120V from both legs



Your argument about meter accuracy appears to be a red herring. The difference between 0.172 and 0.1714 for current is moot anyway.
You say above "I would like to know just how you got the 72.44 degrees" How I got 72.44 is defined at the very top here, did you not see it? It is the angle between force and velocity. If you cannot figure out the arc cos R/Z bit, that is not my fault.
You data will be in error until you accept Bl E/Re as applied force in Beranek's equation, *however* you must understand that Small's electrical input power of E^2/Re is accurate, and how it relates to Beranek's eq.7.1. This you have not done. I have tried to give you a hint or two, but your condescending attitude of the past does not invite a lot of help. Anyway... you say above "Possibly the error is what you have used as input power which is wrong". This is one of *your* errors. Small and Thiele both define input power as E^2/Re = 0.280. I have clarified the power magnitude at the top of 0.244 as dynamic Pin = 0.244, to show the distinction. I have given you reference on Small and Thieles power input before. Some of your errors hinge on you not accepting this. I know what you need to know in order to do so, and to use your words above "your understanding of error sources and analysis is not there". Too bad (for you, that is).
Your approach to back emf is arse backwards, and in fact you have the answer to your dilemma above, but scoff at such a possibility. Eb = Blv. The E-(I Re) is just straight arithmitec and is given by dozens of textbooks. It is *not* wrong. Phase is taken care of on the mechanical side here, not the electrical side. Blv concerns the phase between force F and velocity v. Blv... velocity v, see? And here PF = cos angle = cos 72.44 = 0.3017. Then Blv PF = 11.01*0.0574*0.3017 = 0.1907. This is the real part of *generated* back emf. As I said above "Eb = Blv cos angle = 11.01 * 0.0574 * 0.3017 = 0.1907" and you replied "Incorrect". You are incorrect, not me.
With due respect to your math ability (which is unquestioned by me), you are not as familiar with the dynamics of how parameters relate in what we have discussed. You say "Sheesh, what is the use", and I agree.
BTW, you say above "No, Since, in the part you snipped". This implies at least mild dishonesty, and since I snipped nothing, then any dishonesty is coming from you. Surely you can do better, it is not my responsibility to keep you honest.
Northstar
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says...

--------------much repetive stuff removed------------

----------- Agreed but you were insisting on the accuracy of your data. I agree it is very accurate but not perfect. -------------

------ You have claimed it is the angle between force and velocity but have not shown any calculation of it. Arccos which R/Z? It is not the angle between E and U. that is arctan 32.78/19.5 Y.3 degrees It is not the angle between BlE/Re and U as that is 59.3 and BlE/Re is not the actual force. It is not the angle of the actual mechanical impedance which is 85+ degrees It is not the angle between the actual force (BlI) and U as that is 26.6 degrees. It is not the angle determined from Re/Ze as this can be in the range from 0 to 30 degrees and, in fact, is 26.6 degrees So what is it and how did you get it.? All that I can see is that it is the bugger factor between Eb=BlU =0.632 and your incorrect calculation of E-RI =0.190 ---------->

--------------- A) Berenek does NOT claim that BlE/Re is the applied force. Neither do other references. It simply does not make sense and is an artifact of the model. This is clear in Beranek, Kinsler and the reference from Lahnakoski, as well as from the basic equations 1 to 4 as given below. If you assume BlE/Re as the applied force and Berenak's Zm as the impedance then the phase angle between them is 59.3 degrees., not 72.4 degrees. Even if it was so, the only place you have used it was in calculating an unrelated term Eb. The actual force will be BlE/Re only under blocked coil conditions. Of course, under those conditions, U will be 0 , the mechanical power will be 0 and all the input wil be I^Re loss. B) electrical input power of E^2/Re implies that the input impedance Ze =Re and this situation only occurs for blocked coil conditions. It is not the input power under any condition. It is used for calculation of PAE which is not the actual efficiency. There is no relation to Eq.7.1
Your statements fly in the face of both the physics and the math of the situation and do NOT reflect what your references actually say. ----------- This you have not done. I have tried to give

------------- You are labelling E^2/Re as input power and the actual input power as "dynamic input power" OK but there is no need to do so and the value of 0.280 is an upper limit under blocked rotor conditions. Are you thinking that this power actually exists except under blocked rotor conditions? It appears that Small, Theile are using this as a reference- fair enough - simpler calculations and close enough over the frequency range of interest. Beranek does it differently -using the maximum power theorem as he includes Rg while the others ignore it. Note, as I said before, and you agreed, Input power is always less than this.
Oh, yes, with regard to Small's Eq.31 - I have reconstructed this using Beranek's 7.1 and 7.9 and using a reference power E^2/Re. It wasn't difficult to do so. The interesting thing is that Small's equation contains an approximation for simplicity. Can you find it? One consequence is that it is frequency independent and another is that it is about 10-12% high at the frequency of concern (better at higher frequencies- did he mention that?). The "non-approximate" value is 0.49% which agrees with U^2Rmr/(E^2/Re) and my corrected value. In your use of Beranek the efficiency calculation is based on input power =E^2/Ze which is wrong by about 12 % You are uptight about an unacceptable error of this level, while It appears that it is acceptable to Small. Of course he is an engineer.

