Motor torque and back emf

--------------snip--------------

------------------- Sure they were measured but the phase was not. Since you have not taken into account phase, the result is invalid except when the total impedance is purely resistive. That is the point that I was trying to make. You have to use the real part of IE , not its magnitude. (go to Siskind and check it out- he may have a W,VA, VAR triangle shown) . Consider a resistance (measured) of 3 ohms in series with an inductance of

This current lags the supply voltage by 53.1 degrees so the input power is

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

------------------- This is getting mixed up here. Note that the Z mec above is that seen on the mechanical side but included the effect of Re as I indicated. It is expressed in terms of force/velocity. The power delivered to the mechanical side can be expressed as (v^2)Real part of Z =(0.0493^2)2.23=0.005 watts. This differs from the value of 0.009 that I got before but this value is due to using the difference between two small calculated values and roundoff in calculations wipes out a pretence at accuracy (errors in third digit). E-RI as you have used it doesn't do the job.

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

---------- No problem with the above but the mech power factor you use should be that using D (Rm?) and not including Re as the Zmec above does. The Ze that I calculated {(5.78+0.1) -j1.54

of 0.1+j1.54 ohms at 227.4Hz This corresponds to a mech Z =2.16 +j33.2 N-s/m as compared to 2.23+j33.8 as used above (ignoring Re). Some calculation errors appearing.

-------------- There is no relationship , intuitive or otherwise. Note that the value I got (0.357V ) was calculated above taking into account the phase angle of the current. (I =0.23 @ 14.6 degrees so RI =1.33 @14.6 degrees This leads to E-RI =1.41 -(1.33 @14.67) = 1.41-1.29 -j0.337 =0..11-j0.337=0.355 @-72 degrees (differs from previous answer due to round off used) which is close to the expected Eb

You have done the equivalent of 1.41-5.78*0.23 =0.08

Again you are trying to handle AC circuits as if they were DC. That doesn't work unless the circuit is purely resistive or purely reactive. That is generally not true. The use of phasor models and complex numbers isn't done because EE's like the extra work- it is done because it is necessary.

There's nothing wrong with your data except that there is an implicit assumption in your calculations that the voltages and the current are in phase. That is generally not so.

Ignoring coil inductance, the electrical impedance is not resistive, nor a minimum at 227.4 Hz. With inductance of 1.06mh the impedance becomes resistive at 227.4 Hz (but the minimum doesn't occur until about 250-255Hz and rises at higher frequencies) However, in that case E-RI =Eb is still wrong.

We need to be sure that we are using exactly the right and same data values for Re, Rm, etc as well as coil inductance. Also, possibly, for the sake of others on the newsgroup, we should possibly continue on a direct basis rather than through the newsgroup. My address is easy to decode. I have a little program into which I can toss data and can modify it to give mech side power and electrical input power.

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

I do not believe your analogy is appropriate. We are not dealing with a resistor of xxx ohms in series with an inductance of xxx ohms reactive here. We are dealing with a motor where the power left after the I^2R loss is not zero. If you feel it is zero, then please state where the power comes from that moves the load. It surely doesn't come from the I^2R power of heating the armature coil, since motor heat produces no power on a motor shaft or coil form.

Real power going into heat, yes.

This is where I expressed concern yesterday. Noting that Rms = 2.23 is the mechanical resistance of the driver suspensions, then _ v^2*Rms = (0.0493^2)2.23 = 0.0054 is the power dissipated into the suspensions as heat while overcoming friction. Again we have power being lost to heat, this time on the mechanical side, and this provides no power to move the load, it just heats up the suspensions. To achieve movement of the load, the inertia of the mass still has to be overcome, and this requires mechanical power from the source. Again, I tried to illustrate my view here yesterday, i.e. that your magnitude yesterday of 0.009 and 0.008, (now that today of 0.005) are not the net power moving the load.

Not so, the mechanical power factor relationship may use Zmec. Kinsler, et all "Fundamentals of Acoustics" 3rd ed. p.14 PF = cos angle = Rmec/Zmec.

Morse "Vibration and Sound" 4th ed. p.33 PF = cos angle = Rmec/Zmec.

The correct mechanical impedance includes all mechanical resistance

Per Kinsler, et all, as above at 227.4 Hz

Zmec = sqrt Rmec^2 + (wm-k/w)^2 = 35.5

Sorry but there is. The transformation factor times Blv gives Eb within 0.5% average for the 12 drivers, where Eb = E-(IRe). But it is not very intuitive.

What's wrong with public discussion? I prefer such, as the input (and knowledge exchanged thereby) is not limited to two people. As example, I recall the useful input by Daestrom re back emf polarity, which was greatly appreciated by me. Much of your input has also been appreciated as well.

Bill W.

My address is easy to decode. I have a

------------------- I do not feel the power is 0 nor have I said anything to indicate that. Nor do I think the mechanical power comes from the heat. The analogy was to show the error in applying DC analysis methods to AC systems. I could have replaced the inductor with a series R-L load for which there would be power in this load but I chose to use a simplified approach.

--------- First of all, there is NO average power going into accelerating the mass. If you work with rms quantities then the power you calculate is the average power per cycle. For a mass, a spring, a capacitor or an inductance this is

1. E and I in these elements (or F and velocity) are 90 degrees out of phase. The energy put into the mass in one half cycle is returned to the source in the next half cycle. See for yourself-Plot the product of v(t)*i(t) for sinusoidal v, i when v and i are in phase, 90 degrees out of phase and for some intermediate case- say 45 degrees and see what the instantaneous power does in each case - also estimate the average power in each case. This is basic and related to the concept of power factor. Please refer back to Siskind- I'm sure that he explains this for the electrical elements and the same explanation works mechanically.

As for the numbers, I was not satisfied so I did them again, carefully, and get agreement between approaches. The analysis uses normal AC circuit theory and models This may answer some of your questions:

Starting with E=1.41volts, Re=5.86 ohms, Rms =2.23 N-s/m, M=0.0253kg, K=3425N/m and Bl=7.17 volt-sec/m Nothing else assumed.

The force equation gives F=Zm V where V is the velocity. I have shown this before. Zm = Rms +j(wM-K/w) (note magnitude =root(Rms^2 +(wM-K/w)^2)) as per Kinsler) At 227.4 Hz this becomes Zm =2.23+j33.75 =33.82 @86.22 degrees N-s/m Zm is the mechanical impedance (it behaves as an electrical admittance in the model and this is an advantage.

