120V from both legs

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This is a follow-up to my last post above. Not a "trick" question above, just making sure we keep it straight. We both know the answer is term A. However, your equation C is flawed.

First noting that the the right-hand side gives:

*real* power creating heat in the coil as I^2 R *real* power dissipated in the mechanical resistances of the load as EgIcos(angle between Eg and I)

now substituting E/Z for I into the input side

E E/Z cos (angle between E and I)

gives power in as (E^2/Z) * cos (angle between E and I)

but (and now calling cos (angle between E and I) PF for power factor for simplicity

E^2/Z * Pf = E^2/Z * R/Z = E^2/Z^2 * R = I^2 R = heat dissipated in coil

and we see that the equation provides no power to overcome the mechanical resistances of the load. IOW the power in term on the left-hand side of the equation states the power generating heat, and does not include the mechanical power EgIcos(angle between Eg and I)

Northstar

You have used what I wrote for a "general ohmic Z" in a previous post (not relating to a speaker per se) and misapplied it to the equation that I wrote for a speaker.This is unfortunate as you have taken something out of context (and I could have been at fault for misinterpreting your original question to some extent) You have misinterpreted what I have indicated as Z even though I clearly stated that it was specifically the coil impedance It is not related to the mechanical side. In fact it is the impedance that would be there if the coil was blocked so that it would not move (and Eg would then be 0).

E =ZcI +Eg where Zc is the coil impedance (Zc=root(Rc^2 +Xc^2) at angle tan^-1Xc/Rc (A) Eg =BlU where U is the velocity Mechanical force developed =F =BlI If the mechanical parameters are independent of U and F (small signal model and reasonable for a speaker) Zmech =F/U =[(Bl)^2]I/Eg or Eg =(Bl^2)i/Zm Now (A) is E=ZcI +Eg= [Zc+(Bl^2)/Zm]I =IZtotal Note that the [Zc +(Bl^2)/Zm] term is the total input impedance when the coil is free to move and has two components - the electrical Zc + the equivalent electrical impedance due to the mechanical load Zm which is reflected through Eg.

The equation that I wrote is correct: It does include both terms.

EI =IZtotal =ZcI +(Bl^2)I/Zm =ZcI +EgI

or (I^2)Rtotal =Input power (=E^2/Rtotal) =(I^2)Rcoil +(I^2) (Bl^2)Rmech/(Zm^2)

(sorry - this can't be expressed in terms of E^2/Z as E^2/Zm applied only to the total Z not the individual elements)

This is equivalent to saying that the total power input = I^2Rcoil + (Eg^2)(Bl^2)/Zmech *Rmech/Zmech or EI cos (angle between R and I) =I^2Rcoil +EgI cos (angle between Eg and I) which is what I said.

Note also: looking at the mechanical equation: F=ZmU FU cos (angle between F and U) =real mechanical power But FU =IEg and the angle between Eg and I is the same as the angle between F and U so that EgI cos( angle between Eg and I ) =mechanical power developed.

Nothing that I have said is contrary to the references Kinsler and Beranek (Kinsler's notation is clearer than Beranek's) Nor is it contrary to normal circuit analysis.

Sorry for any confusion from my end.

Phase must be considered at other than mechanical resonance. Should be Eg = BlU cos angle

For net force, phase must be considered at other than mechanical resonance. Would be F = BlI cos angle

In your Zmech = F/U, F should be specified as applied or net, otherwise one might use net force wiich is applied F times cos angle, and the result would be mechanical *resistance*, not impedance

Sorry, but general statements make it impossible to communicate without confusion at least, and frustration at worst.

Phase must be considered for this to work at other than mechanical resonance. Should be Eg =[(Bl^2)i] cos angle / Zm

If Zc is blocked coil impedance as you note above, then in your E=ZcI, I must be blocked coil current as well, but you did not specify such.

In your E = [Zc+(Bl^2)/Zm]I, at other than mechanical resonance, phase must be considered. Should be {Zc + [cos angle (Bl^2)/Zm]} I.

Sorry, but this is confusing at best.

ZcI is a voltage, you need to square the current. Should be Zc I^2 + EgI on the right-hand side.

The last term appears correct, but the first term "(I^2)Rtotal" is actually (I^2)Zelectrical" in conventional expression for impedance. Sorry but this could be confusing.

In view of confusion as above, then starting anew: With reference to a speaker motor, the real *electrical* input power during dynamic steady-state operation is EI * cosine angle between voltage and current, where cosine angle between voltage and current is the electrical power factor PF. Then the real electrical input power during dynamic steady-state operation is

EI PF = E E/R PF = E^2/R PFe (eq. D) or EI PF = E E/Z PF = E^2/Z PFe (eq. E)

where E is applied voltage, I is dynamic current, i.e. current with unblocked coil (motion at steady-state), R is your Zc, i.e the blocked coil impedance as you give above " Zc is the coil impedance (Zc=root(Rc^2 +Xc^2) at angle >tan^-1Xc/Rc ", and Z is the total electrical impedance consisting of R or your Zc + the machanical impedance converted to the electrical side.

Which is it? TIA

Force should be specified as applied force such as Fapp for (angle between F and U) to be included, otherwise using net force would be incorrect.

If you mean IEg as mechanical power, then this should be Fnet*U = IEg. Sorry, but as written the equation is meaningless.

and the angle between Eg and I is the same as the angle between

How do Kinsler and Beranek define electrical input power to a speaker? This would be interesting to me, and I would appreciate seeing the definitions. TIA

IMO, we should restrict analysis to a particular application, and you chose speakers at the outset.

Northstar

To Don Kelly, correction in:

EI PF = E E/R PF = E^2/R PFe (eq. D) or EI PF = E E/Z PF = E^2/Z PFe (eq. E)

Drop the e in PFe, should have been:

EI PF = E E/R PF = E^2/R PF (eq. D) or EI PF = E E/Z PF = E^2/Z PF (eq. E)

I generally make a distinction between electrical and mechanical power factors as PFe and PFm. Here of course PF is electrical power factor. Sorry about that.

Northstar

--------- NO. This is a fundamental relationship -more so than Ohms Law etc. Faradays Law is the basis for this and the relationship for a conductor moving in a steady magnetic field, with the geometry of a voice coil is eg =(Bl)u at any instant in time. For sinusoidal steady state this becomes Eg=BlU where Eg and U are phasors. Since Bl is a scalar, no angle, the phase angle of Eg is the same as that of U. If you insist on an angle between them, it is 0 and cos(0) =1.

-----------

----------- NO. As above, this is a fundamental relationship. At any instant of time, for a current carrying conductor moving in a steady magnetic field as in a speaker, the Lorentz force equation becomes f=(Bl)i and for a sinusoidal steady state current this can be represented as a phasor equation F=BlI where , as above, F and I are in phase as Bl is a scalar. .

