box design has a group delay that is worse compared to the worst closed-box design.
Figure 8 shows the cone velocity plot.
Fig.8. Cone velocity (red). Cone excursion
(blue). Overall SPL (green).
The cone velocity plot has two maxima at (18.5Hz, 0.325m/s) and (71.1Hz, 0.322m/s) while minima occurs at Fb (37Hz, 63.4mm/s). These points correspond to the two maxima points and minima point of the impedance curve above.
Figure 9 shows the port velocity graph.
Fig.9. Port velocity (red). Port "excursion"
(blue). Port SPL (green).
Peak port velocity occurs at 27.5Hz. The vent mach number is equal to 0.0092 and it can be calculated from the following equation.
The port radius used was the absolute minimum, so at high output levels the port linearity will become significantly worse. If the loudspeaker system is going to be used often at high volume levels, better port linearity will be attained with a bigger Av. It is recommended that driver units of this size should have Av sizes of 3 inches or more. The obvious disadvantage of having a larger Av is a longer Lv. Another problem with vents (not just longer vents) is undesirable standing waves in the port.
Figure 10 shows the impulse response of the sample vented-box design.
Fig.10. Overall impulse response (green).
Woofer impulse response (red). Port impulse response (blue).
A 4-volt pulse was applied lasting for only 10us and it has zero rise/fall time. The decay time is about 50ms, which makes it the best alignment, when best transient response is a priority, compared to other vented-box alignments.
One can see the power, accuracy, efficiency and flexibility of the analogous model. SPICE allows for quick design evaluations and modifications. The analogous model can also be modified to accomodate circuit blocks such as crossover networks, notch filters, attenuators, baffle diffraction compensation circuits, impedance compensation circuits and so on. One or more analogous models can be connected in parallel or series combinations to form multi-way loudspeaker systems. Moreover, making the model more accurate or flexible is simply a matter of modifying or adding components to the analogous model. That, when compared to limited commercial software, makes SPICE a versatile, accurate and cheap modelling sofware. However, such extensions to the analogous model is beyond the scope of this paper.
A circuit simulator such as SPICE is a very powerful tool for modeling loudspeakers. "With SPICE, complex systems containing multiple electrical, mechanical and acoustical analogous circuits can be analyzed quickly and efficiently. The four controlled sources of SPICE make it possible to model electro-mechano-acoustic transducers using either impedance analogous circuits, mobility analogous circuits or combiinations of both. This eliminates restrictions imposed on the analogous circuits when the usual transformer is used in modeling transducers." [after Leach].