Motor torque and back emf

---------- Sorry that you have difficulties with normal analysis methods. Lets see, If I cut it down any more, I haven't given you the information. If I cross the T's and dot the i's I am producing a muddle. Respectfully, the muddle is in your mind and i am not sure of the reason except that you have schooled yourself in methods of calculation that are not applicable. ------------snip---

----------- I did my work. I have not tried to hide anything. I have given magnitudes where I did the calculations. I did not use numbers in the development of the equations as all that development was the manipulation of the relationships into a convenient form. The form that I got is useful for any set of data (mass, Rms, Re, E, Bl, , frequency, etc.). I suggest that your references give the equations with or without development and any numbers used are in examples. That is what I have done. The whole basis of the development is shown. Nothing hidden, nothing taken out of context - simply a step by step analysis of a very simple equivalent circuit. All I asked is that you look at these steps. If there is something in the steps which is unclear to you- point at it specifically.

The numerical work uses conventional analysis techniques for AC "circuits" (as this is a circuit analysis). I did not work with magnitudes alone for reasons that I repeatedly explained. I also sugggested that you look at Siskind and AC analysis techniques therein (if not, then get a copy of an appropriate Schaum's outline).

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-------- And as I suggested, true for Siskind, Higginbotham, etc - usual procedure. Develop the model then show a numerical example. That is what I have done.

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----------- It was rude, I admit. for that I am sorry.

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

I agree but either they or you have made some fundamental errors. From what you have quoted, it appears that it is not them who has made these errors. Next week I will possibly be where I can see at least one of you references. As for the model I have developed - I am satisfied of its validity with regard to the dynamics as supported by the only source with meat that I could find so far(who also based his work on Small, etc) has the same model for the driver. This is not a difficult problem. It would be a good exercise for a student in a beginning electromechanical conversion course. The more difficult problem is the actual measurement of parameters and it appears that a lot of short cuts and assumptions are made here.

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----------- It is not a smoke screen. It is contrary to what you believe but you have not presented any sort of rational counter argument to indicate otherwise. Please note that -if you had attempted to read the equations and the development of the equations, you would see where I get the terms. Part of the "muddle" as you state it, is added comments to help clarify this. Total damping does include the (Bl)^2/Re term. My point is that this term is not "actual" mechanical damping. The actual applied force is Fm=(Bl)I. Only part of this is the force acting on the mechanical damping. It all comes down to what is the actual mechanical force and what are the actual mechanical parameters.

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----------- I don't need a simplified explanation- I need a rational argument. Is Re a mechanical element or not? Yes or no. Why not go through the part where I did the development, line by line ("READ THESE STEPS") and tell where you find a problem with those - that would be a start.

--snip------

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No kidding! Since I use this equation in another form, it is reasonable to get it into that form THEN plug the numbers into it. I'm sorry that you have problems with that- it is grade 8 algebra. I am simply laying out the development (i.e. the physics of the situation) first. Is the equation itself correct or not, independent of the numbers used. Does it properly represent the physical situation? Yes or No. Numbers come later. Do your references always give a magnitude in the development of an expression or do they do it later, in an example.? I have a stack of texts that go through the theoretical analysis before plugging in numbers- this is the usual process.

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---------------- What it appears that you want me to do is to plug "only magnitudes" into the equations and from these determine a result - for example: Magnitude of I =0.232, Magnitude of E is 1.41 Volts Magnitude of Eb =0.353 Volts Now you expect me to say that the magnitude of I= E-Eb/Re=(1.41-0.353 )/5.78 =0.183 A OOPS: Now I have I=0.232 and I =0.183 Something is wrong- just as it was wrong when you get 2 values for Eb. GUESS WHAT! THIS SIMPLY ISN"T TRUE AS E and Eb ARE NOT IN PHASE.

You are treating AC calculations as if they were DC. This is a common error of some students in a first circuits course. Common but dead wrong. AS I SAID - this is a place where phase must be taken into account. If I use a CORRECT analysis then I will get

E-Eb =(1.41 @0) -0.353 @ -71.76 degrees DRAW THE VECTORS (phasors to me) =(1.41+j0) -(0.111 -j 0.335) =(1.41-0.111)-j(0--0.335) =1.30+j 0.335

Magnitude of result is sqrt(1.30^2 +0.335^2) =1.34 volts Phase angle of result =arctan 0.335/1.30 =14.48 degrees

Magnitude of Current I =1.34/5.78=0.232 A Phase of current I =14.48 degrees (error 0.02 degrees)

I als get an Eb as calculated from E-RI to be the same as Eb as calculated from V/Bl This is a simple check on the correctness of the number crunching. You had two quite different values for Eb - Do you actually believe that the same physical quantity has two different magnitudes at the same time? That is what your approach gives you. Somewhare I have lost the values you gave me for I, v, etc. but using the parameters you gave me, I do recall that the values I obtained below agree in magnitude.

Thank you. It *is* appreciated, although many magnitudes along the way are still missing, and especially the *derivations* for phase angles.

I never claimed such. My concern was the *relation* between BLv and E-(I Re). Please go back and check it out.

I'll check with Schaum :-)

Are you mixed up, or have I miscommunicated? I stated E Bl/Re was *applied* force, with *NO* motion. I have used cos angle or power factor with applied force to designate net force with motion.

Just quoting Small, and going by this: Connect a driver to an activated amplifier or short the input terminals together. Note the extra mechanical resistance when you push the cone in vs that with the terminals open. This is the back emf and back current **manifesting itself as a physical or mechanical resistance**, and is what Small terms the mechanical resistance of the motor. I don't see this resistance disappearing wnen in operation, and it's my view that power must be expended to overcome it.

My equation is correct re that aspect. I checked it.

????????? You use average velocity 0.0493 *yourself* for power calculation as Pmec = V^2 Rms = 0.0054 in your main argument. My concern was and still is the actual power factor, i.e that an effective magnitude for Rmec may be needed. I'm trying to work this out empirically with the test drivers.

It is intuitive (and gives the correct "average" which really

I do not deny there may be misconceptions. Surely you know that forced vibration, especially *above resonance* and more especially as related to *mechanical* power with *reciprocating* motion is given short shrift in the literature.

rational arguments. - I'm away for the holidays.

Happy Hollidays.

Now... I have stated I came here to learn. Stated it more than once. More specifically, hoping for assistance in some uncharted (at least for me) areas. We seem to be in a fairly polite pissing contest here, your ongoing implications re my needing to learn notwithstanding. Hell man, AGAIN I say, I came here to LEARN. If this bugs you, we can discontinue the discussion. BTW, it's possible you might even learn a thing or two from me..

Bill W.

----- Original Message ----- From: "Bill W." Newsgroups: alt.engineering.electrical Sent: Tuesday, December 23, 2003 4:23 PM Subject: Re: motor torque and back emf

----------- Derivations for the phase angles ?? I used standard comlex number representation as I indicated. If I say v=A+jB then the phasor (not actually a vector) has sides A, B and hypotenuse sqrt(A^2 +B^2) which is the "magnitude" and the angle is inverse tan (B/A) No deriviations other than this general one are needed. I let my calculator do the conversion work as it has the capability to do this directly. As for missing magnitudes -where?. - I gave the magnitudes wherever necessary and in some cases where it wasn't necessary. I did not and will not give magnitudes to equations but only to numerical calculations using those equations.

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----------- E-IRe =Eb =BLv is the fundamental circuit equation that must be satisfied. BLv and E-(IRe) must be the same. If not there is something wrong. You had concerns here. That is why I pointed out that use of phase as well as magnitude information is necessary -and, in that case, the relationship is correct as shown in my calculations.

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----------- I read what you said. I don't think that I am mixed up. With no motion the force is indeed EBL/Re as this corresponds to the electrical equivalent of a short circuit (Eb =Blv=0) The force causing motion -i.e. the actual force is simply (Bl)I or the Lorentz force. The E(Bl)/Re combined with the (Bl)^2/Re is a current source equivalent. I showed how this comes about. It has nothing to do with power factor. There appears to be a terminology problem here. By net force, do you mean that component of (Bl)E/Re that is in phase with the velocity (and contributing to average power) ? That is true. Is it a real force acting on the mechanical system? No. The real force is the Lorentz force and the component of this which is in phase with the velocity is the force producing mechanical power. The Lorentz force is given by (Bl)E/Re -(Bl)^2/Re ( and I showed this)

------------- Small is correct- and power is indeed expended. What is happening in what he describes is that the motion of the cone will induce a voltage in the short circuited coil and this voltage produces a current in the shorted coil (try it). The current will produce a Lorentz force which will oppose the motion. You sense this as an increased mechanical resistance.Nothing new about this - just plain ordinary regenerative braking. The mechanical energy that you put in is being dissipated in the coil resistance - i.e. electrically. It appears as a mechanical resistance because of the force/current and voltage/velocity relationships, and can be treated as such. However, its origin and the energy involved is in the electrical resistance and not part of the actual(as opposed to the manifested) mechanical resistance. That is why I have separated this resistance from the actual mechanical resistance for the purpose of calculating actual mechanical power. To look at it from another viewpoint, Small (J.Audio Eng. Soc. Vol20, p383-95 June 72) and Berenak give models in which all elements are refered to the acoustic side. That doesn't make the electrical and mechanical elements into actual as opposed to equivalent acoustic elements any more than referring them all to the electrical side makes the mechanical and acoustic elements actual R,L and C components. >

---------- your equation is correct for the "average" velocity as calculated from the displacement. However, the average and rms values are different. If rms values are used in one place then rms values must ve used throughout. The difference between the average and rms is a factor of 1.11 for a sinusoid.

Given a sinusoid x= (Xm)sin (wt) where Xm is the peak magnitude and w is

2*pi*frequency What you give for the velocity is 4f*Xm OK The actual velocity is dx/dt =wXm cos(wt) with a maximum velocity 2*pi*f*Xm rms value =(Root(2))*pi*f*Xm =4.44*f*Xm This is the bugger factor. It will be different for a different wave shape.