The actual force is F=BlI This is the total force due to the current in the coil that is applied to the mechanical side - not the "net" which I assume from reading between the lines, is what you are calling that component of current that is in phase with the velocity.

Now, F and U [or I and Eg] will generally not be in phase and F*U so the real part of FU =EgI becomes |Eg||I| cos (angle between Eg and I) which is what I have said. The difference between what you are saying and what I am saying is that I am representing phasor quantities by Eg, U , I and F and these quantities include both phase and magnitude. I would prefer to use bold faced characters to indicate phasor quantities as is normal practice. You are apparently reading Eg, I, F and U as magnitudes alone. Such is not my intention nor is it correct.

----------- >

----------- The problem is that I am giving quantities as phasors which include both phase and magnitude. F/U is the mechanical impedance. The phase is included. F=|F| @ anglef where |F| is the magnitude of F and anglef is the phase Then F=|F| {cos(anglef) +jsin(anglef) ) Of course the F is the total force applied and due to the coil current. This is inherent in the equations used. Again, the phasor form includes the phase as well as magnitude and this is the sense in which I am using F. It is also the sense in which Kinsler and Beranek use it. (see later). If I used (F/U )cos angle= then the result would be Rmech not Zmech and computed results would be correct only at resonance where Xm=0. Zmech has a reactive component which is accounted for in Zm=F/U using phasor quantities. "net" force is of such questionable value that I simply see no point in using it - it is like using net current as the real component of current- limited usage and an approach which can easily lead to incorrect results.

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------ NO: The attempt to use phasor analysis without being able to use proper notation is a pain in the butt and may be the problem here. In all expressions such as Eg=(Bl^2)I/Zm which I have given, Eg , I and Zm are all phasors and include both magnitude and phase terms. Zm =|Zm| @ theta where |Zm| indicates magnitude and @ theta indicates phase angle Then |Zm| = root (Rm^2 +Xm^2) and the phase angle is theta =tan^-1(Rm/Xm) Same format for I and Eg If I=|I| @ 0 then Eg = (Bl^2)| I/Zm| @ -theta

Phase angle is correctly taken into account.

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----------- Certainly so - the current under blocked rotor conditions, with a given voltage applied will be the current at that condition - it should not be necessary to specify that as it is patently obvious. Zc = E/I under the con ditions of a blocked coil where Eg=0.

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-------------------------- NO: Zc phasor has an angle(If Zc =Rc then the angle is 0) . (Bl^2) has no angle- it it is scalar, Zm has angle Zc =Rc +jXc =|Zc| @ zc where |Zc| =root(Rc^2 +Xc^2) and angle zc=arctan (Xc/Rc) Zm =Rm +jXm =|Zm| @ zm where |Zm|= root(Rm^2 +Xm^2) and zm =arctan Xm/Rm {Zc +(Bl^2)/Zm }=[Rc +(Bl^2)Rm/(Rm^2+Xm^2)] + j[(Xc -(Bl^2)Xm/(Rm^2 +Xm^2)] =Rtotal +jXtotal =root(Rtotal^2 +Xtotal^2) @ arctan(Xtotal/Rtotal)

I am simply using the phasor notation Ztotal =Zc +(Bl^2)/Zm which implies all of the above and takes into account both phase and magnitude. By doing so, things can be manipulated more easily before substituting numerical values-- without being incorrect in any way. Numerical work will require the manipulation of both magnitudes and angles.

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-------- Thank you- my error

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--------- See above for Rtotal which includes the coil R and the mechanical R as seen from the electrical side. I did not detail it as Ztotal due to both the electrical and mechanical impedances combined is the total equivalent electrical impedance as seen by the source. While the mechanical impedance is not electrical -it looks like an electrical impedance from the source side - the source is looking into a black box- all it sees is that with a given voltage there is a resultant current and E/I is defined as an impedance. The source doesn't "know" why that particular impedance appears.

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Equation D is wrong.

equation E is right with the definitions that you have used. I would state that: EI PF =|E||I| cos(angle between E and I)=(|E|^2)/ [|Ztotal| cos (angle of Ztotal) nting that |E| is the magnitude of E etc.

---------- I see no reason to do so - You appear to be using a terminology of your own where you imply Fapp is only the part of the force in phase with U - that is the real component of F. Why? The actual applied force is the total force, not just the real component. Something has to shake up the mass (even though the average power to do so is 0). The real part of FU is |F||U| cos (relative angle between F and U) I don't want to apply the cos (...) twice and I didn't.

Note that the mechanical force equation is F =ZmU which, in the phasor notation that I have been using is the total force applied- not just the real part. It includes both the real and reactive part. The phase is included. I could write F =ZmU in the form |F| @ anglef = (|Zm| @ anglez)(|U| @ angleu) and come up with F=|Zm||U| @ (anglez+angleu) = |Zm||U| [cos (anglez+angleu)+j sin(anglez +angleu)] which includes the real part which you want to call Fapp as well as the reactive part.

---------------------

I disagree. I think this is a distortion of the term "net" as "net" implies something less than the total. I have not represented IEg (properly EgI* where I* is the complex conjugate of I) as the real power. It is the complex power= watts + j( vars) as it includes the reactive component as well as the real power component. On the basis of first principles as discussed above, FU =EgI . There is nothing wrong with the equation and the notation is standard practice. Also note that at any instant of time input instantaneous power is ieg =fu where i,eg,f and u are instantaneous values. For sinusoidal steady state- rms values are used Eg,I, F and U which have phase as well as angle information and the power obtained is an average power over a cycle. ------------------

---------- If you have Berenak then Ch3 Eq3.49 ,3.52, 3.55 3.56 and 3.58 might be of interest. Note that Eq. 3.53 says Vector power=Wave +j Qave =e*i It should be noted that he uses "Vector" where modern usage is "phasor" to indicate time phase rather than pysical direction. In 1954 when the first edition was published, this was the terminology. Also he uses e and i to indicate rms (vector) voltages and currents where I am using E and I as terminology even in 1954 was to use lower case values for instantaneous quantities. In addition he uses e*i where e* is the complex conjugate of e - this is a European notation and in North America, EI* where I* is the complex conjugate of I. .All that this does is change the sign of the reactive part of the product. Finally his Wave is the real average power Pin Eq. 3.55 re-expresses 3.53 in terms of (|i|^2)Z* or( |e|^2)Z and extends this to emectromechanical speakers in 3.56. Finally, Eq 3.58 gives an expression for the impedance seen by the electrical source in terms of (Wave-jQave) /|i|^2 and this reduces to what I call Ztotal and from what he says above, Ztotal =E/I as I have written. Kinsler's notation is more modern and clearer than Beranek's but I have returned his book to the owner so I can't quote chapter and verse from it.

