Motor torque and back emf

That was my point.

We can speak of current and voltage, all well and good, but force is what turns the wheels.

My references were, and will be as applied to a loudspeaker motor.

applied electromagnetic force = E BL/R. where E = applied AC electromotive force, generally called applied voltage BL = mechanical force factor, i.e. magnetic field strength times length of motor armature coil wire in the magnetic field R = DC resistance of the motor armature coil

_ v = average velocity

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

Basics.

I came here to learn (it's my thing) :-) , as well as to contribute what little I might in return, and prefer to not be confrontational.

In my equation

IE = I^2 R + I CEMF

where I = current E = applied voltage R = armature DC resistance CEMF = back emf = motor generated voltage (your term)

Go further? I CEMF is the mechanical output, same as your EaIa. This assuming I CEMF and EaIa include all losses other than the I^2R power lost as heat in the armature wire (copper loss).

Reply to
Bill W.
Loading thread data ...

?

To Daestrom and Don Kelly

Looking back through the thread, looks like you two are the nice guys. My apology if I stepped on a toe. The few websites I've visited have these hostile participants who bite newcomers, and I sensed a tad of that here, and was on the defense...

I sincerely appreciate the help from both of you, and in particular to Daestrom for the enlightment on the o-scope polarity bit. I had scratched my head on that more than once. :-)

Nice place here, maybe I'll hang around. Bill W.

Reply to
Bill W.

--------- true but to get force, a current is required. My point is that force and current are intimately linked and voltage and speed are also intimately linked.

---------------------- Ok but I see 3 things.

1) for AC , the average voltage is useless as it is always 0. Are you thinking rms voltage?

2)The above expression, appears to assume that the back emf of the speaker is negligable. This seems to be a handbook approximation based on the low efficiency of a typical speaker (I^2R>>EbackI or IR>>Eback) so that the current is effectively determined by the applied voltage and the resistance and force =BLI becomes (E/R)BL as an approximation.

3)This approximation also appears to consider the force produced at a single frequency in the mid frequency range where coil inductance can be ignored. In addition, the feedback of mechanical load into the electrical side is ignored for the reason in (2).

It can be a useful approach as long as the approximations are valid. That is not always true and one must be aware that it is an approximation.

----------- I am not trying to be confrontational either. If I am coming across that way, I am sorry. By derisively, I meant that you were holding it up as a foolish idea- I agree.

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

---------------- I quote what I said before:

We don't differ here- All I did was to extend this to say EaIa =(k(phi)w)(T/K(phi)) =Tw.

Take care,

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

Reply to
Don Kelly

Does that mean I shouldn't worry about those two bare wires sticking out of the wall with 120 volts between them, since the average voltage is zero? :-)

The zero average for voltage and current reflects the characteristic of a sine function (or cosine), but that doesn't mean energy is not being transferred. _ _ P = I^2R

Even though voltage and current change sign each half-cycle, the square of the current is positive.

My Fluke and Simpson meters read RMS.

The result of the above equation E BL/R gives applied mechanical force, and and is most useful in the mechanical domain. This applied force then is numerically equal to - (velocity times mechanical impedance), where the mechanical impedance includes the total of the mechanical resistance plus the mechanical reactance. I should have noted that coil inductance is negligible below ~ 250 Hz, and thusly is generally ignored, as it is above. Sorry.

Back emf becomes part of the mechanical resistance, i.e. the mechanical resistance of the motor = (BL)^2/R.

Electrical to mechanical efficiency is simply CEMF/E, derived from power out/power in = I CEMF/IE. Yes, efficiency is low, generally under 10%. Mass is under constant acceleration or decelleration, taking a toll on the supplied energy. Note we are not working with constant velocity, and typical motor efficiency.

Note that for a given applied voltage, we have constant applied force that is not frequency dependent.

The mechanical load is included in the equation.

Considering the above, I believe not an approximation.

Sorry, miscommunication.

Bill W.

Reply to
Bill W.

Make that the electromechanical resistance of the motor = (BL)^2/R. Bill W.

Reply to
Bill W.

