Acoustics

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6.3. Electro-mechano-acoustical circuit

Before drawing a circuit diagram for a loudspeaker, we must identify the various elements involved. The voice coil has inductance and resistance, which we shall call L _E and R _E, respectively. The diaphragm and the wire on the voice coil have a total mass M _MD. The diaphragm is mounted on flexible suspensions at the center and at the edge. The total effect of these suspensions may be represented by a mechanical compliance C _MS and a mechanical resistance R _MS = 1/G _MS, where G _MS is the mechanical conductance. The air cavity and the holes at the rear of the center portion of the diaphragm form an acoustic network which, in most loudspeakers, can be neglected in analysis because they have no appreciable influence on the performance of the loudspeaker. However, both the rear and the front side of the main part of the diaphragm radiate sound into the open air.

An acoustic radiation impedance is assigned to each side and is designated as Z _AR = l/Y _AR, where Y _AR is the acoustic radiation admittance. Thus, the mechanical radiation admittance seen by each side of the diaphragm is Y _MR = S _D ² Y _AR, where S _D is the effective diaphragm area. Approximate equivalent circuits for Y _MR and Y _AR are given in Fig. 4.37c and d, respectively.

We observe that one side of each flexible suspension is at zero velocity. For the mechanical resistance, this also must be true because it is contained in the suspensions. We already know from earlier chapters that one side of the mass and one side of the radiation admittance must be considered as having zero velocity. Similarly, we note that the other sides of the masses, the compliance, the conductance, and the radiation admittances all have the same velocity, viz., that of the voice coil.

From inspection, we are able to draw a mechanical circuit and the electromechanical analogous circuit using the admittance analogy. These are shown in Fig. 6.2a and b, respectively. The symbols have the following meanings:

${\tilde{e}}_{g}$ ${\tilde{e}}_{g}$ is open-circuit voltage of the generator (audio amplifier) in volts (V).
R _g is generator resistance in electrical ohms (Ω).
L _E is inductance of voice coil in henrys (H), measured with the voice-coil movement blocked, i.e., for ${\tilde{u}}_{c} = o$ ${\tilde{u}}_{c} = o$ .
R _E is resistance of voice coil in electrical ohms (Ω), measured in the same manner as L _E.
B is steady air-gap magnetic field or flux density in Tesla (T).
l is length of wire in m on the voice-coil winding.
$\tilde{i}$ $\tilde{i}$ is electric current in amperes (A) through the voice-coil winding.
Figure 6.2 (a) Mechanical circuit of direct-radiator loudspeaker; (b) electro-mechano-acoustical analogous circuit of the admittance type; (c) electrical circuit showing static electrical impedance Z _ES and motional electrical impedance Z _EM; and (d) analogous circuit of the admittance type with electrical quantities referred to the mechanical side.
${\tilde{f}}_{c}$ ${\tilde{f}}_{c}$ is force in N generated by interaction between the alternating and steady mmfs, that is, ${\tilde{f}}_{c} = B l \tilde{i}$ ${\tilde{f}}_{c} = B l \tilde{i}$ .
${\tilde{u}}_{c}$ ${\tilde{u}}_{c}$ is voice-coil velocity in m/s, that is, ${\tilde{u}}_{c} = \tilde{e} / B l$ ${\tilde{u}}_{c} = \tilde{e} / B l$ , where $\tilde{e}$ $\tilde{e}$ is the so-called counter emf.
a is radius of diaphragm in m.
M _MD is mass of the diaphragm and the voice coil in kg.
C _MS is total mechanical compliance of the suspensions in m/N.
G _MS = 1/R _MS is mechanical conductance of the suspension in m·N⁻¹·s⁻¹.
R _MS is mechanical resistance of the suspensions in N·s/m.
Y _MR = 1/Z _MR = G _MR + jB _MR is mechanical radiation admittance in m·N⁻¹·s⁻¹ from one side of the diaphragm (see Fig. 4.36). The bold G indicates that G _MR varies with frequency.
Z _MR = R _MR + jX _MR is mechanical radiation impedance in N·s/m from one side of a piston of radius a mounted in an infinite baffle (see Fig. 4.35). The bold R indicates that R _MS varies with frequency.
S _D = πa ² is effective area of diaphragm in m².
${\tilde{p}}_{R}$ ${\tilde{p}}_{R}$ is pressure on the diaphragm due to the radiation load, that is, ${\tilde{p}}_{R} = 2 {\tilde{U}}_{c} / Y_{M R}$ ${\tilde{p}}_{R} = 2 {\tilde{U}}_{c} / Y_{M R}$ .
${\tilde{U}}_{c}$ ${\tilde{U}}_{c}$ is volume velocity produced by the diaphragm, that is, ${\tilde{U}}_{c} = S_{D} {\tilde{u}}_{c}$ ${\tilde{U}}_{c} = S_{D} {\tilde{u}}_{c}$ .