-------- Actually, YOU chose speakers- I originaly dealt with motors in general.

However, in dealing with speakers as we have been doing, the models are all electrical in form and as such follow the rules of normal circuit analysis. Beranek's analysis boils down to application of circuit concepts. Power factor is an electrical circuit concept. How do you deal with a model which is based on a circuit equivalent model without using circuit analysis concepts? (Beranek discusses duality and Thevenin equivalents which are standard electrical circuit ideas.He uses complex number theory as used in circuit analysis since the early 1900's and his diagrams show equivalent circuit models- in fact his book is stuffed with circuit analogies-deliberately) Whether the model is presented as seen from the electrical side or from the mechanical side, the model is being given in terms of circuit equivalent elements and concepts. Why does he do this? It is due to the fact that solution methods established long ago for electrical circuits are applicable to speakers once the mechanical elements and acoustical elements are modelled in this form.

In terms of the original equations as I presented them, (and also presented by Beranek in Eq.3.26, 3.59 and 3.60) noting that we are dealing with rms phasor quantities (and average power, real and reactive), what I have written is consistent and correct. I hope that our differences are in terms of notation - I have tried to stick to "standard" notation as much as possible.

Sorry, you did note the angle.

This is a fundamental relationship -more so than Ohms Law etc. Faradays

YES. You gave no phasor info. My correction was appropriate.

You gave no phase. Sorry, but I can only read what you write, not what you mean.

F/U is the mechanical impedance. The phase is included.

Again, I can only read what you write, not what you mean. I had a poor teacher in mind-reading class.. :)

I disagree. Specifying blocked coil current (for instance as Ib) would not have been innapropriate. After all, dynamic I and static I cannot be the same, and a distinction should be made between symbols. Unless of course one enjoys confusion :)

Zc = E/I under the con

YES. Again you gave no phasor notation. Sorry, but again I cannot mind-read your intent.

You prove my point here. As I said, it is usually referred to as impedance.

NO. Restating eq.E - EI PF = E E/Z PF = E^2/Z PFe This was given before, and phasors are not required.

E^2/Z PFe = E^2/Z*R/Z = (E^2/Z^2)*R but, and perhaps I did not make this clear, E^2/Z^2=I^2 so (E^2/Z^2)*R = I^2*R

Now I^2*R is the heat dissipated in the coil, and during steady- state the equation must contain a power term representing power dissipated into the system resistances. IOW the equation provides no power to drive the mechanical load.

E^2/Z = I^2*R + I*Eg cos angle is OK, but not E^2/Z*PF as input.

Both Small and Siskind use the E^2/R term. It is the only way so far as I know to account for phase between voltage and current. To avoid editing the original, note PF and PFe above are one and the same, as mentioned before.

I did not say or imply that applied force is only part of the force in phase woth velocity. You appear to have misread. Force applied times cos angle = net force. Here the phase consideration is that between force and velocity, where phase angle is zero only at resonance. Only net force, i.e. the real part of the applied force is effective in moving the load.

??? How did you get the impression I implied otherwise? Away from resonance, net force certainly *is* less than the total force applied.

There are many archaic terms floating around. Actually Philip Morse preceeded Beranek by several years with the basics. As noted above, E^2/Ze gives only the real applied electrical power.

You stated on Oct. 27, 2004: "Let's look at speakers: We have an electrical network coupled to a mechanical network and the main variable is frequency" etc etc

OK, however unless we can be specific in equations, as well as staying with the subject at hand, and not rambling all over the place with, uh... diatribes :) , then no point in indulging in confusion, and wasting our time.

Northstar

--------------- What is what you mean by "net" force?. Is it the force acting on the resistive part of the mechanical impedance? I simply note that the actual force produced by the current is F=BlI and since Bl is a scalar, this force is in phase with I - that is a phase angle of 0 or cos(0)=1. We have F=ZmU where Zm is the mechanical impedance and U is the velocity (rms) as phasors. F is a phasor which will have some angle theta with respect to U. FU cos theta will be the real power delivered to the mechanical system. Eg =BlU and is in phase with U. I=F/Bl and s in phase with F so that the phase angle between Eg and I is theta - the same as between F and U This is implicit in the basic equations while "net" force is not.

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--------- Throughout, I have explicitly referred to quantities as phasors. phasors include both phase and magnitude information. This has been re-iterated many times. When I write (and when Beranek does the same) E=ZcI +BlU E is a phasor of magnitude |E| and angle e =|E|{cos(theta) +j sin(theta)} Zc is a phasor Magnitude |Zc| and angle zc =Rc +jwLc =root (Rc^2 +(wLc)^2) at angle arctan wLc/Rc I is a phasor, U is a phasor each with a magnitude and phase. Bl is a scalar (no angle associated)

When doing numerical analysis- each is treated appropriately

For example E= 10 @ angle 0 I = 2 @ angle 30 degrees I* =2 @angle -30 degrees S=P+jQ =EI* =10(2) @ angle (0-30)= 20@ -30 =20 cos (-30) +j 20 sin(-30) =

17.3 -j10 va real power =17.3 watts, reactive =10vars. Saying Z=E/I means Z =(10 @0)/2 @30 =5 @ -30 ohms = 4.33 -j 2.5 ohms note magnitude of Z =|Z| =5 ohms R=4.33 ohms and X =-2.5 ohms (capacitive)

Note also that while we write I^2 Z we really should write |I|^2 Z to get the correct sign on the reactive part. and E^2/Z should be ( |E|^2)/Z* to get the reactive sign right . The power component will be OK in both cases.

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----------- How many times do I have to explicitly say that I am using phasor terminology.? :)

------- In this case there is no need. It is the current into the coil in any case. Specifying Zc =E/I under blocked coil conditions implies that the current is the current under that condition, not the current under any other condition.

--------- Again- E, I , Zc and Zm are phasors as iterated again and again. I am not nor will use magnitudes alone unless specified as such. |E| is a magnitude, E is a phasor. Beyond stating that -what can I do? I could write |E| @ e =[(|Zc| @ zc) +(Bl)^2/(|Zm| @ zm)] |I| @ i where | | means magnitude and lower case e, i etc means angle I have to do this when solving with numerical values but simply writing E= [Zc +(Bl)^2/Zm}I is more efficient for writing equations and relationships.

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----------- How do I prove your point? I have called it total equivalent electrical impedance to clearly specify what it is. I know that it is usually called impedance as it is the impedance looking into the electrical terminals. Since we are dealing with coil impedance, mechanical impedance, input impedance- it is necessary to indicate what impedance is meant. If I had said just "impedance" you would have called that confusing. :)

--------------------

-------- I stated it (see above) using |E| and |I| to indicate magnitudes. In fact what I have written is technically correct. and, if your E and Z are magnitudes , we agree.