To solve for the velocity, I use an equivalent current source which consists of E/Re shunted by Re so that I get the total impedance including Re referred to the mechanical side of

Zme = (Bl)^2/Re +Zm =11.12 +j 33.75 =35.5 @71.76 degrees

The current source converted to a force source becomes 7.17(1.41)/5.78 =1.75 N Then V=F/Zme =0.0492 @-71.76 m/s corresponding Eb =BlV =0.353 @71.76 degrees (volts) The mechanical force is given by VbZm =Fm = 1.664 @14.46 N This corresponds to a current of 0.232 @14.46 degrees (Amps)

All the above agree in magnitude with your values

The mechanical power is 1.664(0.0492)cos (14.46--71.76) =5.4mW That is the mechanical power factor is used.

or more easily Pm=Rm |V|^2 =5.4mW

----- If I look at this from the electrical side then we have E-RI = Eb =Blv The mechanical impedance transfers over to the electrical side as an admittance so I can use the following: E=RI +I /Ym =IZe Note that IYm =Eb

E-IRe =1.41 - (5.78)0.232 @14.46 = 1.41 -1.34 @14.46 degrees =1.41-1.30 -j0.335 =0.11-j 0.335 =0.353 @-71.7 degrees (agrees)

Pin =real part of EI =0.3168W |I|^2Re =0.3111 leaving Pmec =5.7mW as compared to 5.4mW The error in this case is larger as the Pmec is obtained by the difference between two nearly equal numbers. All results have been shown to an "precision" which is phoney in that it doesn't exist.

--------- and Rmec = Rms not (Bl)^2/Re +Rms so I agree with Morse and Kinsler if you follow the above calculations. I would suggest that they originally used the type of analysis that I have used above although they may not have detailed it in their texts.

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

------------ And I have taken it into account correctly using your value for Rms.

Note that I use the complex number notation i.e. Z=3+j4 has a resitance of 3 ohms and a reactance of 4 ohms The magnitude of the impedance is root (3^2 +4^2) =5 ohms and it has an associated phase angle of arctan 4/3 =53.13 degrees For a voltage of 10 V the current is 10/Z =2 @-53.13 degrees and the pf is cos(53.13) =0.6 It appears that you have not used this notation which is a key tool of AC steady state analysis. I repeat: DC methods don't work for AC.

---------------- Blv =Eb and it bloody well should give this. At resonance the mechanical side is purely resistive so that the Eb, I and V are in phase. Then direct subtraction will give the right result for Eb.

HOWEVER, when the mechanical load is not purely resistive then the E-RI calculation must take into account the phase relationships- that is what I have been saying. E and I are not in phase. In this particular case there is a phase shift of 14.46 degrees in the current and considering this gives agreement between E-RI and Eb=BlV. Ignoring it gives a meaningless value. Note that the numerical values that I have found for I, Eb etc agree in magnitude with your values and with each other when different approaches are used. As I said before, Ze being resistive or minimum at 227.4 Hz can occur when the coil inductance is accounted for- otherwise not so. However, in that case the mechanical impedance will not be resistive and also E-RI will still not work.

----------- Nothing wrong with public discussion but when we have been the only ones discussing the subject for some time it may indicate a lack of interest to others. Also the inability to properly present diagrams and math on this newsgroup are a problem. I will not send binaries to this group and ASCII art is pretty nearly hopeless.

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Well, I have been checking in once in awhile to see what's new. Some of this is akin to some vibration theory of rotating machinery I did at one time. Mostly for bearing and noise analysis. Didn't feel qualified though to contribute much. It is interesting how the vibration theory and AC theory 'line-up' in so many ways (including things like resonance).

One of my favorite bits of trivia re: vibration theory is the so-called 'whorling speed' of spinning disk on a shaft. Above this, the forces on the shaft bearings and the vibration of the unit drop. As I understand it, it happens at some multiple of resonant frequency. Always wondered what, (if any) electronic equivalence there would be.

Don, have you run across this or seen anything more on this?

daestrom

------- Sorry, I don't have information on this. I would check with some ME's on this. The only situation that I recall with a lot of vibration was way back in student days when an unloaded shunt DC motor lost its field. Breaker was hit pretty quickly and the motor shook like hell as it slowed down. I guess it didn't have time to shake on the way up. Kept cool -i.e. pants remained dry.

As for your qualifications - from what I have seen, they are more than adequate. -- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Aye, but you had no power, Captain... I considered asking if your "not buying it" was lack of buying power.. :-)

I have not snipped, in order to retain the flow of theory we are into, however it is now down to the nitty-gritty, so please snip if you prefer.

RE the above equation in question:

Pmec = IEb = IE-I^2Re = 0.320 - 0.298 = 0.0222

Although it may require vector analysis to be broken down in detail re power division, I believe it is valid. See: A.E. Fitzgerald et all (Electric Machinery) 6th ed. p.389 on: Eb = E-IR (multiply by I for power) Joe Kaiser (Electrical Power) 3rd ed. p.212: IEb = IE-I^2R

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

As I recall, you noted that the mechanical side is more difficult to analyze. I can visualize this better with *quarter* cycle mechanical motion. Above resonance mass reactance is no longer balanced completely out of the picture. At an end stop of motion, velocity is zero, and energy from the source helps the stiffness reactance get the mass moving. At center equilibrium position velocity is maximum. Kinetic energy is maximum and this energy is returned to the source and suspensions as the mass does work on them, and then slows to a stop at the opposite end point of motion. The process is then repeated in the opposite direction, where the mass is ready to begin another cycle.

You lost me here. How are you defining the terms, and please state the magnitudes. TIA 1. Vb 2. Zm 3. Fm

Now here is where the confusion lies. As I have said, I believe Pmec = Rms |V|^2 = 5.4 mW where Rms = 2.23 and v = 0.0493 is the mechanical power dissipated into the suspensions, but not the total mechanical power being disippated, instead just a part. To get to the point:

Do you feel that Pm = Rm |V|^2 = 5.4 mW is the total mechanical power *dissipated*?

I have not seen this covered in the literature, but am working on a theory. However, I might need your expertise on the math end of it, if you wouldn't mind.

Bill W.

Jump in any time. I might learn something. :-)

Bill W.

Worked once with some 'Sharpel's oil purifiers. Long cylinder spun on its axis, oil fed in one end and out the other with some baffels and such. The centrifugal forces involved would separate any water or sludge out of the oil. Cylinder spun at a nominal 22 000 RPM.

Now, the 'neat' thing about these was the cylinder was suspended at one end and the other end just held against a sort of friction snubber. When they started, the shook quite a bit on the way up (through a couple of critical speeds). But once they were about 3/4 of the way up to speed, the vibration would practically disappear and there was very little wear on the lower snubber.