------------------- I did not imply that. The actual power consists of a constant (average) term and a double frequency term. The use of rms quantities will give the average power (in fact the definition of rms is based average power) and a measure of the varying part (reactive).

----- and the instanteous power will have a constant term plus a double frequency term as above. I'm familiar with AC power/reactive calculations.

---------- Careful - This depends very much on how the meter reads AC voltage. A true rms meter will read correctly for any waveshape but a typical multimeter will only give a correct reading for sinusoids-

------------ By this you imply that R includes a term reflecting the mechanical load as well as the resistance of the coil- fair enough. Given a mechanical resistance Rm =velocity/force then one can move this to an equivalent resistance R=Rm(BL)^2 No problem.

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

------------- Noted- Mass doesn't take a toll on the supplied energy except through increased coil I^R loss. Energy into a mass is conservative just as energy into a capacitor- it can be recovered. The effect of this mass is not included in the equation given.

----------

---------------- Only because of the assumption that the mechanical load is wholly dissipative- i.e. mass and spring effects are ignored in this expression.

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

---------- Only the dissipative term

----------

------------ I disagree. It is still an approximation. Frequency dependence of mechanical elements is ignored. Some one has looked at the dynamics of speakers and their loads and made some assumptions to come up with a simple model. The approximation , as I said, may be pretty good over a range of frequencies. It would be best for very heavily damped speakers (i.e. as originally brought out by Acoustic Research) so that the damping overcomes all other effects (and makes efficiency lousy -say 1-2%). It would be less correct in higher efficiency speakers and enclosures (say a bass reflex where cabinet/ speaker resonances are of concern).

The fact that it is an approximation doesn't make it invalid within the limits assumed just as mid frequency electronic circuit models ignore L and C elements using open or short circuits to replace these as appropriate. That is what this model is doing.

The problem is that it is good to know the basis for this approximation and have this in mind to keep from applying it without thought to situations where it is not valid .

The speaker equations are of the form es =Ri +Ldi/dt +e where e=(Bl)v electrical f =(Bl)i =Mdv/dt +Rmv +1/K(integral of v dt) mechanical es is the supply voltage, i is the current e is the back emf , f is the mech force, v is the mech velocity , M is mass, K is spring compliance and Rm is the mechanical resistance. B is the field flux density and l is the active conductor length perpendicular to the flux This can be represented using an ideal transformer f->

|-----R, L-------| |--------|--------|--------| | i-> Ideal | | | es e Trans v M Rm 1/K | Bl:1 | | | |____________ | |______|______|______|

Ignore L, M and K and transfer the Rm to the primary side to get

|-------R------| | i-> Rm(Bl)^2 es e | |___________|

or i= es/(Req) where Req = R + Rm(Bl)^2 i =es/Rm' and force =Bli =es(Bl)/Req which is your approximation E(BL)/Req If Rm is large, then the electrical resistance R can also be ignored and Req =Rm(Bl)^2 is the mechanical resistance as reflected into the electrical side. To consider such things as the speaker resonance and 3 db points, the M and K must also be taken into account. The model, even with these, is an approximation as the cabinet and room factors are not included (i.e. hang a few more masses coupled by springs and dampers in the circuit)

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

Reply to
Don Kelly

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

My reply this time around is all here between the + lines.

You stated: "Given a mechanical resistance = velocity/force"

Should this not be mechanical impedance = force/velocity ?

How do you define e=(Bl)v electrical (back emf)? Is it just BL times velocity, or are you converting (BL)v, and if so how?

f =(Bl)i could be taken two ways per earlier discussion. Please be specific.

Your diagrams below are scrambled, and not readable with certainty of content. In any event, no point in delving deeper until the above is cleared up.

Bill W.

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

Reply to
Bill W.

Would you please give your results in the following three equations? Let me know if you need values that are not noted below. System compliance Cmt at fc = 0.000292 (stiffness k = 3,425)

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

Yes, you are correct, the reference is a mass and spring, not forced oscillation. For the 12 drivers, the 2 equations a = wv and a = F/m the average difference in magnitude of acceleration at 227.4 Hz is 2.38%, so that F/m gives a pretty close approximation. The downside is that results vary from 0.2% to 10.2%. When an average difference is greater than 1% from what I expect, I look with askance, but I did not check this at the outset here. Sorry.

