Multiple Acoustically in Series Connected Speaker Systems

Dipl. Biol. Ernst Georg Beck, Member AES, 1992

0.Introduction

Most work on loudspeaker systems is done on configurations with one speaker. This is especially true for the development of the basic mathematical models for the different cabinet types [1],[2],[3],[4],[5]. Only very few publications deal with parallel or seriell connected systems. [6],[7].

The "Compound Speaker" of Olson [6] and the "Simulated infinite baffle speaker"of Broadhurst [7] use two acustically in series connected speakers. Basic analog networks had been derived and effects on performance were discussed. It is the aim of this paper to establish the general analog networks and to derive and analyse transfer functions to recieve a mathematical model of the acoustically in series connection of loudspeakers.Furtheron verification of the model is presented.

1.0 General Analysis

Fig.1 presentes the impedance type acoustical analogous circuit of a single speaker (infinite baffle) [1]

Fig. 1 Acoustical analogous circuit of an single speaker

The symbols in the circuit are defined as follows: Pg = egBl/(Rg+Re)Sd eg = open circuit output voltage of the infinite baffle speaker B = Magnetic flux density in driver air gap l = length of voice coil conductor in magnetic field of air gap Sd = effective projected surface area of driver diaphragm Rg = output resistance of source or driver Re = DC resistance of source or driver Cas = acoustic compliance of driver suspension Mas = acoustic mass of driver diaphragm assembly and voice coil + air load Rat = Ras + B2l2/(Rg+Re)Sd2

Ras = acoustic resistance of driver suspension losses Uo = total volume velocity

The acoustical analogous circuit of two acoustical in series connected loudspeakers is shown in Fig.2 :

Fig. 2 Acoustical analogous circuit of two acoustical in series connected loudspeakers The symbols in the circuit are defined as follows: pg1 = eg1Bl1/(Rg+Re1)Sd1 pg2 = eg2B2l2/(Rg+Re2)Sd2 eg1 = open circuit output voltage of the infinite baffle speaker 1 eg2 = open circuit output voltage of the infinite baffle speaker 2 B1 = Magnetic flux density in driver 1 air gap B2 = Magnetic flux density in driver 2 air gap l1 = length of voice coil conductor of speaker 1 in magnetic field of air gap l2 = length of voice coil conductor of speaker 2 in magnetic field of air gap Sd1 = effective projected surface area of driver 1 diaphragm Sd2 = effective projected surface area of driver 2 diaphragm Rg1 = output resistance of source or driver 1 Rg2 = output resistance of source or driver 2

Re1 = DC resistance of source or driver 1 Re2 = DC resistance of source or driver 2 Cas1 = acoustic compliance of driver 1 suspension Cas2 = acoustic compliance of driver 2 suspension Caz = acoustic compliance of air in interchamber beetween driver 1 and speaker 2 Mas1 = acoustic mass of driver 1 diaphragm assembly and voice coil + air load Mas2 = acoustic mass of driver 2 diaphragm assembly and voice coil + air load Rat1 = Ras1 + B1l1/(Rg+Re1)Sd1 Rat2 = Ras2 + B2l2/(Rg+Re2)Sd2 Ras1 = acoustic resistance of driver 1 suspension losses Ras2 = acoustic resistance of driver 2 suspension losses Raz = acoustic resistance of interchamber losses Uo = total volume velocity

Mechanical and electrical connection can be performed as in Fig. 3. If n loudspeakers are in acoustical series connection the following analog circuit is valid (Fig. 4)

Fig. 3 Mechanical and electrical connection of two acustical in series connectes speakers

Fig 4. Acustical anlogous circuit of n speakers acustical in series connected

The element group in brackets is repeated n-1 times. Casn-1,Masn-1, Ratn-1,Cazn-1 and Razn-1 corresponds to the eqivalent elements (see Fig. 2). For low frequencies ,small interchamber volumes Caz and neclectable Raz an interesting simplification can be derived.If Caz and Raz are to be assumed as 0 the network reduces to

