Passive speaker system: theory and calculation

There is another kind of acoustic design of the loudspeaker, which, like the acoustic design described in the magazines “Radio”, 1972, No. 8 and 1973, No. 6, is capable of providing the reproduction of low frequencies by the loudspeaker with a relatively small box size. It has several names, of which the most correct are: a bass reflex with a passive radiator or a FI with a closed hole.

The peculiarity of this phase inverter is that the loudspeaker is placed in a box with an opening near the place of its installation, with a movable system of the second loudspeaker fixed in it without a magnetic system and a centering washer. The diameter of the diffuser of a passive radiator in acoustics is approximately equal to the diameter of a loudspeaker diffuser. The hole in the voice coil is sealed and in this place, an additional weight is attached to the diffuser. The weight of the load depends mainly on the volume of the box and the resonant frequency of the phase inverter.

The principle of operation with a passive radiator is similar to the principle of operation of a conventional phase inverter (see “Radio”, 1973, No. 8). At the resonant frequency of the closed PI, the diffuser of the passive radiator oscillates in phase with the diffuser of the main woofer, providing effective signal reproduction in the low-frequency region. Thus, in contrast to the main phase inverter, here the mass in the hole is replaced by the mass of the movable system of the passive radiator, including the additional weight.

The weight allows more simply than it is done when measuring the size (volume) of the passage in a conventional phase inverter, to adjust the resonant frequency of the phase inverter. With a decrease in the volume of the box of a conventional phase inverter, it is necessary to increase the volume of the passage or reduce the area of ​​the opening, which reduces the efficiency of the phase inverter. A phase inverter with a closed hole is free from this drawback and this is its main advantage.

Frequency response of a column with a passive radiator

Passive emitters increase the emitting surface area. Two diffusers oscillate together, but in antiphase, therefore, firstly, the level in the low frequency range rises, and secondly, the efficiency of the entire speaker system increases, raising the frequency response by half an octave.

For example, consider the generalized frequency response of the acoustic system before and after inserting a passive radiator.

The graph shows that in the presence of a passive radiator, the frequency response of the speaker system increases significantly in the range from 20 to 500 Hz. And this is the low-frequency region, i.e. the same bass.

Both active and each passive radiator has its own resonant frequency. At this frequency, its fluctuations are maximum.

The main difficulty for any speaker system is usually the lowest frequencies, so the resonant frequency is always tried to be lowered. For this, the diffuser of the passive speaker is made more mass.

Calculation of a passive radiator

Another positive quality of a phase inverter with a closed hole is a slightly greater in-phase movement of both diffusers in the resonance region compared to the movement of the air volume in the hole and the loudspeaker diffuser in a conventional phase inverter. The resonant frequency of a phase inverter with a closed hole is (as well as the usual one):

where m_f – the mass of the moving system of the passive radiator plus the air mass that oscillates with it, attached to the diffuser, g; WITH_f – the resulting flexibility (inverse elasticity) of the air volume in the box and the additional moving system, cm / dyn.

The calculation of the phase inverter with a closed hole is performed as follows: by choosing the volume of the box V_f and knowing the effective diameter of the passive radiator diffuser D_eff determine the flexibility of the air volume from the expression:

Here is the volume of the box V_f expressed in cm³, and the effective diameter of the diffuser of the passive radiator D_eff in cm.Recall that the effective diameter of the diffuser is D_eff = 0.85–0.9 D_diffwhere D_diff – full diameter of the diffuser.

The equivalent effective diameter of the elliptical (oval) diffuser is:

where D_b – large, and D_m – small diameter of the ellipse. Since the flexibility of the suspension of the diffuser of the passive radiator C_under much more than the flexibility of the air volume of box C_f, its influence on the total flexibility is extremely small and can be neglected.

Overall flexibility is determined by the formula:

And when Spod >> Cf, Sobs≈Cf.

Taking, as usual, the resonant frequency of the closed phase inverter, equal to the fundamental resonant frequency of the loudspeaker, we find the mass m_fcorresponding to this frequency and flexibility of the selected volume:

As mentioned above, this mass includes the mass of the diffuser of the passive radiator m_glad and the added mass of air vibrating with it Δm, i.e. m_f = m_glad + Δm. The value of Δm depends on the effective diameter of the diffuser and is determined by the expression Δm = 8 * 10^-4D³_eff Thus, the radiator diffuser must have a mass m_glad = m_f – Δm; practically this value will be equal to the mass of the weight that must be installed on the diffuser. To facilitate the necessary calculations, the table gives the values ​​of the flexibility of the volume C_f for boxes with a volume of 20 to 80 liters and diffusers of a passive radiator with an effective diameter of 15 to 22 cm, the value of the added air mass Δm for the same diameters of diffusers is also indicated there.

Vf, l	Flexibility of the volume of the box, cm / dyn 10-6 at Deff, cm
	15	16	17	eighteen	19	twenty	22
Δ m, g	1.7	3.3	3.9	4.7	5.5	6.4	8.6
twenty	0.45	1.35	0.27	0.22	0.17	0.14	0.1
thirty	0.67	0.52	0.41	0.32	0.26	0.24	0.15
40	0.9	0.69	0.55	0.43	0.35	0.29	0.19
50	1.12	0.87	0.68	0.54	0.44	0.366	0.24
60	0.35	1.04	0.82	0.65	0.52	0.43	0.29
70	1.57	1.21	0.95	0.76	0.61	0.5	0.34
80	1.8	1.4	1.09	0.87	0.7	0.57	0.39

The amount of flexibility of the volume of air in the boxes with intermediate values ​​and the effective diameter of the radiator diffuser is determined by interpolation by two adjacent values ​​of flexibility, between which the accepted dimensions are located.

For example, let us determine the mass of the load that must be fixed on the diffuser of a passive radiator with a diameter of D_diff= 22 cm, installed in a FI box of volume V_f= 50 l at a resonant frequency of the FI of 45 Hz. Effective diameter D_eff= 0.87 * D_diff= 0.87 * 22 = 19 cm. We find from the table the flexibility of the volume of air in the box with such an effective diameter of the diffuser; this flexibility is equal to C_f= 0.44 * 10-6 cm / dyn. The total mass of the diffuser must be:

The attached air mass, according to the table, is Δm = 5.5 g. Therefore, to obtain a given resonant frequency, it is necessary to install an additional weight m_glad = m_f – Δm = 28.4-5.5 ≈ 23 g. The additional weight is a steel or copper (brass) disc of thickness h, which for steel, depending on the diameter of the disc d, is

As stated above, the magnet system and centering washer are removed from the loudspeaker intended …