Understanding Thiele/Small Parameters

Although it may pale in comparison to the wonders of the PS3, which can push upwards of 100 million textured, occluded, bump-mapped, shaded anti-aliased polygons per second, it wasn't many decades ago that recorded sound--and the reproduction of the recording--was still pure magic. Heck even now, what's not absolutely incredible about the concept that a conical (well, usually) contraption of metal, paper, plastic and wire can vibrate with such precision that the digital audio processing pathways require 16 and 24 bits? As you might imagine, the design of loudspeakers has benefited tremendously from modern computing power, with FEM solvers and acoustic modeling packages providing super accurate design resources. However, just because we have Cadence doesn't mean we don't need to learn KVL. Furthermore, for hobbyists like me--or more precisely physics morons like me--a complete 3D simulation of a loudspeaker is not only unfeasible it kinda takes the fun out of it.

Before all this simulation business, in the 60's and 70's, a standardized set of parameters was developed which forms the bare minimum of information needed to design a loudspeaker with the driver in question. Think of these parameters as the SPICE "Level-0" model equivalent, with μ_n, C_ox, V_t and the like. What the T/S parameters do for drivers is the same as what the hybrid-π or lumped element models do for circuits; they abstract a very complex mechanism in a way that still has some physical meaning, so that we may design our speakers in an intuitive, methodical manner.


The complexity of a loudspeaker driver is obvious; the magnetic field formed by the magnet and pole pieces is nonlinear, the diaphragm has multiple resonance modes, the surround has nonlinear compliance and damping, the forward and backward radiation impedances are unequal, and the dust cap and cone have a nonuniform radiation pattern, to name a few. The thing is, awesome speakers were designed before we had to think about all these issues, so I maintain it's not absolutely necessary to deal with all the second order effects, and we can do quite well with just T/S parameters. Therefore we ought to draw as much insight as possible from these parameters, and understand what they mean.

I should mention that in the process of organizing my notes to make a coherent blog post, I did some Wikipedia searching to standardize my notation, at which point I discovered that as with most things I do, someone else has already done it better (and written about it). So if you want to see what I would have liked to be able to say I'd written, check here.

Alright so let's start at the beginning. I'm going to look at this from a circuit point of view, since I know most of nothing about physics. We're going to formulate a lumped element model of the driver; that means that things which are supposed to be solid stay solid (woofer cones) and things which are supposed to bend do so linearly (driver suspensions). We'll start with the parameters we know, namely those given by the manufacturer or acquired from direct measurement, then determine the composite parameters and what they mean.

Mechanical Parameters

• S_d (m²): The cone surface area.
• X_max (m): The maximum linear excursion of the woofer, which although goes by different definitions, is generally the maximum excursion in which the voice coil stays in the linear portion of the magnetic field.
• M_md (kg): This is the effective cone mass, which is comprised of the actual cone mass plus the mass of the volume of air surrounding the cone, which is related to S_d.
  
• C_ms (m/N): This is the compliance of the driver suspension system. Compliance is the inverse of stiffness, known best as the "spring constant k, such that f = -kx" in that problem on that exam I failed freshman year.
  
• R_ms (Ns/m): This is the mechanical resistance of the suspension system, it's analogous to the damping factor of the "dashpot" in the standard second order resonator.
  

Electrical Parameters

• R_evc (Ω): The parasitic resistance of the voice coil.
• L_evc (H): The actual electrical inductance of the voice coil. This is difficult to measure, because any current you put through the voice coil will cause the cone to move, thereby inducing a back-EMF from the magnet structure. It's most common to measure this at a very high frequency well beyond the cone resonant frequency, but this leads to other problems as it is a very nonlinear (frequency dependent) parameter.
  
• Z_pk (Ω): Maximum driver impedance, measured at the driver resonant frequency F_s.
  
• B_l (N/A): This is an electromechanical parameter, really. It's the efficiency of the linear motor, the force per amp of the voice coil. It's a function of the number of turns of the voice coil, the area of the voice coil, and the flux strength in the gap.
• P_e (W): This is the rated power handling of the driver.
  

Let's list the mechanical equations which govern the cone motion. If we define x to be the cone displacement (along the axis it is supposed to move) then for the cone mass we have:


which can also be written as

The compliance of the suspension exerts a restoring force on the cone by the function,

But if we differentiate once this becomes:


And finally the mechanical resistance of the suspension resists the cone motion with a force proportional to the cone's velocity:



At this point, we need to decide what sort of analogy we want to make to circuits. The impedance analogy says velocity is equivalent to current, and force is equivalent to voltage. The mobility analogy says that force is equivalent to current, and velocity is equivalent to voltage. The impedance analogy is useful for acoustic circuits, because you get series circuits which have physical correspondence to the real world; the mobility analogy is useful for visualizing mechanical circuits, where there is an almost visual correspondence between KCL, which says all currents into a node sum to zero, and the law of conservation of momentum, which pretty much says the same thing but with different variables.

It goes without saying that all our results should be valid regardless of the analogy we make; as a matter of fact, the resultant network under one analogy can be transformed to the network under the other analogy by taking the network conjugate. Still, for ease of visualization I will choose the mobility analogy. Under this analogy, we can think of circuit branches as linkages through which force is transferred, which isn't too far from what's actually happening:

Under this analogy, the following transformations are made:


Now let's consider how to model the linear motor which drives the cone. The way I think of it is as an ideal electromechanical transformer, with a B_l:1 turns ratio. That is, for one amp of current flowing into the dot on the primary, B_l×1 Newtons of force flow out of the dot on the secondary. Thus the complete driver model looks like this:

Now we are ready to investigate the composite T/S parameters, namely F_s, Q_ms, Q_es, Q_ts, and V_as.