---------- Cute- but all wrong. Eb is not force. In addition, the basic relationships are given , not by Eq. 7.1 but by the following> E-RI (Eq1) F=Zmech*U (Eq2) where Zmec = my Zm' These, for AC are both vector equations- . In general, for AC, E, I and Eb are not in phase. Hence scalar arithmetic is not valid. Go back to your references and see how they handle it for AC.
There are two scalar arithmetic relationships applicable for the geometry of a speaker. They are Eb=BlU (Eq3) F=BlI (Eq4) Note that Bl is a scalar so there is no phase shift between Eb and U. There is also no phase shift between I and F
Eq. 7.1 and the whole basis of the circuit model is based on these relationships. Also - from these relationships Eb/U =Bl .01 and F/I =Bl .01 No power factor term, real or ficticious is involved. --------------- > And here PF = cos angle = cos 72.44 = 0.3017. Then

--------------- Let us assume that your "out of the air" angle is correct. and is the angle between Force and velocity -(which it isn't.) Eb is not force so the angle of 72.44 degrees is NOT the angle between Eb and U. If your angle is between E and U then this angle is also between E and Eb . This would mean that there would be some other angle between E and I , and hence between F and U. Go over my calculations and try to understand them. At least they are consistent. If you are attempting to calculate Eb=E-RI using only the real parts of E,I and Eb, then you will get wrong answers. (oh, yes the real part of Eb is 0.323 Volts ).

------------ Since your errors are mainly in how you do your math, are you admitting that my math is correct. The fact that it fits the physics is also correct. The math must fit the physics and mine does. As for the dynamics of the system- we have only looked at the steady state situation and the models based on the basic equations (1 to 4 above)above). I would suggest that my understanding of the dynamics involved in these very simple relationships is well ahead of yours because of what appears to be a fundamental understanding, on your part of the physics (i.e dynamics) involved coupled by a great ability to quote sources without thinking of what they are really saying.
An example is that you calculate (incorrectly) that the power transferred to the mechanical side is Pin -I2Re =0.0328 = Pmec +Pa But Pmec +Pa =(U^2)(Rms +2Rmr) =(0.0574^2) (1.57+0.83)=0.0079 watts so you are saying 0.0328 =0.0079 Do you hear alarm bells? You have just thrown out the most fundamental principle of physics - "conservation of energy".
Another example is reading a lot more into Beranek's Eq.7.1 than really exists. Instead of replacing Zm by Zm' - try using (BS) instead. Eq.7.1 will then, after manipulation reduce to U(BS)=U(BS) which is true as U=U and BS=BS are both true.
Another is real power Pin=E^2/Ze which is also not true unless Ze is purely resistive.
There are more.
My understanding of acoustics, per se, may be less (but I am beginning to wonder) but so far we have been dealing with a very simple dynamic system model in terms of an electrical equivalent and you have a failing grade in that. It probably doesn't matter as you can simply plug in data into Small's work and turn the crank and large errors won't make much practical difference.
You have made statements that are mathematically incorrect, and others that are physically incorrect and some of the math errors are attempts to patch up the physical inconsistencies. You quote references without understanding them. You may be very good a building speaker systems and may have made it an art. Good for you. Otherwise????
--
Don Kelly
snipped-for-privacy@peeshaw.ca
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Oh sure... Snip out the part about who was dishonest.
Snip out your incorrect efficiency magnitudes.
Snip out my analysis where Eb = 0.1905
Snip out my references.
Snip out my Pmec analysis.
Snip out my Zmec data as related to Beraneks eq.7.1
Snip out my data as related to Smalls Eq.31.
Etc, etc etc.
In short, snip out all my proof you are wrong, then fret with your math for a few days and start another attempt to rationalize your errors, along with a fresh group of imprompt remarks (I call them insults).
Regarding the angle of 72.44 degrees, I tried to tell you early on that an effective mechanical resistance Rmec was needed, and you responded positively, but could not refrain from tossing in more insults as usual.
Suffice it to say I was consultant for both Harman International and their subsidiary group for 19 years, as well as manufacturing (a small company) on my own. Dr. Small is also employed at Harman, and I have worked with him before. We found the subject we have discussed difficult to analyze in terms of back emf, however I pursued this aspect on my own off and on over the years. I came here hoping to get insight into an aspect or two that was not quite clear to me (Daestrom was helpful) if you recall. My mistake was in not dropping the dialogue with you early on.
Now... I have solved the dillema you face to my satisfaction, and you have yet to do so, not even being able to derive the correct magnitude to use regarding Rmec as a start. Also your comment regarding conservation of energy below gives little hope for any analysis you come up regarding what we have discussed ever being correct. To your credit, you did suggest private email for the discussion, however having found out your ad hominem oriented character, I'm glad I chose not to go that more "personal" route.
Finally... If you want to go back and reinstate ALL the pertinent data you clipped by using my last reply UNEDITED, and fitting in your UNEDITED comments below, and apologize for your imprompt remarks (if not insults) below, I shall consider responding to your future technical remarks.
Northstar.

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says...