Fig. 5 Reduced acustical circuit of n speakers acustical in series connected if Caz and Raz =0

In this circuit the elements are   Rats = Rat1 + Rat2 + ... Ratn 1/Cass = 1/Cas1 + 1/Cas2 + ... 1/Casn Mass = Mas1 + Mas2 + ... Masn

This means that Rat and Mas will be increased and Cas decreased. For identical loudspeakers

	Rats = n x Rat1
	Mass = n x Mas1
	Cass = 1/n x Cas1

The circuit is identical to Fig. 1 .Compared to a single direct radiator identical acoustical in series connected speakers can be looked as one speaker with modified circuit elements namely

1 speaker	2 speakers	4 speakers	8 speakers	16 speakers
Rat	2x Rat	4 x Rat	8 x Rat	16 x Rat
Cas	1/2 x Cas	1/4 x Cas	1/8 x Cas	1/16 x Cas
Ma	2 x Mas	4 x Mas	8 x Mas	16 x Mas

and so on.

The resonance frequency of the several systems defined as fs=1/(2 p(CasMas)^1/2) are identical as the Q of the driver resonant circuit Qms,the electrical Q, Qes and total Q, Qts defined as [1]

for one speaker

for two speakers

Qms = 1/(2 pfsCasRas)	Qms2 = 1/(2 pfs1/2 Cas2Ras)
Qes = 2 pfsReMasSd2/(B2l2)	Qes2= 2 pfszRe2xMasSd2/(2B2l2)
Qts = QmsQes/Qms+Qes	Qts2 = Qms2Qes2/Qms2+Qes2

 
z = 1/2 if electrical parallel
z = 2 if electrical in series

Reference efficiency is defined as m =( 4 p²/c³⁾ fs³ Vas/Qes and for 2 speakers m =( 4 p²/c³⁾ fs³ 1/2Vas/Qes and so on.

To take a closer look to such acoustically in series connected systems the simplest form , a system with 2 speakers will be analysed. Furtheron to simplify nomenclature instead of Multiple Acoustically in Series connected Speakers the acronym MASS is used. MASS2 for 2 speakers, MASS4 to name 4 speakers etc.

1.1 Analysis of a MASS2-System

Circuit of Fig. 2 is the basic simplified network to describe such a system. Methods of analysis are those of [1] and [2]. Defining 1/Ca2= 1/Cas2 + 1/ Cab ,Ra2=Rat2+Rab and substituting Ca2 for Cas2 this circuit also represent a closed MASS2. (Cab= acoustic compliance of box volume behind Speaker 2).
For simplification Raz and Rab are neclected.

Fig. 6 Acustical anlogous circuit of a MASS2 closed box system

If Cz is neclected because of very low value (equivalent Volume can be made between 2-5 liters) the circuit reduces to the one in Fig. 5. So it´s possible to achiev differnt values of Thiele/Small-parameters according to what speakers are connected in such a system.

If both speakers are identical (real speakers may have fabrication tolerances up to 20%) and if analysis of (2) is valid :
a_Mass2=Cas1/(Cab*2). To get a_Mass2=a Cab must be halved. This means that it is possible to design closed cabinets with half the enclosure volume and same response as single speaker closed box system. The price to pay is for a second identical speaker.

The simplified electrical equivalent circuit of the closed MASS2 can be found as a dual of Fig.6. Relationship between acoustical and electrical elements is defined by Ze= B²l²/Sd²Za where Ze is the impedance of an element in the electrical circuit and Za is the impedance of the corresponding element in the acoustical analogous circuit.Fig. 7 presents the dual of Fig. 2.

 

Because eg1 and eg2 are identical and Rg1 and Rg2 too it exists serial and parallel connection presented in Fig. 8.