It should be noted that the coil inductance L _E is highly nonlinear. In practice, the reactive coil impedance does not rise linearly with frequency but is roughly proportional to the square root of frequency. A more accurate model [4] can be made by adding a second inductor with a resistor in parallel with it, but in this text we shall use the simple model with a single inductor.

The circuit of Fig. 6.2b with the mechanical side brought through the transformer to the electrical side is shown in Fig. 6.2c. Hence, this represents the circuit as seen from the input terminals. It is important for several reasons. Firstly, we have to take the electrical impedance into account when considering the load presented by the loudspeaker to the amplifier driving it. The loading effect will also modify the frequency response of any passive crossover network that may be used. In addition, we can calculate the parameters of a drive unit by measuring the input impedance, as will be explained in Section 6.10, without the need for an anechoic chamber. The mechanical admittance

Y_{M 1} = {\tilde{u}}_{c} / {\tilde{f}}_{c}

$Y_{M 1} = {\tilde{u}}_{c} / {\tilde{f}}_{c}$

is zero if the diaphragm is blocked so that there is no motion

({\tilde{u}}_{c} = 0)

$({\tilde{u}}_{c} = 0)$

but has a value different from zero whenever there is motion. For this reason, the quantity Z _EM = B ² l ² Y _M1 is usually called the motional electrical impedance. A quantity often found on data sheets is the electrical suspension resistance R _ES = B ² l ² G _MS = B ² l ²/R _MS. This resistance is in series with the coil resistance R _E at resonance. When the electrical side is brought over to the mechanical side, we have the circuit of Fig. 6.2d.

The circuit of Fig. 6.2d will be easier to solve if its form is modified. First, we recognize the equivalence of the two circuits shown in Fig. 6.3a and b according to Norton's theorem (see Fig. 14.4). Next, we substitute Fig. 6.3b for its equivalent in Fig. 6.2d. Then, we take the dual of Fig. 6.2d to obtain Fig. 6.4a. (See Figs. 3.41 and 3.42).

Figure 6.3 The electrical circuit (referred to the mechanical side) is shown here in two equivalent forms (a and b) according to Norton's theorem. The circuits are of the admittance type. (Note: The generator in (b) is of constant flow type.).

Figure 6.4 (a) Low-frequency analogous circuit of the impedance type with electrical quantities referred to mechanical side. Z _MR is given by Fig. 4.35. The quantity ${\tilde{f}}_{c}$ ${\tilde{f}}_{c}$ represents the total force acting in the equivalent circuit to produce the voice-coil velocity ${\tilde{u}}_{c}$ ${\tilde{u}}_{c}$ . (b) Single-loop approximation to Fig. 6.4a valid for X _MR ² ≫ R _MR ².

The performance of a direct-radiator loudspeaker is directly related to the diaphragm velocity. Having solved for it, we may compute the acoustic power radiated and the sound pressure produced at any given distance from the loudspeaker in the far field.

Voice-coil velocity at medium and low frequencies

The voice-coil velocity

{\tilde{u}}_{c}

${\tilde{u}}_{c}$

, neglecting ω ² L ² compared with (R _g + R _E)², is found from Fig. 6.4a,

{\tilde{u}}_{c} \approx \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) (R_{M} + j X_{M})}

${\tilde{u}}_{c} \approx \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) (R_{M} + j X_{M})}$

(6.1)

where

R_{M} = \frac{B^{2} l^{2}}{R_{g} + R_{E}} + R_{M S} + 2 R_{M R}

$R_{M} = \frac{B^{2} l^{2}}{R_{g} + R_{E}} + R_{M S} + 2 R_{M R}$

(6.2)