--------- They are fine once corrected for the Pa from one side. ------------------

------- Your analysis is based on a calcuation which is correct only when E and I are in phase. That isn't true so your results are hogwash. This has been pointed out to you before- That is - your analysis has already been shown to be crap. You are trying to use "DC" analysis for an AC situation. It doesn't work If any of your references indicate this - burn them. --------

----------- I looked at the references I had available- including the one on the net. They don't support you. ----------

----------- Again, you don't satisfy conservation of energy and the initial basis is incorrect. I pointed this out previously. Pa +Pmec should be the same coming from the Pin-I^2Re as from (Rms +2Rmr)U^2. I get this but you don't. -----------------

-------------- Sorry, This was also considered. ---------

---------- Right- have you found the approximation in that equation? Or have you even tried to do so? -------------

--------- You have presented no proof. You have presented incorrect math along with incorrect understanding of basic circuits (and remember that this is what we have been dealing with). Blatent errors, often repeated, are not proof. ----------------

--------- All I ask is that you show where you got this particular angle - you have blustered but not produced. If you have a sound basis for this - show me. If (Bl^2)/Re .1 and Rms +2Rmr = 1.57 + 0.83 according to your data, the sum is 19.5. You also give wMmd 5.91 and Cms =0.00025 so wMmd -1/wCms 2.78 This gives an angle of arccos 32.78/19.5 Y.26 degrees. Ignoring Cms gives an angleof 61.5 degrees Ignoring Rms +2Rmr as well as Cms gives 64.5 degrees Ignoring (Bl^2)/Re and Cms give 86 degrees Ignoring (Bl^2)/Re gives 85.8degrees
SO: WHERE THE HELL DO YOU GET 72.44 degrees????? --------------------

him
terms
---------
Did you ever try to run your analysis by Dr Small? And get agreement? Surely he didn't have a problem with back emf. From what I can gather he understands phasor analysis but you don't.
How is it that when I do the math , including phase correctly, and showing the steps., I get consistent results while you don't.? The problem is not mine. ----------- I came here hoping to get insight into an aspect or two that was

-------------- Oh dear, you are still thinking along the line that was discredited before. It appears that you still think that (Bl^2)/Re is an actual rather than an equivalent mechanical resistance. I really doubt whether Small would agree with you. ----------- Also your comment regarding

--------- Why? It is fundamental. ------------- To your credit,

------------- I did that for two reasons: a) it is easier to discuss when proper diagrams and non-ascii equations can be used. b) To save you some public humiliation as, whether or not you believe it, I have no desire to "score points" from you. You can still do so if you wan't to unlearn some serious misconceptions. -----------

--------- Since, all that was said before has become repetitous - there is no point going over it. It is on record. I simply summarised my points. I doubt that my repeating it again will produce a response which will have some meat. For example, you have avoided the consideration of obvious mathematical errors that you have made and also have avoided detailing how you got the magical angle of 72.4 degrees. You have also not attempted to find the approximation in Small's Eq.31 nor have you explained how Pin =E^2/Ze is true if Ze is not purely resistive. You have also avoided the problem of using scalar aritmetic for vector quantities.
You can think, try it- it doesn't hurt.
As for apologies- I apologise for some comments and for those mistakes that I have previously acknowledged. I am not going to apologise for disagreeing with your more blatent errors in math or physics.
I won't be bothering you any more so you can enjoy your Christmas.
--
Don Kelly
snipped-for-privacy@peeshaw.ca
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Thank you.

No need, since you made the blantent errors, as we shall see below.

Nah... You offer hogwash and crap, not me. Been saving this in case you lose your shitte, which you now have:
First, to reinstate my specs, which you clipped:
Bl = 11.01 Re = 7.09 Rms = 1.57 2Rmr = 0.83 w = 1,278 f = 203.4 v - 0.0574 Mmd = 0.0281 (includes mass of the air load, such that *total* mass reactance wMmd+2Xms = wMmd = 35.91 Cms = 0.000250 E = 1.41 Rg = 0 I = 0.172 during steady state motion Sd = 0.0216 area of cone Ze = 8.198 Zmec = 32.99 includes Zr n = 0.0055 = 0.55 % electrical to acoustic efficiency angle between force and velocity = 72.44 deg. angle = 72.44 = phase angle between force and velocity cos angle = cos 72.44 = 0.3017 Eb = 0.1905 Dynamic Pin = 0.244 Pmec+Pa = 0.0328 Pa = 0.001367 n = 0.0056 Pmec + Pa = 0.0328 Cl = 0.2098 copper loss, i.e. I^2R power lost as heat in coil
Now the equivalent circuit per Kinsler 14.10a and Beranek 3.44b, and note we add cos angle, as the frequency is not at mech. resonance:
--------------Re------------------------ + : 1.41 v RMS input : : --------Rl=[(Bl)^2/Zmec+Zr] cos angle---
Re = 7.09 Rl = Zmot = (11.01^2 / 32.99) * 0.3017 = 1.109 Ze = Re + Zmot = 7.09 + 1.109 = 8.199 as measured
Reference on Zmot, Kinsler 14.48b and Beranek right-hand side of 3.62. Note Zr, the air load impedance, was included in Zmec in the specs. Reference on Ze, Kinsler 14.48a and Beranek 3.62.
Now also, Daestrom on Nov. 17 2003 (per Google) noted for a similiar circuit, and although for a DC source, the principle holds for AC (figure as instantaneous with the source positive as shown) "If we draw a 'flowpath' around the circuit from battery, through armature resistance, through CEMF source and back to battery," we see that the voltage drop across Rl may be viewed as CEMF or back emf, which is also depicted as such in the analysis of AC motors by many authorities.
Voltage drop across Re = I Re = 0.172 * 7.09 = -1.219 Voltage drop across Rl = I Rl = 0.172 * 1.109 = -0.1907 Then by Kirchhoffs voltage law: 1.41 + (I Re) + (I Rl) = zero
Just as I said, back emf Eb = 0.1905.
Now you stated two posts back: "oh, yes the real part of Eb is 0.323 Volts". You are off by around 70%, a blantent error.
Now for the fun part :)
Power generating heat in the coil is Pheat = I^2 Re = 0.172^2 * 7.09 = 0.2098 watt Pmec+Pa = I^2 Rl = 0.172^2 * 1.109 = 0.0328 watt Total dynamic power out at steady state is Pout = Pheat + (Pmec+Pa) = 0.2098 + 0.328 = 0.2426 watt Dynamic Pin = E^2/Ze = 0.2425 watt
So (contrary to your criticism) conservation of energy is conserved.
But... the main point here is that mechanical power + acoustic power must be 0.0328, and you said two posts back:
"But Pmec +Pa =(U^2)(Rms +2Rmr) =(0.0574^2) (1.57+0.83)=0.0079 watts"
You are in error by over 400%, a grossly blantent error. Such a discrepency does not befit one who understands the basics.
----------------------------------------------------

