*The method allows the determination of parameters in saturated systems, and the algorithm is adapted to saturation level.*

*Parameter’s determination on saturated systems is a complex task. Sometimes you can determine these parameters by calculation but building errors must be checked to verify the final values of them. You can use accurate off-line models but the time of process could be excessive for a factory process. In this article we describe a recursive algorithm to calculate the parameters in real time and buried in a generic measurement system made to test three phase inductors.*

*By Ramon Bargalló Perpiñà, Electrical Engineering Department, Manuel Roman Lumbreras, Guillermo Velasco Quesada and Alfonso Conesa Roca, Electronic Engineering Department. Polytechnic University of Catalonia and Jaime González Salmerón, Grupo Premo*

In order to have a fast measurement of the parameters the flux linkage must be known. It is, however, expensive and difficult to measure the flux. Instead, the flux can be estimated based on measurements of voltage and current. After discussion we present a Luenberger’s observer for the flux and a recursive algorithm to calculate the parameters of the considered model. Also the measurement systems calculate the total losses on the inductor and using our model we can separate it on joule and magnetic losses.

### Flux estimation and observation

There are some possibilities to estimate the flux. The following table summarizes the most commons.

These estimators have some problems:

a) If the resistance (r) is not correctly determined the estimated flux is different to the correct one.

b) If the inductance (L) is not correctly determined the estimated flux differs to the correct one.

For these reasons it is better to use a closed loop observed. The proposed Luenberger’s observer is:

For values of γ > 1 is absolutely stable and the convergence of the observed values (^) to the real ones is guaranteed.

In the above expression, I_{N} is the current that produces the flux Ψ_{N}; it isn’t the nominal current of the tested coil or transformer.

The unknown parameters are: r, a, and b. Using the above expression it is possible to determine the inductance:

We use the above observer in addition to a recursive estimator to obtain the unknown parameters. Figure 1 shows the structure of that.

### Discrete System

By discretization the equation 1 becomes:

With T in the sampling period and k represents the k sampling interval (actual time could be calculate by t = k*T) To obtain the unknown parameters we solve recursively the above equation.

### Least Squares Recursive Method

The recursive least squares method (RLS) tries to determine the unknown parameters of the following matrix equation:

y(k) are the measured set on instant k, ϕ^{Τ} (k) are known functions, e(k) are the measurement errors (unknown) and θ are the unknown parameters. The RLS method calculates estimation for the actual time (k) based on the measurements until k and the estimation for the earlier time (k-1). The whole algorithm is:

For the penalizing factor λ the suitable values are from 0.95 to 0.995 and the losses function to be minimized will be:

As a result of approximation of (10) at the surround of λ = 1. T is a pseudo-time constant

Suggested values for λ are 0.95 to 0.995 and these values indicates values for T for 20 to 200. It is accepted that values older that 2*T aren’t any influence for the final result.

The only limitation of this method is that the parameters must be expressed as coefficient of known functions. In our case this is true because the exponent (n) for the current-flux function can be considered known (they can be determined by early test and usually it is 5, 7 or 9 depending to the saturation level of the material.

There is another limitation of this method but this is due to the characteristics of the applied signal: It must be a called persistent excitation of order N, and the following conditions must be guaranteed:

These limits exists

These matrix be positive definite

These expressions are true if the harmonic component of the applied signal contains at least N/2 harmonics.

### Experimental Test

Figure 2 shows the experimental setup. The whole system is made with the following parts:

Current controlled PWM three phase converter. Current and voltage sensors connected to a data acquisition card. Data acquisition card: 16 differential channels; 1.2Msamples. PC with labview v8.2 to show and calculate the interested values (current, voltage, power losses, parameters, etc.) The user could select what data is shown.

The following pictures show the results for a one-phase transformer (1.3 kVA, 220/380 V, 5.9/3.5 A, PJ = 40 W, Po = 18 W, R1 = 0.942 W, R2 = 1.202 W, YN = 0.99 Wb, IN = 1.41 A) with PWM alimentation.

The magnetic material of this transformer has an exponent n = 7 (these value was calculated earlier) Final values for a and b are 0.48 and 0.52 respectively; they are agree with the values calculated by off-line methods. The following picture shows the comparison between the experimental saturation characteristic and fitted equation using the above values for a, b, and n.

Figure 7 shows the comparison between calculated inductance using experimental measurements, calculated inductance using expressions (2) and (3) and by using 2D FE calculation.

**Conclusion**

The explained method allows the determination of parameters in saturated systems, and the algorithm is adapted to saturation level.

For sinusoidal applied voltage the results are bad: the final values for the parameters a and b have offset. These are due the limited harmonic component of these signal (1 harmonic only allows to determine only 2 parameters under theorist assumptions: no measuring errors, etc.). As a conclusion it is necessary to apply a signal with some harmonics to avoid or minimize measuring errors).

**Acknowledgment**

This work was partially granted by FIT-030000-2007-79 (MITYC) and Grupo PREMO S.A.

**References:**

1) Ljung. System Identification: Theory for the user. Prentice Hall, New Jersey, 1987.

2) R. Bargallo. Aportación a la determinación de parámetros de los modelos en la máquina asíncrona para una mejor identificación de variables no mensurables incidentes en su control. PhD dissertation. UPC, Barcelona, 2001.

3) Justus. Dynamisches Verhalten Elektrischer Maschinen. Vieweg Publishing, Braunschweig/Wiesbaden, 1991.

4) B. Peterson. Induction machine speed estimation. Observation on observers. Doctoral Dissertation. Lund Technical University. 1996. Lund. Sweden.

5) O. Nelles. Nonlinear System Identification. Springer. Berlin. 2001.