Application note

have a large effect on the performance and hence the efficiency of the motors and ultimately that of .... The machine used in the simulation results is a cage rotor.
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Application note 2009

Parameter estimation of 3-phase induction machines at standstill using Matlab software Farah-Mahad Mohamed Ismail Polytech’Clermont-Fd Email: [email protected]

Abstract This application note presents a solution to estimate electrical parameters of 3-phase induction machine at standstill. These parameters are the stator resistance , the stator inductance and the rotor time constant . The proposed estimation procedure was carried out for induction machines with an error less than 5% by analyzing the transfer function and the steady-state equivalent circuit. Based on this analysis, the signals and models to estimate the electrical parameters of the machine are proposed. The simulation results obtained using Matlab are shown in this paper. Keys words: induction machine, electrical parameters, estimation, model, least square

Contents Abstract ................................................................................................................................................... 1 Figures ..................................................................................................................................................... 2 Tables ...................................................................................................................................................... 2 Variables .................................................................................................................................................. 2 Introduction............................................................................................................................................. 3 Principle ................................................................................................................................................... 3 Model of induction machine ............................................................................................................... 3 Parameters estimation ........................................................................................................................ 5 Application description ........................................................................................................................... 7 Simulation set-up ................................................................................................................................ 7 Estimation’s algorithm ........................................................................................................................ 8 Simulation results .................................................................................................................................... 9 Conclusion ............................................................................................................................................. 12 References ............................................................................................................................................. 12 Appendix................................................................................................................................................ 13 Appendix A.

’s estimation algorithm ............................................................................................. 13

Appendix B.

’s estimation algorithm ............................................................................................ 13

Appendix C.

‘s estimation algorithm .................................................................................. 17

Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

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Application note 2009

Figures Figure 1 : Simulation set-up for estimation........................................................................................ 7 Figure 2 : Simulation set-up for estimation ............................................................................ 7 Figure 3 : Estimation of with steady-state model .............................................................................. 9 Figure 4 : Estimation of with steady-state model .......................................................................... 10 Figure 5 : Estimation of with dynamic model ...................................................................... 11

Tables Table 1 : Parameters' motor .................................................................................................................... 9

Variables The following list shows the different variables used in this application and in the equations and schemes presented here Stator currents transformed by Park’s transform Stator voltages transformed by Park’s transform Stator resistance, rotor resistance Stator inductance, rotor inductance, mutual inductance Rotor time constant Leakage factor Pulsation’s excitation signal, machine speed Slip which is Mutual inductance Stator resistance,

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Application note 2009

Introduction Induction Machine is widely used in industrial applications and hence the estimation of its parameters plays a pivotal role in vector control methods, which is used for achieving highperformance torque control. Indeed, during its use, motor parameters change considerably with change in temperature, frequency and saturation. These changes can greatly affect the performance of the drive system. Therefore, if the motor control is designed with the wrong parameter values, then the drive system will not perform efficiently. So, these parameters have a large effect on the performance and hence the efficiency of the motors and ultimately that of the system is greatly affected in the long run. There are a lot of parameter estimation techniques. But, in this paper, one method which estimates the stator resistance , the stator inductance and the rotor time constant will be presented. This method is based on the dynamic model of the machine and the steady-state model. We estimated the parameters with an error of less than 5%. The principle is described as well as the simulation results on Matlab are analyzed.

Principle Model of induction machine A reasonable model of an induction motor must first be identified before the parameters for that model can be deduced. Thus, the dynamic model and the steady-state model are presented. Both of them serve to estimate the parameters. Machine dynamic model By using the machine electrical equations and Park transformation, the dynamic model of induction machine is given as [1]:

To keep it simple we have worked in the stator reference frame: 3

Note that

are zero because rotor is a cage so it is short circuited. Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

Application note 2009 Based on this dynamic model, the transfer function

can be written as [4]:

Where and are the stator current, and the stator voltage vector, respectively, with representing the Laplace transform constant. The continuous time model (1) can be approximated by its discrete-time equivalent (2) by using the discrete transform with the sampling rate sufficiently high compared to the system dynamic [4].

Where

is the delta operator, defined as

.