The way it was explained was that operating far above the natural frequency, when any imbalance from sludge/debris would create an unbalanced force, it would be out-of-phase from the dominant vibration (that was at the natural frequency). Not sure I got all that straight, but I *think* that's what the manual said ;-)

Anyway, the 'durn thing worked. Just shook a lot on the way up and down, but not once running.

daestrom

------------------- No problem with using E=Ri +Eb for an AC situation as long as the quantities are treated as phasors( once called vectors). I did this: E-IRe =1.41 - (5.78)0.232 @14.46 = 1.41 -1.34 @14.46 degrees

and the resultant Eb is the same as that obtained from a mechanical analysis and with your data. However, your references above are dealing with the DC situation. (At least Fitzgerald is -and I am quite familiar with his books. I hope also that Kaiser is also dealing with a DC situation - if not, I suggest that you burn

his book) I did the following which is valid for AC

Your approach of using EI-RI^2 =EbI treating the products and subtraction as scalar operations is not correct. Repeat- it is invalid for AC except for purely resitive circuits (which this isn't). For AC the complex power is defined as S=E(phasor)I* where I* is the complex conjugate of I. Then S =P+jQ where P is the real power and Q is the reactive "Power" We can also say P =|E||I|cos (angle between E and I) =|E||I|(pf) where |E| indicates the magnitude of the phasor E (similarly for I) treated as a phasor.

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

----------- Note that you indicate that energy is taken from the source in part of the cycle and returned at another part. (This is true for a mass even without the spring). That is in agreement with what I said. However, the av rage power into the mass from the source is 0. The same thing happens for a spring, a capacitor or an inductor.

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

------------ Vb - sorry -this should have been V - the mechanical velocity. Zm is the mechanical impedance -mechanical elements only as I have indicated above. Fm is the actual mechanical force (F is the driving force as a source treating it as the equivalent of a current source I=E/Re in parallel with e -moviing to the mechanical side the result is F=BlE/Re in parallel with an equivalent mechanical resistance (Bl)^2/Re ) This allows calculation of the velocity but to get the actual mechanical force it is necessary to look at the velocity V times the actual mechanical impedance Zm (or use F-((Bl^2)/Re)V (using phsor arithmetic, not scalar).

All that I am doing that you haven't done is to use phasor methods for AC analysis. When your reference discusses a mechanical power factor - there is an implication right there that phasor methods or their equivalent are being used.

-----------

---------- Yes- with the model used so far which really deals with the driver in a closed box and nothing else as far as I can see.

For a closed cabinet Rms also may include the cabinet losses(Ludwig.) The model in the reference (Ludwig) that I gave you is much more complete and includes further elements including resistive elements which include resistive terms. Neither of us have considered these elements at this point and the model so far includes only the data that I listed above (and only data which you have provided). It doesn't include cabinet venting, and all the other goodies. Including these elements modifies the circuit and the results. All the circuit model does is to transform the mechanical system into an equivalent electrical system which can be analysed using well established methods. Also as you know the model is valid only for small signals as nonlinearities can cause some of the parameters to vary with velocity or pressure.

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Before I reply to your last post, would you please give the magnitudes of your terms as used above, i.e: V Zm Fm F I=E/Re e F=BlE/Re (Bl)^2/Re F-((Bl^2)/Re)V (using phasor as you note)

Not nit-picking, just trying to correlate terms, and thanks again for all. TIA

Bill W.

----- Original Message ----- From: "Bill W." Newsgroups: alt.engineering.electrical Sent: Tuesday, December 16, 2003 1:51 PM Subject: Re: motor torque and back emf

calculation. I

---------- I have no problem with this magnitude. All I have done is to include phase as in some cases it is important.

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

---------------------- I did do the full development. but I can repeat it *once* more. See the end of this message.

------------- The trees and the expressions used by Kloss, Small and others are based on the math- the same math that I am using. Without the math you would not have a model to give you information or the nice equations that are given by your references. The math is but a tool. I am simply using the appropriate tool for the job. It is very hard to hammer nails with a screwdriver- it is the wrong tool. Math applied to the basic physical relationships is all that I ,or your references, are doing -why- because it is the right tool. So far you haven't given anything from your references that I disagree with. What you have given is your interpretations which I disagree with. You haven't given any reason why the mechanical equivalent of an electrical element becomes, somehow, a real mechanical element with real mechanical losses. I am waiting. . -------------------

--------- Yes I do - Key word is BLOCKED which is the short circuit current of the source expressed in mechanical terms Ishort circuit =E/Re which results in a blocked coil force (Bl)E/Re . That is implicit in the model and math that I am using. Look up "current source" in Siskind.

You also state that this is *not* the net force when motion occurs (and I agree wholeheartedly with that.) but when you calculate losses you are USING it as the force in the case of non-zero velocity whereas I am not. I take care of the difference between this "source" force and the *actual

Did you bother to even look at what I did, step by step above? It appears not.

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

--------- ?????? 1.75 dB? 0.137 nanoseconds? How do you get these values? At this time the current and force will still be 0.

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

------------------------ a)If it is the blocked coil force then it is the force that will exist as long as the coil is blocked. It has nothing to do with the instant at the start of motion. b)We have been dealing with rms quantities which are absolutely meaningless in the conditions you describe. c) the DC resistance of the coil is not the actual resitance of the coil at frequencies even as low as 60Hz. It is close enough for government work so that the error in using the DC resistance is not worth worrying about d)Physically at the instant of applying the voltage, the current will be 0 because there is inductance in the winding and this is very important in the starting transient. The nature of this transient will depend on the inductance Not until this transient is over will the DC resistance limit the current and the force. Your visualisation simply flies in the face of the physics involved and is unnecessarily cumbersome. Clamp the coil, apply voltage, measure the current and calculate (or measure) the force after giving the RL transient time to die down (the transient in this case has a time constant dependent on L/Re where L is the coil inductance. I would estimate that the transient would be nearly gone in about a millisecond. (not a fraction of a nanosecond) on the basis of a coil inductance of about 1mH which is consistent with the low Z point at or near

source model (see > >Siskind

------------------- I used the values of Bl and Re that you gave me. (If I used BL instead of Bl that is a typo) If you looked at what I did above, and apparently you didn't, this development is quite apparent (along with the development of the (BL)E/Re term.

Look where I have marked ******************** as I cannot keep repeating things

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

--------- In fact, I looked at Ludwig but the model that I have used, while the same for the driver, is one that I developed myself using the basic equations for the electrical and mechanical system. If you have a problem with these equations then say so. It is very obvious that your reference sources also used the same equations and used the same mathematical development- then simplified it for general "handbook" use.