---------- If the maximum deflection of a sinusoid of frequency f is Xm, its rms magnitude is Xrms =Xm/root(2) and the rms magnitude of the velocity is Vrms =2*pi*Xrms (phase shift of 90 degrees with respect to displacement). Similarly the rms acceleration will be 2*pi*Vrms (180 degrees out of phase with the displacement). Isn't this an easier way to go ad ensuring all values are rms.

------------- I did all calculations on the basis of rms values. I never use average values for calculating power in an AC situation -it doesn't work except for square waves or DC. -That is part of the reason why the rms concept was established. The only reason that I can think of for the agreement in numerical values is that this average velocity or some other average value was used in determining Bl . To calculate power due to a current in a resistor, The use of I^2R for DC where I is the average current, is correct. In the case of AC, it was recognised (about 100 +years ago) that the use of average power didn't work. One could (correctly) integrate the instantaneous power over a cycle and take the average. This led to an average of [R(Imax)^2]/2 Out of this the rms concept was developed. Irms =Imax/root(2) so that the average power was given by (Irms^2)R. That is the rms value is the current which produces the same heat as a DC current of the same magnitude. It is applicable to all waveforms (different bugger factors) The power factor concept came about when it was recognised that only the component of current which is in phase with the voltage produced average power. The remainder was called volt amps reactive (var) Because of the nature of the in-phase/out of phase components of currents and voltages, an existing mathematical technique was adapted for calulations - that is a double barrelled or complex number arithmetic - and further on into Laplace transforms signal processing methods. Just tools but useful tools.

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----------- Understood. My contention is that the difference between the actual mechanical impedance and the equivalent mechanical impedance must be clear even if they are combined for some calculations .>

--------- Go to a general engineering dynamics text. One of these should cover it - likely using free body diagrams rather than the equivalent circuit approach which Small and Berenak apparently used. It is pretty straight forward- as you have indicated, the mass becomes a dominant factor.

By the way Bullock, T.S. gives an analysis which parallels work by Olson and Locanthi(sp?) AES 1986 preprint 2841 (H-2) It is shown on

There is an error where Sd rather than Sd^2 is used in the Re term referred to the acoustic side. As the U of A library was closed over the holiday period, I was unable to get at some references that I wanted. In addition, all copies of Kinsler were out until the 15th of Jan. Where I live is too far from a library with the desired journals, etc. I'll check with a friend in town who taught mech eng'g. and he may have a reference. >

-------- It doesn't bug me. I do not have your practical experience in the area of loudspeakers and acoustics. I bow to you in this respect and can learn from you in this regard -not so much the theory but the application of the parameters per se in practical enclosure design. I do know circuit analysis and electro-mechanical energy conversion (which this is) and, in these areas and I have reason to believe that I have a better understanding of such analysis and modelling than you do.

Some of what we have been at loggerheads about is terminology but some is fairly fundamental.

Take care

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

Thank you for the references, and we shall see what we shall see. :)

We will have only hassles until we settle the role of (Bl)^2/Re = 8.89. You stated in effect that no power is required to accelerate the mass, and your mechanical power equation says so, where you give mechanical power at 227.4 Hz as Pmec = 1.664 * 0.066 * 0.0492 = 0.0054 watt

or (as equated to my notation of power dissipated into the suspension resistance)

Pmec = v^2 * Rms = 0.0493^2 * 2.23 = 0.0054 watt

Note Rms = mechanical resistance of the suspensions, so your equation shows the power dissipated into this resistance, with no power shown as dissipated into the (Bl)^2/RE = 8.89 resistance. Note Small's term for (Bl)^2/RE is Rme. My view is that (Bl)^2/RE *must* be added to Rms to calculate the mechanical power, i.e Pmec = v^2 * [ Rms + (Bl)^2/RE ] = 0.0493^2 * (2.23 + 8.89) = 0.027 watt

agreeing with Halliday and convention

P = F v cos angle = 1.75 * 0.0493 * 0.313 = 0.027 watt

where angle = arc sin Xmec/Zmec = 71.78 then cos angle or PF mec = 0.313

Now hang on here please, as 0.313 may not agree with your current magnitude (as I recall it was your original magnitude however), but *bear with me* without getting into this aspect at the moment, i.e. "wait for it" :)

First, try looking at the physical driver so: The neck of the coil former equates to the usual motor's shaft. In other words, in the usual motor, the driven armature is attached to the shaft, and the shaft drives the load. Here the driven coil/armature is attached to the coil former and it drives the load, i.e. the cone and air mass. The point being, the cone and air mass ***is*** the load, it is not a part of the motor. As such then, at 227.4 Hz the mass load will require power to be accellerated, since the mass gets a free ride ***only*** at resonance, not at 227.4 Hz.

Now the net force magnitude in your above equation is

Fnet = 1.664 * 0.066 = 0.11 or v *Rms = 0.11

I maintain net force is v * [Rms + (Bl)^2/RE] = 0.548

giving Pmec = Fnet * v = 0.548 * 0.0493 = 0.027 watt

and not (as you note) 0.11 * 0.0492 = 0.0054 watt.

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Noting respect for Halliday, I shall now stay with him, and show that net force is indeed 0.548 and not 0.11. Starting, with scalar (energy-based) magnitudes:

Halliday:

Work = delta KE

Where kinetic energy KE is

KE = 1/2 m v^2 = 0.5 * 0.0253 * 0.0493^2 = 0.0000307

i.e

Work = delta KE = 1/2 m v^2=0.5*0.0253*0.0493^2 = 0.0000307

with mass and velocity measured, and starting from zero velocity.

Halliday:

Fnet = work/distance = w/d = 0.0000307/0.0000543 = 0.566

for distance traveled during 1/4 cycle, note.

This is close enough to convention, and to that which I have given earlier as

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

************

Note carefully between *********** here, please

*** The net force expression *includes* (Bl)^2/Re so ***

F net=(v Rms)+(v (Bl)^2/Re)=(0.0493*2.23)+(0.0493*8.89)=0.548

This can be written as

Fnet = v * (Rms + (Bl)^2/Re) = 0.0493 * 11.12 = 0.548

Showing the damping coefficient or damping constant is based on Rms ***plus*** (Bl)^2/Re at 227.4 Hz, so that again, at 227.4 Hz

Pmec = v^2 * [Rms + (Bl)^2/Re] = 0.027 watt

again agreeing with Halliday and convention

P = F v cos angle = 1.75 * 0.0493 * 0.313 = 0.027 watt

and not your mechanical power of

Pmec = 1.664 * 0.066 * 0.0492 = 0.0054 watt

**************

Now to Halliday again:

Halliday (and Serway) state ***regarding forced oscillations***

F max x max = --------------------------------------------- sqrt [ m^2 (w^2-wo^2)^2 + (b^2*w^2) ]

solving for b (damping coefficient)

| F max^2 | | ----------- - [ m^2 (w^2-wo^2)^2 ] | | x max^2 | b = sqrt| -----------------------------------| = 11.05 | w^2 | agreeing with, Rms + (Bl)^2/Re = 11.12

We see the damping term is not just Rms, but instead must include (Bl)^2/Re, as I have noted all along.

Then, yet again, giving mechanical power at 227.4 Hz of

P mec = v^2*[Rms+(Bl)^2/Re] = 0.0493^2*11.12 = 0.027 watt

and not your

Pmec = 1.664 * 0.066 * 0.0492 = 0.0054 watt

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Since you appear to respect Small as well as Halliday, note that he gives

(Bl)^2/Re Rat = Ras + ------------ Sd^2

These are acoustic magnitudes. Converting to mechanical by multiplying by Sd^2 gives

Rmec = Rms + [ (Bl)^2/Re ]

Again the damping term must include (Bl)^2/Re and again mechanical power is

Pmec = v^2 * [ Rms + (Bl)^2/RE ] = 0.0493^2 * (2.23 + 8.89) = 0.027 watt

If this doesn't convince you, I can show you the same thing with Serway, Kinsler, and Beranek as I have previously done.

Another term is used to make a power equation workable at resonance. Also the resistive term (Bl)^2/Re can be related to the the retarding force of the load and back emf. We may consider these later if this is settled, as this has likely been a bit frustrating for both of us. Oh yes, one more thing, as Columbo used to say... Could you please be explicit and give magnitudes in your response? Or as Jack Webb on Dragnet said.. "Just the facts Mam, just the facts".

Bill W.

---------- Since (Bl)^2/Re is directly a result of the coil's electrical resistance, power in this term is not mechanical power. This is a representation of the electrical resistance as seen from the mechanical side. It does have an effect on the behaviour of the system. It is also very clear where it comes from. I have shown this. As to what Halliday actually says- I do not know. What you have quoted, so far, of Halliday and others does not counter my argument.

There is no fundamental difference between the speaker as a motor and a conventional motor except that the armature in the speaker moves linearly in and out while that of the conventional motor rotates. (It is also possible drive a DC motor with a low frequency AC source and have oscillatory motion in which mass plays an important part). There are other devices than speakers which use linear motors. Even a doorbell is one such device. The principles of operation are the same.

All I am working with is the defining equations of the device. If these are wrong, you were asked to say where they are wrong.

Mass will be accelerated - definitely- at all frequencies unless the cone is clamped so it wont move. However, the component of the force associated with the acceleration of the mass is 90 degrees out of phase with the voltage. There will be instantaneous power accelerating the mass but at

another part of the cycle the energy stored in the mass is returned. and the average power into the mass over a cycle will be zero, zilch, nada. V*Xmec will be 90 degrees out of phase with V*Rmec The situation is similar to that of an inductance or capacitor - look in Siskind. This doesn't mean that there is a free ride. It does mean that there will be higher losses in the electrical resistance and a higher total force required because of the Xmec. Electrically, EI will be greater than P - hence the origin of the whole basis of the concept of power factor and its importance.

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------------- You are dealing with a change of KE of a mass Something appears to be lost or over-simplified. Delta work = delta KE will be a better way to do this Delta simply meaning a small change or 1/2 m (v+delta v)^2 - 1/2 mv^2 = delta work. An avoidance of calculus is evident. Fair enough.