Small clearly states his assumptions that would cause any approximation, and considering them he is right on the money.

he he ...

Right.
Now don't get excited.. I told you where and how to get 72.44 degrees. It is not my fault that your phase angles are off-target. Also (Pmec+Pa) is just as important in deriving 72.44 degrees, and you are off by a factor of 4+. This has been a problem for you, and I have pointed it out in many ways, but to no avail.

You are consistently wrong? I am constant as the northstar :)

It appears it may take years of work for you to figure out the role of (Bl^2)/Re.

If you are truthful on b, I'm sorry for *your* humiliation.
Northstar
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-(3.7 @ -85)(0.172 @ 26.5) =0 (NB: You have a sign problem in your statement of KVL if you are using conventional current ) That is, use vector quantities or use two equations for real and reactive components. The circuit consists of an impedance and a resistance in series- your circuits text references should show you how to handle this.

**** and also it is 0.632 volts from (11.01)* 0.0576

*****Sorry, it is not my error. It is your use of "DC methods" for AC conditions. Note that the equation : 1.41 -7.09I =BlU can be broken into two equations on the basis that I=0.172 @ angle i and Eb .01(0.0574) @ angle u =0.632 @ angle u Then: 1.41-(7.09)(0.172)cos (i) =0.632 cos(u) Real part 0-(7.09)(0.172)sin (i) =0.632 sin (u) Reactive part. Both have to be satisfied. You have used the first one only and have assumed i =0 and u r.44 so you get 1.41 -1.2195 =0.1905 which corresponds to 0.632 cos 72.44 Now the second equation becomes -1.2195*0 =0.632sin -72.44) =0.6025 or 0.6025 =0 OOPS! Suppose that I is not at angle 0 then sin (i) = 0.6025/1.2195 =0.494 and angle i ).6 degrees now 1.41 =1.2195 cos (29.6) =0.35 but it should be 06025 Maybe u is wrong so try cos(u) =0.350.632 =0.554 and u is -56.4 degrees, not -72.4 degrees. We can repeat this until the results converge, satisfying both equations. This will be close to i&.57 degrees and u =-59.65 degrees Now Pin =EI cos 26.57 =0.2169 watts I^2 =0.2098 watts Pmec +Pa =0.2169-0.2098 =0.0071 which should be 0.0079 so more refinement on the angles is needed. (note that calculating Pa +Pmec from Pin-I^2Re is subject to larger errors than using (Rms +2Rmr)U^2- from basic error analysis) correspondingly 1.41-ReI =1.41 -1.2195(cos 26.57+j sin 26.57) =(1.41-1.091)-j0.544 =0.3200 -j 0.544 with a magnitude of 0.632 and a phase of 59.65 degrees. Note that Beranek's Zm has an angle of arctan (32.78/19.5) Y.26 degrees
Now I don't expect you to follow the above, based on past experience, but the upshot of it is that a)The angle between force and velocity, which is not the angle between back emf and velocity (which is 0) nor is it the angle between E and U (59 degrees) is definitely NOT 72.44 degrees as that leads to nonsense values. b) The current has a phase angle and this must be taken into account when calculating voltages or when calculating power or the phase of Zin.
I stand by my original calculations as attached below.

**OK
****Based on an incorrect Zmot. Try (0.172^2)* 3.7 cos 85.3 degrees =0.0095 as a rough approximation. compare to (0.0574^2)(Rms +2Rmr) =0.0079 Now take a look at the values that I calculated based on data you provided. There is a check between the values as calculated from U as found by Berenak's 7.1, as well as the current from BlI=Zmec U and the calculation of both Eb and the powers from E and I. Your calculations fail that check because you have made inappropriate assumptions and calculations for vector quantities. ------

--------- Certainly you got these values as you have done what is effectively DC calculations. you still ignore the fact that I^Re +(U^2)(Rms +2Rmr) should give you the input power. (0.172^2)(7.09) +(0.0574^2)(1.57 +0.83) =0.2098 +0.0079 =0.2177 watts which is not 0.2426 watts. There is an problem as if your calculations were right, there shouod not be this difference. You also use Dynamic Pin =E^2/Ze As I have said before, this is true only if Zin is real. Otherwise it is not true E^2/Ze =Pin +jQin where Qin is the reactive vars input.