----------------- Please look at what they have written, NOT the numbers YOU have chosen to put into their equations. What do they define each of these terms , F, Zmec, Rmec, angle, v to be? That might be of interest -what is the background for their statements. Just plugging numbers into an equation doesn't prove a thing. All their equations will give my result if their F is my Fm and and my Zm and Rmec They fit my contention as well as yours. I have no problem with their equations. You have not proven anything by the above listing. Again you are quoting formulae without the reasons behind them and apparently plugging numbers into them blindly. You

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

--------------- And using my Fm, Zm, Rmec =Rms these equations will give my answer of 0.0054 W. So far you haven't given me any reason, on the basis of quotes or formulae, that you are right. On the basis of knowledge of the basic circuit analysis as well as knowledge of motors, I feel quite confident in my position.

---------- Actually the above doesn't prove a thing. If I use my Fm=1.664 and the pf associated with my Zm (0.66) along with your velocity P=Fm*d cos(angle)/t =1.664*0.0000543*0.066/0.0011 =0.0054 W

I can also use your data to transfer the Rms to the electrical side as an

__________________________________ Development, Again. starting parameters known E=1.41Volts, Re =5.78 ohms, Rms=2.23 Ns/m, M=0.0254 Kg, K=1/Cmt=3425 N/m Bl=7.17 N/A Variables: Eb =back emf (volts), Fm =actual mechanical force (N), V =velocity m/s, Zm =actual mechanical impedance due to mechanical elements, w=2*pi*frequency Basic: I =(E-Eb)/Re (1) In steady state, Fm =ZmV (2) where Zm =Rms +j(wm-K/w) =sqrt (Rms^2 +(wM-K/w)^2) @ angle arctan (wm-K/w)/Rms

Fm =BlI and Eb =BlV substituting in(1) Fm =(Bl)E/Re -(Bl)Eb/Re but this can be written as Fm =(Bl)E/Re -[(Bl)^2/Re]V Then (2) becomes (Bl)E/Re -[(Bl)^2/Re]V =ZmV or (Bl)E/Re = [(Bl)^2/Re +Zm]V (3)

--------- While it is not necessary to know this, from a circuit point of view we actually have a voltage source consisting of E behind a resistance Re. The equivalent current source (see any circuit book) consistes of a current E/Re shunted by Re. E/Re is the "short circuit current (output voltage 0)(If the coil is held stationary then V=0 and the actual locked coil force is (Bl)E/Re). This current source when expressed in mechanical terms becomes a "force source" F=(Bl)E/Re shunted by the equivalent mechanical resistance (Bl)^2/Re. We can treat this element as a mechanical element as long as we remember it is not an actual mechanical element and as it is an internal part of the equivalent source, any power calculated in this element does not have a real world meaning in that it cannot be equated to any real mechanical or electrical loss. It's use is that a simpler model results.

----------------- From the above we can find V =[(Bl)E/Re]/[(Bl)^2/Re +Zm] (4) Knowing V we can find Eb. Knowing V and Zm we can find the force Fm =ZmV (2) From Fm we can find I=Fm/(Bl) Now the results can be checked by calculating Eb=E-RI and comparing it to the value found from V. The actual mechanical power is the real part of FV* where V* is the conjugate of V This becomes FmV(pf) where pf = cos of phase difference between Fm and V Alternatively it can be written as Pmec =|V|^2Rms The term |V|^2 (Bl)^2/Re is, as I indicated above, isn't a true mechanical power and unfortuantely it isn't the I^2Re loss either. It is simply an internal part of the "force source"

That is the analysis part: now plug numbers Note magnitude @ angle form will be shown At 227.4 Hz (Bl)E/Re =7.17*1.41/5.78 =1.75 N w=2*pi*227.4 =1429 rad/sec Zm =2.23 +j(0.0254*1429 -3425/1429) =2.23 +j33.75 =33.82 @86.22 degrees [mag 33.82] (Bl)^2/Re =8.89 Ns/m Then from (4) 1.75 (@0degrees reference) =(8.89 +2.23 +j33.75)V or 1.75 @0 =(11.12 +j33.75)V =(35.54 @71.76 )V [mag 35.54] V=1.75/(35.54 @ 71.76)=(1.75/35.54) @ -71.76 =0.0492 @ -71.76 [Mag:

If I assume F =1.75 (@0 as E is assumed at 0) , this corresponds to a current of 0.244 A @0 Then E-RI =1.41 -0.244*5.78= 0.00 !!! This seems to differ from what Eb actually is - it is inconsistent -i.e something's wrong. This inconsistency does indicate that something is wrong with using this force as the actual mechanical force produced by the current in the coil. However, it is consistent with the 0 velocity or locked coil case which we appear to agree upon. ------------ If I do the same analysis at resonance then I will get V=1.75/11.12=0.157 m/s Eb =7.17*0.157 =1.128 Volts Fm =0.157*2.23 =0.350 N I =0.350/7.17 =0.049 A Eb=1.41-5.78*0.049 =1.128 V checks

IF I use your force of 1.75 N as the actual mechanical force and the impedance at resonance of 11.12 Ns/m the velocity and Eb will be the same but the current will still be 0.244 A Frequency doesn't affect the current. !!!! Is this true? Obviously not. There is an inconsistency again. Again,at this current Eb =0 which implies the locked coil condition.

Sorry- I keep trying to send his directly Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Please excuse the top posting here, but the muddle in this thread has become impossible to wade through.

Also.. please excuse the duplicate posts yesterday. Got "editor memory error" upon clicking on post, then Supernews or whoever didn't show the post for over a half hour. Excuses, excuses, but blame *must* be laid.. :-)

I'll likely move on shortly. My email is at the end of the post here, if anyone wants to contact me.

------

Mr. Kelly

Thank you for your data, etc below.

Now... One thing I've learned is to appreciate simplicity. Einstein said things should be as simple as possible, but no simpler... Since you've taken my joking as serious, this may prove good advice here... Anyway along this line, and I don't want be derisive, and you won't like what I'm about to say, but I believe answers and points made can usually be made simple. You get into so many side issues with your dissertations, that frankly it's hardly worth the effort to wade through and try to ferret out the needed info. In fact I have wondered if you were intent on causing confusion, but have gone along so far, assuming you are well intended. I'll try below to ferret out the points of contention and address them as concisely as possible.

-------

You stated "Small and others are based on the math - the same math that I am using."

No. As example Kinsler, Beranek, and others state:

Zmec = F/v = 1.75/0.0493 = 35.5

This is just v = F/Zmec rearranged. Simple and intuitive. More force - go faster, more impedance - go slower. Basic. Requires no decoration. Yet you give the result as 33.82. The same math as Beranek and Kinsler would give 35.5.