F_s, Driver Resonance Frequency

The cone mass and suspension compliance form a second order resonator, and as such it has a resonant frequency. We know the resonant frequency of an LC tank, it's given by the following equation:


Q_ms, Driver Mechanical Q

Loosely speaking, the Q of a system is a measure of how lossy it is. To measure this quantity, you need to know both how much energy it can store, and how much energy is dissipated. Note that nowhere in this definition is the requirement that the system is any sort of oscillator or resonator; an RC or LC has a perfectly well defined Q. Of course, the value of Q will be frequency dependent, since L and C impedance is a frequency dependent parameter. For resonators, we have a more specific definition of Q, since resonators are designed to operate at a particular frequency, the Q is defined as the energy stored by the energy lost over one period at the resonant frequency.

For a parallel RLC, the Q works out to be defined as

where ω_s is the resonant frequency in radians^-1. Looking at the mechanical portion of our circuit (ignoring the transformer action and back-EMF for a moment), we have the following expression:

However, there are often times that R_ms is not given, as driver datasheets are notoriously non-standardized, and some manufacturers think an illegible handwritten chart is enough. There is a way to compute R_ms using what we know about resonance and transformers. At mechanical resonance, the inductor and capacitor cancel each other perfectly, so the circuit looks like this:
But using the impedance transformation properties of the transformer, we redraw the circuit as
So the value of the impedance at the resonant frequency is simply Z_pk = R_evc + 2πf_sL_evc + B_l²/R_ms. With woofers which have resonant frequencies in the 100Hz range and lower, the impedance from the inductor is negligible, so we simply have two resistance terms. It's very easy to measure Z_pk with a multimeter and frequency generator, and R_evc as well; therefore, an alternative formulation for Q_ms is as follows:


Qes, Driver Electrical Q

Just as the suspension losses dampen the cone oscillation, as the cone oscillates a back-EMF is generated in the voice coil, which is then dissipated in R_evc. Since we assume that we are driving the voice coil with a voltage source (not always valid once the crossover network is introduced) the circuit can be redrawn as follows, this time focusing only on R_evc and ignoring the effects of R_ms, which we have accounted for with Q_ms.
Again we can use the transformer to derive the equivalent resistance seen at the mechanical resonator side, resulting the the following simplified circuit:
From this, we can see immediately that


Q_ts, Driver Total Q

The formula we're given for Q_ts is
But what does this mean? We'd expect Qts to be the driver Q accounting for both mechanical and electrical damping, let's see if this is true. We substitute in expressions for Qms and Qes:

This is the identical expression as if we'd computed the Q of this circuit directly:

Which makes intuitive sense.


V_as, Air Volume Equivalent to Driver Compliance

I'm not sure what the "as" stands for, but I would guess it's "acoustic suspension". Way back in the day, speakers were designed such that the drivers were mounted in sealed enclosures large enough so that dominant restorative force on the driver cone is provided by the suspension of the driver, not the springiness of the air pressing against the cone. This is called an "infinite baffle" enclosure. Then in the 50's Henry Kloss and a few others turned it around, and built the cabinets smaller so that the air pressure on the cone dominates over the suspension force (hence the "acoustic" in "acoustic suspension) which has the main advantage that it is a much more linear spring, thus lowering distortion levels. Anyway the V_as value is the volume of the cabinet such that if it were any larger the speaker would be infinite baffle, and if it were any smaller it becomes an acoustic suspension. So let's derive the expression for V_as.

Starting with atmospheric pressure which is ρc² where ρ is air density and c is the speed of sound at normal conditions, we know that for well behaved gasses (air, at STP)


So if we move the cone by Δx, that displaces a volume of S_dΔx. This results in a pressure change inside the cabinet of

which when multiplied by the area of the driver gives the total force on the cone:


But, we want that to be equal to the suspension restorative force, which is given by


So if we equate the two, we end up with:


So how is all this useful? Well, I want to eventually write a program to do loudspeaker design, like LEAP. This is the starting point, now that we have a model of the loudspeaker driver it just takes some adjustments to add crossover networks, cabinet effects, passive radiators, etc.. That is, provided I don't have to work on my thesis.

Bibliography

1. Beranek, Leo L., "Acoustics", New York : McGraw-Hill 1954
2. Thiele, A.N., "Loudspeakers in Vented Boxes, Parts I and II," J. Audio Eng. Soc., vol. 19, pp. 382-392 (May 1971)
3. Thiele, A.N., "Loudspeakers in Vented Boxes, Parts I and II," J. Audio Eng. Soc., vol. 19, pp. 471-483 (June 1971)
4. Small, R.H., "Direct-Radiator Loudspeaker System Analysis," J. Audio Eng. Soc., vol. 20, pp. 383-395 (June 1972)
5. Dickason, Vance, 'The Loudspeaker Design Cookbook", Audio Amateur Press: New Hampshire 1995
6. Leach, W. M. Jr, "Introduction to Electroacoustics & Audio Amplifier Design", Dubuque: Kendall Publishing, 2003
   
7. Image taken from: http://www.djsociety.org/Graphic/speakerdriver.gif