****** No. If conservation of energy was conserved then I^2Re +U^2(rms +2Rmr) should be equal to the power in. Since your calculations are wrong-

***** Sorry, the error is on your part. You have proven that you do not understand the basics of simple AC analysis. You haven't even tried to make any valid checks on to your results. You have Eb =0.1905 by your "DC type" calculations where Eb =0.632 by BlU You have applied a power factor to this to make up the difference but the basic equation is not BlU*pf and you have not given any reason for applying this particular pf to the back emf which is not determined by the phase angle between force and velocity. You have calculated power on the premise that E and I are in phase and give no justification for assuming that they are so. You then simply ignore the fact that calculation of the powers can be done using U and I magnitudes as well as the parameters Rms, Rmr and Re. and that the results of these calculations disagree with what you claim.
Please note that I laid all steps of my calculations out step by step and also ran checks to detect errors in calculations. This is something that you haven't done.

the
-***- In fact you have done no such thing. You have waved your hands but not given the basis. Show me. If you are right then it will be obvious, if not, and all indications are that it is not right, then I can see where it is wrong. You still haven't said how to get this angle. There is no relationship between this angle and that of Zm, Zm' I, or anything else that I can see. Otherwise I wouldn't have asked. My phase angles are correct. You have given no evidence to the contrary or pointed out just how they may be in error. Sorry to disappoint you. ------

showing
an
** I definitely know the role of that term. No problem on my part. I also did, more than once, show just how it appears. In fact, it appears that I have a better grasp on it than you do.

can
I
**** Oh, I was being truthful and I am not humiliated. All your calculations are based on incorrect math and to some extent, a lack of understanding of the various physical relationships behind the math. Might I suggest again, that you do try to learn basic AC circuit analysis. You have references and Beranek, Kinsler, and, I hope, Small, do it correctly. Sorry to disappoint you.
--
Don Kelly
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I don't think you should mess with Smalls efforts too much, Pal.

You shot yourself in the foot here.. Again.
And again, this is for the record, please leave intact and reply below the ***** TIA
Efficiency per Beranek is:
Beranek Pin = E^2/Ze = 0.2425 His eq. 3.53 Beranek Pout = v^2 Rmr = 0.001367 His eq. 7.5 n = 0.001367 / 0.2425 = 0.0056 = 0.56%
You see the references, use them.
Now Smalls efficiency: His eq.31
n = (po/2 pi c) * [(Bl)^2/Re] * (Sd^2/M^2)
n = 0.000544 * 17.097 * 0.5909 = 0.0056 = 0.56%
You see the reference, use it.
*****
That is exact correlation of a lot of terms, as well as tying Beranek to Small exactly, along with showing the accuracy of my data. To disagree with power in, efficiency, etc as you do above does not speak well at all for you.
Northstar
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degrees. OK * agrees.

------------------- Since I got his results independently, all I am doing is confirming for myself, how he got there.

by
based
not
----------- Have I? Show that Zin is real and why? . Show that your angle of 72.44 degrees has a basis in fact ( I have shown the contradictions that this does produce)? Tell me how E^2/Ze is Pin when the voltage and current are not in phase (and Zin is not real)?

---------- Beranek's Eq. 3.53 gives E^2/Ze =Pin +j Qin.= Real power in + reactive "power" in
If current and voltage are not in phase then Pin is NOT E^2/Zin as Zin will not be real. You will have to use (E^2/Ze)* cos [angle between E and I (which is the same as the angle of Ze)] You haven't so your pin is incorrect.

----------- Again, you have an incorrect Pin. The equation is correct- your math is wrong

----- I did- correctly.

------------ I did correctly - this is based on a reference power of E^2/Re so it is an approximation to PAE

--------------
I disagree with your power input. That is the main difference. Beranek gives actual efficiency while Small doesn't. You have two separate results. They shouldn't correlate as PAE is not the actual efficiency. If your Pin was correct, then you would not get an efficiency which matches the PAE.. I used a PAE with Beranek using (U^2)Rmr /E^2/Re to get a PAE of 0.49%.You should also get the same result. I used Small as written to get 0.56% I also used Small corrected to w^2Sd^2/Zm in place of Sd^2/M^2 ( as would be found by use of Rmr as above) and got 0.49% Small's value is a PAE which includes an approximation Note that it doesn't use Pin at all.
I used (U^2)Rmr/Pin with the correct Pin and got 0.7% You used an incorrect Pin to get 0.55%
Your value is supposed to be an actual efficiency which should always be higher than the PAE. Correlation between the two is a a fiction as they are two different things. The value should be different than Small's PAE.
Go back to Small and others and read the text. Do the same with circuits texts and learn how to handle vectors correctly. That is all I ask. I am tired of correcting the same errors over and over again. I 'm sorry that you don't appear to have the experience to analyse a very simple circuit model or to run proper checks on your values, preferring to bugger factor the results to fit.
Go and do your homework. I have better things to do
Bye
--
Don Kelly
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Typo..sorry. in the following delete the word cos:
cos angle = arc cos (Wl / pi w A^2 Zmec) = arc cos 0.3017 = 72.44 degrees
Giving:
angle = arc cos (Wl / pi w A^2 Zmec) = arc cos 0.3017 = 72.44 degrees
Northstar
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** Let's see, I questioned your interpretation and math.? I also got snide remarks. I repeatedly suggested you drop the condescending

** You were no more polite than I was. I recognised your abilities but also recognised lack of abilities. Should I have said, "Oh, that's fine, thank you", when you made blatent errors, and didn't even try to check them out?
You repeatedly used your self-proclaimed

---------- I snipped nothing. I interjected comments at points and marked these. This doesn't detract from your errors but simply points them out. ---------

real
not
efficiency.
around
**Why?- the example was that of a series R and X. However, it makes no difference. I am interjecting here with ** to point out what I agree with and disagree with.