-------

You stated: "You have not given reasoning for your contention. You have made quotes but the one you have given re (BL)^2/Re supports my contention- not yours. I have stated that it is the electrical resistance term referred to the mechanical side - as such, even though it is an "equivalent mechanical resistance" it is not an actual mechanical resistance." I really shouldn't waste my reasonong here... My derivation and logic behind

(Bl)^2/Re = 7.17^2/5.78 = 8.89

If we call the quantity R or Z the impeding quantity Iq, than taking the logical equation for velocity above

v = F/Zmec

Then from the force applied Fapp side

v = Fapp/Iq

transposing

Iq = Fapp/v

sub in for Fapp

E (Bl)/Re 1.41*7.17/5.78 Iq = ------------ = ---------------- = 35.48 v 0.0493

Then from the load or retarding side, logically we use back emf Blv instead of driving emf E

Blv (Bl)/Re (Bl) v (Bl) Iq = -------------- = ------------ = (Bl)^2/Re = 8.89 v v Re Just as I said (which you scoffed at), (Bl)^2/Re is the resistance the motor encounters in overcoming the back emf BLv, or as Small and Kloss said, it is the ***mechanical*** resistance of the driver motor.

Three pages of aside or unrelated equations coming up? :-)

----------

You stated "If it is the blocked coil force (and it is) then it cannot be used as the actual mechanical force when motion exists."

How many more times do I have to tell you I am ***not*** doing this. In how many more ways can I state that net force equals applied force*cos angle.

Fnet = Fapp cos angle = E (Bl)/Re * Rmec/Zmec

= 1.41*7.17/5.78 * 11.12/35.5 = 0.548

Sorry but I think you don't understand mechanical that well. Think of it this way. Push a sliding door direct in its line of motion, i.e. from zero angle. cos angle or power factor = cos zero = 1 .

Fnet = F*PF = 1.75*1 = 1.75

All your apploed force is useful in moving the door. Now push from an angle of 45 degrees.

Fnet = F*PF = 1.75*cos 45 = 1.75*0.707 = 1.237

Now push (you do it) :-) from an angle of 90 degrees

Fnet = F*PF = 1.75*cos 90 = 1.75*0 = 0

None of your applied force is used to move the door. (you may stop pushing now..) :-)

Bottom line, net force is Fnet = Fapp cos angle = E (Bl)/Re * Rmec/Zmec

= 1.41*7.17/5.78 * 11.12/35.5 = 0.548

You claim 1.664. Quote "That is why I calculated the net mechanical force acting on the mechanical impedance (1.664 N)"

Maybe two pages here? :-)

-------

You stated "Then Pmec =1.664*0.0492*0.066 =0.0054 W" Halliday and Morse: P = F * v * cos angle = 1.75*0.0493*0.313 = 0.027 Beranek and Villchur: P = v^2 * Rmec = 0.0493^2*11.12 = 0.027 Colloms: P = F^2/Zmec^2 * Rmec = 1.75^2/35.5^2*11.12 = 0.027 Kinsler: P = F^2/Zmec cos angle = 1.75^2/35.5*0.313 = 0.027

or

Restating my net mechanical power equation:

Using my *measured* amplitude A or d (distance the mass travels from equilibrium position to the end point of motion) at 227.4 Hz of 0.0000543, and the power equation P = work/time, then for 1/4 cycle Pmec net = Fnet*d/t = 0.548*0.0000543/0.0011 = 0.027 Now, if you don't agree that 0.027 is net mechanical power, instead of your magnitude of 0.0054, then WHICH OF THESE DO YOU DISPUTE?

a. F net = F cos angle = 1.75*0.313 = 0.548

b. d or A is *measured* amplitude, i.e distance the mass travels during 1/4 cycle at 227.4 Hz = 0.0000543.

c. Time for a quarter cycle is T/4 = 1/f divided by 4,

And another question.. why do you get a different result than Halliday, Morse, Beranek, Villchur, Colloms, and Kinsler?

--------

We need to get on the same page or wrap this up. We'll never get to acoustic power, etc this way.

--------

My email is snipped-for-privacy@mounet.com (divide 8 by 2)

Bill W.

I submit that Kinsler etc said : Zmec=F/v (and agree) I also submit that they did not say " =1.75/0.0493 ..." You have used the values F=1.75 and V=0.0493 ----They didn't! I can use the same formula with F=1.664 and my v of 0.0492 the same v to get

------------------ Obviously you haven't read what I said. I have no problem with what you have done although you have made it over complex. (why substitute another term Iq?) I said I=(E-Eb)/Re and substituted BlV for Eb and F/(Bl) for I to get

(Bl)I =(Bl)E/Re -(Bl)Eb/Re =(Bl)E/Re - ((Bl)^2)V/Re =ZmecV

Do you see the (Bl)^2/Re term?? This looks more like one line than 3 pages :) I'm sorry that you have problems with the equations. They are really quite simple and are definitely related. I agree that the term (Bl)^2/Re is the equivalent resistance of the driver coil expressed in mechanical terms. This is not, as you imply, a factor of the back emf.

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

----- Ah, but this is not what you said. In addition, the Fnet has nothing to do with the power factor. The force that is involved in producing power is involved. What you really said is: " Applied force means just that, it means the gross

-------- This is fine - note that power factor is not involved. What I observed is that you have used this "applied force" in the case where motion does exist. I answered, and you did not address this, that this force is the actual force under the condition that there is no motion and , as you stated, is not the force that exists with motion. However, in spite of what you have said, you have then turned around and used it as if it was the actual force. You can't have it both ways. In addition, use of this as the actual force does lead to contradictions and anomalies in terms of the currents, etc. You have not adrresed these.

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

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

---------------- Wow, grade school stuff. However, while well intentioned, your analysis is not germane to the issue. You are looking at force vectors -i.e. force not in the direction of motion. That doesn't apply here. The net force acting on the mechanical elements is BlI - now the part of this force that acts on the mechanical resistance is related by F*pf. No problem there but it is only a part of the force. Should it be called "net" force -that is questionable. What I contend and have shown is that this force is not the (Bl)E/Re force but something less unless the coil is stationary. You have said as much yourself.

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

------------- This force is the component of force acting on the mechanical resistance. If you now want to call it net force, be my guest.

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

------------ This assumes that the applied force =1.75 and also that the actual mechanical resistance includes the "equivalent " due to electrical resistance. Since this is due to coil resistance, it is not a "real" mechanical element- I have said all this before and shown the development. These two things are the crux of our disagreement.

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

----------- Nah. Fmec =BlI =Zmec*V

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

---------- You don't get it:

Halliday P=F*v*cos angle Beranek: P=v^2Rmec Colloms:P=F^2/Zmec^2*Rmec Kinsler: P = F^2/Zmec cos angle

I dont, as I said, dispute any of these. They are all quite OK If I put my Fm and Zm and Rmec in, I get my result.