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

--------

(and you affirm it above), you correctly treat the "R's" as

---------

Zmec at fc = Rmt = 11.12. 23 is Rec

-------- This should be of help:

Electrical impedance:

Z = E/I = Re + Rec @ fc = 28.78 @ RP = 6.21

Rec = Eb/I = Z- Re @ fc = 23.0 @ RP = 0.43

where: RP = "resistive point" = low Z point above resonance Z = electrical impedance, as above E = applied voltage = 1.41 I = current @ fc = 0.049 @ RP = 0.227 RE = DCR of coil = 5.78 Rec = resistance of electrical impedance other than Re as noted above. I may have noted as Res earlier.. Eb = back emf = E-(I Re) @ fc = 1.127 @ RP = 0.098

-----------

Mechanical impedance: Noted as ratio of force to velocity Kinsler eq. 1.30

_ _ Zmec = F/v @ fc = 11.12 @ RP = 35.50

otherwise 1 Zmec = sqrt [Rmt^2 + (wm - -------)^2] @ fc = 11.12 w Cmt @ RP = 35.50

where: Zmec = mechanical impedance = as above _ F = average applied force = E (Bl)/Re = 1.75 _ v = average velocity @ fc = 0.157 @ RP = 0.0493 Rmt = total mechanical resistance = Rms + (Bl)^2/Re = 11.12 Rms = system mechanical resistance = 2.23 (Bl)^2/Re = Rme = mechanical resistance due to back emf = 8.89 Bl = force factor = 7.17 wm = mass reactance = 36.15 Cmt = system compliance = 0.000292 = 1/4 pi^2 fc^2 m

Xmec at RP agrees with the 33.7 figure you note above:

Xmec = sqrt (Zmec^2 - Rmt^2) = 33.7

Xmec at fc = 0

Bill W.

Reply to
Bill W.

I will use f=mdv/dt +Dv +K(integral of v.dt) and the following values in this single equation m=0.0253 kg, K=3425 N-s/Meter, I assume that the values you gave are the mechanical values nt the values reflected to the electrical side. Bl =7.17 I don't know D please give me a value. I will refer the Z to the mechanical side. and use a damping term D+(Bl)^2/Re = 11.12 ohms? You use a different term than D

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

no problem

As I said, at 227Hz, the mech impedance is very nearly completely due to the mass so the variations may well be due to variations in between drivers of the same type.

------------ It is.

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

-------- Define please. The term is new to me.

There are more terms and numbers coming in. At what frequency is this low z point? I think I know what point you are referring to but I want to be sure. Looking at a curve there is a minimum impedance as some point above resonance and then Z rises again. However, it is my understanding that above this point the coil inductance has an effect. I can see the interaction between this inductance and the spring or mass producing another resonance.

check this: is it sqrt {Rmt^2 +( wm -1/wCmt) ?

------------ This is not due to back emf - it is simply the coil resistance transferred to the mech side as with a transformer. Back emf is simply the voltage produced by the coil motion However, I am nit picking.

------------ Good - at least we are getting closer. It appears that the RP is the only thing I want clarification on.

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

Reply to
Don Kelly

Yes.

Only my term Rms fits your D, i.e. the mechanical resistance of the suspensions Rms = 2.23.

Such that at fc Zmec = Rmt = Rms + (Bl)^2/Re = 11.12

where Rmt is the total mechanical resistance.

But how does this relate to a result for your above eq. ?

f =mdv/dt +Dv _K(integral of v.dt)

It would be very helpful if you would include the resultant value of your equations. TIA.

No, only one driver of each type (model) is used, IOW no two drivers were the same type. However as I noted, all are 8 inch.

Above resonance and with increasing frequency, the impedance has a depression (decreases to a low value, then increases again). At the low point the impedance appears as resistive, in this case at 227.4 Hz. I define RP as the frequency of this low point in the impedance, i.e. 227.4 Hz.

