Starting from the equivalent model given by
Thorburg, Unruh, and Struck:
The impedance of the speaker is the sum of the electrical resistance in the voice coil,
Re, the inductance of the voice coil,
Le, and the driver's electrical equivalent RLC network
Rp,
Cp, and
Lp. Here, the mechanical properties of the loudspeaker are modeled as a mass-spring-damper system as described previously (
here,
here, and again
here). Denoting the mass, mechanical damping, and effective spring stiffness as
m,
c, and
k, the RLC parameters are related to the mechanical parameters through the magnetic coupling (
B*l) as:
Cp = m / (B*l)^2
Rp = (B*l)^2 / c
and
Lp = (B*l)^2 / k
Furthermore, one can right the systems natural frequency
w0 and damping ratio
z in terms of the electrical (
Rp, Cp, Lp) or mechanical parameters (
m, c, k). The dynamics are exactly the same.
w0 = sqrt(
k/m ) =
1 / sqrt(
Lp*Cp)
and
z =
c / (2 * sqrt
( k * m ) ) = sqrt(
Lp / Cp ) / (
2 * Rp)
Therefore, we can write the impedance of the equivalent
parallel RLC network
Zp describing the dynamics of the mechanical system as (in s-domain):
Zp(s) = ( s * (B*l)^2 / m ) / (s^2 + 2*w0*z*s + w0^2)
Therefore, the total impedance for the complete driver equivalent circuit
Zt is the
sum of the impedances of the three components in the circuit,
Re, Le, and
Zp.
Zt(s) = Re + s*Le + Zp
->
Zt(s) = Re + s*Le + ( s * (B*l)^2 / m ) / (s^2 + 2*w0*z*s + w0^2) [ * ]
For steady-state AC sinusoidal analysis (as occurs when measuring the impedance of headphones using a sweep of frequencies), the complex impedance of the driver can be found by replacing the complex frequency
s with the angular frequency
j*w. Rearranging equation [ * ] into the real and imaginary parts, one obtains the complex impedance of the driver
Zt(j*w) =
Rt + j*Xt
where Rt is the real part of the impedance (i.e., the resistive component) and Xt is the imaginary part of the impedance (i.e., the reactive component). These are given as
Rt = Re + ((B*l)^2 / m) * ( 2*z*w0 * (w / w0 )^2 ) / (1 - (1 + 4 * z^2 * w0^2)*(w / w0 )^2 ) [ ** ]
and
Xt = w*( Le + ((B*l)^2 / m) * ( 1 - (w / w0 )^2 ) / (1 - (1 + 4 * z^2 * w0^2)*(w / w0 )^2 ) ) [ *** ]
These expressions show us how the complex impedance of the driver varies as a function of frequency
w, where the traditional frequency in Hz is given by
f = w / (2 * pi ).
There are a few things to note
1) Regarding the
magnetic coupling, highlighted in red in equations [ ** ] and [ *** ] above. If the magnetic coupling is weak (
B*l is small) or if the effective mass of the driver diaphragm is large (
m is big ), then the contribution of the driver mechanics to the total electrical impedance becomes vanishingly small and the total driver impedance is dominated by the voice coil resistance and inductance. Namely, the impedance simplifies to
Zt ~ Re + j*w*Le
when
(B*l)^2 /m is small
Furthermore, if
w*Le is small throughout the audio range (i.e.,
w*Le / Re << 1), the impedance can be further simplified to
Zt ~ Re.
2), Regarding the
frequency dependent contribution of the mechanical system, highlighted in purple in equations [ ** ] and [ *** ] above. When the natural frequency of the system is much higher than any frequency of interest (namely,
w/w0 << 1 for all
w of interest) , the contribution of the mechanical system to the resistive impedance vanishes (the
purple term in [ ** ] becomes vanishingly small for all
w of interest), and the contribution of the mechanical system to the reactive impedance is a constant (specifically, 1; namely, the
purple term in [ *** ] is essentially equal to 1 for all
w of interest). Namely, the impedance simplifies to
Zt ~ Re + j*w*(Le + ((B*l)^2 / m) )
when
w/w0 << 1
Furthermore, if
w*(Le + ((B*l)^2 / m) ) is small throughout the audio range (i.e.,
w*(Le + ((B*l)^2 / m) )/ Re << 1), the impedance can be further simplified to
Zt ~ Re.
3), Regarding the
frequency dependent contribution of the mechanical system, highlighted in purple in equations [ ** ] and [ *** ] above. When the natural frequency of the system is much lower than any frequency of interest (namely,
w/w0 >> 1 for all
w of interest) , the contribution of the mechanical system is constant (the
purple terms in [ ** ] and [ *** ] are independent of
w for all
w of interest). Namely, the impedance simplifies to
Zt ~ Re - ((B*l)^2 / m) * ( 2*z*w0 ) / (1 + 4 * z^2 * w0^2)
+ j*w*(Le + ((B*l)^2 / m) * 1 / (1 + 4 * z^2 * w0^2) )
when
w/w0 >> 1
Furthermore, if
w*(Le + ((B*l)^2 / m) * 1 / (1 + 4 * z^2 * w0^2) ) is small throughout the audio range (i.e.,
w*(Le + ((B*l)^2 / m) * 1 / (1 + 4 * z^2 * w0^2) )/ Re << 1), the impedance can be further simplified to
Zt ~ Re.
So, here I've derived the complex impedance for a speaker model in terms of the dynamics of the mechanical (and equivalently its electrical counterpart) system. The resistive and reactive components of a general dynamic driver are given by equations [ ** ] and [ *** ], respectively. In general, both the resistive and reactive components of a dynamic driver's impedance are nonzero and functions of the excitation frequency
w.
I've examined three situation where the complex impedance simplifies to a constant, real impedance over a limited frequency range. Any other special cases I should take a look at?
It should be clear that there are multiple ways in which a *dynamic driver can exhibit a constant, resistive impedance over a limited frequency range, each with a different physical interpretation of the driver mechanics.
The next step is to relate the total system impedance (including the amplifier's damping factor into
Re) to electrical damping and understanding how the
mechanical system is affected by these external circuit parameters.
Cheers