X_{M} = ω M_{M} = ω M_{M D} + 2 X_{M R} - \frac{1}{ω C_{M S}}

$X_{M} = ω M_{M} = ω M_{M D} + 2 X_{M R} - \frac{1}{ω C_{M S}}$

(6.3)

Voice-coil velocity at low frequencies

At low frequencies, assuming in addition that X _MR ² ≫ R _MR ², we have from Fig. 6.4b that

{(X_{M})}_{low f} = ω (M_{M D} + 2 M_{M 1}) - \frac{1}{ω C_{M S}}

${(X_{M})}_{low f} = ω (M_{M D} + 2 M_{M 1}) - \frac{1}{ω C_{M S}}$

(6.4)

where

M_{M 1} = 2.67 a^{3} ρ_{0}

$M_{M 1} = 2.67 a^{3} ρ_{0}$

(6.5)

is the mass in kg contributed by the air load on one side of the piston for the frequency range in which ka < 0.5. (See Table 4.4). The quantity ka equals the ratio of the circumference of the diaphragm to the wavelength.

The voice-coil velocity is found from Eq. (6.1), using Eqs. (6.2) and (6.4) for R _M and X _M, respectively, so that

{\tilde{u}}_{c} = \frac{{\tilde{e}}_{g}}{B l Q_{E S}} β_{c} (f)

${\tilde{u}}_{c} = \frac{{\tilde{e}}_{g}}{B l Q_{E S}} β_{c} (f)$

(6.6)

where β _c(f) is a dimensionless frequency response function given by

β_{c} (f) = \frac{j \frac{f}{f_{S}}}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}

$β_{c} (f) = \frac{j \frac{f}{f_{S}}}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}$

(6.7)

The suspension resonance frequency f _S is given by

f_{S} = \frac{1}{2 π \sqrt{M_{M S} C_{M S}}},

$f_{S} = \frac{1}{2 π \sqrt{M_{M S} C_{M S}}},$

(6.8)

where M _MS = M _MD + 2M _M1 is the combined diaphragm and air-load mass, and

Q_{T S} = {(\frac{B^{2} l^{2}}{R_{g} + R_{E}} + R_{M S})}^{- 1} \sqrt{\frac{M_{M S}}{C_{M S}}} .

$Q_{T S} = {(\frac{B^{2} l^{2}}{R_{g} + R_{E}} + R_{M S})}^{- 1} \sqrt{\frac{M_{M S}}{C_{M S}}} .$

(6.9)

When f = f _S, the real terms in the denominator of Eq. (6.7) vanish, and we see from Eq. (6.9) that the total Q value of the suspension resonance equals Q _TS where Q _TS is the reciprocal of the effective resistance in the mechanical circuit multiplied by the square root of the ratio of the mass to the compliance of the diaphragm. If we define f ₁ and f ₂ as the frequencies at which the velocity is 3 dB below its peak value, then Q _TS = f _S/(f ₂ − f ₁). Therefore, increasing the Q value increases the height of the resonance peak while decreasing its width. At f _S, the inertial and static reactances in Fig. 6.4b cancel each other so that the velocity

{\tilde{u}}_{c}

${\tilde{u}}_{c}$

is simply the driving force (first term in Eq. 6.6) divided by the total resistance in the loop, as shown in Fig. 6.6b. The total Q can be separated into two parts

Q_{T S} = \frac{1}{\frac{1}{Q_{E S}} + \frac{1}{Q_{M S}}} = \frac{Q_{E S} Q_{M S}}{Q_{E S} + Q_{M S}}

$Q_{T S} = \frac{1}{\frac{1}{Q_{E S}} + \frac{1}{Q_{M S}}} = \frac{Q_{E S} Q_{M S}}{Q_{E S} + Q_{M S}}$

(6.10)

namely the electrical Q

Q_{E S} = \frac{R_{g} + R_{E}}{B^{2} l^{2}} \sqrt{\frac{M_{M S}}{C_{M S}}}

$Q_{E S} = \frac{R_{g} + R_{E}}{B^{2} l^{2}} \sqrt{\frac{M_{M S}}{C_{M S}}}$

(6.11)

and the mechanical Q

Q_{M S} = \frac{1}{R_{M S}} \sqrt{\frac{M_{M S}}{C_{M S}}}

$Q_{M S} = \frac{1}{R_{M S}} \sqrt{\frac{M_{M S}}{C_{M S}}}$

(6.12)

