How to Derive constant of total back/counter EMF for DC motors?

[Oz Alikhan]

Hi,

I have been reading around about back EMF and their derivations for simple DC motors. However for some reason, the step between obtaining the total emf of the motor from summing the individual emf of the coils is not very clear. For example:

Induced emf due to single coil: [itex]e = d∅_c/dt[/itex]

Since the flux linking a coil is: [itex]∅_c = ∅ Sinωt[/itex]

Therefore the induced emf is: [itex] e = ω∅ Cosωt [/itex]

Since there are several coils all around the rotor, each one has a different emf due to its position (i.e each one has a different flux change through it). Therefore, total emf is the sum of the individual emfs.

This means, [itex]E_b = K ∅ ω_m [/itex] <<< How?

First of all, I don't understand how the [itex]Cosωt[/itex] disappears to obtain the final expression. Secondly, what parameters define the constant K? In some place I read, [itex] K = 2 N/\pi [/itex], while in other places it stated [itex] K = 2 N R B L [/itex]. Why is it different for each case and how is it derived in the above proof?

The above derivation is from end of page 5/start of page 6 of the following source: http://vlab.ee.nus.edu.sg/~bmchen/courses/EG1108_DCmotors.pdf.

Warm Regards,
Oz

 

[NascentOxygen]

I guess the idea is to have sufficient windings so that no sooner are we past the peak of one sinusoid, then the commutator turns to deliver the upcoming peak of the next coil's sinusoid. So the output becomes a series of peaks, not perfectly flat.

I'll leave your more searching question for someone better placed.

 

[jim hardy]

Author skipped over some clever algebra.
I think you'd sum values of a series of sine functions, one for each turn in the armature winding.

First of all, I don't understand how the Cosωt disappears to obtain the final expression.

Commutation makes it disappear.
The brushes pick off a segment of each winding's cycle near the sinewave peak, so voltage at the brushes is not sinusoidal but unipolar..

Nine minutes into this excellent old Army film is a graphical representation of that commutation.

Six minutes in shows from whence comes that velocity term.

I wouldn't attempt at this late hour to derive that formula for the commutated sine wave. I guess that's why your author skipped it.
I don't see how he could call equation 5 the result of integrating equation 4. Book editor should have called him on that shortcut, i'd say.

In my class we determined product term K[itex]\Phi[/itex] empirically by no-load dynamometer tests on a machine.

 

[Oz Alikhan]

Hmmm, I see. Very well, that clears a lot; Thanks to both of you .

P.S. Old school videos are indeed the best

 

[alphy]

Hello Oz,

Thank you for your question. The derivation of the constant of total back/counter EMF for DC motors can be a bit confusing, but I will try to explain it in a clear and concise manner. The equation you have mentioned, E_b = K ∅ ω_m , is known as the back EMF equation for DC motors. Let me break down this equation and explain each term in detail.

E_b: This represents the total back EMF of the motor, which is the sum of the individual EMFs of all the coils in the motor.

K: This is the constant of proportionality that relates the back EMF to the speed of the motor and the magnetic flux. It is also known as the motor's back EMF constant.

∅: This symbol represents the magnetic flux, which is the measure of the strength of the magnetic field.

ω_m: This is the angular velocity of the motor, which is the speed of the motor measured in radians per second.

Now, let's look at how this equation is derived. The back EMF is generated in a DC motor when the armature (rotor) rotates in the magnetic field created by the stator. As you mentioned, the induced EMF in a single coil can be calculated using the formula e = d∅_c/dt. This is because the magnetic flux through the coil changes as the rotor rotates, and this change in flux induces an EMF in the coil.

Now, when we sum the individual EMFs of all the coils in the motor, we get the total back EMF, which is represented by E_b. Mathematically, this can be written as:

E_b = ∑ (d∅_c/dt)

Since the magnetic flux through each coil is different due to their positions, the total back EMF is the sum of all these individual EMFs. Now, let's look at how the constant K is derived.

The constant K is dependent on the number of turns in the coil (N), the radius of the coil (R), the magnetic field strength (B), and the length of the coil (L). These parameters are used to calculate the magnetic flux through the coil. The value of K can vary depending on the specific motor being used, which is why you may have seen different values for K in different sources.

To simplify the equation, the constant K is often written as K = 2 N/\pi .