Exactly, as I note above. We have discussed this before but without a defining term, now being RP = 227.4 Hz.

But our concern here is at RP and below.

No. Square the mechanical reactance term, wm-1/wCmt

Zmec = sqrt [Rmt^2 + (wm -1/wCmt)^2]

otherwise as

Zmec = sqrt [Rmt^2 + (wm - k/w)^2]

The total mechanical resistance of the motor at resonance, as noted above. Rmt = Rms + (Bl)^2/Re = 11.12 as noted above

Likely we are on the same page. Looking in the mechanical domain, since in effect wm and k/w cancel at resonance, this leaves resistance to be the load. Rms is the resistance due to suspensions, so this leaves (Bl)^2/Re to be resistance encountered in overcoming the back emf.

Good. Regarding power factor, I have not dropped it by the wayside, but as you suggested, am reviewing. This has been a sticky issue for me for some time, as was the back emf polarity bit. I have values for electrical power factor, i.e. Re/Z as well as a term for mechanical power factor, that although may not be definable as power factor in this case, are *very* useful and accurate as related to a contradiction in back emf terms. Regarding this would you please:

  1. Define power factor as you view it in this case.
  2. State if you view electrical input power and mechanical output power as scalars or vectors in this case.
  3. State if you view power factor in this case as: PF = 1 means 100% of input energy becomes useful output PF = .9 means 90% of input energy becomes useful output PF = .5 means 50% of input energy becomes useful output

etc. TIA

Bill W.

Reply to
Bill W.

In terms of steady state sinusoidal rms quantities the above becomes Z =F/V =D+j(wm-K/w) This Z flips to an admiitance when referred to the electrical side Then I/Eb =Z/(Bl)^2. which is 1/Zelect as I see it. Including Re changes the term in Z from D to D+(Bl)^2/Re With the values given resonance is at w =root (K/m)=378 or fc=58.6 Hz (ignoring the small effect of the resistance in shifting the resonance slightly. ). At this frequency the measured Z should be Rms +(Bl)^2/Re=11.12 as wm-K/w =0 At 227.4 Hz the Zmec above will be Zmec = 11.12+j(1429*0.0253 -3425/1429) =11.12+j33.76 =35.5 at angle 71.8 degrees

The above does not include any enclosure modeling.

--------------- The problem that I see is just above this you indicate a Z=35.5 at this frequencywhich I agree with. It is dominated by the mass as you originally indicated and My solution confirms. Now you say that this is the low point in the impedance (above resonance). If I look at my expression then I see Z=

0 at f=0 and also at f=infinity and peaking somewhere in between. Note that I am looking at the mechanical Z which, as seen from the electrical side becomes an admittance (f/v =(Bl)^2(i/e). I haven't explored the values of e/i but this should be easy to do as Eb/I =(Bl)^2/Zmec ignoring Re (D only) and the electrical impedance is similar with the D +(Bl)^2/Re term To get the low point above fc implies that onemust consider the inductance and its effect at this frequency as with the driver parameters alone - this dip doesn't exist.

--------------- Then , in the case of my calculation above, the effect of L has been ignored.

-------- right -proofreading booboo>

--------------- I wouldn't put it in those terms as I see it as the effect of resistance transfromed to the mechanical side. The transformation factor is (Bl)^2 E/I =(Blv)/(I/Bl) and fits conservation of energy. The back emf is generated by the speed and it is no more nor less than that..

Anyhow, I have to run- out of time.

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

Reply to
Don Kelly

Back to finish reading:

Power factor basically was originally defined electrically in terms of the ratio of the real power to the apparent power -that is Watts/volt-amps. It can be expressed as cos of the phase angle between current and voltage. From this , for an impedance R+JX where R is the real part and X is the reactive part, and the total Z is (R^2+X^2) at a phase angle of arctan X/R The cosine of this angle is R/(R^2+X^2) and this too is the power factor. Lag if current lags voltage., Lead otherwise. At resonance, the X term is 0 so the power factor becomes R/R =1

Power is a scalar but complex power is a phasor with both a real and reactive component. In most electrical analysis, the complex power (S) is used where S=Vtimes the conjugate of I =P+jQ

None of the above. It is only related to the phase angle as indicated.