 

The symbols in Fig. 7/8 are defined as follows:

Res1 = electrical resistance due to driver1 suspension
Res2 = electrical resistance due to driver1 suspension
Cmes1 = electrical capacitance due to driver 1 mass
Cmes2 = electrical capacitance due to driver 2 mass
Lcez = electrical inductance due to air in interchamber beetween driver 1 and speaker 2
Lces1 = electrical inductance due to driver 1 compliance
Lces2 = electrical inductance due to driver 2 compliance

Presented circuits are only valid for frequencies within piston range of the system divers. Element values are assumed to be independent of frequencies within this range. Due to (2) voice coil inductance and radiation load resistance are neclected. Analysis is simplified by defining a number of parameters:

Front driver parameters are

	(1) Ts1=1/ ws1=Cas1Mas1=Cmes1Lces1;
	(2) Qms1= ws1Cmes1Res1=1/(ws1Cas1Ras1);
	(3) Vas1=p0c2Cas1;
	(4) Qts1=1/( ws1Cas1Ras1).

Back driver parameters are the same substituting 1 to 2. Eq. (1) defines the resonance frequency of the front driver1 ( ws=2 pfs). In Eq. 10 p₀= density of air (1,18 kg/m³) and c is the velocity of sound in air (345 m/s). Vas1 represents the acustic compliance of the driver suspension in terms of a volume of air (standard condition s of temperature and pressure) which has the same acustic compliance. It is assumed that the value of Mas1 includes the effect of the airload on driver. It is conveniet to define the following system paramters:

(5) K=pg1/pg2=(B₁l₁Sd₂eg₁Re₂)/(B₂l₂Sd₁eg₂Re₁₎; R_g=0

For series connection this reduces to

(6) Ks=B₁l₁SD₂/(B₂l₂Sd₁). For parallel connection there is K=Kp (Eq. 5).

(7) Qtc₂=1/( wc₂Ca₂Rat₂)

(8) Tc₂²= 1/ wc₂²=Ca₂Mas₂

(9) L= Ca₁/Caz

(10) P=Ca₂/Caz

(11) y=Fc₂/fs₁

Following the method of (1), analysis of the circuit of Figs. 6 and 7 and substitution of the parameters defined above yields the functions which describe the system. The response function is:

(12) G(s)=s⁴Ts₁² Tc₂² + s³Ts1²(Tc₂/Qtc₂) + S²Ts₁²(1+P((K+1)/K)) / D(s)

where D(s) = s⁴Ts₁² Tc₂² + s³(Ts1(Tc₂/Qtc₂) + Tc₂²(Ts₁/Qts₁) + s²(Ts₁²(1+P) + (Ts₁Tc₂)/(Qts₁Qtc₂) + Tc₂(1+L)) +

s(Ts₁(1/Qts₁)(1+P) + Tc₂(1/Qtc₂)(1+L)) + (1+L+P) and s= s+j w is the complex frequency variable.

Literature:
1) Small,R.H.,"Direct Radiator Loudspeaker Analysis", J.Audio.Eng.Soc., Vol.20, pp.383-395, (1972 June)
2) Small,R.H.,"Closed Box Loudspeaker Systems", J.Audio.Eng.Soc., Vol 20, pp.798-808, (1972 Dec.);  Vol 21,pp. 11-18 ( 1973 , Jan./Feb.)
3) Small,R.H.,"Vented Box Loudspeaker Systems", J.Audio.Eng.Soc., Vol 21, pp.363-372 (1973 , June);  pp. 438-444 (1973 July/Aug.); pp. 549-554 (1973 Sept.); pp. 635-639 (1973 Oct.)
4) Small,R.H.,"Passive Radiator Loudspeaker Systems", J.Audio.Eng.Soc., Vol. 22, pp.592-601 ( 1974 June); pp.683-689 (1974 Nov.)
5) Keele,D.B.,"Low Frequency Horn Design using Thiele/Small Driver Parameters",J.Audio.Eng. Soc., Preprint No. 1250 (K-7), 57th Convention 1977
6) Olson,H.F.,"Sound transmitting Apparatus",U.S.Patent 2688373, application 1971,patented 1974
7) Broadhurst,A.D.,Loudspeaker Enclosure to simulate an infinite Baffle, Acustica, Vol.39,1978