Steady-state model The induction machine equations in the steady state [1] are of the following form

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Application note 2009

Parameters estimation The model (3) contains the parameters , and their simultaneous estimation in the linear regression model derived from (3) requires the use of special excitation signals. Nevertheless, the parameters are estimated with many errors. That is why other models that can estimate the parameters with better accuracy are proposed. We want to estimate all parameters with the machine at standstill, i.e. speed is considered zero, so we have only four parameters to be estimated. In order to reduce the parameters to be determined with the dynamic model (3), we estimate and using the steady-state model (4). Once and are known, we use model (3) to estimate . Finally, the 3 parameters are estimated with a better accuracy. To estimate the parameter using the Least Squares algorithm [2] it is necessary to rewrite each model in the form of a linear regression such as

Where y(t), H(t), and θ are the prediction vector, the regression matrix, and the parameter vector, respectively. Estimation of

and

By analyzing the input impedance of the steady-state equivalent circuit (4) of an induction machine, it is possible to choose a model that can adequately be used for parameter estimation under certain constraints.

If we consider

Where

and

(i.e. constant stator voltage), the model (6) becomes:

are stator phase voltage and current.

So, we estimate by using the voltamperemetric method which consists of sending a voltage step function on one phase of the machine. The machine stays at standstill and by applying Ohm's law we obtain . 5

Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

Application note 2009 Otherwise, if the frequency (i.e. ) is high as compared to approximated by (7).

, the model (6) can be

This model is shown in (7)

This model can be used to estimate and at = 0 rad/s by supplying the machine with a sinusoidal signal, the magnitude of it been sufficiently small so that the machine stays at standstill, and its frequency must be in the range [4] so that the approximation (7) is valid. We consider only . Estimation of If and are known, it is possible to derive a model that should provide a better estimation of when compared to the estimation done using model derived from (3) if appropriate excitation waveforms are selected. It must be noted that at zero speed the model represented by (3) can be simplified. The following model can be used at zero speed if an appropriate signal is employed for the estimation:

Where So by using a sinusoid with a low magnitude and a low frequency in the range [4] in order that the model rests valid, a better estimation of the parameters with good precision at standstill is possible.

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Application note 2009

Application description The application was developed on Matlab.

Simulation set-up We take a machine model represented by its electrical equations. Its inputs were voltage on each phase and we observe output currents on each phase and the torque1. Internal parameters: stator resistance, stator inductance, rotor time constant, number of even pole, inertia’s moment. Estimation of

Figure 1 : Simulation set-up for

estimation

Estimation of

7 Figure 2 : Simulation set-up for 1

estimation

This output doesn’t use.

Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

Application note 2009

Estimation’s algorithm The estimation carried out by stator voltages and currents. The 3 estimation algorithms used are referred to in the appendix. Algorithm:  Generate an appropriate excitation on each of the three phases of the machine - For ’s estimation, it’s a constant voltage. - For ’s estimation, it’s a sinusoid of 20V at 100 Hz. - For ‘s estimation, it’s a sinusoid of 20V at 1 Hz. Acquire N stator voltages and currents - For ’s estimation, only voltages and currents on a single phase are enough.  Solution of the model equations - For ’s estimation, it’s ( - For ‘s estimation, it’s - Estimation using mean estimator for ’s estimation and least square for other parameters.

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Application note 2009

Simulation results The proposed models were studied by simulation on Matlab where it was assumed that the measurements were corrupted by an additive white noise term w(t) to simulate the presence of noise such as in an output error model. The noise sources were distributed with a variance of fv*max(V2) for voltages, and with a variance of fc*max(I2) for the currents. The values for fv=0.1 and fc=0.05 have been chosen to represent a relatively large noise contribution. The machine used in the simulation results is a cage rotor. The parameters are presented in Table 1. Sampling time is 0.1ms and the number of sample for the estimation is 1000. Parameters’ motor (Ω) 6 (H) 0.104 σ 0.0858 (s) 0.075 Table 1 : Parameters' motor

The error must stay lower than 5% for each estimated parameter. In order to do so, the estimation has been repeated one hundred times and the error was calculated for each and every value. Fig. 4 shows the estimation of , as obtained using the steady-state model (6) with the machine at standstill for an input of 20V step-function. To estimate this parameter, we used a mean estimator. Note that the error is lower than 5% for all estimations.