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----------------- Do you use a speaker only for a quarter cycle? Is the v that you are using the average v, the rms v or the peak v? If it is peak v then this will be the work in that quarter cycle. If it is something else, it is meaningless. Note that in the next 1/4 cycle, from the peak velocity to 0, the work will be of the same magnitude but will be returned to the source from the mass (mass will decelerate dumping stored energy back into the system) Net work in the half cycle will be 0. A mass, a spring, a capacitor and an inductance, are energy storage elements- they do not dissipate energy and stored energy on part of the cycle will return to the system on another part of the cycle. This is the case whatever way you want to calculate it. This ties right back to power factor etc. Secondly, this energy into and out from the mass has nothing to do with the resistance or the average power per cycle (which is what is considered power unless specified otherwise). .

------------- See above. Is this what Halliday actually says or implies or is it your interpretation? If the velocity is sinusoidal, then the force is sinusoidal and the Fnet that you give based on the work in a 1/4 cycle is what?. The force that exists will change in each small increment of time. It will be different at x =0+ than at x=Xmax. Work vs distance is varying. Which value of force should be used? Quarter cycle "force" or average force is not of much use -particularly for power calculations and for the type of analysis involved here.

-------------- You are apparently, although you have not said so explicitly, defining Fnet as the real component of the force. Please note that this has nothing to do with the mass. The component of force driving the mass is not included in this. Again, I have a problem. It is that Fnet as you indicate is non-existent .It is an artifact of the model. The term (BL)^2/Re is a part of the force source model - this falls out of the equations. The component of force associated with this is simply the difference between (BL)E/Re and the Lorentz force BlI. I have shown this.

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--------------- There is no problem here- the term (Bl)^2/Re does represent the effect of the coil resistance on damping. That is why I used (Bl)E/Re = [(Bl)^2/Re +Zmec ]V to determine the velocity V. The resistance of the driving amplifier and any corrective electrical networks will also affect damping- that doesn't make them mechanical except in terms of their equivalent effects. It doesn't mean tha tpower lost in a resistor or its mechanical representation is actual mechanical power.

--------- Tell me, does Halliday use this information in any way. Remember that I haven't seen Halliday. If Halliday does consider that Pmec including the effect of Re is the actual mechanical power, then he is wrong. Pure and simple. Note that an equation that I used for calculating Pmec is of the same form with different numbers. In using P=Fv cos angle =1.664*0.0492 * cos (86.22 =1.664*0.0492*0.066 =0.0054 watts I am using the same equation - with different numbers-that you are using. The Lorentz force and the actual mechanical impedance and power factor. Halliday may use your values but are you sure that he claims that this is the actual mechanical power or does he simply use this value to determine Xmec as you once showed.?

------------------- They have simply taken methods used in electrical circuits and expressed the damping coefficient in a somewhat cumbersome "plug in and turn the crank formula". However, this formula is intended to be convenient for use with measured quantities. No problem there. Much of the formulae you have given are of this form- If they do the job- fine - but don't try to assign a physical significance on the basis of these. There be dragons in those woods.

****Certainly the Re term does enter into damping. I have never said or implied otherwise.**** The electrical and mechanical sides are coupled and not independent so that electrical elements have an effect on the mechanical side An energy absorbing element in the model appears, as seen from the mechanical side, as an energy absorbing element . No big deal. I have no problem with that. ****That doesn't mean that it is an actual mechanical element****- In the math, it is treated as a mechanical element- no problem. The problem occurs when the difference between the math and the actual device is forgotten. When I calculate the velocity, I include this term. However, when calculating mechanical power, I recognise that this term, while it affects velocity and the damping, is not an actual mechanical resistance. That is - I am keeping the physical device in mind rather than simply turn the arithmetical crank. As an example, the mechanical load of an AC induction motor appears from the electrical side as an electrical resistance. One can calculate the electrical power in this resistance and equate it to the mechanical shaft power. There would be no point in calling it an actual electrical resistance as that would be in defiance of fact, even though my electrical measurements "see" this as a resistance.

Please note that I can also check the calculation for mechanical power. Use the current and source voltage as well as the power factor angle between them to determine the source power. Then subtract the I^2 loss and what's left over is the mechanical power (by the conservation of energy principle which predate myself, Small, Halliday, etc and is well established in both fact and literature).

Source power =E*I*cos angle =1.41*0.232 *cos(14.46) =0.3168 watts I^2Re ={(0.232)^2}5.78=0.3111 watts Source power - electrical power lost =0.3168-0.3111=0.0057 watts =power delivered to the mechanical system. (Authority --conservation of energy)

Note that my calculations do check without contradictions. There is likely an error in some of the data that you gave me and I used (again I indicated specifically what data I used) I had previously asked and was tole that your values were rms. Obviously your velocity was an average value so using this with other values as rms is like comparing apples and pears- related but not the same- Did you use the velocity in determining Bl and other parameters?

--------- And this is consistent? do you get consistent values when relating back to the source voltage and current, etc? You have made a claim. You have not backed it up. I have given a full development which indicates where the (Bl)^2/Re term comes from - I asked you to look at these few equations and pick them apart. Do you see a flaw in my going from

E=ReI+(Bl)V and (Bl) I =ZmV to get (Bl)E/Re-{(Bl)^2/Re}V =ZmecV where Zm is based only on the mechanical elements M, K and Rms Zm =Rms +j(Mw-K/w) =sqrt[Rms^2 +(Mw-K/w)^2] at an angle inverse tangent of (Mw-K/w)/Rms

{At 227.4 Hz (1429 rad/sec) Zm=2.23+j(0.0254*1429 -3425/1429) =2.23 +j33.75 magnitude of Zm=|Zm|=sqrt(2.23^2 +33.75^2) =33.82 Ns/m phase angle inverse tangent (33.75/2.23) =86.22 degrees These are values I gave before.}

If these are OK and the conclusion that V=[(Bl)E/Re]/[(Bl)^2/Re +Zmec] {Magnitude 1.75/35.54 =0.0492 Phase of V is -71.76 See below for equivalent impedance magnitudes} where (Bl)^2/Re +Zmec =[(Bl)^2/Re +Rms] +j [Mw-K/w] magnitude sqrt [((Bl)^2/Re +Rms)+((Mw-K/w)^2] at angle inverse tangent of (Mw-K/w)/((Bl)^2/Re +Rms) That is - what you call Zmec and I call an equivalent Z which includes the effect of the coil resistance. For this equivalent Z the magnitude is sqrt((8.89+2.23)^2 +(33.75)^2)= .

35.54 and the phase is 71.76 degrees

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------------ Actually, you haven't - you have given a formula with no development or background of that formula and plugged your numbers into it. If you take F*V* cos angle and convert this to I*Eb cos angle (there will be no change in the power found) then calculate E*I cos angle (different angle) less Re*I^2 do you get agreement? If not, then one of the most important physical laws is violated. Try it.

Again, I have no problem with the Re term as part of the damping. You are the one calling this term part of the mechanical power. If you dont use it except for evaluation of the reactive component of the force, it makes no difference in practice in the long run. How do they use this? At one time you implied a use of sqrt(1-(pf)^2) in determination of the reactive component of force. For this it is OK. However, there is no way that the Re term is actual mechanical power and I suspect that your various sources, on careful reading, do not actually imply that. (their papers would have been shot down by the referees of the Journals). It simply doesn't stand up to physical interpretation of the parameters involved.

--------- The equations that I gave are general- ***they work at any frequency**** for which the parameters are valid and can include coil inductance and be expanded to include the source in the model. I did do the same calculations for resonance. The term (Bl)^2/Re, is, I agree, an equivalent mechanical resistance which must be taken into account. It falls out of the coupling between the primary equations (as does E(Bl)/Re). It does affect damping and the frequency response but where I disagree with you is that it is not an actual mechanical loss element as it is due to the electrical resistance. It does affect total power but it is an artifact of the model and power in it is really meaningless as it is interior to the source model used and such a model is correct only beyond it's terminals. That is why I had to calculate the input power and actual I^2 Re loss from the electrical side. This is something that is recongnised although possibly not always clearly expressed in circuit analysis texts.

---------------- Since I have done very little numerical work in this response, there are really few magnitudes to express but I aim to please. I have given magnitudes in previous numerical work so the values given are no more than a re-iteration of that work. I have also explained that magnitudes alone are facts but only part of the facts. I gave you the all important but often insufficient magnitudes and the complete basis for the calculation of various quantities well as step by step numerical work. I am quite happy to have you go over these step by step, with questions.

If you want other references try this one (if you can find it -out of date but dates seem to be around the time of Small's work). Gourishankar V., & Kelly, D.H.; Electromechanical Energy Conversion, second edition, Intext, New York, 1973 ISBN 0-7002-2404-1 Library of Congress number 72-84132

Particularly chapters 5 and 6

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

So it's you against Halliday, Serway, Kinsler, Beranek, Colloms, and Small, otherwise my measurements are in error, eh? Sure, sure... But my measurements are not in error, I remeasured, and they are absolutely correct. At 227.4 Hz: Source voltage (voltage applied) E = 1.41 volts Current I = 0.227 amps DC resistance Re = 5.78 ohms AC Impedance Ze = 6.21 ohms

You stated in your last post:

"Source power = E*I*cos angle = 1.41*0.232 *cos(14.46) = 0.3168 watts "

You stated previously:

" In this particular case there is a phase shift of 14.46 degrees in the current"

Now I ask previously:

"How did you arrive at 14.46 degrees"

You replied:

"I calculated the velocity from BlE/Re =Zme V (Zme is the total impedance referred to the mech side. Then I looked at Fm=VZm where Zm is the mechanical impedance. and found the orresponding I Alternatively, I get the same result from E-RI =BlV =ZmeI where Zme is the mechanical mpedance transferred to the electrical side. (or E=ZeI where Ze =Re +Zme) These calculations were given.The latter is 1.41 =(Re+(Bl)^2/[Rms+j(wm-K/w)]I =(5.88-j1.517)I giving I =0.232 @14.46 "

This is a *non-answer*, just bull and more bull. Your angle of 14.46 is wrong. The correct phase angle between voltage and current is

Phase angle = arc cos Re/Ze = arc cos 5.78/6.21 = 21.44 deg.

It is not 14.46 degrees as you state. I use here ***measured values*** of Re and Ze, and you have altered them to support your argument. This is bull and is not acceptable.

Bill W.