What I dispute is your interpretation where you use the force (in your own words) which is the actual force ONLY in blocked coil conditions, in a situation where there is a velocity. This is a contradiction. Following this contradiction back to the current results in a constant current and force at all frequencies and (correctly) a back emf of 0 corresponding to a velocity of 0, although you do have a correct velocity. How can the velocity be 0 and 0.049m/s at the same time? Isn't this intuitively a problem? Is the current from the source independent of frequency? How can Eb calculated one way be different from Eb calculated another way? if both ways are based on the same equations. You have these inconsistencies and intuition as well as common sense say that "something's wrong" I don't have these inconsistencies. Again you are not recognising the nature of the current or force source used. Also, how can the electrical resistance become an actual rather than "equivalent" mechanical resistance. Again, is this intuitively correct? it sure as hell is not physically correct.

Now you take an expression for the real or power related part of the force and try to support your view by plugging in numbers. You then start to call the force Fnet by bringing in the power factor. You are getting more, not less, inconsistent.

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

And what does the repetition of the above (b), (c) have to do with (a)?

I also would be interested in what you used to measure the amplitude to an accuracy of 1/1000 mm ? Also this amplitude will be dependent on just where in the cycle the starting position would be. It is interesting that, for the velocity of 0.0493 m/s at 227.4 Hz, the calculated peak excursion would be

--------- As I indicate above. I have no problem with what they say (at least as far as you have quoted them).

I DO have a problem with the values that YOU use and YOUR interpretation . (or are you saying that they ,each and everyone, gave an example using the data that you have given for a particular driver?).

Do you see the difference?

I also have a problem in that you have apparently not even tried to see the development that I have shown nor attempted to answer the questions that I have raised about inconsistencies.

--------------- You are right with regard to that.

Thank you

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Mr. Kelly

Thank you for your development.

Would you be so kind as to fill in your magnitudes where I have noted ***magnitude =

Sorry, but I did ask for the magnitudes in my request.

Thanks in advance.

Bill W.

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

Development, Again. starting parameters known

E=1.41Volts, Re =5.78 ohms, Rms=2.23 Ns/m, M=0.0254 Kg, K=1/Cmt=3425 N/m Bl=7.17 N/A

Variables:

Eb =back emf (volts), *** magnitude = Fm =actual mechanical force (N), *** magnitude = V =velocity m/s, *** magnitude = Zm =actual mechanical impedance due to mechanical elements, *** magnitude =

w=2*pi*frequency ***magnitude =

Basic: I =(E-Eb)/Re *** magnitude = (1)

In steady state, Fm =ZmV (2) where Zm =Rms +j(wm-K/w) =sqrt (Rms^2 +(wM-K/w)^2) ***magnitude =

@ angle arctan (wm-K/w)/Rms ***magnitude =

Fm =BlI and Eb =BlV ***magnitude =

substituting in(1) Fm =(Bl)E/Re -(Bl)Eb/Re ***magnitude =

but this can be written as Fm =(Bl)E/Re -[(Bl)^2/Re]V ***magnitude =

Then (2) becomes (Bl)E/Re -[(Bl)^2/Re]V =ZmV ***magnitude = or (Bl)E/Re = [(Bl)^2/Re +Zm]V ***magnitude = (3)

While it is not necessary to know this, from a circuit point of view we actually have a voltage source consisting of E behind a resistance Re. The equivalent current source (see any circuit book) consistes of a current

E/Re ***magnitude =

shunted by

Re. E/Re ***magnitude =

is the "short circuit current (output voltage 0)(If the coil is held stationary then V=0 and the actual locked coil force is (Bl)E/Re). This current source when expressed in mechanical terms becomes a "force source"

F=(Bl)E/Re ***magnitude =

shunted by the equivalent mechanical resistance

(Bl)^2/Re. ***magnitude =

We can treat this element as a mechanical element as long as we remember it is not an actual mechanical element and as it is an internal part of the equivalent source, any power calculated in this element does not have a real world meaning in that it cannot be equated to any real mechanical or electrical loss. It's use is that a simpler model results. From the above we can find

V =[(Bl)E/Re]/[(Bl)^2/Re +Zm] ***magnitude = knowing V we can find Eb. ***magnitude =

Knowing V and Zm we can find the force

Fm =ZmV ***magnitude = From Fm we can find

I=Fm/(Bl) ***magnitude =

Now the results can be checked by calculating

Eb=E-RI ***magnitude =

and comparing it to the value found from V. The actual mechanical power is the real part of FV* where V* is the conjugate of V This becomes

FmV(pf) ***magnitude =

where pf = cos of phase ***magnitude =

difference between Fm and V ***magnitude =

Alternatively it can be written as

Pmec =|V|^2Rms ***magnitude =

The term

|V|^2 (Bl)^2/Re ***magnitude =

is, as I indicated above, isn't a true mechanical power and unfortuantely it isn't the

I^2Re ***magnitude =

loss either. It is simply an internal part of the "force source" That is the analysis part: now plug numbers Note magnitude @ angle form will be shown At 227.4 Hz

(Bl)E/Re =7.17*1.41/5.78 =1.75 N

w=2*pi*227.4 =1429 rad/sec

Zm =2.23 +j(0.0254*1429 -3425/1429) =2.23 +j33.75 =33.82 @86.22 degrees [mag 33.82]

(Bl)^2/Re =8.89 Ns/m

Then from (4)

V=1.75/(35.54 @ 71.76)=(1.75/35.54) @ -71.76 =0.0492 @ -71.76 [Mag:0.0492]

The corresponding

Eb =7.17*0.0492 =0.353 @ -71.76 volts [mag: 0.353]

Fm =ZmV ={33.82*0.0492) @ (86.22-71.76) =1.664 @ 14.46 [Mag: 1.664N]

I =Fm/Bl = (1.664/7.17) @ 14.46 =0.232 @ 14.46 [Mag: 0.232A]

Pmec =|V|^2Rms =((0.0492)^2)* 2.23 =0.0054 Watts. If the power factor approach is used then we have the angle between Fm and V is 86.22 degrees corresponding to the angle associated with

Zm cos 86.22 =0.066

Then Pmec =1.664*0.0492*0.066 =0.0054 W

To check that the I and Eb are correct

E-RI =1.41 -5.78*0.232 @14.46 =1.41 -(1.34 @14.46) =1.41-1.30-j0.335 =0.112-j0.335 =0.353 @ 71.6 degrees [Mag: 0.353 ]

This checks. If the Fm was wrong, then I would be wrong and this check would show an error. If I assume

F =1.75 (@0 as E is assumed at 0) ,

this corresponds to a current of 0.244 A @0

Then E-RI =1.41 -0.244*5.78= 0.00 !!!