A PFof 1 implies a purely resistive impedance or load A PF of 0.9 implies real power/volt amps =0.9 eg. |S|=root(RI^2 +XI^2 ) and P=RI so that in this case pf =P/S =R/root(R^2

+X^2)

The useful output to input ratio is effectively the ratio of useful output/(useful output + losses) =efficiency.

The power factor that you use is different from the power factor as considered electrically and that is is a potential source of confusion. I haven't looked into it carefully but it appears that you could model the loss in Re and that in the mechanical impedance as parallel resistors and come up with a power division and this appears to be what you are using.

It appears that, at one time, the basic theoretical work was done for speakers and from this parameters and methods of measuring and using these parameters were developed. In addition there is a "in house" usage of terms which sound like but are different from those that are used in circuit analysis, with just enough overlap at times to cause problems.

Note that in my last reply the Zm that I gave was F/v On the electrical side this transforms to an admittance.

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

Reply to
Don Kelly

Mr. Kelly, I want to express my appreciation for your generous efforts and time expended in this thread. You have been of considerable help. Thanks.

I can't get the numbers to fit on the transformation factor. Values? Explanation? TIA

becomes R/R =1

Yes, cosine angle = R/sqrt(R^2+X^2) = R/Z = .931 at 227.4 Hz.

Rec/Z = Pmec out/Pin = electrical to mechanical efficiency

@fc = 0.799 @RP = 0.0692

Rec Pmec = Pin ------- @fc = 0.0552 @RP = 0.0222 Z

I'll get back to you on the back emf contradiction, after sorting out a thing or two. Again, Thank you.

Bill W.

Reply to
Bill W.

Continuing re back emf:

Measured values

E = 1.41 volts = applied sinusoidal voltage Re = 5.78 ohms = DCR of armature coil I @ fc = 0.049 amps = current at resonance I @ RP = 0.227 amps = current at 227.4 Hz fc = 58.6 Hz = resonant frequency RP = 227.4 Hz = 227.4 Hz "resistive point"

The motor formula is

E-Eb I = ------- R

where R in this case = Re = DCR of coil/armature I = current E = applied sinusoidal voltage Eb = back emf

so that

Eb = E-(I Re)

Eb @ fc = 1.127 Eb @ 227.4 Hz = 0.098

-------- With measured values again of

Bl = 7.17 _ v = 0.0493

the other basic equation for generated back emf gives Eb = Blv

Eb @ fc = 1.126 Eb @ 227.4 Hz = 0.353

Note the two equations agree close enough at fc, i.e. at resonance, but not above resonance. Measured generated _ back emf voltage on the dual motor driver agrees with BLv, so apparently Blv is the generated back emf and E-(I Re) is the net or effective back emf. This because one must use E-(I Re) = 0.098 to obtain the correct mechanical power at 227.4 Hz of I Eb = 0.0222 watt, which agrees with power in = power out

IE = I^2 Re + I Eb = 0.320 watt

where I^2 Re = 0.298 is the power lost as heat in the armature coil (copper loss), and I Eb = 0.0222.

My reasoning for the lack of agreement is that at resonance, the reactance due to acceleration of the mass is out of the picture, being in effect canceled by the reactance of suspension stiffness. However, I can not come up with _ a simple or intuitive expression to relate E-(I Re) to Blv.

Bill W.

Reply to
Bill W.

that is:

_ v = 0.0493 at 227.4 Hz _ v = 0.157 at 58.6 Hz

Reply to
Bill W.

I'm not sure what you are getting at but remember that at resonance the current is resistive and then E, I and Eb are in phase so your measured E-RI =Eb

However, at 227.4 Hz the impedance that I get is (5.78+0.1) -j1.54 =6.08 @-14.6 degrees I=1.41/6.08 =0.23A @ +14.6 degrees E-IR = becomes, when accounting for the phase angle, 0.357 V (at phase -69.8 degrees) which agrees closely in magnitude to your value of 0.353.