**so far so good

**incorrect. There is no net power in the reactance. This is fundamental in all texts. Instantaneous power is in during part of the cycle and returned in another part of the cycle. In this case EI* =E^2/Zin = Pin +jQin because you must account for the phase angle. The real power is due to the component of current that is in phase with the voltage--Hence the concept of power factor. That is EI cos 45 =2(0.707)*(0.707) =1 watt The reactive volt-amps in is, in this case, 1va and is a measure of the energy that is being shuttled into and out from the reactance.
.

**the above two lines are OK

**Pin =EI*pf =2*(0.707)*(0.707) =1 watt Pin-I^2 =Pmec =0

---------- Your misunderstanding of the example shows your error. That is basic power calculations for an R-X series circuit. This is the root of the problem. Real power and apparent power are two different things. As for a motor or speaker, sure there is power transferred to the mechanical side- I never denied that. I simply have pointed out the error that you repeated in the example above. Note also that there is reactive input to the speaker and this also gets transferred to the mechanical side but does no work. If you had tried to follow my calculations based on your data, I gave the power and the reactive transferred as well as gave check sums working from U^2(Rmec +jXmec) and these agreed. You don't have this agreement. Why? Because of your erroneous assumptions. Sorry- go back to all your references and READ THEM! Try to understand them. The first is Siskind as it is AC analysis which is the main problem (as shown by your incorrect answer to the example.
Please note that the magnitude of EI =E^2/Ze =I^2(Ze) is the magnitude of the input volt amps- what is sometimes called "apparent power" and the real power is EI(cos angle between E and I). The reactive "power" EI(sin angle between E and I ) does no real work whether in an R-L circuit or in a motor.
Your understanding of this is seriously flawed.
-----------

** even here you are making the mistake of dealing with things as if they were DC. Zin and R are not in phase. Arithmetic subtraction is incorrect. you use Z= root(R^2 +X^2) to get root (4+4) =2.828 because R and X are at right angles Then you simply do arithmetic subtraction and cal the result a resistance. I terms of directions substitue East for R and North for X 2 units east and 2 units north gives 2.828 units NE No problem there but now you want to say 2.828 units NE -2 units E = 0.828 units E That is exactly what you have done. Does it make sense? NO.

------------------- As I said, you won't find what you say in any references. You have given your own interpretation to such references based on what appears to be a desire to use DC analysis for AC situations. If you wish, we can start from the time varying quantities and develop from there- however, surely Siskind does this.

** Crude: If Vrms = 0.0574 then A should be 0.0574(root(2))/1278 =0.0000635 (Vrms*1.414/w) In addition, you are assuming that the energy input in a cycle is 4 times that in a 1/4 cycle. Sorry- that isn't true- nor is your analysis. You have also tried to use a simplified and incorrect approach to the power by only considering 1/4 cycle when, it is obvious, as I once pointed out to you, that that is not correct as things are different in the next half cycle as could be seen by drawing the waveforms of e and i with different phase angles, and plotting the product and eyeballing the average.
If e =Emax cos(wt) and i=Imax sin(wt+a) the product ei =(Emax*Imax) cos(wt)cos(wt+a)=(2Erms*Irms)*cos(wt)*[(cos(wt)*cos(a)) -(sin(wt)*sin(a))] =(Erms*Irms)*[cos(a)*cos^2(wt) + cos(wt)*sin(wt)*sin(a)] Over a period or more, Pave =Erms*Irms*cos(a) This is covered in your Siskind (or should be) and he does explain power factor. The reactive component neither deilvers or draws an average power. -------------

** OK

.
------

** This is, as I suspect, based on your erroneous concept of the real power input ***

E/I
**
** ** You have again gone around in circles. Main problem is that you have assumed Zin is real (and how that can be so when E and I are not in phase, is a quandary) and you have calculated from your Zmech value, an equivalent VA which , in briefer terms is U^2(32.99) =0.1087. Now you use your incorrect E^2/Ze -I^2Re =0.0328 (and that is what you have done above -just thrown in a factor T and worked over 1/4 cycle) So you get 0.0328/0.1087 =0.3017 and call this a pf and carried out your calculations. You have conveniently ignored the fact that if E and I are in phase, Zin will not be real. Also you have ignored the fact that if Zmech is not real, then Zmot is not real and Zin cannot be real. You have calculated Pin on the basis that Zin is real (E and I in phase). You have then decided that the real part of Zin must be 8.198 -7.09 =1.108 ( accuracy not warranted) but (11.01^2/32.99 =3.674 so you have come up with a phase angle of 72.44 degrees to make it fit. The problem is that Zin is not real and the phase angle of Zmot is not 72.44 degrees. I have shown you calculations based on your own data, that give Zmec, Zmot, Ze as well as calculated I that is in agreement with your I.

cannot
-----------
---------- This mechanical impedance includes the value of Rms as an "equivalent mech resistance as seen by the mechanical system" No problem there. It is used to determine U. However, it is clearly not the mechanical resistance (i.e. the open circuit resistance) which is Rms +2Rmr. Are you saying that your Rms is in error?

