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15th February 2007, 01:22 PM  #1 
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T/S conversion from acoustic to electrical lumped model?
I've been studying the ThieleSmall papers. In “VentedBox Loudspeaker systems Part I” Small “presents” a conversion routine from acoustic model to electric model. Small describes this as “taking the dual of” acoustic model “and converting all impedance elements to their electrical equivalents by the relationship” Ze=Bl^2/(ZaSd^2).
It looks like all series connections are converted to parallel connections and all parallel connections to series connections. Also, C converts to L, and vice versa while R stays the same. English is not my native tongue so I’m not quite sure what Small meant by “taking the dual”. The papers do not explain this process either as it’s explained (in detail) in some of the reference articles – which I do not have. Could someone answer why the conversion process goes this way with more “common folk” terms or direct me to a link of an article explaining this. I am not a total newbie on T/S parameter stuff so I know which parameters represent enclosure and which parameters represent the driver. Small also ignores the effect of voice coil inductance so it’s not shown in the figures. What I’m generally interested is not the details but the theory of why the conversion process goes this way. 
15th February 2007, 01:51 PM  #2 
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The best book I have ever seen that explains these conversions between different types of circuit models (mechanical, acoustic, and electrical) is Beranek's Acoustics text. He does a great job of explaining the advantages of these circuit manipulations, if you read his text the ThieleSmall papers become much easier to understand.
If I remember correctly, taking the dual of a circuit is done by placing a node in each loop and then connecting them to form the outline of the dual circuit. This is a concept that can be found in any undergraduate Electrical Engineering AC circuits text. Every circuit element is inverted so that inductors become caps and the caps become inductors while resistors are 1/resistance. I think voltage sources become current sources also. The advantage of doing this is that the equivalent mechanical circuit uses velocity as a current and force as a voltage while the equivalent acustic circuit uses volume velocity as a current and pressure as a voltage. I have probably butchered the explanation, I am writing from memory based on reading the text about 20 years ago. If you are going to buy one book to learn and understand acoustics and speaker design buy Beranek's Acoustics text. To understand the detailed derivations in Thiele and Small's papers in my opinion it is required reading. Hope that helps, 
15th February 2007, 02:08 PM  #3 
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If you find Beranek confusing (as I did my first time through), it may be worthwhile to start with Harry Olson's "Dynamical Analogies." Once I understood that, Beranek became much clearer.
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15th February 2007, 02:29 PM  #4  
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15th February 2007, 02:35 PM  #5 
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Sy, that is a great book too. You can download a copy of Olsen's text at :
http://www.pmillett.com/tecnical_books_online.htm When I read Beranek I came with a background in computer modeling and simulations so it clicked for me right away. It is what I do for a living. 
15th February 2007, 09:51 PM  #6 
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The dual transformation originates in the analogy that is made between the mechanical and electical worlds. A mechanical impedance is defined as Zm=F/v, ie force divided by velocity. The similarity with an electrical impedance is obvious where ohm's law states that Ze=U/I. From this, it is natural to make an analogy between the mechanical and electrical entities. Force corresponds to voltage and velocity to current, and (surprise) impedance to impedance. Masses become inductors, springs become capacitors and mechanical resistance (viscous losses) become resistors.
However, looking at the conversion from electrical to mechanical by the loudspeaker, the force is instead proportional to current via the force factor. It is "the other way around", and this results in that the transducer can be seen as a "dual transformer" or a gyrator. It turns out that an impedance on one side of the gyrator is seen as T²/Z, ie impedance seen on the electrical side is Ze=T²/Zm. Testing this equation on a mass, which has Zm=jwM results in the electrical impedance Ze=T²/(jwM), which is identical to the impedance of a capacitor with C=M/T². Likewise, the electrical impedance seen from a spring, with the mechanical impedance 1/(jw*Cm) results in an electrical impedance of Ze=T²*jw*Cm, ie it looks as an inductance L=Cm * T². If the impedance is a series connection Zm=Zm1+Zm2, then the resulting electrical impedance is Ze=T²/(Zm1+Zm2)=1/(1/(T²*Gm1)+1/(T²*Gm2)), ie a parallel connection of two admittances T²*Gm1 and T²*Gm2. In a similar way, a parallel connection gets transformed to a series connection. ...and it all originates from the relation Ze=T²/Zm. It is possible to make admittance analogies as well. In these, the voltage corresponds to the velocity and the current corresponds to force. In this case springs correspond to inductors and masses to capacitors and no dual transformation is nessecary. The methods are equivalent, but for some reason almost everywone uses the first method. And yes, there is a transformation from mechanical to acoustical too, that is not a dual transformation but an ordinary one, which corresponds to a multiplication/division by Sd². HTH 
16th February 2007, 12:04 AM  #7  
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16th February 2007, 12:26 AM  #8 
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“Masses become inductors, springs become capacitors and mechanical resistance (viscous losses) become resistors.”
This much I figured. And to continue, in acoustic model air volume becomes capacitor, duct with gas inside becomes an inductor and air resistance becomes resistor. According to my view on Olson’s book it seems the circuit on the right (the attachment of my first post) is a combination of electrical, rectilinear (mechanic) and acoustic models. Beranek’s book shows a similar model but with transformers in between the different analogous elements (fig 3.43 p. 81). The book also describes a concept of impedance inverting transformer but the circuit in concerned figure doesn’t seem to invert impedance as the transformers can be omitted with certain component value conversions (fig 3.44 p. 84). Am I right? Mechanic and acoustic analogies seem to be “interchangeable” with each other but what Beranek’s book also introduces are “mobility” and “impedance” type analogies where the impedances seem to be inverted. Am I right that the described impedance inversion is needed when converting between these two analogies? Is this what is done for the lumped circuits shown in the attachment of the first post and if yes, why? This is the part, which I find extremely hard to follow in this whole concept. If my questions seem stupid try to bear me, this stuff is pretty new to me. 
16th February 2007, 12:59 AM  #9  
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Hope that helps, 

16th February 2007, 05:31 PM  #10 
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I wrote a bit about this in the techdoc of Basta!. Maybe this image clarifies a bit, it shows the three domains in one diagram. Electrical to the left, a gyrator (or "dual transformer", = the speaker motor) mechanical in the middle, a transformer (=the membrane) and the acoustic domain, here without any loading impedance.
Scroll down to "Loudspeaker" on this page: http://www.tolvan.com/basta/Basta!TechDoc.htm 
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