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----------- You have made a fundamental error here. You have an expression showing that(E^2/Z^2)R =I^2R which is correct using magnitudes. no problem there. However, you have now turned around and said that the R is the coil resistance. It is not. The mechanical impedance is reflected to the electrical side so that there is a term relating to this. To be correct the Z that you use should be Z=Ztotal ={Rc +(Bl)^2/Zm} = [{Rc +[(Bl/|Zm|)^2]

+J{Xc -[(Bl/|Zm|)^2]}] and the power becomes |I^2| (Rc + {(Bl/|Zm)^2})Rm = coil loss + power transferred to the mechanical side.

---------------

----------- Now we are down to the term Z It appears that your claim is based on the input impedance Z having no resistive component other than the coil resistance. In that case your statement is true. However that condition only occurs when the coil is locked and the velocity is 0. There is no problem accounting for phase between voltage and current. Use phasors- that is why they are standard in AC analysis. They make life a lot easier.

What I have said, all along, includes this. If you want to use E^2/R then use R =Rc +R' If you want to use E^2/Z then do it properly- Z is not the impedance of the coil alone but includes the effect of the mechanical side (after all this is what is actually happening)..

----------

--------- Sorry, that is not true. The load is not resistive. If it was then Net force might be useful. It is true that this component of the force is the only component at any frequency to deliver average power but the whole motion of the cone etc depends on the total force. The equation of motion on the mechanical side is F=BlI = (Rm +jXm)U=ZmU Xm exists. Now you can say that Fnet |F| cos angle between F and U = Rm|U| but you still have to use the whole F in the basic equation. No problem here. However, you must solve to find F before finding Fnet. This generally means solving the complete pair of equations. E=ZcI +BlU BLI =F =ZmU Doing this with phasors is a piece of cake and that is exactly what Kinsler and Beranek do.

---------------

It is just the emphasis that you put on net force where i see it as simply one component of the total force. The total force determines the mechanical motion.

-------- And I have no problem with this provided that instead of "power" you say "complex power" or va and the Ze includes the effect of the mechanical system and is not just the coil impedance.

You used to quote Beranek as being the main authority. Beranek, in 1954, started using the circuit analysis models. Before that there was little done in this way. Certainly Beranek does have it right even though his terminology is somewhat convoluted. Kinsler is clearer.

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I believe this is an example given in response to something else.

Agreed. I am willing to carry on and in what I write, I will use E, Eg, I, U, Z etc as phasors with both magnitude and phase information. The reason is that it generally saves a lot of typing. For example:

so E =|E| @ e where |E| is the magnitude and e is the angle

Complex power = S=P +jQ where P is the real power and Q is the reactive S=EI* =II* (R+jX) where II* is (|I| @ i)(|I| @-i) =|I|^2 (that is I^2 as often used ) I would prefer to use bold E,I etc for phasors and unbolded for magnitudes but I am sticking to normal ascii text Note that S= |E||I|{cos (e-i) +jsin(e-i)} I will also try to be sure to indicate when I am using magnitudes alone In addition, unless otherwise specified, I will use rms values of all variables. Since I have a copy of Beranek on hand, as do you, that should be the reference text for both of us.

To Don Kelly

--------- snipping ---------

Thank you for making your position clear. I shall now do the same.

Neville Thiele spoke at the July 2004 AES meeting in Singapore stating that the Thiel-Small parameters were first described by himself in 1961, later republished in the AES journal in 1971, then during the next three years expanded upon by Richard H. Small. He further stated that over the last thirty years they have been used widely, almost universally, to characterize loudspeaker drivers (this of course is true, they are the reference standard today). He went on to state that it has become conventional to express a speakers sensitivity in terms of its power available sensitivity (PAE), i.e. the ratio of its acoustic power output to the "available" electric power input that it would have absorbed if it had presented to the input a pure resistance equal to its DC resistance Re. Further stating the parameters apply at frequencies where the wave number ka is less than one, where ka=wr/c, for a 12 inch woofer being 360 Hz.

IOW the inductance of the coil is neglected for low frequencies per the above, and since that is what is almost universally done, it is what I have done, and shall continue to do. This means R or Re is equal to the DC resistance of the coil, and Ze is the electrical impedance of the coil *when in motion*.

Obviously per the above, one cannot accept Beranek as the sole reference as you have suggested. Sorry. I believe the above is at the heart of the difference we have regarding input power (note Thiele puts the word available in quotes).

Now regarding our difference:

E^2/Re in my power input equation equates to "available" power as above (in fact Small states input power as E^2/Re, with the coil inductance neglected). Then if E^2/Re = IE = available power or apparent power or volt amps the power equation is

E^2/Re PF = I^2 Re + I Eg PF

PF on the left is electrical power factor Re/Ze. PF on the right is not needed if Eg is replaced or calculated with E - (I*Re).

Now, regarding phasors which you seem to enjoy referring to so much, when I use PF for power factor, the phase angle is of course arc cos PF, and angle is evident. As to the use of phasors per se, this is not done regarding speaker parameters by Thiele, Small, or Morse. Siskind even notes rotor power input (RPI) at start-up simply as (E^2/R^2)*R, which is of course E^2/R. Beranek does mention vectors as being commonly used in electrical circuit theory up front in his book, but I don't believe he considers phase per se in the main analysis equations (chapter 7), except in eq.1 where mechanical impedance = Rm+jXm, and even so, in equations following he uses sqrt Rm^2 + Xm^2.

The point being that while you are quite proficient in vector analysis, the above mentioned have not used vector notation to any degree (pun intended), and neither shall I, nor shall I consider vector notation unless I consider it is required. My further reasoning is: The power factor term (cos angle) provides more simplistic presentation. In fact it is very intuitive if viewed as the portion or fraction of the applied or whole part that is real and/or usable. Accordingly, sorry but I shall not use or consider equations with vector (or if you prefer, phasor) notation. Where appropriate,I shall use the power factor term. IOW I shall stay with conventional speaker notation as outlined by the above mentioned, and consider only the same.

Northstar

"Northstar" wrote in message news: snipped-for-privacy@news.supernews.com...

---------- I have no problem dropping L at low frequencies. Having done this then the Ze as you describe it would then be Ze =Re +(Bl)^2/Zm where Zm is the total mechanical and acoustic impedance and is complex. Is this how you see it?.