The normalized velocity is plotted in Fig. 6.5 using 20 log₁₀|β _c|. It is a universal resonance curve. Below the resonance frequency, it has a slope of +6 dB per octave of frequency. Above the resonance frequency, it has a slope of −6 dB per octave. The acceleration is given by the first time derivative of the velocity

Figure 6.5 Normalized voice-coil displacement, velocity, and acceleration. The solid line is for $Q_{T S} = 1 / \sqrt{2}$ $Q_{T S} = 1 / \sqrt{2}$ . The dashed line is for Q _TS = 2.

{\tilde{a}}_{c} = j ω {\tilde{u}}_{c} = \frac{2 π f_{S} {\tilde{e}}_{g}}{B l Q_{E S}} α_{c} (f)

${\tilde{a}}_{c} = j ω {\tilde{u}}_{c} = \frac{2 π f_{S} {\tilde{e}}_{g}}{B l Q_{E S}} α_{c} (f)$

(6.13)

where α _c(f) is a dimensionless frequency response function given by

α_{c} (f) = \frac{- \frac{f^{2}}{f_{S}^{2}}}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}

$α_{c} (f) = \frac{- \frac{f^{2}}{f_{S}^{2}}}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}$

(6.14)

The displacement is given by the first time integral of the velocity

{\tilde{η}}_{c} = \frac{{\tilde{u}}_{c}}{j ω} = \frac{{\tilde{e}}_{g}}{2 π f_{S} B l Q_{E S}} γ_{c} (f)

${\tilde{η}}_{c} = \frac{{\tilde{u}}_{c}}{j ω} = \frac{{\tilde{e}}_{g}}{2 π f_{S} B l Q_{E S}} γ_{c} (f)$

(6.15)

where γ _c(f) is a dimensionless frequency response function given by

γ_{c} (f) = \frac{1}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}

$γ_{c} (f) = \frac{1}{1 - \frac{f^{2}}{f_{S}^{2}} + j \frac{1}{Q_{T S}} \cdot \frac{f}{f_{S}}}$

(6.16)

The normalized displacement and acceleration are also plotted in Fig. 6.5 along with the velocity. We see that when f/f _S ≤ 1/3, the displacement is virtually constant. This is the stiffness-controlled range in which the displacement is simply the static deflection as determined by Hooke's law, that is, the product of the driving force and the compliance:

{{\tilde{η}}_{c} |}_{f \leq 1 / 3 f_{S}} \approx \frac{{\tilde{e}}_{g} B l}{R_{g} + R_{E}} C_{M S}

${{\tilde{η}}_{c} |}_{f \leq 1 / 3 f_{S}} \approx \frac{{\tilde{e}}_{g} B l}{R_{g} + R_{E}} C_{M S}$

(6.17)

The displacement curve is that of a second-order low-pass filter with a 12 dB/octave slope when f/f _S ≥ 3. As is seen from Eq. (6.16), the displacement in this range is proportional to 1/f ², and the equivalent circuit is that shown in Fig. 6.6a. When f/f _S ≥ 3, the acceleration is virtually constant. This is the mass-controlled range in which the acceleration is simply the driving force divided by the mass in accordance with Newton's second law of motion

{{\tilde{a}}_{c} |}_{f \leq 3 f_{S}} \approx \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) M_{M S}}

${{\tilde{a}}_{c} |}_{f \leq 3 f_{S}} \approx \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) M_{M S}}$

(6.18)

The acceleration curve is that of a second-order high-pass filter with a 12 dB/octave slope when f/f _S ≤ 1/3. As can be seen from Eq. (6.16), the acceleration in this range is proportional to f ², and the equivalent circuit is that shown in Fig. 6.6c.

Figure 6.6 Simplified forms of the circuit of Fig. 6.4a valid for very low frequencies (a), at principle resonance frequency (b), above principle resonance frequency (c), and at high frequencies (d).

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Table of Contents for Acoustics

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6.3. Electro-mechano-acoustical circuit

Voice-coil velocity at medium and low frequencies

Voice-coil velocity at low frequencies

Table of Contents for
Acoustics