If you compare Machowski book you shared and the reference two (cypress) you can see that here the reference vector a rotated to become reference vector q and b -> d, but in cypress Application note a -> d and b -> q. changing this will straighten up the matrix.

So actually there is no difference in the expressions, but they are written in different ways.

Hm.. But if a denotes the real part (and b the imaginary part) of the grid voltage, the two different dq-transormations will not end up in the same results.

It is common that you have machine data for reactances and resistances in d,q and then it is important that the tranformations is done correctly - I presume?

Think of it like this; in Machowski literature a and q are real parts and b and d are imaginary parts (refer Fig 3.36) but in cypress literature a and d are real parts and b and q are imaginary parts. so it does not matter which notation you follow.

What wrong in the Eq 3.167 is the order of vector [Ed; Eq] if you change the order [Eq; Ed] you will have to change the rows of the transformation matrix too and that will give you the same matrix as given in cypress application note.

Okay, assume that a,q denotes the real parts, and b,d the imaginary parts - as in the Machowski model.

Does that then mean that machine reactances from an arbritrary data sheet, containing for example Xq and Xd, still can be used without knowledge about how the guy that produced the data sheed defined the relations between a,b,q,d?

Or must one know how the guy of the data sheet defined the relations between a,b,q,d? If they relate to the the "real q" or the "imaginary q"?

Regarding Machowski using [Ed; Eq] instead of [Eq; Ed]. It is actually a clever thing. Then he makes the transformation matrix involutory - which is practical in a sense.

https://en.wikipedia.org/wiki/Involutory_matrix

[ Does that then mean that machine reactances from an arbritrary data sheet, containing for example Xq and Xd, still can be used without knowledge about how the guy that produced the data sheed defined the relations between a,b,q,d? ]

I think what it means is that you should keep an eye on the text, as different authors do these type of tricks, like you mentioned and written in text between 3.166 and 3.167 and not claimed in other texts. Infact they called parks and inverted park transform.

[Or must one know how the guy of the data sheet defined the relations between a,b,q,d? If they relate to the the "real q" or the "imaginary q"?]

Thats right. Beware of mixing text books :) I think that authors should change the notation when doing these things, like instead of d,q if one has used e,r (or something else) there was less chance of confusion.

Consider a simplidied synchronous machine of reactances X_d and X_q. Those are not dependent on the transformation method (power-invariant, amplitude etc.). However, there exists an importance of the real physical meaning of the reactances in the dq system to avoid computational errors. Consider a PM synchronous machine of 8 poles. The d-axis is conventionally defined along the pole centre of each pole, and the q-axes exists in between. It means that the PM machine has 8 d-axes and 8 q-axes distributed spatially along the airgap of the machine. In the electrical domain, mechanical degrees are transformed to electrical degrees. There exists a repetetive rotating magnetic waveform along the air gap. In the electrical domain, the phase windings distributions of each phase are connected together. You will end up with a single d-axis and a single q-axis. In generator models, the permanent magnet flux is on the d-axis, and the no-load induced emf is on the q-axis (90 degrees lagging). In natural conditions, the armature reaction causes the terminal voltage to lag with the q-axis, causing a negative d-axis component of the terminal voltage to be negative, which also defines the rotor angle. If in turns the load current lags the terminal voltage, a significant part of the current will be a negative d-axis current, which demagnetize some of the permanent magnet flux in the positive d-axis direction, but this also produces reactive power. By replacing the PMs with a field winding, the voltage-current relationship can be modfied and one can control the reactive power from the field winding. Shorter air gap over the poles makes X_d > X_q for field-wound synchronous machines. Not true for PMs, as they are as air.

Maybe my question was too general.

In the case I have in mind - I do not think about PM machines, but salient pole synchronous machines. The rotor windings are fed by some kind of DC-current source.

Active power is not always totally invariant of (d,q) according to the equation attached from p455 in the already mentioned book by Machowski et al.

The equation you refer to will allways hold, but it is strictly coupled to the sign convention from the dq-system it was derived. Therefore the sign will not allways look the same.

In a motor convention of the dq-transform, the positive q-axis will lead the positive d-axis with 90 degrees.

However, in a generator convention of the dq-transform, the positive q-axis will lag the positive d-axis with 90 degrees.

Allso, your equation implies power-invariant transformation, which makes it easier to calculate the power. Moreover, it is dominantly used for field-wound synchronous machines because it is the only way you will end up with a reciprocal dq-equivalent circuit, for the interaction between the rotor field winding and the d-axis.

For permanent magnet machines, a amplitude-invariant transformation is more common, as the rotor magnetic field is "passive" in the equivalent circuit. Also, the u_dq voltage and the i_dq current will obtain a magnitude which turns out to be equal to the amplitude of the actual three phase voltages and currents.

You mean that you might end up in the wrong signs of P and Q if you don't know what you're doing while making the variable transformation?

So the Machowski definition of the relation between a,b and d,q is for generators, but the other examples in the original post for motors?

I couldn't quite grasp your fourth paragraph above.

• With Power-invariant you mean that the transformation matrix is a unitary one?
• Do you mean that other a,b to d,q transformation matrices are used for different kinds of machines?