-------- I note that Beranek defines PAE (in 1954, reprint with added material in some chapters in1986) in terms of E^2/Rgenerator which is the well known maximum power condition for a resistive source. Maximum power occurs when the load resistance equals the (non-zero) source resistance Note that this is simply impedance matching. Note that B uses the source resistance and it is true that the maximum power delivered to the speaker will be when the input resistance is the same and there is no reactive component (i.e. at resonance). This is nothing new or startling. Are you absolutely sure that Theile was speaking of "power available sensitivity" in terms of the coil Re or was it in terms of E^2/Rsource as indicated by Beranek (section 7.7 and the power expressions are based on prior material in this section and in ch.3) Note the terminology "maximum power available sensitivity" is also given by Beranek. What Beranek writes makes sense. Use of Re does not make sense unless Theile was redefining the PAE in terms of the power input which would exist when the coil is locked and Ze=Re = coil resistance. This will not be useful unless the source impedance is 0. However, this "maximum power available " , etc is derived from the circuit equations and relations that we have been discussing, not the other way round. To go from, say , Beranek's Eq.7.14 and assume that this is available power under all conditions, as you appear to have done below (A) is invalid (it is also invalid if you use Re).

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------------- You have indicated above that Re is the coil DC resistance. If the available power input is E^2/Re then this implies that all the power input is loss in the resistance of the coil. That is the impedance Ze seen by the source is Re. This available power input is actually only available under the condition that the coil is locked. I suggest that Theile intended to say that the reference max available power is defined as Beranek did it - check his notation. I don't have any reference on hand at the moment In any case, except under the reference condition E^2/R? where R? is Rg (Beranek), Re =Rdc (coil) or some Re including the mechanical part of the impedance (and at resonance) , there is no way, for the input power to be E^2/Re. That is: a) E^2/Re is not IE The equation you should be using is Pin = |E||I| cos (angle between E and I) =|I|^2 Re +|Eg||I|cos (angle between Eg and I) In your terminology IE pf1 = I^2 Re + IEg pf2 where pf1 is the pf seen at the terminals ={(Re+Rm((Bl/Zm)^2)}/Zm using magnitudes It will be Re/Ze only if no power is transferred to the mechanical side - locked coil condition (as Ze includes a resistive component. pf2 is the mechanical power factor Rm/Zm in magnitude where Zm=root (Rm^2

+Xm^2)

If Re is the coil DC resistance as you have indicated, then E^2/Re =I^2Re If Re =Ze as at resonance then E^2/Re =I^2[Rcoil +(Bl^2)/Rm] It doesn't hold at any other condition.

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--------- He uses them in chapter 3 etc where he first establishes his models. He assumes that the reader is familiar or can go back to earlier chapters as needed. Note also that since Eq 7.1 the u that he establishes is a phasor quantity. Note that in eq 7.5 he uses |uc|^2 - that is uses the | | to indicate a magnitude. In Eq. 7.7 and following he does the same. In 7.7 he does use eg but he is also using eg as the source voltage reference at angle 0. In this chapter he is establishing some useful "turn the crank" relationships and is interested in only the powers per se. I notice that he doesn't use pf either. The equations that you refer to are initially established from phasor relationships. Note that he establishes the fundamental equations for the model in ch3 and these are in phasor terms. These are the same equations that I am using. I don't know what Theile does but I note that the Siskind expression for rotor power input at startup apparently refers to a DC machine. It is certainly not true for an AC machine. The reason that I, as well as Beranek and Kinsler use phasors is basically because it is a lot easier to manipulate things It can be done using pf and magnitudes - i.e you can carry along magnitudes and phase information if you are careful. Use of phasors is not magic, it simply makes life easier. Note the confusion you expressed when I expanded a simple equation in phasor form to the detailed listing of magnitudes and phase angles >

------- I have no problem with the use of power factor. Use it but use it correctly. The problem that I see with your equation (A) above is not that it uses power factor but that it makes an untrue assumption. In addition pf =Re/Ze is true if Re is the total resistive part of Ze -not just the DC coil resistance. Otherwise not. Don't add Re +(Bl)^2/Zm as scalars as that is not valid. Example suppose Re = 2 ohms and (Bl^2)/Zm is 5 ohms with a phase angle of Zm = 60 degrees Ze is not 7 ohms and the pf is not 2/7 Ze = (2) +(5 @60) = (2+j0) + (2.5+j4.33) = (2+2.5) +j(0+4.33) =4.5 +j4.33 =6.24 @43.9 degrees pf =cos 43.9 =0.72

Suppose that I do it this way: real part of (Bl)^2)/Zm =5 cos 60 =2.5 pf of this term =0.5 reactve part =5 sin 60 =4.33 Now to get the total we add the real parts and reactive parts separately real part of Ze =2 +2.5 = 4.5 reactive part of Ze =4.33 magnitude =root (4.5^2 +4.33^2) =6.24 ohms pf =4.5/6.24 =0.72

The thing is that I can crank the numbers with most scientific calculators (or even slide rules) more easily than I can do it using roots and pf even though it is essentially the same thing.

To say that E=ReI +BlU gives I =(E-BlU)/Re and then note that I =F/(Bl) =ZmU/(Bl) to get ZmU/(Bl) =E/Re -BlU/Re and from this get U= (BlE/Re)/(Zm +(Bl^2)/Re) will be correct for phasor quantities E,I, Zm and U. It will not be correct if only magnitudes are used. You must account for the phase of all these quantities including Zm

+(Bl)^2/Re . If the phase of E is taken as 0 then it is easier but one still cannot add Zm and (Bl^2)/Re as magnitudes and get meaningful results. What I am doing is looking at the basic equations and seeing how they can be re-expressed and what results can be gleaned from doing so. Hence I write the basic equation: E=RI +Eg multiply by I* =mirror image of I around real axis I=|I| @ angle i, I* =|I| @ angle -i EI* =R|I|^2 +EgI* Take the real part Real part of EI* = |EI|cos (angle between E and I*)=R|I|^2 +|EgI|cos angle between Eg and I* or Pin = |EI|cos (angle of E-angle of I) =coil loss + real power transferred. Done: angle information is carried along. However, If you want to use magnitudes and pf's - Ok not a problem per se. Just that I have been using phasors since about 1953 so they are intuitive to me.

I'll get back to you shortly.

Northstar

I have further thoughts (in place of sleeping) re Theile and his use of E^2/Re . This is reasonable on the basis that he is likely assuming that E is constant (where Beranek considers the internal source voltage as constant.) Theile is looking at the speaker independently of the source -which we have done in our correspondence. In that case the maximum possible power that can be delivered to the speaker input is E^2/Re. Use of this in consideration of PAE (which is not real efficiency) is valid as it doesn't change with frequency and allows easy comparison of relative efficiency at different frequencies with a minimum of calculation . However, this "maximum available power" is not actually available ( terminology "available" can be misleading) or even an indication of input power under any condition other than blocked coil conditions. During my attempt to sleep I was able to determine that E^2/Re is always greater than the actual input power (except under blocked rotor conditions where the speaker becomes nothing more than a heater). If you wish, I can show this Note that this will, naturally, support my position. :)

Sorry, you are wrong, it is true for an AC machine. Siskind is referring to an *AC motor* in his book "Electrical Machines" vol. 2, (eq. 80), where he states:

RPI = (Ebr^2 / Rr^2 * Xbr^2) * Rr

RPI = rotor power input Ebr = voltage applied to blocked rotor Xbr = reactance with blocked rotor Rr = Rotor resistance

note we are dropping L, thusly Xbr, giving here

Power applied = E^2/Re

Now this is not nice... since the confusion is coming from your end, as we shall see.

