## 3 Broadband Stator Model for Data Storage
### High-Frequency Modeling of PMSM
#### 3.2.1 PMSM HF Electrical Equivalent Circuit Model  
This HF approach of EEC models is similar to conventional low-frequency circuits, but the proposed EEC models are much more complex and take into account a number of phenomena that are considered parasitic and which are neglected in the standard approach. Thus, the complex association of electrical components (resistance, inductance,  
Figure 6: Example of Bode diagram depicting CM (**a**) and DM (**b**) impedance behaviors—taken from [25]. and capacitance) connected in series or in parallel means that the following phenomena can be taken into account:  
* Interturn capacitive couplings;
* Phase-to-ground capacitive couplings;
* Leakage inductances in the first turns of the windings;
* Magnetizing inductances;
* Iron and eddy current losses. The model complexity is determined by the purpose of the modeling and the accuracy required. The first approaches identified in the literature are based on \(\pi\) modeling of each machine phase, similar to what is used to describe the behavior of transmission lines [49]. For example, we can identify the HF EEC models developed in [34; 43; 50] and presented in Figure 7.  
Figure 7(a) compares the impedance frequency response estimated using the model presented in Figure 6(a) with its experimental profile for the three parallelized phases of the 750 W induction machine studied in [50]. Figure 7(b) compares the experimental and estimated frequency response characteristics, using the model described in Figure 6(b) of the CM (curve with the lowest mean impedance over the considered frequency range) and DM (curve with the highest mean impedance over the considered frequency range) impedances, of the 15 kW induction machine tested in [34]. Finally, Figure 7(c) compares the experimental and estimated CM (left curve) and DM (right curve) impedance profiles calculated using the model shown in Figure 6(c) for the 2.2 kW electric machine studied in [43]. It should be noted that the accuracy of the various models presented remains rather limited and does not perfectly characterize the impedant frequency behavior of these machines over the entire frequency range considered. Building on this postulate, more recent work has sought to construct more complex HF EEC models capable of taking into account more HF phenomena. This is particularly the case in [44; 51], whose models have been improved in [45]. Figure 9 shows the complete HF EEC model developed in [45] to characterize the impedant frequency behavior of an 8 kW, star-connected PMSM. Note the following in this model:  
* \(R_{g}\) is the resistance of each phase. * \(C_{g_{i}}\) represents the parasitic capacitances between the windings and the stator frame. * \"EMF\" blocks describe the electromotive forces generated in each of the machine phases during operation. * \"Part 0\" blocks characterize the CM and the DM behavior around resonance and anti-resonance frequencies at very high frequencies (between one and a few tens of megahertz). * \"Part 1\" blocks characterize the CM and the DM behavior around resonance and anti-resonance frequencies at high frequency (between 10 kHz and 1 MHz). Figure 7: Examples of \(\pi\)-based HF EEC models of machine phases—taken from [50] (**a**), [34] (**b**), and [43] (**c**). * \"Part 2\" blocks model magnetizing inducances and eddy-current losses in the ferromagnetic core. * \"Part 3\" blocks characterize the DM impedant behavior at high frequencies (between 10 kHz and 1 MHz). * \"Part 4\" blocks model the effects of leakage inductances in the first turns of coils. Figure 10 compares the experimental and estimated frequency response of the CM (a) and DM (b) impedances of the machine studied in [45], using the model shown in Figure 9.  
Figure 8: Experimental and modeled impedance frequency response of three parallel phases in [50] (a), of CM and DM in [34] (b), and of CM and DM in [43] (c). Figure 9: Example of a complete HF EEC model of a star-connected PMSM—taken from [45].