Current (A)

Voltage (V)

Voltage Echelon on one phase 30 20 10

0

100

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500 600 Sample Current response

700

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900

1000

0

10

20

30

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3 2 1

error (%)

2 1 0

40

50 60 Experience

Figure 3 : Estimation of

2

70

with steady-state model

Stator voltages and currents of 3-phase of the machine

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Application note 2009 Fig. 5 shows the estimated value of as obtained using a steady-state model with the machine at standstill. The input signal for this case was a sinusoid with a magnitude of 20V at 100 Hz. Note that the LS algorithm expects the regression matrix H(t) and the noise w(t) to be uncorrelated [2]. In the others terms, the regression matrix of the motor model mustn’t contain the measurements. Otherwise, the estimation could become biased. So, as the measured currents are present in this matrix (as we can see in (8)), the noise doesn’t add on to these currents. We can see it in fig. 5. We observe that the error is lower than 5% for all estimated values. The mean error is 0.5%.

Voltage on A-phase

Voltage (V)

50

0

-50

0

100

200

300

400

500 600 Sample Current response

700

800

900

1000

0

100

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Current (A)

5

0

-5

500 600 Sample Estimation's Error on Rs

error (%)

1.5 1 0.5 0

40

50 60 Experience

Figure 4 : Estimation of

with steady-state model

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Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

Application note 2009 To estimate , the problem is to simulate the presence of noise because of the least square algorithm. As seen, the noise must be uncorrelated regression matrix [2]. So, either the voltages or the currents mustn’t present in the regression matrix to avoid a biased estimation. It is not the case; both of them are present in this matrix. Fig. 6 shows the estimation of and we notice that this estimation is biased when we add a noise on voltages and currents. Consequently, we can see that the ordinary least square method are not suitable for estimation when we want to simulate the presence of noise on the measurements. A remedy for this problem is to use a least square extended algorithm [3]. Voltage on A-phase

Current response

25

4

20

Current (A)

Voltage (V)

3 15 10 5

2 1

0 -5

0

500 6

15

x 10

1000 Sample

1500

0

2000

0

Estimation's Error on Ls

500

1000 Sample

1500

2000

Estimation's Error on Tr 100.2

10

error (%)

error (%)

100.15

5

100.1 100.05 100 99.95

0

0

20

40 60 Experience

80

100

Figure 5 : Estimation of

99.9

0

20

40 60 Experience

80

100

with dynamic model

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Application note 2009

Conclusion This paper discussed a procedure to estimate the parameters of induction machines by using its steady-state model and the transfer function . We have canned study its robustness and evaluate the noise’s influence of the estimation. It’s important to respect the excitation signal so that the used models rest valid. Using a combination of steady-state model and dynamic model with an appropriate signal, we proved that this method is capable of accurately estimating machine parameters. Nevertheless, we also saw that the least square method is not appropriate for estimation because of bias problem and how it was tackled using the extended least square method.

References [1] Baghli, L. (2005). Modélisation et commande de la machine asynchrone. IUFM Lorraine - UHP . *2+ Chapuis, R. (2008). Théorie de l’estimation. Polytech'Clermont-Fd . [3] G.BINET. (2004). La méthode des moindres carres. Université deCaen . [4] R.M. Moraest, L. A. (2003). Parameter Estimation of Induction Machines By Using Its steady-State Model and Transfer Function. IEEE Explorer .

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Application note 2009

Appendix Appendix A.