** Nonsense. You are still going around in circles based on an incorrect Pin and assumptions which are not warranted.

------ **But the phase angle given for U, using Beranek is arctan (32.78/19.5)Y.26 degrees which isn't 72.44 degrees. Better tell Beranek that he was wrong. Oh, yes, this isn't the value to use as Beranek's Zm is not Zmec for which the values using your data becomes 32.87 at a phase angle of 85+ degrees. This gives Zmot =3.69 @ -85.81 and Zin =8.23 @ -26.56 degrees. You have 8.198 @ angle 0 based on 1.41/0.172 and this magnitude agrees well considering all measurement and parameter errors. Note that I calculated from U is 0.1714 @ 26.56 degrees which agrees well with 0.172 for magnitude. ----------

-------------- **Oh yes, you have calculated an R based on a phase angle of 72.44 degrees and then used this R to calculate the phase angle. try again.

** I have no problem at all with that.

** Again, look at 3.53 E^2/Ze =Wave +jQave The real power is Wave. and you have failed to read beyond that point. Qave is the reactive power which is neither loss nor useful power but is simply a measure of energy shuffled into and out from storage. This is why the concept of power factor arose- to differentiate between the apparent power and the real power. See Siskind. You have been using the wrong real input power all along. Even if you were using the correct input power, the result would differ from Small's reference power as a)Small doesn't use the actual power input but a "nominal power" (which is a better term than "applied power" as power isn't "applied". b) Berenak's 7.5 =U^2Rmr (one side) leads to an actual efficiency of U^2Rmr/Pin where Pin is the actual input power divided. Hence exact agreement should not be expected as the "Power reference" is different. c)In addition, Small does use an approximation as I pointed out. This will also make a difference Change the Pin in Beranek to E^2/Re and change Zm to wM and you will get Small's equation. Try it.
This approximation along with your erroneous input power produces agreement by coincidence so "exact "agreement between your value and another which SHOULD NOT agree is nonsense.

----------- **The agreement between your magnitudes of I as measured, U as measured, Zmec magnitude as measured and magnitude of Ze as measured, with my calculated values from your data, is better than one could normally expect. That is good. The agreement between your two efficiency figures says volumes- but not in the way that you mean.
Much of your argument and "proof" is based on circular calculations based on incorrect premises. I have pointed out the inconsistencies which you want to ignore (and in the long run, the main thing that happens is that the efficiency is somewhat different than what you think it is). . I have also pointed out the error in assuming Pin =E^2/Ze if E and I are not in phase. You have a phase angle for Zmec which you can't justify and applied it where it shouldn't apply. You have tried to explain it away by coming up with some ficticious R = about 10 ohms by hand waving based on the wrong angle that you have obtained. The facts are that E and I are not in phase, Zin is not real and your phase angle doesn't apply to E=BlU. In addition, your phase angle along with your calculation of Zin as real, leads to serious contradictions.
The truth is that you do not know how to handle vector quantities, or even simple circuit analysis. The model is basically a simple transformer with resistance on one side and a RLC circuit on the other side. It is not difficult. You quote equations but have no real idea of what they mean or how they are obtained. Outside of that , you are good with your meters.
No answer is needed, I've had my say and if you don't even try to think, there is no point going on.
--
Don Kelly
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You stated"
" we have a vector U @ a and a corresponding vector Eb @ a The two are in phase and the magnitude of Eb =(Bl)* magnitude of U That is back emf in a nutshell- there is really no more to it. Assuming a power factor does not give you Eb but gives the real component of it - IF - you have the angle correct. "
My calculation for real back emf is Eb = Blv PF = 11.01 * 0.0574 * 0.3017 = 0.191 where PF = cos angle = cos 72.44 = 0.3017
Please give your magnitude, power factor, and angle for real back emf, as I don't recall seeing your magnitude, at least. TIA
I'll get back to you on your last post after Santa leaves.
Northstar
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degrees with respect to the reference. This indicates that the PF of EbI is cos(59.26+26.56) =0.0729 On this basis IRe +Eb = (0.1714 @ 26.56)(7.09) +0.632 @ -59.26 =1.41 @ 0 =E applied as it should do.
You have been given all this information.
All that I can add, is that the magnitudes and phase angles should be rounded off to no more than 3 significant digits as the base data doesn't warrant a greater "precision"