Again, not nice.. Why be impolite? Anyway...... we shall see below who uses power factor, or cosine angle, or phasors correctly. (I prefer phasors set on stun, note).

I have taken the liberty to number your steps for clarity.

You have the correct magnitude (explained oddly, if not wrong) for electrical impedance Ze in step 3, that is Ze = 4.5 Then you go on to add a reactance to Ze, giving Ze in error as Ze = 6.24. Then you contradict yourself on power factor due to your error. You state the angle is 60 degrees, and clearly mechanical power factor is the cosine of the angle

PF = cos angle = cos 60 degrees = 0.5

But your error on Ze causes another error where you get PF = 0.72

Now regarding your error on Ze: What the source sees is what's called the the electrical impedance Ze, which is the DC resistance of the coil plus the an electrical resistance (or equivilant electrical resistance if you prefer), often called the motional impedance, noted as Zmot by Kinsler and Zet by Beranek (eq.3.62), which is due to the mechanical impedance of the load Zmech. The conversion factor from mechanical to electrical is (BL)^2, but to work at other than

*mechanical* resonance, we must consider phase (phasors if you prefer) or (power factor as I prefer). The electrical impedance Ze then as seen by the source and as given by Beranek, Kinsler, Morse, Olson, and others is

Ze = Re + [(Bl)^2/Zmech] * PFmech

Ze = 2 + (5 * 0.5) = 4.5

This is basic and has been the source of confusion in your phasors since day-one. Impedance is a primary parameter, either electrical or mechanical, and if erroneous then all falls apart.

Sorry, but apparently you understand that cos angle or power factor must be used to get the real part of Zmech, but don't know how to manipulate the term, at least regarding what we are discussing.

Here you have taken Beraneks Eq.7.1, and removed the system reactance term, such that it works only at mechanical resonance. The correct equation for all frequencies is (assuming Rg = zero)

U= (BlE/Re)/(Rm+jXm)

Beranek defines Rm and Xm following eq.7.1

Why take out the reactance? What is the point or purpose here? Sorry, but I am about to think you don't understand the *role* of phasors at frequencies other than at mechanical resonance.

Northstar

Fire at will with your phasors, but as noted, I prefer them set on stun :)

Really.. no need, I realize that E^2/Re is always greater than the real input power.

Northstar

----------- firstly - what Siskind wrote is correct for the rotor of an induction motor under "blocked rotor" conditions. However, this is not true under other conditions. Nor is it what you originally "quoted".

Now for the speaker the power is E^2/Re under locked coil conditions. It is the maximum possible power that can be delivered to the speaker terminals. Unfortunately it is not a measure of the actual input power that is delivered to the speaker terminals. That is why I say that the input power under normal operating conditions is NOT E^2/Re .

-------------------- For the speaker analogy>

------- I have no intention of being polite but when I say "use it properly" I am referring to statements such as those you have made below where it is not being used properly.

------------ Please explain what happens to the reactive component of (Bl)^2/Zm? I note below that you accuse me of dropping a reactive term (which I didn't) It exists and can't be simply wished away. You have ignored this part of Zm but it is part of the electrical input impedance. What I have done is correct, although the notation is different from Beraneks (see below) As to the mechanical power factor, please read line 1 (pf of 1/Zm = pf of Zm) where I use cos 60.

------------------ Your last sentence is correct. The rest is incorrect.

If you said Ze=Re +[(BL)^2/Zmech] = root {[Re +((Bl^2)/Zmech)cos angle of Zmech)]^2 +[((Bl^2)/Zmech)sin angle of Zmech)]^2} which is what I have said, you will have, essentially, what Beranek says (Eq. 3.62) Berenak defines his Ze as the "blocked coil " electrical impedance - which, ignoring coil inductance is simple Re = DC resistance of the coil. He Uses Zm +Zl as what we are calling Zm. and gets total electrical impedance Zet =Ze +(Bl^2)/(Zm+Zl) NB This is a "phasor or vector (as he calls it) equation.

. There is no use of pf in this equation as it is a vector (i.e. phasor in 1954 terminology). Kinsler, as I recall also uses a phasor form. In addition, reading beyond Eq.3.62 he notes that "the electromagnetic transducer (eg a speaker) is an impedance inverter. By an inverter we mean that a mass reactance on the mechanical side becomes a capacitive reactance when referred to the electrical side ...... well illustrated by the circuit in Fig .3.47" In other words, there will be a reactive component reflected to the electrical side and this reactive component appears as part of the electrical input impedance. This corresponds to what I have said above. Your approach ignores this and also twists the basic meaning of power factor as defined (basic definition is in terms of real to apparent power which is related to the angle between the voltage and current and secondarily to the phase angle associated with the impedance . You have added two resistances to get an "impedance" then taken the ratio of Re to the sum of the resistances to come up with what is actually a votage divider - not power factor by anybody's definition.

impedance =Re +(Bl^2)/Zm Complex number or phasor In terms of magnitudes :

Real part of (Bl^2)/Zm = (Bl^2)/|Zm| *pf where pf =Rm/Zm reactive part of (Bl^2)/Zm =(Bl^2)/|Zm| sin angle of Zm =-(Bl)^2/|Zm| |Xm/Zm|

This leads to the results that I have given and is in full agreement with Beranek and Kinsler. Note also that Zm is defined below Eq3.64 and Eq 3.66

---------- If you look at what I did do, I used the pf to get the real part of the Zmech I also found the reactive part which is necessary to determine the total input impedance rather than just the real part of it.

Suppose that you walk 1 mile east and then walk 1 mile NE How far have you gone from your original location? By your reasoning you have gone E by 1.707 miles ignoring the northward component I would say that you have gone 1.707 miles East and 0.707 miles North and are 1.85 miles from the starting point at an angle of 22.5 degrees frome E Eastward component is 1.85 cos 22.5 =1.707 miles Eastward component /total distance =pf=0.92 That is the basis of our differences. As to not understanding pf and how it arises, or how to use it, I'd have had quite a different career if I hadn't learned that over 50 years ago since I have used it most working days since. Ditto with circuit analysis which is all that we have been dealing with- nothing complex such as acoustics. All I have been trying to say is that the reactive component cannot be ignored. You have done that.

-------------- Sorry, I didn't even refer to Eq 7.1 in the above. I only used the basic equations (Beranek gives them in 3.59 and 3.60 but I knew these before seeing Beranek ) Note that Beranek's Eq. 7.1 is a phasor equation as is mine.