This seems to differ from what Eb actually is - it is inconsistent -i.e something's wrong. This inconsistency does indicate that something is wrong with using this force as the actual mechanical force produced by the current in the coil. However, it is consistent with the 0 velocity or locked coil case which we appear to agree upon. ------------ If I do the same analysis at resonance then I will get

V=1.75/11.12 = 0.157 m/s

Eb =7.17*0.157 = 1.128 Volts

Fm =0.157*2.23 = 0.350 N

I =0.350/7.17 = 0.049 A

Eb=1.41-5.78*0.049 = 1.128 V checks

IF I use your force of 1.75 N as the actual mechanical force and the impedance at resonance of 11.12 Ns/m the velocity and Eb will be the same but the current will still be 0.244 A Frequency doesn't affect the current. !!!! Is this true? Obviously not. There is an inconsistency again. Again,at this current Eb =0 which implies the locked coil condition.

Good question. I figured it was coming.

While awaiting the magnitudes in your development, I'll go ahead and answer this question, as this may prevent confusion in the magnitudes.

As I noted earlier, the equilibrium (center of excursion) to end-point of motion was measured at resonance = 58.6 Hz. Not too difficult at all, since the cone is moving along pretty well here. I measured peak to peak and divided my 2, giving the amplitude A or distance traveled d by the mass during 1/4 cycle as

d = 0.00134/2 = 0.00067

As an aside, *average* velocity at resonance is distance/time _ v = d/t = A/t = 0.00067/0.00427 = 0.157

where d = A = distance mass travels in 1/4 cycle at 58.6 Hz t = time taken for mass to travel 1/4 cycle at 58.6 Hz where t = 1/fc divided by 4 = 1/58.6 divided by 4 = 0.00427 Qtc was *measured* at resonance as 0.82 (response down 1.74 dB or so as I recall, would have to calculate it out). Now if we we adjust response at resonance to flat, then we can use the relationship where amplitude is proportional to frequency squared and get amplitude at 227.4 Hz. This is just the amplitude at resonance divided by Qtc so that amplitude at 227.4 Hz Arp is

Arp = Afc/Qtc * (fc/frp)^2 = 0.00067/0.82 * (58.6/227.4)^2 = 0.0000543

Note your magnitude above of 0.000049 adjusted for Qtc is

Which is off by 0.37% from my calculation. A little lax, but we may get somewhere here yet... ;)

Oh by the way, I like to start from zero, less to lose that way.

Bill W.

---------- OK - You are calculating an average velocity in the quarter cycle. However, in all the calculations rms quantities have been used (when I asked about this way back when, you did indicate that rms vaules were in use. (it really doesn't matter *except for power calculations* as long as one knows what is being used and the use is consistent and the basis is made explicit ). There is a difference which for a sine wave is 11% so , if rms values are used as generally done- then this is an approximation which is about 11% too low.

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

----------------- OK- In the above you have failed to define Qtc, (Arp, Afc and Arp I can figure out from context but all these terms are jargon associated with the field )- and the rationale behind amplitude proportional to square of frequency ( It appears that you are assuming mass is the dominant part of the mechanical impedance -Yes? No? ) Also,why the sqrt (Qtc)?

I calculate a Q in the electrical circuit sense of (2*pi*fo/((Bl)^2/Re

----------- The 0.37% is a figure of the imagination. Don't fret it. Starting with a magnitude of 0.00067 the best accuracy that you will get is 2 figures- The difference between 0.0000541 and 0.0000543 is meaningless ;) If the accuracy is actually that good I would be surprised :>O

The figure of 0.000049 that I calculated is found from the velocity as calculated from data you supplied. There is no adjustment for Qtc in this and no need for adjustment as it is based on conditions at 227.4 Hz without reference to resonance or the evaluation of Qtc. I'm glad that it checks :>) However it will reflect the errors and or approximations involved in the measurements from which you obtained the data. It appears that there are more of these than I thought.:>{ Are they important? Probably not in practice so convenience is the main thing However, when taking measurements and determining parameters from these measurements, the use of Qtc etc makes life easier. Once the data is known reasonably well (M, Rms, K, Bl, etc then these can be used directly.

----- Thank you

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer

Mr. Kelly Again, I have been forthright with my driver data and equations, giving magnitudes for terms as well as the result. If you have nothing to hide, then courtsey requires the same from you. The questions are repeated at the end of the post for your convenience. Again, I request the magnitudes of *both" the terms within the equation and the magnitude of the result. In the interest of offering the olive branch, let me say I am not trying to be aggressive here, just trying to get to the bottom of our difference re net force, mechanical power, and and mechanical resistance. If you refuse to offer your data, then your objective clearly is other than honest communication, that might allow us and others to learn in areas that have had little or no prior discussion. Extending the olive branch even further, note that it has never been my intention to "piss you off". Again, an example of what I request. _ _ v = F/Zmec = 1.75/35.5 = 0.0493

TIA

Bill W.

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

E=1.41Volts, Re =5.78 ohms, Rms=2.23 Ns/m, M=0.0254 Kg, K=1/Cmt=3425 N/m Bl=7.17 N/A

Variables:

Eb =back emf (volts), *** magnitude = Fm =actual mechanical force (N), *** magnitude = V =velocity m/s, *** magnitude = Zm =actual mechanical impedance due to mechanical elements, *** magnitude =

w=2*pi*frequency ***magnitude =

Basic: I =(E-Eb)/Re *** magnitude = (1)

In steady state, Fm =ZmV (2) where Zm =Rms +j(wm-K/w) =sqrt (Rms^2 +(wM-K/w)^2) ***magnitude =

@ angle arctan (wm-K/w)/Rms ***magnitude =

Fm =BlI and Eb =BlV ***magnitude =

substituting in(1) Fm =(Bl)E/Re -(Bl)Eb/Re ***magnitude =

but this can be written as Fm =(Bl)E/Re -[(Bl)^2/Re]V ***magnitude =

Then (2) becomes (Bl)E/Re -[(Bl)^2/Re]V =ZmV ***magnitude = or (Bl)E/Re = [(Bl)^2/Re +Zm]V ***magnitude = (3)

While it is not necessary to know this, from a circuit point of view we actually have a voltage source consisting of E behind a resistance Re. The equivalent current source (see any circuit book) consistes of a current

E/Re ***magnitude =

shunted by

Re. E/Re ***magnitude =

is the "short circuit current (output voltage 0)(If the coil is held stationary then V=0 and the actual locked coil force is (Bl)E/Re). This current source when expressed in mechanical terms becomes a "force source"

F=(Bl)E/Re ***magnitude =

shunted by the equivalent mechanical resistance

(Bl)^2/Re. ***magnitude =

We can treat this element as a mechanical element as long as we remember it is not an actual mechanical element and as it is an internal part of the equivalent source, any power calculated in this element does not have a real world meaning in that it cannot be equated to any real mechanical or electrical loss. It's use is that a simpler model results. From the above we can find