Yes that is a true representation!

The case for a synchronous machine is that if one uses amplitude invariant transformation for generator transformation, then:

d-axis flux linkage is:

phi_d = Lafd*i_fd - Ld*i_d

rotor field winding flux linkage assosicate with the d-axis:

phi_fd = Lfd*i_fd - (3/2) Lafd*i_d

Clearly, the mutual inductances between the field winding and the three-phase armature is not reciprocal. The d-axis current is coupling (3/2)*Lafd with the field winding, whereas the field current is coupled only Lafd with the stator armature.

If one multiplies with sqrt(3/2):

sqrt(3/2)*phi_d = sqrt(3/2)*Lafd*i_fd - sqrt(3/2)*Ld*i_d

And defines power-invariant flux linkage and d-current:

phi_d' = sqrt(3/2)*phi_d

i_d' = sqrt(3/2)*i_d

You end up with:

phi_d' = sqrt(3/2)*Lafd*i_fd - Ld*i_d'

phi_fd = Lfd*i_fd - sqrt(3/2)*Lafd*i_d'

Where the flux linkage coupling is reciprocal which makes an equivalent circuit possible, with a shared mutual inductance path.

Lmd = sqrt(3/2)*Lafd

In addition to my answer above:

There will exist different abc-to-AlphaBeta-to-dq conversion matrixes, depending on the chosen positive d- and q-axes.

With amplitude transformation power equals to 3/2 times the current*voltage

Power-invariant power equals "sqrt(3/2)*current times sqrt(3/2)*voltage", or " current' times voltage' "

where

current' = sqrt(3/2)*current (amplitude)

voltage' = sqrt(3/2)*voltage (amplitude)

The voltage in the power-variant turns out to be the line-to-line voltage in rms. Divide the amplitude by sqrt(2) to get the rms, and multiply with sqrt(3) to get line-to-line...

Sure. But the initial issue was about consequences of different transformations.

Do you mean that it does not matter how you define d,q related to a,b as long as they are somehow related by

n*pi/4 +/- delta,

where n reprsents an integer, and delta the load angle of the machine? It that the conclusion?

Thank you Lars. I dont know what you mean with the posted equation. Maybe I missunderstood you? I try below to answer according to my interpretation.

The load angle of the machine I typicallly defined as the angle between the terminal dq voltage and the no-load voltage, e_q (u_d=0 under no-load). Yielding:

tan(delta) = u_d/u_q

Accounting for the phase shift (phi) between the armature current and the terminal voltage:

tan(delta+phi) = i_d/i_q

In generator operation, the d-axis current is conventionally defined positive when it opposes the d-axis field winding flux linkage. Then POSITIVE d-current is accounted on the negative d-axis. It is important to note that a positive d-current in this convention, demagnetizes (flux weakening) the terminal voltage. This is what calculated in FEM when you draw a potier diagram of the machine.

I separate my response into a number of smaller posts.

Load Angle:

I follow the (common?) definition of power systems.

P_G = E*V*sin(delta)/X

where delta is the angle difference between stator and rotor.

In my original post in this question, it is this load angle that is used in the two different definitions I found of the relations between stator voltages, in the complex plane, Ureal + jUimag, (a,b), and the synchronous machine EMFs Ed + jEq, (d,q).

I cannot see that the angle delta in Figure 3.36 in my original post (I repost it here for convenience) follows the definition that you give in the above:

"tan(delta) = u_d/u_q"

Rather that delta is (according to my second attached figure from Machowski, as you see there are alternative definitions as well which confused me in the beginning) the angle between E_q and and V_s.

One thing that then confuses me with Figure 3.36 is that is seems like he there defines delta as the angle between E_q and the real part of V_s, lets call it V_s_a. (But maybe that is because the angle of V_s is a part of delta following the definition that delta = delta_r - delta_s?)

I am trying to show how your equations are consistent in the framework presented in the included screenshot...

To me it seams like the delta angle in the ab to dq transformation is not the load angle. It is either a transformation angle between to rotating frames.

If ab is the alpha-beta voltages, they are rotating, then delta is a continously changing angle 2*pi*f*t. The above analysis was for steady state. It must be expanded for transient analysis...

It is of course a transformation angle between the rotating frames.

It is however common that you in the power systems world neglect the variations of stator frequency.

The book I refer to is (among other things) for dynamic analysis.

The thing is that since you assume that dws/dt=0, you do the dynamic computations inside the machine.

I'm not a deep expert in the interaction between system and machines, but it is approximately as follows:

• Then for the next time step of load flow you know the new value of the rotor angle deltar[t], and the stator angle should (if no net loads have changed, which they have) only be deltas[t]=deltas[t-1]+wsDeltaTime[t].
• Then by the rotor angle deltar[t], and the stator angle deltas[t] one should be able to get the load angle delta[t]. If the active power of the machine computed in the dynamic computation corresponds to the one involving the load angle delta[t], everything is "fine". Else. Some kind of iterative procedure has to take place, I assume.

I hope this clarified the picture somewhat. But as I said, I am not an expert in the "classical approach" of interconnecting grid load flows with machine dynamics. Not yet.

Jonas:

I am not sure if I'm familiar with your equation (97) above.