I did NOT take out any reactive term and the equation that I wrote above is valid at any frequency, not just at the resonant frequency. In what I wrote, Zm is complex: Zm=Rm +jXm (I've told you this before). where I did not break down Rm and Xm into components as B does in 7.2 and

7.3 (He uses 7.2 and 7.3 to make 7.1 and some of the following equations simpler - i.e so that they fit in a line on the page. ) If I did, then my Rm would be Rms +2Rmb as in 7.2 except for the Re term which I included separately. and my Xm =wMmd +2Xmr -1/wCms same as Beranek's so (Bl)^2/Re +Zm =[ (Bl^2)/Re +Rm] +jXm = [((Bl^2)/Re) +Rms +2Rmr) +j Xm ] which is what Beranek uses. Nothing left out. It is amusing to be accused of leaving out a reactive term which is exactly what you have done in your input impedance calculation.

Then why did you say this? "Then if E^2/Re = IE = available power or

-------------------- There is an implication that E^2/Re =IE If there isn't then why did you even say this? If you said E^2/|Ze| PF = then it is OK as long as the PF is not the PF used in IEg PF

-----------------

Interesting- reducing to E^2/Re PF =IE which really is meaningless. E^2/|Ze| magnitude =|IE| magnitude (E/Ze)^2 (Re +real part of the (Bl)^2/Zm) is the real power input. The pf in this case is [Re +Real part of Bl)^2/Zm)]/[magnitude of Ze]

If Ze is real (i.e. 4.5 ) and Re is real (i.e. 2) then Re/Ze=2/4.5 is not a power factor -it is simply a ratio unrelated to power. Go back to Siskind and see what is meant by power factor and what is essential to the concept.

I am not trying to insult you but am simply trying to be helpful as I see some problems with the circuit concepts involved.

combining posts:

--------

Sorry, you appear confused. E^2/Re=IE, if I=E/R May I ask about your background? TIA

Of course. E^2/Ze PF is the heat or copper loss, where PF is the electrical power factor.

Eg = I [E-(I*Re)]. One and the same, since Eg = Blv cos angle

We discussed this. The (Bl)^2/Zm) term belongs on the output side. See the mechanical impedance term there? the input is *electrical* power.

Sorry, but This is where your confusion lies. In your example:

Ze = Re + Rec = 2 + 2.5 = 4.5 is the total electrical impedance seen by the source. Re = 2 as you noted Rec = (term as used by Small, often called motional impedance) = 5*PF = 5*cos 60 degrees = 5*0.5 = 2.5

That's it. You cannot add anything else to it. Note that PF times 5 gives the *real* part of 5, no need to consider the reactive part any longer. If you want to call Ze the real impedance, I suppose that is an accurate description.

No need. I stated that power factor is the portion or fraction of the whole quantity that is real and/or usable. To be more specific, the part that can do work.

Yes thatk you, but I believe not from my end, if you'll pardon me.

------------------------------------------------------------------

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Here is what I said verbatim "Siskind even notes rotor power input (RPI) at start-up simply as (E^2/R^2)*R, which is of course E^2/R." Exactly what I said again above ***in speaker terminology*** we are using. Sorry, but apparently just attempting to use on your part. Sorry, but I am about to get pissed off here. Either you are grasping at straws with silly nitpicking or simply do not understand. Which is it? TIA Let me put the ststements side-by-side, original first:

(RPI)at start-up simply as (E^2/R^2)*R which is of course E^2/R. RPI = (Ebr^2/Rr^2 * Xbr^2) * Rr = E^2/Re

And I *did* note Xbr was dropped since we neglect L of the coil.

Have it your way with yourself.

Freudian slip, imo. Proof? You state above "I have no problem with the use of power factor. Use it but use it correctly." This when you are the one misusing it.

We throw it away. Discard it. File 13... It must be eliminated so we can move on with the real part. It is bad, bad, bad. Reactive provides no power, Captain. It just bounces up and hits us on the arse.

Yes... that's the problem. You should have dropped it and not added it back to the actual electrical impedance the source sees, i.e. Ze = 2 + 2.5 = 4.5.

It was part of the MECHANICAL IMPEDANCE -- sorry -- but this may be your confusion. Look at it this way:

(Bl)^2/Zmech converts the *mechanical impedance* to the electrical side, then to find the real part if this *impedance*, i.e. the resistance, we multiply by the *mechanical* power factor

Electrical *resistance* seen by the source due to the mechanical

*impedance* then is

Rec = [(Bl)^2/Zmech] * PFmech = 5*cos 60 deg. = 5*0.5 = 2.5

then Ze = Re + Rec = 2 + 2.5 = 4.5

That's *it*. We have added the real electrical resistance of the coil Re and the real part of the mechanical impedance together to get 4.5. That is what is called the electrical impedance. Note I did say electrical *impedance*. It is actually the electrical *resistace* - two resistences added, or if you wish the real resistance Re and "equivilant" resistance Rec. Don't look at me however. I didn't give it the name impedance, imo this is a misnomer. End of sermon, put whatever you can afford in the plate. TIA

No no, not at all. See above.

Good thing, we wouldn't want to use cosine angle twice there. Can't get *too* real in these things... Might get ugly.

the ratio of Re to the sum of the resistances to come up with

Hurrah!!! Now you're getting close. See above. Yeah yeah!!!!!

Trust me, your U= (BlE/Re)/(Zm +(Bl^2)/Re), works *only* at mechanical resonance where mechanical impedance Zm = Rmech.

Amusing? Hardly. You have not addressed mass reactance properly for your equation to work above resonance. Your mistake is not amusing to me. BTW, since you would find other peoples mistakes amusing, is that why you're here, to amuse yourself with the mistake of some poor soul less informed than yourself? BTW again, who would this poor soul be? :)

Northstar snipped-for-privacy@hotmail.com remove the high card to reply

this

-------------- But under the conditions of a speaker, this is true only if the load is purely resistive and equal to Re. This is not the case except at resonance or under blocked coil conditions. If ths is true than what you are implying is that IE is constant under all conditions. The confusion is not mine.

As to background- I gave you that before. It does include use of circuit analysis almost daily for over 50 years (in industry as well as acadamia) and a heavy concentration in electromechanical energy conversion for over 40 years. During those times, any errors that I have made in these areas have been challenged by many who are sharper than you and I put together. I will happily put my circuit analysis capability/experience up against that of Kinsler, Beranek and Siskind. This problem as posed is not a "speaker" problem but a very basic circuit analysis problem. I do not claim expertise in acoustics ( I have a friend who has such expertise and it is his books that I have borrowed.) but am quite comfortable with "circuit equivalents" of mechanical systems.