++++++++++++++++++

of the driving amplifier and any corrective electrical

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

-----------

Good for you. Please note some small errors that you have made as indicated below and the failure to understand the large effect these small errors can have on calculated values. I am not against Halliday, etc. So far you have not given me any thing of theirs which I can say is wrong. What I disagree with is your interpretation of what they have said. Some of what you claim that they say IS wrong. Some of it may be a poor usage of terms. Whatever. I have also questioned the use of average values in conjunction with rms values or peak values or for calculation of power and if you have done so in determining of parameters then these parameter calculations are in error unless you have made appropriate corrections. Note that nowhere have you given the way that measurements were made except that maximum displacement was measured at resonance, and how you calculated an average velocity. I hope that you didn't use this average velocity to find an average acceleration.

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

----------- . I previously indicated the parameters that I used for the calculation. If there are errors then they are in this data for M,K, Bl, E, Re and Rms that YOU provided. All values that I calculated are based on this information and the theory was shown first then the calcualtions were made step by step. The whole calculation setup and the basis for each step was given clearly.

-----

------------- Your contention is the bull. I did not change any values and you know it. All steps in calculations were clearly laid out. Note that the Rms referred to the electrical side is 0.1 ohms. 5.78 +0.1 =5.88 . I did not use the value you measured for Ze but calculated Ze from the values you provided for the parameters. Sorry that you couldn't follow it. Try reading the "non-answer" slowly and carefully and maybe you will get it.

In detail I calculated Fm=(Bl)I=ZmV where Zm =2.23+j33.75 Ns/m (mechanical components only) Fm= 1.664 @14.46 degrees (N) Any problem with this value in phase or magnitude?

I =Fm/Bl =0.232 @14.46 No shift in phase with respect to Fm. If you want to question I then you must question Fm

Eb=BlV =0.353 @-71.76V All this previously shown. Ze=Re +Eb/I =5.78 +[(0.353/0.232) @ -71.76-14.46 ]=5.78+(1.52 @ -86.22)

Please note that (Bl)^2/Zm =(7.17^2)/(33.82 @86.22) =1.52 @-86.22 ohms =0.1+j1.517 ohms as a check It does check. (as said before a mech impedance translates to an electrical admittance).

Then Ze =5.78 +0.1-j1.517 =5.88+j 1.517 which leads to a magnitude of root (5.88^2 +1.517^2) =6.07 ohms and a phase angle of arctan 1.517/5.88 =14.46 degrees

Or , using arccos (5.88/6.07) =14.36 degrees.

The arccos calculation is less accurate than the arctan calculation (it is based on arctan along with a few additional calculations which lose accuracy for small angles). In addition you have also ignored the small contribution of Rms to the value of Ze- while small (0.1 ohm) it will affect the angle calculation, particularly using arccos. (deliberately doing this using arccos 5.78/6.07 gives an angle of 17.8 degrees) If I use your value for Ze then , using Xe=root(Ze^2-Re^2) (ignoring the 0.1 ohm factor as you have done) then Xe =root(6.21^2 -5.78^2) =2.27 ohms which will correspond to an Xmec =(7.17)^2/2.27 =22.6 Ns/m where we have both calculated a value of 33.8 Ns/m---OOPS! Doesn't this indicate that something is rotten in Denmark? It should check reasonably well but it doesn't. Note that my value does check. Note also that errors in phase would lead to errors when I plug back into E-RI and compare the result to the back emf BlV. I did this and showed you that the results checked. Have you attempted any such checks yourself? I suggest that you do. Now any error in the Ze that I obtained, as compared to the value you measured is not in the calculation but in the data on which that calculation was based- the data you gave me-- or it is an error in the Ze that you measured. In either case it is in your data. Note also that the difference in magnitude between your Ze and mine is 2.3 %. That is pretty good. Using arccos for small angles leads to relatively large phase errors. (the difference between your R/Z and mine is about 4% leading to a phase angle change of nearly 8 degrees). You have ignored Rms in your calculation which accounts for about 3 degrees and also used arccos for calculating the angle which gives about another 5 degrees error. I would respectfully submit that you have compounded some errors in you attempt to show me wrong and, further, you have not made even the most rudimentary checks of your own data and calculations.

Just as a further question; Do you have any idea of how Halliday got the following expression? I do.

Hint: it requires only a couple of lines to get this from equations that I gave (and the final conclusion that b =(Bl)^2/Re +Rms is essentially the same as saying 2=2). Hint 2: what is the relationship between Xmax and Vmax for a sinusoid?

sweetness and light to you too,

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

It's unfortunate you choose projection, i.e. accuse me of error rather than admit your own *major* error that invalidates your analysis. Apparently I over-estimated you, and you have drawn the "battle line". Too bad.

Yes you are. It is proven below by Hallidays equation explicitely.

Right, it is you that's wrong.

This is aversion. What Halliday, Beranek, etc say is not even needed re your major error regarding the plase angle between voltage and current. This is just basic electronic theory..

This is a smoke screen. As I said, there was little use in going deeper until the issue at hand was resolved.

Another smoke screen, but I'll address it anyway.. I told you early on that I measured velocity at resonance of 58.6 Hz. Sorry, but if you cannot extrapolate to ~ two octaves up to 227.4 Hz, then you better go back to high school. Regardless, I gave you the equation for extrapolating, and you did not reject it. Regardless again, the magnitude of velocity is *not* required to obtain the phase angle of voltage and current, therefore my comment re smoke screen.

No. Your error involves the manipulation of my data and invalidates your analysis, making it useless.

Clearly wrong, that is.

This is just more bull without any backup theory giving specific magnitudes for all terms re how you obtain this 0.1. Re is *measured* accurately at 5.78, and *that is that*. Your saying it is 5.88 is nothing but bull to support your error. Otherwise prove Re = 5.88 and not the measured 5.78. Yep, just prove it as in P R O V E.

I'm going to give our exchange a chance at new life here. Also hopefully someone else will join in here. Daestrom, where are you? :) To answer the above, I *do* have a problem with your Zm, etc. In fact several problems. Zm = 11.12+j33.75 = 35.53 not 33.82 as your equation gives. Here is your chanch to correct one of your errors. In my notation and reference below ( Halliday and others as noted) the resistance *must* include (Bl)^2/Re, i.e.

Rmec = Rms + (Bl)^2/Re = 11.12, not just Rms = 2.23.

HERE IS THE KEY TO THIS.....................................................

Your equation using Rms alone works ***ONLY*** at resonance. Halliday works at ***any*** frequency. I have been through this with you earlier and will not go through a bunch of your erroneous equations below, attempting to dispell them. The theory can and has been put in a straightforward manner. Again see the Halliday amplitude and retarding force equations below. And yet again........ you ***must*** include (Bl)^2/Re in the mechanical resistance term..

DID YOU EVEN PLUG IN THE NUMBERS INTO THE HALLIDAY EQUATION, AND SEE THAT IT WORKS AT ANY FREQUENCY, THEN TRY YOURS WITH Rms ONLY AT 227.4 HZ? I have given you a powerful set of tools with my VERY ACCURATELY MEASURED data. Why don't you apply these tools appropriately?

Now, is the above re phase an attempt to confuse, or just error on your part? Phase on the mechanical side between force and velocity is

phase angle mec = arc cos Rmec/Zmec = arc cos 11.12/35.53 = 71.76 deg.

Mechanical power factor then is cos angle = cos 71.76 = 0.313

On the electrical side, phase angle between voltage and current is phase angle elec = arc cos Re/Ze = arc cos 5,78/6.21 = 21.44 deg.

Electrical power factor then is cos angle = cos 21.44 = 0.931

*** Your angle of 14.46 deg. is wrong.***

RE your notation above of Fm= 1.664:

Net force is applied force times cos angle or power factor PF, which is: Fnet = F PF = E (Bl)/Re * PF = 1.75 * 0.313 = 0.548

This too I have shown before, and besides it is BASIC, no "high level" phase manipulation is needed, .

Now there is no point in my addressing the following where you again claim various magnitudes that do not match ***measured*** data. ***Surely*** you know the reference is that which is measured, and certainly on the basics above of voltage E, DC resistance Re, impedance Z, and current I. Note these basic magnitudes are all that is needed to show your error in phase angle of voltage and current that invalidates your analysis.

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

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

Yes, but in essence you get 3. :)

Really, then why do you call b = Rms only at 227.4 Hz?

I assume you mean x max, i.e. displacement from center equilibrium, or amplitude?

Mr. Kelly, I regret you are confrontational and inflexible re your error, but that is your right. I had hoped for better than that which is transpiring and tried to avoid confrontation, but you leave me no other choice than to walk away. I have stuck around in the hopes we may eventually get to another stage where I would appreciate input. However, it appears you feel you operate with a direct line from the Heavens... Why do you not just admit your major error, redo your analysis, and move on? Or is it that you have a problem with BLI at 227.4 Hz? :)

Bill W.

---------- Sorry, the over-estimation was on my side- I expected more sophisticated mathematical and physical understanding.

-----------

As I asked - did you figure out how Halliday got this equation? Apparently not. I don't disagree with it as have checked it and it is absolutely correct. In addition, this equation, proves nothing. It is derived directly from an equation which I gave.

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

-

------------- True and I have used basic electrical (not electronic) and mechanical theory. CORRECTLY. I worked directly with the starting equations and laid out all steps in the analysis. Did you even try to follow these steps. Can you use comlex numbers?

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

------------ Did you measure displacement or velocity.? I saw the equation for extrapolation - fair enough, but with the parameters Re,E, Rms, K, M,Bl there is no need to use this equation. I see the basis for your

extrapolating equation and didn't reject it. I simply didn't need to use it. The equations that I use are, in spite of your contention, valid at all frequencies (for which the parameters are valid). As to velocity - my comments still stand. What is the peak velocity or the rms velocity- more important and useful than average velocity. No- it's average magnitude is not needed. In fact you have not indicated where you use this average velocity. If you don't need it , then why calculate it? It cannot be compared directly to rms values nor is it meaningful to use it for power calculations. Is it a smoke screen to propose such questions? What I consider a smoke screen is the avoidance of any answers to questions when these questions involve understanding of the engineering, physics and math and the differences between these. .

