## 3 Broadband Stator Model for Data Storage
### High-Frequency Modeling of PMSM
#### 3.2.2 Transfer Function Approach to PMSM Behavior  
In the interest of early diagnosis of interturn short-circuit faults in PMSM, [PERSON] proposes in [52] to develop a transfer function multi-model to accurately model the behavior of a machine stator winding over a wide frequency range from a few kilohertz to over ten megahertz. The idea is to determine the transfer function that matches the experimental behavior of the winding. The method is based on a graphical model that uses the experimental admittance plot (inverse of the impedance) to determine the linear, time-invariant transfer function characteristic of the machine behavior. The transfer function of such a system is described by the Equation (2) below:  
\[H(s)=\frac{a_{n}s^{n}+a_{n-1}s^{n-1}+\dots+a_{0}}{b_{m}s^{m}+b_{m-1}s^{m-1}+ \dots+b_{0}} \tag{2}\]  
where \(n\) and \(m\) are the order of the numerator and denominator polynomials, respectively, while the coefficients \(a_{i}\) and \(b_{j}\) that characterize the frequency response are calculated by solving an optimization problem.  
In the example developed in [52], the optimization problem is solved using a genetic algorithm. This type of algorithm uses the principle of natural selection to ensure that the initial population of the problem evolves towards the best possible solution after a given number of generations \(N_{g}\). To find an optimal solution, the order of the transfer function is increased if the transfer function of the best individual from the previous generation does not satisfy the system frequency response. The solution is adopted when this condition is satisfied. It should be noted that if the optimization problem is solved by a relatively high-order transfer function, it is possible to split the study into two different  
Figure 10: Comparison of experimental and modeled CM (**a**) and DM (**b**) impedance frequency behavior—taken from [45].  
frequency ranges in order to obtain simpler solutions that describe the desired behavior as accurately. The resulting multi-model transfer function can then be stored and reused in a diagnostic process.  
Another approach is to use vector fitting techniques to extract a generic transfer function for characterizing the CM impedance of an electrical machine [53]. This transfer function is defined by the relation (3).  
\[H(s)=A+B.s+\sum_{k=1}^{N}\frac{r_{k}}{s-p_{k}} \tag{3}\]  
where \(A\) denotes a resistive behavior, \(B\) refers to an inductance type of behavior, \(N\) is a natural integer that characterizes the order of the polynomial, and \(r\) and \(p\) are the various residues and poles of the transfer function, respectively.  
To go further, ref. [35] proposes to use this transfer function as a basis for an equivalent circuit model, where the number of parameters to be estimated depends on the order of the resulting transfer function. Two approaches are then considered to evaluate these parameters:  
* A constrained optimization, to minimize the error between the model and the experimental measurement of the machine CM impedance [35];
* An analytical calculation of the various parameters, taking into account the type and design of the winding as well as the material and method of impregnation of the conductors [24].  
The author argues that for the machine studied, a 4 th-order transfer function satisfies the conditions to accurately model the CM impedance behavior. The resulting model is compared in Figure 11 with the impedance frequency response measured on the machine.