-------------

------------- This is not true. It is the total power input - part of which is lost in copper loss. The rest is converted to mechanical power (where some more will be lost). >

I replied "Freudian slip, imo.", and you didn't follow up. One generally reaps what one sows. I see your modus operandi so: Rather than admit an error, you first throw out a barrage of generally accurate but unrelated equations, etc as distraction. When that is ignored, you then take the insult route.

The Thiele-Small parameters from the 1961-1972 era tied together the former work of Raliegh, Morse, Olson, Beranek, Kinsler, and others. This was over

30 years ago, and they are the standard reference works. Yet you have not apprised yourself of them, and continue to espouse your ignorance. This in spite of their common usage and ready access. Had you done so, you would know that using only E^2/Re as the applied electrical power to a loudspeaker gives the correct electrical to acoustic efficiency. This is made clear from Smalls eq.2 along with eq.31, otherwise from the basic efficiency expression power out/power in and Beraneks and Morses work. Had you been open-minded rather than obstinate in your ignorance, you might have learned something. Accordingly, you give the impression of an internet grandstander.

So where to now? I realize that my knowledge is limited, that I shall never live long enough to learn even 0.01 % of the knowledge available in the world today. This grieves me, but does serve to remind me to be more open-minded. The downside is that when I encounter someone of your apparent ability, but who is close-minded, it dissapoints me. I came here with hopes of finding intelligent and open-minded discussion, but then nothing gained-nothing lost, I suppose.

BTW, I said "since you would find other peoples mistakes amusing, is that why you're here, to amuse yourself with the mistake of some poor soul less informed than yourself?" That you did not respond perhaps explains the situation.

-----------

I'll note here your last error (likely a wasted effort), where you said your equation was correct and I was the one in error:

You stated " U= (BlE/Re)/(Zm +(Bl^2)/Re) "

Beranek gives velocity in his eq.7.1 as

. Bl E U = -------------------- (Rg+Re) (Rm+jXm)

which in your form above is

BLE/(Rg+Re) U = ------------------ (Rm+jXm)

where U=velocity, BLE/(Rg+Re)=force, (Rm+jXm)=mech impedance Zm

you have ignored Rg, which is OK as modern-day amplifiers have negligible source resistance Rg. However, Beranek defines resistance Rm in the next eq.7.2 as

Rm = (Bl)^2/(Rg+Re) + Rms + 2 Rmr

This means (Bl)^2/(Rg+Re) is already *included* in mechanical impedance Zm, so you cannot add it to Zm again. Hooting about phasors as you did doesn't cut it. Phase enters the picture in adding Rm and jXm, not in the addition of resistances. This is sophomoric at best, or conniving at worst. Only you know, and I no longer give a crap.

Northstar snipped-for-privacy@hotmail.com drop the high card to reply

grandstander.

------------ I thought that I had answered you before. I wish that I did have Small's work and his Eq2 and Eq31 which you have mentioned without either quoting them or indicating the region of applicability. I note that when I had Kinsler, you jumped to Beranek as a reference and when I got Beranek, you jumped to someone else. I really doubt that Kinsler and Beranek have any fundamental circuit model or analysis difference from Small or Morse. I also note that the Beranek version is the

1986 revision and the Kinsler was revised sometime in the 90's. It may be that Small, in your references, is dealing with operation near the minimum impedance point. If so, you omitted this factor. In that case then the input impedance will be roughly approximate to Re +(Bl^2)/Zmech.-at that particular frequency but not in general.

As to the use of E^2/Re in the efficiency calculation. Please note that , as you have admitted, the input power is less than E^2.Re If you want actual efficiency, you would have to use the ratio of (acoustic power out/actual input power). This is obviously NOT the PAE, and, in fact, is higher than the PAE. The PAE, as I previously said, is a convenient measure-allowing quick calculation of relative efficiency at different frequencies by ony calculating the power by Eq. 7.15. This is a turn the crank process which is a hell of a lot easier than solving for the input power as well as output power at any given frequency.

-------------->

------- I came with the same objective. The fact that I disagree with you on basic circuit principles seems to be a problem. The fact that your basic understanding of the principles is less than you think is also a problem. Have you ever had a formal AC circuits course? It appears that you are self taught and nothing is wrong with that except that you have had nobody to point out and clear up misconceptions which then become engrained. You have made conceptual mistakes that any sophomore student would soon have explained and corrected. I hark upon circuit analysis because that is all that has been involved so far in the models. Please do not take this as a comment on your intelligence or ability. It is not a comment on your ability to put together a damned good speaker system. It is a comment on an area of knowledge where your understanding is weak. You want to be open minded- part of this is questioning yourself. You are having a problem with someone who is not a speaker expert nor pretends to be, is questioning you- not on speakers per se but on interpretation of the models and equations that we have been dealing with. I happen to have a strong circuits and machines background which is the basis of my disagreement. Part of the problem may be that I do not necessarily use exactly the same terminology asd Beranek uses (Partly because his terminology is relatively clumsy- not his core expertise).

-------------------

--------- I responded. Did you not read it?

---------

------------- I wonder why you put Eq. B i a different form than A? Actually it is correct but for comparison , B can be written as . Bl E

which is the same form as you used for A. The difference, which I explained before, is that I used Zm as the actual mechanical impedance ( Berenak in Eq 7.1 to 7.3 does not name it as mechanical impedance or anything else- his Rm and Xm are simply shorthand notation which is common practice to keep equations easier to follow. Nothing more.) His Rm does include (Bl^2)/Re. My Rm doesn't, as I explained more than once, and on the basis of the development from scratch. As a result [(Bl^2)/Re +Rm] is the same as Beraneks Rm. I could have used Beraneks notation but I started with E=ZeI +BlU F=BLI =ZmU where Ze =Re +j0 ignoring inductance and source resistance Zm the mechanical and acoustic impedance=(Rms+2Rmr) +j[(wMmd +2Xms-1/wCms] Thus I have my (Bl^2)/Re + Zm =[ (Bl^2)/Re +Rms +2Rmr ] +j[(wMmd

+2Xms-1/wCms] which agrees with Beranek's Zm . There is no difference. The addition is done correctly. (Think: Ze =Re=Re +j0) I kept the (Bl^2)/Re term separate only to emphasise that it is electrical in origin, not mechanical. Beranek's Fig. 7.4 emphasises this. There is no difference between what I say and what Beranek or Kinsler say. We are working by the same rules of analysis and get the same results even if somewhat different terminology is used along the way. Surely you can, given the definitions for my Rm and Xm, relate them to Beranek's results- it isn't difficult. The "error" is not mine.

At least I try to do my work independently from basics and check as far as I can against other references available- taking into account changes in notation. It appears that you simply plug into a formula without understanding what is behind it and then try to defend it on faulty reasoning or false arguments. Pity. I think you are capable of more.