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

----- Pleas review and READ what I have said. You have made no attempt to show errors in what I have done but just claimed that I am in error. I have laid out all the numbers that I used and how I did my calculations. Now, rather than say that I manipulated, look at what I have done and show where, in this "manipulation" that I have made errors.

-----

----------

------------ You obviously did NOT read what I wrote, or if you did read it you have made no attempt to try to understand it. At least I have had the courtesy to read and understand what you have said. Please note that I did NOT say that Re =5.88 ohms What I DID say was that Re + the resistive portion of the mechanical impedance reflected over to the electrical side is 5.88 ohms. Using Z =F/v as you properly corrected me on the corresponding Zem=Eb/I which corresponds to units of velocity over force - the inverse of the mechanical Z. Note that you did not question these values in a much earlier correspondence. That is where the 0.1 comes from as indicated above and shown below. (calculated two ways as a check) where I marked it with HEYHEYHEY

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

NO. Zm is as I gave it. What I call Zm is the ACTUAL mechanical impendance which does not include Re I have repeatedly said that. Fm is BlI which is the actual force. I also do use Re in calculation of velocity as Re does affect the system behaviour (i.e. damping) However, representing it as a mechanical element for calcualtion purposes doesn't make it a mechanical element - to do so is letting the math displace common sense.

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

-------------- Please note that I said that the effect of Re is part of the damping. I have never said otherwise. Note that In calculation the velocity, I have always used (Bl)E/Re =[((Bl)^2)/Re +Zm ]V where Zm is due to the mechanical elements only. This by the way leads directly to Halliday's equation. That is why I said that it is equivalent to saying 2=2. Halliday's equation is unnecessary unless one is using it to as part of a calculation of the value of Rms. Please note that, at resonance as well as any other frequency, I don't use Rms alone in finding velocity. Where did you get that idea? It certainly wasn't from anything that I wrote.

-------- PLEASE READ WHAT I ACTUALLY SAID. I used the basic equation and shuffled it to get Halliday's expression for b - big deal. I did this symbolically so the relationship immediately becomes clear.

As I indicated to you, when I asked if you knew where Halliday got this equation, I had aready effectively used this in the basic equations that I gave you. . It follows directly from the equation used to find V as cited above and used by both of us. Halliday's equation for b is a trivial reworking of this. That is why I suggest that you explore the relationship between Vmax and Xmax for a sinusoid. ------

Your values are acc rate - you have Z electrical =6.21 ohms while I have the value of 6.07 ohms based on your mechanical data. The difference is 2.3% which is actually damned good considering the total difference which can occur in the various measurement errors (I have less faith in meter accuracy than you do). I did point out that, with your values and calculating X from root(Z^2-Re^2) then determining the equivalent X on the mechanical side. This leads to an Xmec =22.6 compared to an Xmec of 33.8 Ns/m as calculated from the mechanical parameters. Hence the OOPS! Things don't jibe. It is not that Ze is inaccurate in magnitude but it does imply that the phase angle you get for Ze is in error. I also indicated that small errors in Re and Ze can lead to large errors in arccos Re /Z As to the tools???? They are rote applications without showing any apparent desire to show or know the basis of the equations quoted.

The values that I use are consistent. Both basic equations (E=RI +BlV and BlI =ZmV (my Zm) ) are satisfied. Try your values. You will not get E-ReI =BlV. Does this mean that the basic equations are wrong ? No. It means that some calculation is wrong. That is why I do make these consistency tests.

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

------- Neither _------ Phase on the mechanical side between force and velocity is

Including the Re term in this - fine. However, that is where you are wrong. You have kept the math but thrown out the physical significance.

-------

Again, the crux of the matter is the treatment of Rmech = 11.12 as a true mechanical impedance. I see you haven't tried to understand the basic equations that Ipresented before where the source of the (Bl)^2/Re term is quite apparent. I have pointed out inconsistencies and you have simply ignored these. Do you not think that when you start from a basic set of equations, that the results obtained for given quantities by different approaches should agree with these equations?

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

------------ E(Bl)/Re is a force which exists only under velocity = 0 conditions. Kinsler (see below) refers to it as an "apparent" force. I have shown and again I ask you to read, that F=BlI =(E(Bl)/Re -V(Bl)^2/Re You have not questioned how I got this or how I went on to use this to find V and then used V to find BlI and I , etc. The equations are there -step by step.

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

---------- Golly Gee. Sorry- I have clearly pointed out where I got the phase angle and how small errors in your measured Z and Re of nearly the same angle can translate to large errors in phase. In addition I pointed out discrepancies in Xm or in E-RI that I don't have.

HEYHEYHEY

-------- I take that as a No- you don't know.

--------

------------- I Don't - Please read what I have written, rather than be in a state of denial. I have always recognised that Re is a factor in the damping due to the coupling of the electrical and mechanical equations. You have failed to see what I have repeatedly said about this and also have ignored the fact that I used it correctly. .

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

------ Yes.

-----

---------- I have no problem, I 'm happy. As for the error - I have given my viewpoint and the reasoning for it. That is more than what you have done. As for inflexibility- I have reviewed my material repeatedly, did calculations and also did check sums which indicated the correctness of my answers. Please try to do the same. Punch your values into the basic electrical and mechanical equations (only 2) and see if they are consistent and reasonable. Is that too much to ask? .

Please note again that I have laid out each step of my analysis. Nothing hidden, Nothing based on quotes from references that you don't have. You can look at these steps- both the theory and the example and question specific points- fair enough. However, please don't attribute to me things that I have not said and also please distinguish between what your references actually say and what you think they say.

I now have borrowed a copy of "Fundamentals of Acoustics" 3rd edition 1982, by Kinsler, Frey, Coppens and Sanders. .(Formerly Kinsler & Frey - both are now dead ) This book is intended for advanced undergraduates or graduate stucents in science and engineering and was so used by the friend who loaned me the book. A small section of this book, Chapter 14, deals with transducers and after general theory , deals with the moving coil loudspeaker. I note a few points.

1) Zm used is definitely the mechanical impedance. In fact it is defined in terms of the open circuit impedance (I=0) . He also attaches basic acoustical R,X as referred to the mechanical side but this doesn't change the picture. Re is definitely electrical and left on the electrical side. It is very clear that my conception of the mechanical power agrees with this text . what is used is V=(Bl)I/Zm where Zm is the mechanical impedance. He also then uses Ze (electrical) =Zeb +(Bl)^2/Zm and Zeb =Re +jwLe This is in line with what I have said but pproached from the electrical side rather than the mechanical side. Note that in this case I=E/Ze and from this BlI can be found leading to V. I started with finding V and then finding I. The two approaches will lead to the same result and both include the mechanical and electrical resistances in the damping term as they should. 2) the complex number analysis approach that I have used (and that you pooh-pooh) is also very much a part of the analysis used in this text. All the Z's etc are expressed as complex numbers. Why? - it is simply a more rational approach to analysis than use of power factors which are not used in this text (possibly because the earlier editions date back to 1950 and 1962 and common electrical circuit analysis methods (including two-ports) have since spread to mechanical and electromechanical system analysis because it works well. 3)There is an emphasis on various equivalent circuit models - of which several exist. I use a particular visualisation and a model which leads to the same relationships as obtained in this text.

I am very confident in the correctness of what I have said. This reference simply provides back up support for my views. I hope to borrow a copy of Beranek from the same source when he can dig it out of storage. In the meantime, I will use this book to study the acoustic side of things.

Thank you for your time.

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

No, it is from Hallidy, Resnick, and Walker Fundamentals of physics, sixth edition, and given re forced oscillations. Serway and others give the same, and proves you wrong. Here it is again, along with your reply, which *** says it all ***.

| F max^2 | | ----------- - [ m^2 (w^2-wo^2)^2 ] | | x max^2 | b = sqrt | -----------------------------------| = 11.05 | w^2 |

agreeing with, Rms + (Bl)^2/Re = 11.12

You replied:

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

I said:

You replied:

Really... I said I measured, you reply asking ask if I measured... We can take a break if you wish, Mr. Kelly.

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

Regarding accuracy of measurements, you stated:

Mr. Kelly, we have a rather extensive set of lab instruments. This includes 3 Fluke, 2 Simpson, and 2 HP voltmeters, factory calibrated. These were checked against each other to insure accuracy at the start of my measurements. They all read the same, except one Simpson was very slightly off and of course was not used. The data was taken with *two* meters monitoring (a Fluke and a Simpson), and a minimum of five measurements made for *each* magnitude, to insure repeatibility and therefore accuracy. I hope your medical doctor uses this kind of precaution and care. :).

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

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

Mr. Kelly, please consider the following CAREFULLY and with an open mind:

Also if you will, please leave the following between the ++++++'s all together (reply below the second +++++'s) as I have stated my case herein. TIA

++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++

You stated:

Without complex notation, in the most simple form:

Ze = E / I = 1.41 / 0.227 = 6.21

EXACTLY AS MEASURED, not 6.07 as you calculate.

Now you ask if I can do complex numbers. So then, to please me' Lord. :)

Ze = Re + j Xe = sqrt (5.78^2 + 2.27^2) = 6.21

Again agreeing with measured impedance of 6.21.

where Xe = Re (tan angle) = 5.58 * .393 = 2.27

and angle = arc cos Re/Ze = arc cos 5.78/6.21 = 21.44 deg.

Again.. this uses accurately measured magnitudes, and is clearly basic theory re Fitzgerald. Note also that your incorrect phase angle re voltage and current of 14.46 degrees invalidates your analysis.

Now............ if you continue to use your above theory, i.e. "Ze =5.78 +0.1-j1.517 = 5.88+j 1.517 which leads to a magnitude of root (5.88^2 +1.517^2) =6.07 ohms and a phase angle of arctan 1.517/5.88 =14.46 degrees Or, using arccos (5.88/6.07) =14.36 degrees." please outline and give ALL magnitudes and ALL equations for EACH and EVERY term, as I have done. In other words, please return the courtesy I have shown you here, as I am getting ***VERY*** weary of seeing your numbers without accompanying substantion beside them. That's as in ***VERY*** weary.