’s estimation algorithm

%% -------------Algorithme de calcul de Rs ----------------------% % Julien Rosele et Farah-Mahad Mohamed Ismail % % % Polytech'Clermont-Ferrand % % Génie Electrique (GE5) % % Octobre 2008 % % Projet 08AB05 : Détermination automatique % % des paramètres électrique de moteurs asynchrones % % % % ####### Détermination de Rs ####### % % Procede: Modele de la MAS en regime permanent par phase % % - Excitation avec un signal d'échelon de 20V % % - Utilisation de l'estimation moyenne % % % % Ce programme calcul (Nexp fois )le paramètre Rs de la machine % % en fonction du courant et la tension d'une phase du moteur et % % on observe à chaque estimation l'erreur commise % % Les propriétés suivantes : % % - Bruitages des signaux % % - Estimateur moyenneur % % - Calcul sur Ne points de mesure % % % % ATTENTION : probleme avec le modele mathematique, % % il faut diviser par 2/3 le resultat % %-------------------------------------------------------------------------% %% debut du programme Rs=6; varC=2.22*0.05; 0,05*MAX(Is) varV=20*0.1; 0,1*MAX(Vs) % Echantillonnage Te=1e-4; Ne=1000; t=[1:Ne]; Nexp=100;

% Modèle (ohm) % bruit blanc sur les courants de variance % bruit blanc sur les tensions de variance

% % % %

période d'échantillonnage nombre d'échantillons Temps nombre d'expériences

%% Experience for k=1:Nexp %% Acquisition du courant et de la tension dans une phase %On garde les Ne dernieres valeurs Iacq=Ia([length(Ia)-Ne+1:length(Ia)],1); Vacq=Va([length(Va)-Ne+1:length(Va)],1); %% Bruitage

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Application note 2009 N=length(Vacq); %Courant : Ia_b=Ia bruitée b= varC*randn(1,N); Ia_b=Iacq + b'; %Tension : Va_b=Va bruitée b= varV*randn(1,N); Va_b=Vacq + b'; %% Loi d'ohm Rs = Va / Ia Rs_est=Va_b./Ia_b; %% Estimateur moyenneur Rs_est=sum(Rs_est)/Ne; Rs_est=Rs_est*(2/3); %problème du modele mathematique normalement on doit prendre Rs1/2 error(k)=abs(Rs-Rs_est)*100/Rs; end

%% Representation graphique figure(1); subplot(311); plot(t,Va_b); title('Voltage Echelon on one phase'); xlabel('Sample'); ylabel('Voltage (V)'); subplot(312); plot(t,Ia_b); title('Current response'); xlabel('Sample'); ylabel('Current (A)'); subplot(313); plot(error); title('Estimation''s Error on Rs'); xlabel('Experience') ylabel('error (%)') % fin du programme %%

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Application note 2009

Appendix B.

’s estimation algorithm

%% ---------------- Algorithme de calcul de sigma*Ls ---------------------% % Julien Rosele et Farah-Mahad Mohamed Ismail % % % Polytech'Clermont-Ferrand % % Génie Electrique (GE5) % % Octobre 2008 % % Projet 08AB05 : Détermination automatique % % des paramètres électrique de moteurs asynchrones % % % % ####### Détermination sigma*Ls ####### % % Procede: Modele de la MAS en regime permanent % % - Excitation avec un signal de 20V et de 100Hz % % - Utilisation de la methodes des moindres carres % % % % Ce programme calcul (Nexp fois ) Sigma*Ls de la machine % % en fonction des courants et des tensions (repere de Park)et % % on observe à chaque estimation l'erreur commise % % Les propriétés suivantes : % % - Bruitages des signaux % % - Calcul sur Ne points de mesure % %-------------------------------------------------------------------------% %% debut du programme close all; % definition des variables fs=f; % fréquence de pulsation Ws SigmaLs=sigma*Ls; % Valeur de sigmaLs (modele) varV=20*0.1; % bruit blanc sur les tensions de variance 0,1*MAX(Vs) % Echantillonnage Te=1e-4; Ne=1000; t=[1:Ne]; Nexp=100;

% période d'échantillonnage % nombre d'échantillons % Temps % nombre d'expériences

%% Experience for k=1:Nexp %% Acquisition des courants et des tensions %On garde les N dernieres valeurs Va_acq=Va([length(Va)-Ne+1:length(Va)],1); Vb_acq=Vb([length(Vb)-Ne+1:length(Vb)],1); Vc_acq=Vc([length(Vc)-Ne+1:length(Vc)],1); Ia_acq=Ia([length(Ia)-Ne+1:length(Ia)],1); Ib_acq=Ib([length(Ib)-Ne+1:length(Ib)],1); Ic_acq=Ic([length(Ic)-Ne+1:length(Ic)],1); %% Bruitage N=length(Va_acq); % Courant %---> Attention de ne pas bruiter les courants sinon apparition de biais % (cause courant present dans la matrice d'observation (de regression))