Also note that neither IE nor EbI is real power because of the phase shift. Taking this into account gives:

The real part of IE is 1.41*0.23*cos14.1 =0.314 W I^2R =5.78(0.23^2) =0.305 So Pmech =0.009 W Real part of EbI =0.357*0.23* cos (14.6--69.8)=0.008W which agrees within the accuracy used for the calculation and the accuracy of the measurements.

Electrically, the model gives Ze =5.78 +(Bl)^2/(D +j(wm-K/w)) =5/78 +Zme where Zme is the mechanical impedance as seenfrom the electrical side. This results in Ze=Re+ {D -j(wm-K/w)}(Bl)^2/{D^2+(wm-K/w)} At resonance this becomes Re +(Bl)^2/D=28.8 ohms at angle 0 At 227.4 Hz it becomes

5.78+(2.23-j33.3)(51.4)/{2.23^2 +33.3^2) =5.88-j1.54 =6.08 at angle -14.6 degrees Nearly real but not quite and that's the problem. When assuming real quantities, then this assumption will lead to discrepancies such as those that you have found.

These are rough calculations and my value for | Z| is slightly different from yours but within reasonable bounds. The important difference is that, in general , for AC, EI is not real power. The calculations must take into account both magnitude and phase. However, since, in real life, distortion, non-ideal conditions etc come into play, the approximations are such that they make life easier and give results close enough in practice.

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

Reply to
Don Kelly

Yes, I did. Thank you.

Bill W.

Reply to
Bill W.

Tying it together:

Power out must equal power in minus power lost.

At 227.4 Hz:

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

where I^2Re is power lost as heat in the armature coil.

This value for mechanical power Pmec of 0.0222 must be valid, since I, E, and Re were measured.

Mechanical power factor = Rmec/Zmec = 11.12/35.5 = 0.313

agreeing with your:

"Zmec = 11.12+j(1429*0.0253-3425/1429)=11.12+j33.76=35.5 at angle 71.8 degrees"

from which PF = cos angle = cos 71.8 = 0.312.

then as analogous to P true or P net = IE PF = IE cos angle, and with Eb = E-(IRe) = 0.098, and I = 0.227

Pmec net = IEb PFmec = 0.227*0.098*0.313 = 0.007 watt

Agreeing closely with your value above of 0.008 watt So it appears IEb is applied or apparent mechanical power,` and must be factored by the mechanical power factor for net mechanical power.

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

However the jury is still out on the transformation factor for Blv to E-(I Re) above resonance. That is:

At 227.4 Hz Eb = E-(I Re) = 0.098 _ At 227.4 Hz Eb = BLv = 0.353

where E = 1.41 I = 0.227 Re = 5.78 BL = 7.17 _ v = 0.0493

As noted, I cannot find a simple or intuitive relationship for 0.098/.353 = 0.278 or 0.353/0.098 = 3.60.

Bill W.

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

Reply to
Bill W.

Mr. Kelly.. Upon further thought, and looking from a mechanical viewpoint, I'm not certain of our above values for net force. Looking from a mechanical viewpoint:

Per H & R P = F cos angle v = 1.75*0.313*0.0493 = 0.027

Note that force here, is applied force such that when voltage is applied from standstill, if we freeze time before motion starts and before any reactive elements are in play, i.e. with "blocked rotor", the magnitude is

E Bl/Re = 1.41*7.17/5.78 = 1.75.

Then in operation we consider power factor such that net force = Fnet = 0.548, and net, not applied force time velocity determines net power. This is a fair amount off from

F = I Eb = 0.227*0.98 = 0.022

however it is with the assumption that resistance is constant with frequency at 11.12. With a resistance of 9.13, mechanical power factor is

PFmec = cos angle = Rmec/Zmec = 9.13/35.5 = 0.257

and _ Pmec net = F PF v = 1.75*0.257*0.0493 = 0.022

What do you think here?

Bill W.

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

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

Reply to
Bill W.

PolyTech Forum website is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.