++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++

Again, please leave the following between the ++++++'s all together (reply below the second +++++'s) as I am now stating my case re the mechanical side herein. TIA

+++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++

You stated:

Your Zmec of 33.82 and angle of 86.22 are wrong, again in most simplistic terms:

Zmec = Fapplied / velocity = 1.75 / 0.0493 = 35.50

Then in complex notation, again to please me' Lord :)

Zmec = Rmec + j x mec = sqrt (11.12^2 + 33.75^2) = 35.53

where Xmec = Rmec (tan angle) = 11.12 * 3.03 = 33.74

close enough to our agreed x mec of 33.75

where angle = arc cos Rmec/Zmec = arc cos 11.12/35.53 = 71.76 deg.

And again, if you continue to use your above theory, i.e.

please outline and give ALL magnitudes and ALL equations for EACH and EVERY term, as I have done above. In other words, please return the courtesy I have shown you here, as again I am getting ***VERY*** weary of seeing your numbers without substantion beside them.

++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++

You said:

I assume here you that by Re you mean (Bl)^2/Re. If so, sloppy, but it does not matter *what* you call (Bl)^2/Re, it is part of the resistive damping, i.e.

Rmec = Re + (Bl)^2/Re = 11.12.

Can you not see this, and that including (Bl)^2/Re in your Zme (my Zmec) which you state as

gives the correct Zm or Zmec as noted above of

Zmec = Rmec + j Xmec = sqrt (11.12^2 + 33.75^2) = 35.53

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

I ask:

"DID YOU EVEN PLUG IN THE NUMBERS INTO THE HALLIDAY EQUATION, AND SEE THAT IT WORKS AT ANY FREQUENCY, THEN TRY YOURS WITH Rms ONLY AT 227.4 HZ?"

You replied

Yep, about as clear as mud. Again you are yakking without support of numbers. Here, do this please for 227.4 Hz: Show Halliday's damping equation plugging in the numbers and show what you get. Do it as I have done, i.e. "outline and give ALL magnitudes and for EACH and EVERY term"

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

You stated:

I believe you ask earlier about this, and I've been there and done that. At 227.4 Hz:

v max = w A = 2 pi f A = 1428.8 * 0.0000543 = 0.0775 v max = pi/2 * delta x / delta t = 3.14/2 * 0.0000543 /0.001 = 0.0775 v max = F max / Z mec = 2.75 /35.53 = 0.0774 v max = pi/2 * v avg = 3.14/2 * 0.0493 = 0.0774 v max = a t = 70.44 * 0.0011 = 0.0775 v max = a max /w = 110.65 / 1428.8 = 0.0774 v max = pi/2 sqrt (2 KE/m) = 3.14/2 sqrt (2 * 0.0000307/0.0253) = 0.0774

finally, per Mr. Newton, with F = ma = 0.0253 * 70.44 = 1.78

v max = F/m * t = 1.78 / 0.0253 * 0.0011 = 0.0774

Does that suffice?

Note all correlate at 0.0774 to 0.0775. Are you impressd now with my measurements? I am. :)

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

You said:

Sorry, but your Ze of 6.07 is based on your misapplication of my data, as proven above. No one who reads the above would agree with your magnitude of 6.07, rendering it as incorrect. Bull? :)

I suppose you heard on your direct line from the Heavens that if the calculations don't match tha measurements, then you should change the measurements?

Reminds me of the Boyles method of weighing a hog. You place the hog on a see-saw plank, pile rocks on the other end until balanced, then carefully guess the weight of the rocks. :)

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

You said with reference to me describing my measured data as tools:

Finally I see what may be your problem here. No, they are where the rubber meets the road. They are reality, that which is indisputable, if measured properly. That which proves an equation valid or not. Apparently to you they are secondary to your calculations. I now see why you have trouble accepting a result you get when plugging numbers into, say Halliday's equation.

-----------

You said:

There you go again...

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

You said:

You are welcome. Thank you for your time as well.

Bill W.

Sorry, a typo in my last post:

5.58 should be Re of 5.78 as noted above and below.

Bill W.

Sorry, another typo...

0.001 should be 0.0011 for delta t, as it was elsewhere below.

I'm through now... sorry. :)

Bill W.

------------ That depends on what F is used. For example, If the cone is driven mechanically and the Fmax and Xmax are measured, the damping can be calculated using this expression. If for that situation, the electrical circuit is open- then the damping will be strictly due to the mechanical resistance (i.e. Rms). If the coil is shorted, then the effect of the electrical side will come in and the damping will be Rms +(BL)^2/Re. When energised from the electrical side and current produces the force- then the latter term will apply. I agree with this.

Somehow, I don't see where something that I agree with proves me wrong.

As for the development of this expression for the damping term. Again you are simply quoting equations without consideration of their physical basis- so and so says such and such. They are correct but you can only assume that. Rather than just assume (as you did) that the expression was correct, I derived it. That is what I was getting at. I can simply use F' =(BlE/Re) =[(Bl)^2/Re + Zm]V where Zm= Rms +j(wM-K/w) This is an expression I have shown before - it is useful. Then noting that wo^2=K/M and that, for a sinusoid, Vmax=wxmax where xmax is the maximum excursion. F'max/xmax = F'rms/xrms = w F'/V Now do some algebraic manipulation and the Halliday expression results with (Bl)^/Re +Rms replacing b Halliday , Serwick, etc simply did this very thing. Nothing new or exciting-just a re-arrangement of an equation. You can do that as an exercise.

------------- > You replied:

--------- I stand corrected.

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

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

I'm glad that you used such care. However, what is the accuracy of each of these instruments. I do not deny that you have been careful in measurements. However, The measurements are only as good as the accuracy of the meters which is ?? All I note is that calculations based on data from your measurements do show some differences. Considerinig the kind of measurements that you have taken (eg velocity) the agreement between the electrical impedance as measured and the impedance as calculated from the parameters that you have given are within 3% in magnitude. That is good.

I used Zm =Rms +j(wM-K/w) =2.23+j 33.75 =33.82 @ 86.22 Ns/m (This is the impedance due to the mechanical components only as I have clearly stated on many occasions. It does not include (Bl)^2/Re and is simply based on the primary equation F=BlI =ZmV ) This is then reflected this back to the electrical side From the electrical side the result is (Bl)^2/Zm =(7.17)^2/33.82 @86.22 =1.520 @-86.22 =0.1 +j1.517 Then the total Ze =5.78 +0.1 +j1.517 =6.07 @14.46 degrees. I gave you this information if you had cared to read it. This is also in line with Kinsler, et al. I noted two things: a) that the 0.1 ohm value comes from the reflection of Rms to the electrical side. b) small errors in values of the magnitude of Z lead to large phase angle errors. In fact the difference between your Z and mine is about 2.3% but the angle difference is about 50% For small angles, it is difficult to get good phase angle data from Z and R measurements. As for the other equations. I did give, step by step the procedure: That is calculation of V using (BL)E/Re =[(Bl)^2/Re +Rms +j(wm-K/w)]V Knowing V one can use BlI =ZmV to find I The current so obtained had a phase shift of 14.46 degrees. I have given the calculations for these in detail so I wont repeat them here. Now please note that I have calculated Ze as above using the actual mechanical resistance reflected to the electrical side and adding Re. I also calculated Ze using this I and the Eb determined from BlV.

Now run checks: a) E-ReI =1.41 -5.78(0.232 @14.46 ) =1.41-1.34@14.46 =1.41-1.30-j0.335 =

0.353 @-71.76 volts This should be the same as Eb= BlV As previously calculated V= 0.0492 @-71.76 so BlV =0.353 @ -71.76 degrees Agreement so that at least E-ReI =Eb is satisfied. OK

Try this with your data which for Ze =6.21 @-21.46 leaads to I =0.227 @

21.45 degrees Then E-ReI =1.41 -1.31 @ 21.45 =0.515 @68.55 degrees. We both calculate V the same way so BlV is still 0.353 @ -71.76 degrees Two values for Eb - OOPS there is a discrepancy. Something is wrong as this cannot be. The discrepancy in the magnitude of the current is not that large - what is left- the phase angle of the current must be wrong. Where are the errors if all calcuations have been done correctly (and they have been ) ? They must be in the measured values use to find the parameters.

Now lets try another check. I used Ze=5.78 +(Bl)^2/Zm and the componet of this which is 0.1+j1.52 is directly related to 2.23+j33.75 (Zm) My value for Ze was based on the Zm which has a reactive term Xm =33.75Ns/m Now lets try your value of Ze and the associated phase to find Xm You have Ze=6.21 @ -21.45 degrees This Ze reflected back to the mechanical side by (Bl)^2/Ze leads to 8.27 @

21.45 =7.07 +j3.03 = a problem. Suppose we simply reflect back the Xe as you have ignored the effect of Rms so I'll also do that Then Xm =(Bl)^2/Xe =(7.17)^2/27 =22.64 Ns/m for a value which should be 33.75 Ns/m OOPS AGAIN. Results don't agree. There appears to be a problem with your values in terms of inconsistency. These inconsistencies indicate that something is wrong with your values. Mine do not suffer this inconsistency. Hence, I would say that your EXACT values are wrong or that parameters EXACTLY measured are wrong. Either way it is your problem. Please note that you cannot use two numbers as you have done to show that my numbers are wrong- all that you can say is that the numbers that you use give a different result.. That is why further checks are needed - That is what I have done and what you have failed to do. Please don't quote Fitzgerald to me -I passed that level of circuit analysis sophistication over 50 years ago.- what you have quoted simply justifies nothing as what you have said boils down to calculating an X from Re and Ze then using Re and X to find Ze -showing only that you did the arithmetic correctly and having nothing to do with the accuracy or validity of the Re and Ze. You have simply done the complex equivalent of 2+3 =5 so therefore 5-2 =3. Whoopee. I am not arguing against your calculation procedure. I am pointing out that for the claimed accuracy there is a large discrepancy - that is your values do not check. I will stay with my conclusions as you have simply not made your case.

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

deg.

