Does anybody know how to find the formulas to the current components in the d and q axis that can make the torque production maximum in the two field weakening areas?

recommend you a paper: maximum torque control of an induction machine in the field weakening region.

Thanks for the suggestion.
I've seen it, and although it is a good article it did not help me because in the formulas that displays the stator resistance is neglected and I needed the formulas considering the stator resistance, as shown in the article "Current Control of Induction Machine in the Field-weakening Region ". This last article has what I needed however I can not get them myself in order to confirm its accuracy.

If anyone could help me I would appreciate

I would assume you are talking about max torque per ampere in the context. When the induction machine is in the field weakening region, the field component of the current is no longer a degree of freedom, thus the d component is fixed. So how do you define a max torque under such condition?

In the field weakening the value of two components of stator current that maximizes the torque is given by the intersection of constraints and voltage supply current in the case of Constant Power Zone and by the formula of the tension restriction and the torque formula in the case of the last zone.
My problem is that I am using the Wolfram Mathematica and I am trying to solve the equation (Rs-ids * we * L's * IQS) ^ 2 + (Rs + QSI * we * Ls * ids) ^ 2 ^ 2 == Umax and trying to maximize the binary formula: T = (3/2) * P * (Lm ^ 2/Lr) * ids * Iqq, and unfortunately do not know how.

I don't really understand the voltage model you're using...

However, I'd try substituting Iqs and Ids with I*cos(gamma) and I*sin(gamma), respectively. This reduces the thing to a single variable equation, since of course nominal current must be considered for maximum torque.

The model I'm trying to is similar to the one in the paper "maximum torque control of an induction machine in the field weakening region" but I'm considering the stator resistance and in the last Region of the Capability Curve of the Induction Machine the nominal current is no longer available and i can't find the way to obtain the formulas of the iqs and ids with the resistance

Some quick suggestions: 1. in field weakening region, especially in the region where the constraint is only the bus voltage, motor speed is rather fast. In this case, the resistor-caused voltage is generally smaller than inductor caused. That means the former can be neglected and that's also what is done in many papers. 2. if you do have to consider it but cannot derive the explicit expression, try some numeric ways.

The problem is that I can't despise the stator resistance because I have a 3 kW induction machine. Trying which numeric ways?

Thanks for all the help and suggestions.

maybe you should start from the basics since not all here have the access to the paper. are you trying to achieve max torque with respective to what variable?

Is there any way to put here the paper that I'm talking about?
What I'm trying to achieve is the formulas of the ids and iqs that maximize the production of the torque in the two field weakening areas. In the first field weakening area (Constant Power Zone) I achieve the formulas by solving the system of this two equations:
ids^2+iqs^2==Imax^2
(Rs*ids-we*L's*iqs)^2+(Rs*iqs+we*Ls*ids)^2==Umax^2

But in the second I only have the equation:
(Rs*ids-we*L's*iqs)^2+(Rs*iqs+we*Ls*ids)^2==Umax^2,
and in the paper it says that the formulas are achieved by taking this formula and maximizing the formula of the torque: T=(3*P/2)*(Lm^2/Lr)*ids*iqs and I tried with the Wolfram Mathematica but I was not successful. Probably it was something wrong that I made...

If you really want the paper, you have to buy it or get it in appropriate way. I don't think a paper is right to be put here. It's all about copyright.

I have the paper. I was just asking if I could share it to try explain what is my problem...

You could briefly REWRITE the most important formulas and explanations (with your own words) and link to them (e.g. as an image).

In the torque equation, when considering all the motor parameters are constants, the max torque is achieved at ids = iqs. With this in mind, combined with voltage equation, cannot you get the ids, iqs and max torque?

I don't think that for maximizing the torque in FW region, any current formulas are a good choice. For high-speed torque producing, motor voltage has to be achieved as high as possible. That's just physics. It is a reson why I would recommend some voltage controller, not any predefined formulas which are dependent on motor parameters (which are not constant in fact). Sean is right with the ids = iqs formula, but it is fact only for a linear magnetizing curve (which is a fiction :) ). For real motors with nonlinear mag. curve it is an optimization task. But in FW region it is easier. Voltage as much as you can. And third region - limit the usd not to "eat" all the voltage, good choice is usdmax = Usmax/sqrt(2).