V =[(Bl)E/Re]/[(Bl)^2/Re +Zm] ***magnitude = knowing V we can find Eb. ***magnitude =

Knowing V and Zm we can find the force

Fm =ZmV ***magnitude = From Fm we can find

I=Fm/(Bl) ***magnitude =

Now the results can be checked by calculating

Eb=E-RI ***magnitude =

and comparing it to the value found from V. The actual mechanical power is the real part of FV* where V* is the conjugate of V This becomes

FmV(pf) ***magnitude =

where pf = cos of phase ***magnitude =

difference between Fm and V ***magnitude =

Alternatively it can be written as

Pmec =|V|^2Rms ***magnitude =

The term

|V|^2 (Bl)^2/Re ***magnitude =

is, as I indicated above, isn't a true mechanical power and unfortuantely it isn't the

I^2Re ***magnitude =

loss either. It is simply an internal part of the "force source" That is the analysis part: now plug numbers Note magnitude @ angle form will be shown At 227.4 Hz

(Bl)E/Re =7.17*1.41/5.78 =1.75 N

w=2*pi*227.4 =1429 rad/sec

Zm =2.23 +j(0.0254*1429 -3425/1429) =2.23 +j33.75 =33.82 @86.22 degrees [mag 33.82]

(Bl)^2/Re =8.89 Ns/m

Then from (4)

V=1.75/(35.54 @ 71.76)=(1.75/35.54) @ -71.76 =0.0492 @ -71.76 [Mag:0.0492]

The corresponding

Eb =7.17*0.0492 =0.353 @ -71.76 volts [mag: 0.353]

Fm =ZmV ={33.82*0.0492) @ (86.22-71.76) =1.664 @ 14.46 [Mag: 1.664N]

I =Fm/Bl = (1.664/7.17) @ 14.46 =0.232 @ 14.46 [Mag: 0.232A]

Pmec =|V|^2Rms =((0.0492)^2)* 2.23 =0.0054 Watts. If the power factor approach is used then we have the angle between Fm and V is 86.22 degrees corresponding to the angle associated with

Zm cos 86.22 =0.066

Then Pmec =1.664*0.0492*0.066 =0.0054 W

To check that the I and Eb are correct

E-RI =1.41 -5.78*0.232 @14.46 =1.41 -(1.34 @14.46) =1.41-1.30-j0.335 =0.112-j0.335 =0.353 @ 71.6 degrees [Mag: 0.353 ]

This checks. If the Fm was wrong, then I would be wrong and this check would show an error. If I assume

F =1.75 (@0 as E is assumed at 0) ,

this corresponds to a current of 0.244 A @0

Then E-RI =1.41 -0.244*5.78= 0.00 !!!

This seems to differ from what Eb actually is - it is inconsistent - i.e something's wrong. This inconsistency does indicate that something is wrong with using this force as the actual mechanical force produced by the current in the coil. However, it is consistent with the 0 velocity or locked coil case which we appear to agree upon. ------------ If I do the same analysis at resonance then I will get

V=1.75/11.12 = 0.157 m/s

Eb =7.17*0.157 = 1.128 Volts

Fm =0.157*2.23 = 0.350 N

I =0.350/7.17 = 0.049 A

Eb=1.41-5.78*0.049 = 1.128 V checks

IF I use your force of 1.75 N as the actual mechanical force and the impedance at resonance of 11.12 Ns/m the velocity and Eb will be the same but the current will still be 0.244 A Frequency doesn't affect the current. !!!! Is this true? Obviously not. There is an inconsistency again. Again,at this current Eb =0 which implies the locked coil condition.

--------- Your challenge completely ignores the fact that I have repeatedly given you the information you request. If you had read as far as my "Pissed off " comment then you have read through the results that I have obtained including the magnitudes. I really find it hard to comprehend why you cannot relate an equation to the numerical results when reference is made to the original equation. It appears that you want me to show the results at 227.4 Hz and attach these results to an equation which is also valid at other frequencies as well. The reason for doing it the way that I have done is simply because analysis- determining the equations - is separate from plugging numbers into them. However, to lead you by the hand and give you the results for the specific equation at the specific frequency of 227.4 Hz. I will take what I had already done and interpose these results directly after the applicable expression. I will do the calculations correctly -using magnitudes alone where applicable and using phasors where magnitudes alone are not applicable( and not by erroneous use of magnitudes alone where not applicable as you have done) and indicate (as I already did) the resultant magnitude specifically. If that doesn't work, I am not sure what I can do except suggest that you study study Siskind (circuit analysis) , and a beginner text on mechanics (free body diagrams, etc)- not to get formulae but to see the why behind the formulae.

HERE IS THE CRUX OF OUR DIFFERENCES

All the (Bl)^/Re term is is a transfer of the electrical resistance term to an equivalent mechanical resistance. It does NOT mean that it is now an actual mechanical resistance. Any attempt to construe it as such is wrong. It is a mathematical convenience and simply accounts for the electrical resistance on the overall response. If you had attempted to follow the development below - the source of this term is very apparent.

---------------------------------------------------------- Starting information

This expression doesn't care a hoot about the source of Eb - it is simply saying that there is a voltage differential across a resistor carrying current.

This expression doesn't give a hoot with regard to the source of the force Fm. It could be due to amorous mice on the cone. It simply says that the response to an applied force is a velocity which depends on the force and the impedance due to the MECHANICAL elements.

Fm =BlI and Eb =BlV THESE are the relationships coupling the electrical and mechanical sides and requiring that the above (1,2) are linked together.

by V ****************READ THESE STEPS******

Then substituting for Fm in Fm=ZmV

, , (Bl)E/Re = [(Bl)^2/Re +Zm]V (3)

The above is simply manipulation of the equations into a more convenient form in that it brings all the terms in V together.

This is a model which brings the electrical side over to the mechanical side as equivalent force and equivalent resistance. The force (Bl)E/Re is equal to a real mechanical force ONLY when the cone is not allowed to move. While the math treats them as mechanical parameters, the math is a tool and if you assume that they are real mechanical parameters then you are letting the math overcome common sense. Whether or not you believe me, I simply ask you to think, not parrot sources out of context (if not out of context then I seriously question the sources) . ------------

simply an artifact of the model and really has no meaning. It is definitely not a mechancal loss

=2.23 +j33.75 =33.82 @86.22 degrees

Z' = (Bl^2/Re +Zm =8.89 +2.23 +j33.75) =11.12 +j33.75 =35.54 @71.76

corresponding

W****> Note that Zm cos (71.76) =(33.82) cos (86.22) =2.23 =Rms

-- Don Kelly snipped-for-privacy@peeshaw.ca remove the urine to answer