I believe your equation from the equivalent circuit should have a 0.5 coefficient. There is a good derivation of this equation in Krishnan's book Switched Reluctance Motor Drives on pages 9-11.

Hope this helps.

Dear Dave, Hi

Thanks a lot for your kind response.

Where does this .5 coefficient come from? What is my mistake?

Rather than me type the equations out, you can see the derivation I referred to on Google Books (see link).

http://books.google.co.uk/books?id=mW2zm2mvQRYC

The answer to your paradox is this: 

The first equation (based on voltage times current) provides the power flow from the electrical side into the coupling field of your device. For a magnetically linear (hence, conservative, lossless) coupling field such as in this case, a power balance argument quickly leads to the conclusion that this power can be: (i) stored into the coupling field, as magnetic energy, and/or (ii) transferred to the mechanical side via the electromagnetic torque.

In your case, as you correctly say, the input electric power can be written as:

i*L*pi + 1/2*i^2*dL/dtheta*omega +  1/2*i^2*dL/dtheta*omega

The first two terms describe the change in magnetic energy storage. This is due to two things changing, namely, a change in current and a change of geometry (reflected by a change in theta). To convince yourself that this is true, just evaluate the time derivative of field energy Wf = 1/2*L*i^2. The third term represents the power transferred to the mechanical side, and is equal to torque times omega, with torque evaluated as the partial of coenergy with respect to theta, keeping current constant. 

I hope this helps.  

Mr Dionysios C Aliprantis

Do you have what explains your answer ? ebooks or Courses ...

thank you for your contribution

A textbook that explains the basics of electromechanical energy conversion, which we use at Purdue for the graduate-level introductory course on machines, is:

Analysis of Electric Machinery and Drive Systems, by Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steve Pekarek

http://www.amazon.com/Analysis-Electric-Machinery-Drive-Systems/dp/111802429X/

Dear Dr. Dionysios C Aliprantis

Thanks warmly for your informative answer. You are right. I am completely convinced with your answer. Would you please also contribute to my other question asked in researchgate about the Kron's primitive machine model?

I would be glad to look into this question as well, however, I do not have the book you are referring to, so I am afraid I cannot.

This questions comes regarding the speed voltage terms in the d-q axis model of the unified machine (Kron's machine). I want to know if the coefficients are inductances dealt with also in transformer voltage terms? Dr. Bimbhara believes that the speed voltage coefficients are different, although practically they can be well dealt with using the proper inductances in transformer voltage terms.

My second question is regarding the exact book you have in hand (Dr. Krause). In the d-q model of the separately excited DC machine, we face a term in Vqr which is the voltage of armature. This voltage comes to be Md*wr*If. My question is: Md is the inductance between DS and DR windings. So, if we believe that in the rotor these is just the qr and not the dr winding, then what does this Md coefficient have to do?

Moreover, how can be analytically derive Md? Is it just something hypothetical? can you prove the relation the Md coefficient has with the K coefficient (Ea=K*phi*wr).

You are humbly appreciated for your kind help.

Like I said, I do not have the Bimbhara book, so I cannot speculate as to what he is referring to.

Re the question from Krause's book: in the latest version (3rd edition), dc machines are treated in chapter 10. Please specify the exact formula(s) you are referring to, because I am afraid I cannot find an equation that uses the symbols you are using.

That is on page 384 of the 3rd version.

Please refer to Eq. (10.3-1). Do you agree that if we are to replace the DC machine with a primitive machine, we should have DS and QR windings? So Laf is what I asked regarding Md.

To make it clearer, look at this statement in Krause's book:

where L FF and L AA are the self-inductances of the fi eld and armature windings, respectively, and p is the short-hand notation for the operator d/dt . The rotor speed is denoted as ω r , and L AF is the mutual inductance between the fi eld and the rotating armature coils. (Bottom of Page 384).

If we believe that the armature reaction mmf is cross magnetizing (perpendicular to the field winding axis), then what does this mutual inductance mean?!

I see. This L_AF should not be interpreted as the physical mutual inductance between two coils. Rather, you should interpret this as a parameter (with units of inductance) that relates the (average, neglecting harmonics) open-circuit voltage in the armature (through commutator action) to the field current (times speed). As you correctly state, if you consider two sinusoidally wound coils that are 90 degrees magnetically displaced with each other, their mutual inductance would be zero. But, in this case, this L_AF represents something else. A little insight into this phenomenon is found in Section 10.2 of Krause. More details can be found in other machine textbooks. I am not in the position to relate this to the "primitive machine" of your other book.

I hope this helps. 

Dear Dr. Dionysios C Aliprantis

Thanks a lot for the very informative response. One question still remains in my mind: Can we calculate Md in a direct manner (rather than equating Md*wr*if to K*phi*w)?

Can we assume this Md is an inductance after all in a manner to be able to relate it to armature turns, magnetic path reluctance, etc. so that we derive the relation between Md and the physical parameters and geometrical dimensions?

Yes, of course you can calculate this parameter. It will depend on design parameters such as the ones you mention, as well as the commutator design. 

I exactly point out the method of calculation. One is the indirect way: just equate the voltage from the relation having Md (Md*wr*if) and the relation having the K factor (l*phi*w). But I want to know the direct way.

Can you derive Md taking it into account as an inductance?

Moreover, I criticized this statement: " L AF is the mutual inductance between the fi eld and the rotating armature coils" .I believe that the mutual inductance is zero for the reason you correctly mentioned.