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Application note 2009 %Tension : Vi_b=Vi bruitée b= varV*randn(1,N); Va_b=Va_acq + b'; b= varV*randn(1,N); Vb_b=Vb_acq + b'; b= varV*randn(1,N); Vc_b=Vc_acq + b'; %% Transformation de concordia % Courant Isd=sqrt(2/3)*(Ia_acq - .5*Ib_acq - .5*Ic_acq); Isq=(Ib_acq-Ic_acq)/sqrt(2); % Tension Vsd=sqrt(2/3)*(Va_b-.5*Vb_b-.5*Vc_b); Vsq=(Vb_b-Vc_b)/sqrt(2); %% Estimation par la methode des moindres carres Y1=[Vsd;Vsq]; %Vecteur de prediction Y H1=[Isd -2*pi*fs*Isq; %Matrice de regression H Isq 2*pi*fs*Isd]; %resolution de Y1=H1(t)*teta teta1=inv(H1'*H1)*H1'*Y1; %Vecteur de parametres SigmaLs_est=teta1(2); error(k)=abs(SigmaLs-SigmaLs_est)*100/SigmaLs; end

%% Representation graphique figure(1); subplot(311); plot(t,Va_b); title('Voltage on A-phase'); xlabel('Sample'); ylabel('Voltage (V)'); subplot(312); plot(t,Ia_acq); title('Current response'); xlabel('Sample'); ylabel('Current (A)'); subplot(313); plot(error); title('Estimation''s Error on SigmaLs'); xlabel('Experience') ylabel('error (%)')

%% fin du programme

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Application note 2009

Appendix C.

‘s estimation algorithm

%% ---------------- Algorithme de calcul de Ls et de Tr ------------------% % Julien Rosele et Farah-Mahad Mohamed Ismail % % % Polytech'Clermont-Ferrand % % Génie Electrique (GE5) % % Octobre 2008 % % Projet 08AB05 : Détermination automatique % % des paramètres électrique de moteurs asynchrones % % % % ####### Détermination de Ls et Tr ####### % % Procede: - Modele dynamique de la MAS % % - Excitation avec un signal de 20V et de 1Hz % % - Utilisation de la methodes des moindres carres % % % % Ce programme calcul (Nexp fois ) les paramètres Ls et Tr % % en fonction des courants et des tensions (repere de Park) % % Les propriétés suivantes : % % - Bruitages des signaux % % - Calcul sur 150 points de mesure % %-------------------------------------------------------------------------% %% Determination de ls/Tr et 1/Tr (approche par moindres carrées) % Connaissant Rs et SigmaLs close all; clc; % definition des variables Rs; % Resistance SigmaLs=sigma*Ls; % Valeur de sigmaLs varC=3.125*0.05*0; % bruit varV=20*0.1; % bruit % Echantillonnage Te=1e-4; Ne=2000; t=[1:Ne]; Nexp=100;

% période d'échantillonnage % nombre d'échantillons % Temps % nombre d'expériences

%% Experience for k=1:Nexp %% Acquisition des courants et des tensions %On garde les N dernieres valeurs Va_acq=Va([length(Va)-Ne+1:length(Va)],1); Vb_acq=Vb([length(Vb)-Ne+1:length(Vb)],1); Vc_acq=Vc([length(Vc)-Ne+1:length(Vc)],1); Ia_acq=Ia([length(Ia)-Ne+1:length(Ia)],1); Ib_acq=Ib([length(Ib)-Ne+1:length(Ib)],1); Ic_acq=Ic([length(Ic)-Ne+1:length(Ic)],1);

17 %% Bruitage N=length(Va_acq); % Courant b= varC*randn(1,N);

Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

Application note 2009 Ia_b=Ia_acq + b'; b= varC*randn(1,N); Ib_b=Ib_acq + b'; b1= varC*randn(1,N); Ic_b=Ic_acq + b'; %Tension b= varV*randn(1,N); Va_b=Va_acq + b'; b= varV*randn(1,N); Vb_b=Vb_acq + b'; b= varV*randn(1,N); Vc_b=Vc_acq + b'; %% Filtrage f=abs(fft(Ia_b)); %% Transformation de concordia % Courant Isd=sqrt(2/3)*(Ia_b - .5*Ib_b - .5*Ic_b); Isq=(Ib_b-Ic_b)/sqrt(2); % Tension Vsd=sqrt(2/3)*(Va_b-.5*Vb_b-.5*Vc_b); Vsq=(Vb_b-Vc_b)/sqrt(2); %% Estimation par la moindres carres % definition Usd et Usq Usd=Vsd-Rs*Isd; Usq=Vsq-Rs*Isq; % %derive de Usd Usd1=[0;Usd([1:size(Usd)-1],1)]; dUsd=(Usd-Usd1)/Te; % %derive de Usq Usq1=[0;Usq([1:size(Usq)-1],1)]; dUsq=(Usq-Usq1)/Te; % %derive de Isd Isd1=[0;Isd([1:size(Isd)-1],1)]; dIsd=(Isd-Isd1)/Te; % %derive de Isq Isq1=[0;Isq([1:size(Isq)-1],1)]; dIsq=(Isq-Isq1)/Te; % %derive de dIsd dIsd1=[0;dIsd([1:size(dIsd)-1],1)]; d2Isd=(dIsd-dIsd1)/Te; % %derive de dIsq dIsq1=[0;dIsq([1:size(dIsq)-1],1)]; d2Isq=(dIsq-dIsq1)/Te; %% Echantillonage b=b([length(b)-Ne+3:length(b)]); Usd=Usd([length(Usd)-Ne+3:length(Usd)],1); Usq=Usq([length(Usq)-Ne+3:length(Usq)],1); dUsd=dUsd([length(dUsd)-Ne+3:length(dUsd)],1); dUsq=dUsq([length(dUsq)-Ne+3:length(dUsq)],1); dIsd=dIsd([length(dIsd)-Ne+3:length(dIsd)],1); dIsq=dIsq([length(dIsq)-Ne+3:length(dIsq)],1); d2Isd=d2Isd([length(d2Isd)-Ne+3:length(d2Isd)],1);

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Application note 2009 d2Isq=d2Isq([length(d2Isq)-Ne+3:length(d2Isq)],1); %% Methode des moindres carres %Vecteur de prediction Y Y2=[d2Isd - (1/SigmaLs)*dUsd; d2Isq - (1/SigmaLs)*dUsq]; %Matrice de regression H H2=[-(1/SigmaLs)*dIsd (1/SigmaLs)*Usd; -(1/SigmaLs)*dIsq (1/SigmaLs)*Usq]; % Vecteur de parametres teta=pinv(H2)*Y2;

%

% Biais bi(k)=Ls/Tr-teta(1); bj(k)=Ls/Tr-teta(2); %bi(k)=1/Tr-teta(1); %b1(k)=0;b2(k)=0; b1=- (1/SigmaLs)^3 * mean (b'.*(Usq-Usd) ./ (-dIsd.*Usq + dIsq.*Usd)); b2=- (1/SigmaLs)^3 * mean (b'.*(dIsq - dIsd) ./ (-dIsd.*Usq + dIsq.*Usd)); % Parametres Tr_est=1/(teta(2)+b2); Ls_est=(teta(1)+b1)*Tr;

% Erreur d'estimation error_Ls(k)=abs( Ls - Ls_est)*100/Ls; error_Tr(k)=abs( Tr - Tr_est)*100/Tr; end %% Representation graphique figure(1); subplot(221); plot(Va_b); title('Voltage on A-phase'); xlabel('Sample'); ylabel('Voltage (V)'); subplot(222); plot(Ia_b); title('Current response'); xlabel('Sample'); ylabel('Current (A)'); subplot(223); plot(error_Ls); title('Estimation''s Error on Ls'); xlabel('Experience') ylabel('error (%)') subplot(224);

Parameter Estimation of 3-phase induction machine at standstill | Polytech’Clermont-Fd

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Application note 2009 plot(error_Tr); title('Estimation''s Error on Tr'); xlabel('Experience') ylabel('error (%)') figure(2) plot(f); %% fin du programme

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