Since I had previously (more than once) given you a detailed analysis and all the calcuations that you ask for in earlier posts , I will not repeat what you didn't read or understand at that time in the hope that you would now read and understand it. It is all there - all steps shown. I used F=BLE/Re =1.75 as well as the equivlent impedance (Bl)^2/Re +Rms

+j(wM-K/w) to get 1.75 @0 =[11.12 +j33.75]V in order to calculate V =0.0492 @-71.76 Details of the steps were given. Of course the effect of Re is included there. This falls out of the original electrical and mechanical equations. What you fail to realise is that I distinguish between the Zm due to the mechanical components and the effect of Re as an equivlent mechanical resistance. I also gave details on how I arrived at this. Simply put what you call Rmec includes (Bl)^2/Re This is a terminology problem. What I call Zm depends only on the mechanical side parameters. This is what Kinsler,Frey etc call the open circuit mechanical impedance (I=0). My Zm is then 2.23 +j33.75 =33.82 @86.22 degrees It depends on the mechanical elements

Your Zmec =8.89 +Zm =11.12 +j 33.75 =35.54 @71.76 This reflects the damping effect of Re as an equivalent mechanical resistance. I have always used it this way. What I have done is then used BlI =ZmV to find I. This is perfectly valid and does reflect the basic mechanical equation correctly.

------------ I see what you have done- I have also repeatedly given the basis for my approach. I also developed, not copied, the expression leading to the BlE/Re etc terms . READ, THINK The problem is that while convenient for calcuation of a force velocity relationship, the force BLE/Re is not the actual force seen by the mechanical system nor is the (BL)^2/Re an actual mechanical resistance. These apparent (and Kinsler,Frey use this word) and useful. Hence I use them where applicable The result is that when I calculate power I look at P =I^RRe +RmsV^2 in which the mechanical power and the electrical loss are separate. This also agrees with Kinsler & Frey To calculate V^2[(Bl)^2/Re +Rms] is wrong. I have also given the reasons for that. In addition, I have shown calculations for power in terms of the EI-I^R =Pmec and this also checked with the value of Pmec from (V^2)Rms. Again- there are ways to check. THe checks come down to the following E-RI =BlV and BLI=ZmV are the basic equations and these must be satified. If not there are errors either in arithmetic or concept. My values satisfy these equations. Yours do not. Case closed.

-------- If you cannot follow analysis that is your problem. Simply plugging in numbers will not do the job. Also simly getting a number that is the same as your Rmec means SFA . Aren't you interested in why the Halliday expression leads to the value for b? If I use Halliday's expression and use the coulomb force the resulting b will be 2.23. Whooptedo (and this is putting in as many numbers as you did) I used the expression using the symbols as then I can see exactly where each term has its effect. Please note that your references do the same. That is pretty standard procedure. Also note that somewhere up above I did show how this halliday expression was obtained. I left the details to you as an exercise. I am not interested in simply plugging numbers into an cookbook expression - I want to know the basis for this expression before I trust it. >

Thank you Vmax =wXmax would have sufficed.

The numerical correlation is also nice but since many of the values for Vmax are calculated on the basis of values found from a given small set of measurements the correlation should be there. For example (1), (2)(4) are based on the same values and assumptions Sice I do not know how you found a, M, and a few other things, I will reserve my judgement on the correlation - as said above using 2 +3 = 5 to show that 5-2 =3 doesn't mean anything.

------------------- I have answered this above. If you still cannot follow it- thats your problem.

I've heard that before. No- If measurements disagree with the calcuations check the calcuations. However, if the results of one set of measured values disagree with the results of another set of values which shoudl agree - that is a different problem. That is the problem that occurs here:)

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

------------- I have no problem with Halliday's expression or the result that you have obtained - Please read what I have said instead of assuming right off the top that it is wrong. . I have determined for myself where it comes from and that satisfies me. Your throwing it at me doesnt do that- particularly as I can apply it in different circumstances and get a different but correct result for that circumstance. It shoudl give the right result with the right input. However, the equation doesn't give a hoot as to the input. You have given the right input. The result for B needs not have any numbers put into it. It simply falls out. As for the tools - you are using rote applications. They work and that is al you need care about. The problem is that I want to know whay they work and the basis for these expressions. That is information that you haven''t given me. I want to go beyond the number plugging to the rational behind the plug in expressions. Small did in the development of his methods. Kinsler does. These people do the analysis and from their work, they produce the rote equations which you use. The fact that you may not fully understand the basis for these equations doesn't interfere with your ability to use them. Fair enough. Please note that I used your measurements and have not violated any of the equations that you like to use.. . >

----------- Example: R Z angle cos angle X aprox Xmec 5.78 6.07 17.8 0.952 1.86 5.88 6.07 14.46 0.969 1.52 *

33.8 5.78 6.21 21.5 0.931 2.27 22.6 5.88 6.21 18.76 0.947 2.00 The largest differece between Z values is 2.3%. The largest difference in R values is

You didn't leave my statement intact.. Your original answer to the above was "sweetness and light to you too"

Anyway.. You have NOW finally obtained one of the defining texts on the subject, and found a way to try and sidestep your mistake. OK... do it the hard way.

Really. *NOW* you say that if the cone is driven mechanically at open circuit, the damping is strictly due to Rms, but if energised electrically the damping is Rms+(BL)^2/Re. Since you have rejected the Rms+(BL)^2/Re term, why didn't you tell me long ago that you employed mice (or whatever) to push the cone back and forth. In other words, for three weeks or more, I try to convince you that in normal operation damping is Rms+(BL)^2/Re, not just Rms, so now that you understand better (thanks to Kinsler et all), you agree with me.

So then... you are correct with your analysis when mice push the cone. Feel better now? :)

Sure, you agree NOW.

Darn it, Halliday didn't consider mice pushing the cone.

Right off, confusion (lack of magnitudes). This can be construed as (using your angle of 86.22, and your Zm of 33.82)

F' = 1.75 = .548

Which of course is wrong. Want to give magnitudes and prove yourself correct?

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

Also, I'm sure you gave this before, but would you be so kind as to do a worked example with my data on these two simple equations - just for the record? TIA

E-RI = BlV

BLI = ZmV

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

Would you please state the *total* amplitude of power you see being transfered to the mechanical side? Is it 0.0054 watt?

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

Fluke, Simpson, and HP meters factory calibrated are not accurate? We even measured current through precision resistors and the voltage across them to check agreement with I = E/R. Again, since your calculations don't match measurements, one questions measurements?

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

As I told you, I am weary of your equations without detail re magnitudes, and you have again failed to provide magnitudes below, so I'll hold up on answering the remainder here, except for an item or two. I gave you my most nifty set of maximum velocity equations as a test. Shucks, there might even be an original. Who knows?

v max = w A = 2 pi f A = 1428.8 * 0.0000543 = 0.0775 (1) v max = pi/2 * delta x / delta t = 3.14/2 * 0.0000543 /0.0011 (2) = 0.0775 v max = F max / Z mec = 2.75 /35.53 = 0.0774 3) v max = pi/2 * v avg = 3.14/2 * 0.0493 = 0.0774 (4) v max = a t = 70.44 * 0.0011 = 0.0775 (5) v max = a max /w = 110.65 / 1428.8 = 0.0774 (6) v max = pi/2 sqrt (2 KE/m) = 3.14/2 sqrt (2 * 0.0000307/0.0253) = 0.0774 (7)

finally, per Mr. Newton, with F = ma = 0.0253 * 70.44 = 1.78 (8) v max = F/m * t = 1.78 / 0.0253 * 0.0011 = 0.0774

I added this comment: "Note all correlate at 0.0774 to 0.0775. Are you impressd now with my measurements? I am. :)

Now I would have been delighted at one time, had someone given me these, but you.... nooooooooooo you nit-pick and complain like an old maid.... You even assigned them numbers, so you can do a more effective job of nit-picking. Your reply:

using 2+3 = 5 to show that 5-2 = 3 does mean something too. It means you got an equation right. :)

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

No, as I said earlier, over 1% deviation is suspect. 3% is *very* suspect.

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

Fine if you keep the mice well fed :)

Bill W.

Because of disagreements, I would like to know the following information - which is really what I wanted in the first place :). a)All measured values- magnitudes etc, at what frequency, whether rms, peak or average and preferrably the method of measurement. b)the method used in deriving parameters such as Rms, Bl, M, K, etc from these measured values. c)Resistance of meters on voltage and current scales would be of interest (if digital the meter impedance for voltage measurements will be so high that they don't matter but, on current measurement the meter impedance may be of importance, and the relative location of a voltmeter and an ammeter can make a difference. Meter impedances and rated accuracy (i.e. +/-1 digit

+/ y% of full scale.

In addition, I believe that, at one time you mentioned that the 227.6Hz measurement was at the point where the input impedance was a minimum. As far as I can see (and Kinsler, etc also say this) that will imply a series resonance between the coil L and the cone mass. My calculations ignored L but it may be wrong to do so. . Including L, the input impedance will have a minimum at which it should be nearly resistive. Input measurements will include the effect of L as well as any variation of the coil resistance with frequency.

I am more confident than ever in the basis of my calculations- it isn't rocket science. They are consistent on back checks. I also truly believe that you have done a good and careful job of making measurements.

If the calculations are correct and the measurements are correct then what is left is possible approximations that exist in the determination of parameters from measurements and approximations/assumptions that are needed (and they do exist) to make this determination practical. This is why I want to see the approach that is used for such determination. It is simply an exercise of interest. I also think that results within 10% are probably pretty good, considering that it is assumed that everything is linear which is not necessarily true.

So far -playing around with different L leads to fairly flat (ie. over about

10 Hz range) minimum where the magnitude minimum and the phase = 0 are at somewhat different frequencies.This is to be expected as R will drop to 5.78 ohms at high frequencies and X will be 0 at some lower frequency (as well as at resonance) - This behaviour is shown by Kinsler and Frey and my results show the same behaviour.

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

Mr. Kelly

Returning the extra mile :)

Since we are at an impasse, and in the interest if finding agreement, I have gone over your post of a couple days ago in detail (where you give magnitudes). If we cannot achieve agreement, there are a couple of options to settle the question of accuracy beyond doubt regarding measurements, test instruments, and calculations, and I suggest we use one of them. But for now...