Fair enough. All the best to you and yours in this season. Right now, all the above is really unimportant.
--
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This is to correct 2 mis-wordings at *** in my analysis, where at one point I wrote electrical power factor PFe (should heve been mechanical power factor PFmec), and at the other point I wrote PFe (should have been PFmec). The magnitudes are correct, just mis-labeled. Sorry. --------------
Looking at the denominator in Beranels eq. 7.1, which i call electromechanical impedance Zemec, and noting it the electrical effect as well as the >mechanical, we can factor out the electrical part (Bl^2)/Re with the electrical power factor Re/Ze = 7.09 / 8.198 = 0.8648 leaving the *mechanical* impedance as
Zmec = Zemec * PFe = PFe * sqrt {[(11.01^2 / 7.09) + 1.57 + 0.83]^2 + [(1278 * 0.0281) - (1 / 1278 * 0.000250)]^2} = 32.99
= 0.8648 * 38.142 = 32.99
Now... this is an impedance and we find the real part Rmec with the cosine angle between force and velocity = cos 72.44 = 0.3017 *** = mechanical power factor PFmec
*** Rmec = Zmec PFmec = 32.99 * 0.3017 = 9.953
Otherwise derivable just as
Rmec = Zemec PFe PFmec = 38.142 * 0.8648 * 0.3017 = 9.953
or as net force / velocity, as it *must* be
Rmec = Fnet / v = 0.5714 / 0.0574 = 9.955
Mechanical power Pmec than is
Pmec = v^2 Rmec = 0.0574^2 * 9.953 = 0.0328 watt
Conservation of energy is satisfied as Pin = Pout, where dynamic power in is E^2/Ze and power out is that into heat I^2 Re + mechanical power v^2 Rmec
E^2/Ze = I^2 Re + v^2 Rmec = (0.172^2 * 7.09) + (0.0574^2 * 9.953) = 0.2425
Pin = E^2/Ze = 1.41^2 / 8.198 = 0.2425
for exact agreement.
Now to calculate acoustic power on one side of the cone and efficiency:
Acoustic resistance Rmr for one side of the cone is given as
Rmr = (po / 2 pi c) w^2 Sd^2 = 0.000544 * 1633284 * 0.0004665 = 0.415
where Sd = cone area = 0.0216
Giving acoustic power Pa of
Pa = Pmec * (Rmr/Rmec) = 0.0328 * (0.415 / 9.954) = 0.001367 watt
or
Pa = v^2 Rmr = 0.0574^2 * 0.415 = 0.001367 watt
and efficiency of
n = Pout / Pin = 0.001367 / 0.2425 = 0.00564 = 0.564 %
In agreement with the Thiele/Small reference efficiency of
n = (po / 2 pi c) * [(Bl)^2/Re] * Sd^2 / M^2 = 0.000558 * 17.097 * 0.5909 = 0.00564 = 0.564 %
where po = 1.21 at static pressure 0.770 m Hg, and c = 345 m/sec
agreeing with Beranek's power out / power in as
n = Pout/Pin = v^2 Rmr / E^2/Ze = 0.001367 / 0.2425 = 0.00564 = .564%
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snide
out?
This
but
a
power
---------------- Actually the analogy is apt. You have a basic misconception. \ The only average power input is equal to the power dissipated in the resistance, the mechanical resistance and in the air load - that is in Re, Rms, and Rmr. There is no average power into the mass or the spring. These are energy storage elements as are inductance and capacitance of an electrical circuit.
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Morse and Lahnakoski explain how there is an additional resistance, due to electrical parameters. Morse wrote 7.1 in 1936, 18 years before Berenak, and his book would be valuable to anyone as enlightment into 7.1. Morse helped make MIT into an acoustics research center early on. Beranek was there also and he integrated the electrical and mechanical in his equation. It's not immediately apparent, but it's there.
I sense a measure of respect between us and let's not say you are in error per se, but rather that the problem is your power input magnitude is low to match your output power magnitude. This is because you are working with the mechanical, and most all your data is correct, so far as the mechanical goes. I see this as a loss to both of us, as well as to anyone who has been following our exchange, and there is little point in us continuing to claim the other is wrong. We simply are working two different theories, without integration.
I'll summarize very briefly here. You give input power as 0.217, but per Beraneks eq. 3.53 dynamic power is
Pin = E^2/Ze = 1.41^2 * 8.198 = 0.2425 watt
Then noting we agree that acoustic power out is
Power out Pa = v^2 * Rmr = 0.0574^2 * 0.415 = 0.001367
Then per Small as power in = power out / efficiency where his eq.31 gives efficiency n = 0.00564 (and this efficienct magnitude agrees with Beranek)
Pin = Pa/n = 0.001367 / 0.00564 = 0.02424 watt
These two magnitudes of power input are not equal by coincidence.
I'm sorry we are at impasse, and perhaps you may want to look into the works of Morse, Lanakowski, and the Thiele/Small works.
You are welcome to get back to me at any time, and again, best to you and yours in the Holliday season.
Northstar
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Here is a link to Morses' credentials (you have to see them to believe them) which I intended to include in my last post, but overlooked:
http://www.informs.org/History/Gallery/Presidents/ORSA/philip_morse_1.htm
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degrees. U=0.632/11.01 =0.0574 (@ 59.65 degrees) No discrepancies. Some relative errors in Zmec and the Pmec +Pa but close enough to confirm that you did a good job of measurement- which we both knew to be true.
Effectively, all I have done is take into account the existence of Xmot and used vector rather than arithmetic sums. I have also differentiated between real power and complex or apparent power. Doing this means reduces the discrepancies to small errors due to relative data errors and no need to put in correction factors or postulate a resistance with no physical basis. My numerical work is no more accurate than yours, but it does fully take into account the vector relationships.
I hope this is of help. When I get a chance, I will look at Morse, etc. However, nothing that I have seen in Beranek or Kinsler is contrary to what I have said. Small's equation is easy to develop. Beranek's 7.1 is even easier. The analysis is based on a rather simple circuit model, using well established analytical methods.
--
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