As I noted before, we are into an area mostly uncharted, with enough pitfalls to accomodate both of us. RMS, peak, average, resonance or above, mec ohms, vs electrical, angle of electrical vs mechanical, net vs applied re volts, power, etc etc. The point being it is easy to screw up (I speak first-hand). I believe if you take your statement re (Bl)^2/Re of a couple days ago where you note it is part of the damping for the driver when *energized electrically* (in operation) with force applied due to voltage and current) and follow through we can achieve agreement, certainly on the calculated vs measured impedance (the rest should follow, note). Therefore I have taken your analysis with magnitudes and "revamped" it, considering the above.

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

Here is your analysis and statement from a couple of posts ago.

BEGIN QUOTE:

"I used Zm =Rms +j(wM-K/w) =2.23+j 33.75 =33.82 @ 86.22 Ns/m

This is then reflected this back to the electrical side From the electrical side the result is

(Bl)^2/Zm =(7.17)^2/33.82 @86.22>=1.520 @-86.22 =0.1 +j1.517

Then the total

Ze =5.78 +0.1 +j1.517 =6.07 @14.46 degrees.

------- For example, If the cone is driven mechanically and the Fmax and Xmax are measured, the damping can be calculated using this expression. If for that situation, the electrical circuit is open- then the damping will be strictly due to the mechanical resistance (i.e. Rms). If the coil is shorted, then the effect of the electrical side will come in and the damping will be Rms +(BL)^2/Re. When energised from the electrical side and current produces the force- then the latter term will apply. I agree with this. "

END QUOTE.

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

Note that usage of caps below is not intended to be derisive, but for emphasis. So then, the analysis (slightly different procedure from yours, but valid) with the driver IN OPERATION, where as you note, TOTAL damping or resistance = Rms +(BL)^2/Re, giving Zm (or Zmec) as

Zmec = Rmec + jXmec

Zmec = sqrt (Rmec^2 + Xmec^2)

where

Rmec = Rms+(Bl)^2/Re = 2.23+(7.17^2/5.78) = 11.12

Xmec = (wm-K/w) = 36.15-2.40 = 33.75

Giving Zmec than as

Zmec = sqrt (11.12^2 + 33.75^2) = 35.53 @71.76 degrees

where phase angle = arc cos Rmec/Zmec = 71.76 degrees

As check with Kinsler eq. 1.30 and Beranek eq. 1.11

Zmec = F/v = 1.75/0.0493 = 35.50 @ 71.76 degrees

In lieu of 33.82 @ 86.22

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

Then to convert Zmec = 35.53 MECHANICAL ohms into ELECTRICAL ohms Rec

Rec = (Bl)^2/Zmec = 7.17^2/35.53 @71.76 deg = 1.44 @71.76 deg =1.44 * cos 71.76 = 1.44 * 0.313 = 0.45 electrical ohms

Then total CALCULATED electrical ohms or impedance Ze is Re of 5.78 due to DC resistance of the coil, plus Rec of 0.45 due to mechanical or motional impedance is

Ze = 5.78 + 0.45 = 6.23

Agreeing with MEASURED value of 6.21 within 0.3% Which is excellent agreement indeed.

Hope this is a start for better agreement of calculations and measurement.

Bill W.

Please see addendum below.

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

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

Upon further calculating, the (Bl)^2/Re term needs to drop out at resonance (since mechanical reactance is canceled at resonance) for the equation above to work at different frequencies. Then since one is dealing with both the mechanical and electrical, logically one should consider the electrical as well as the mechanical power factor (only the mechanical PF of 0.313 is considered above. It appears the multiplier in the equation is the ratio of the mechanical power factor to the electrical power factor, i.e. at 227.4 Hz,

multiplier rp = PFmec / PFelec = 0.313 / 0.931 = 0.336

and at resonance of 58.6 Hz multiplier fc = PFmec/PFelec = 1 / 0.201 = 4.98

Then taking the Rec equation above at it's 1.44 * 0.313 point and changing the multiplier of PF = 0.313 to 0.313 / 0.931 = 0.336 gives at 227.4 HZ Rec = 1.44 * 0.313/0.931 = 1.44 * 0.336 = 0.484 CALCULATED electrical impedance at 227.4 Hz then is

Ze = 5.78 + 0.484 = 6.264

Off from MEASURED Ze of 6.21 by 0.87%. Not quite as good as above, but within my preferred 1% tolerance :)

---------

Then at resonance of 58.6 Hz Rec = (Bl)^2/Zmec * PF mec / PF elec = 7.17^2/11.12 * 1 / 0.201 = 23.00

Giving CALCULATED electrical impedance at resonance of

Ze = 5.78 + 23.00 = 28.78

Matching MEASURED electrical impedance at resonance of 28.78 exactly. Modesty prevented my not capatilizing the word "exactly" there, note :)

If I didn't have a brain fart here, the above should be correct. I'll double check the relationships on the other test drivers.

Bill W.

---------------- Absolutely so. I have said this several times. I use different terminology but use F'=(BlE)/Re ={(Rms +(Bl)^2/Re +j(wm-K/w)}V This becomes 1.75 ={11.12 +j 33.75}V =(35.54 @71.76 ) V The Z in this case is your Z The resulting V =0.0493 @ -71.76 degrees I use the Zmec that you use to find V given F. So far we are in agreement.

I have written Zm =33.82 @86.22 as the Z due to the mechanical elements only. and have done that deliberately because I want to use BlI =ZmV to find I.

--------- Here is where we differ. If we do as you indicate then not only the resistive part of your Zmec must be brought over but also the reactive part. You did this but then have ignored the latter in your calculation. That is (Bl)^2/Zmec =(7.17)^2/35.5 @71.76 =1.44 @ -71.76 as you say Converting this Rec =0.45 as you say and Xec =-1.37 Then | Ze| =root{(5.78+0.45)^2 +(1.37)^2 =6.38 at a phase angle of 12.4 degrees

In addition- as you have included Re in the equivalent mechanical impedance you have already taken it into account. In effect you have used it twice in your calculation for Ze. Not using Re gives a Ze of 1.44 which is ridiculous. The closeness that you got for Ze is coincidental. Suppose that, instead of initially moving Re to the mechanical side as the mech equivalent (Bl)^2/Re, I bring the original mechanical impedance back to the electrical side as an equivalent electrical impedance Bringing Rms +j (wM-K/w) = 33.82 @86.22 back to the electrical side gives Zem =(7.17)^2/33,82 @86.22 = 1.520 @-86.22 and Zem cos 86.22 = 0.100 Zem sin 86.22 = 1.517 Adding the Re gives Ze =5.78+0.1 +j1.517 =6.07 @14.46 degrees

Kinsler uses this approach - bringing all values over to the electrical side and his Zmot =Zem. I chose to convert do as you did- convert E and Re to a current source and E/Re in parallel with Re and transfer them to the mechanical side where he brings the mechanical elements to the electrical side and leaves the source as E in series with Re. Different approaches- giving the same result.

He also includes Le in his Ze as it will have an effect at this frequency. Hysteresis and incremental magnetisation in the core and gap will be included in measured values of Ze but these terms should be very small.

However, There does exist a real problem and that is that I think Bl=7.17 is low. You have said average V=0.0493 where the V calculated above is an rms value of the same magnitude -that is a problem. This is not a calculation method error as we both used the same values for Z and F to do this. There we agree.

From what you have given when showing different ways to calculate Vmax, you have given A =5.54x10^-5 and Vmax =0.775 for which Vrms =0.0548 not 0.0493. Your F of 1.75 calculated from E is an rms value and corresponds to the Fmax you give as 2.47 I suspect Bl should be about 7.17*1.11 =7.96 That is why I wanted to see more on the measurements and calcuation of parameters from measurements - something is not quite kosher. Note that if Bl=7.96 is used Ze =5.78 +0.12 -j18.7 =6.19 @-17.6 degrees which is closer to your measured

6.21

In addition, I am getting some problems from Halliday- I have proven to my satisfaction from first principles that it is correct but numerical values lead to b^2 negative - not physically possible- I can't simply flip the sign and evaluate b =11+ . I will redo the calculations yet again. Again- this expression is such that small numerical errors are magnified (differences between two large and nearly equal squares). This also points toward Bl

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

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

----------- This is something that I don't agree with. (Bl)^2/Re is a resistive not a reactive or frequency dependent term. At resonance, the reactive part (wm-K/w) =0 leaving only the resistive part. The equation (Bl)E/Re ={ (Bl)^2 +Rms +j(wm-K/w) }V is valid at any frequency of concern on the basis of Le=0. The equation that you refer to Rec=(Bl)^2/Zmec should actually be Ze=Re +(Bl)^2/Zm where Zm = Rms +j(wm-K/w) (both magnitude and phase informatin must be preserved. This form will be valid at all frequencies including resonance

The mechanical equivalent impedance at resonance is 10.12 +j0 Now using my Zm at resonance Zm=2.23+j0=2.23 Converting this to an electrical impedance give Zem=(7.17)^2/2.23=23.05 ohms Ze =Zem+5.78 =28.83 ohms purely resistive.

-------------------- Then since one is dealing

--------- As I previously indicated, this value ignores the reactive component that exists at 227.4 Hz so is only the real part of Ze. For example, you have a pf electrical of 0.2 at resonance (pf mec is fine). My understanding of PF is the same as Siskind's. cosine of Real power/apparent power which becomes cos (arctan R/X) or the cos of the phase angle of the current with respect to voltage (or velocity with respect to force) At resonance Zmec is purely real and Ze is purely real -both have related pf =1. F,V E and I are in phase at resonance so these pf's are not according to the normal pf definitions.

---------- This does give me a better idea of what you are calling pf electrical - it appears to be the ratio of Rms to Rms+(Bl)^2/Re Not the normal definition Note 11.12*2.23/11.12 =11.12*0..2005 =2.23 Is it necessary to go this way rather than simply saying (Bl)^2/2.23 =23.05 as I did above?

--------- Be modest- your measurements are excellent but note that the value of 23.0 is not thesame as 23.00 as your value of 0.201 is good to 3 figures, not 4. Modesty says 28.8 +/_ 0.1 but you're allowed:) >

-------------- Without going into further analysis and it is too late to do it now, use of these multipliers leads may be misleading you into extra work. Again, how are you defining power factor in each case.?

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