To increase the efficency of electric drives, the working frequencies are getting higher, bringing serious Electromagnetic Compatibility (EMC) problems, as well as insulation stress and higher bearing currents. Hence, it is important to have an electrical machine electrical equivalent circuit model to predict the electromagnetic interference levels. This review summarizes the current state of the art in electrical machine modeling and analysis in high frequency. The main analysis tools as Finite Element Methods, analytic and measurementbased tools are compared. Then, different machine highfrequency models are reported, detailing their individual features. Additionally, the influence of the machine design parameters in EMC behavior is outlined for future analysis. This is a short version of the full article.
Typically, an electrical drive consists of a power source, an inverter that applies the desired voltage and frequency to the electrical machine, usually with Pulse Width Modulation (PWM) technique and the electrical motor, as shown in Figure 1. The straight arrows indicate the radiated noise caused by the switching circuits of the converter.
The dashed line represents the differentialmode EMI, which is going through one phase and returns across the other phases to the DC bus of the power source. Finally, the dotted line is the commonmode EMI, which goes across the common ground, connecting all elements to the disturbance [1].
Even if PWM technique is the most used control technique, its pulses generate a Commonmode voltage (CMV) in the output connection to the motor causing diverse problems such as leakage current, shaft voltage and bearing currents. The rapid voltage changes (high du/dt) also deteriorate the winding insulation and can cause protection failures in the case of shortcircuit or contact defects ^{[2][3][4]}[2,3,4]. Moreover, the generated EMI can affect the power source grid and the elements connected to it, for instance, sensors and safety systems in electric cars [5].
Specifically, bearings are affected by the CMV variations, as leakage currents flow through them ^{[6][7]}[6,7]. Those currents damage the bearings, and together with the winding insulation damage due to voltage stress, the reliability and lifetime of the complete electric drive are decreased.
Some solutions are proposed to mitigate this EMI in electric drives. Different inverter typologies and modulation algorithms are proposed to reduce CM voltage ^{[1][6][8]}[1,6,8]. Filters and other types of addons, such as shields or insulation are also widely used to avoid EMI flowing to the power grid as shown in ^{[6][9][10][11]}[6,9,10,11], but they increase the cost of the drive, as well as the volume and the weight as they are bulky and heavy. Hence, those factors can reduce considerably the competitiveness of the final product on the market, for example, in the electric car.
Thus, to reduce the development cost of the product, an EMI strategybased design should be added early in the product development cycle to reduce costs and achieve the best solution. Consequently, understanding and predicting EMI noise using highfrequency models during the design stage is essential to manage EMI problems.
According to the literature, the most extended modeling techniques for representing the behavior of electrical machines in highfrequency are based on electrical equivalent circuits or Lumped Parameter Models (LPM). These models consist of electrical circuits that comprise several resistances, inductances, and capacitances. For example, to analyze the influence of the winding placement and the winding connection on the bearing currents ^{[12][13]}[32,57], to analyze the commonmode currents ^{[14]}[73] or to analyze the overvoltages at motor terminals due to modulated supplying voltages ^{[15]}[61]. Therefore, it is identified a lack of a comprehensive review about the analysis tools and methodologies for the study of electrical motors in high frequency, identifying the advantages and drawbacks of the main proposals that can be found in the literature.
As stated before, the final objective is an EMI reduction strategybased design for electrical machines. To reach that objective, in this paper a comprehensive review of highfrequency behavior of electrical motors is proposed, covering the highfrequency phenomena, the tools available to analyze them accurately and the different existing highfrequency models. Thus, a review of the influence of different design parameters on the highfrequency behavior of the electrical machine is also collected in this paper. It is also found that the most influential design parameter is the winding placement and the impregnation amount, at least for CM currents.
In lowfrequency operation range the parameters such as resistance and inductance hardly depend on the frequency, so they can be considered to be constants. However, in high frequencies, some new phenomena arise leading to variations in these parameters. Thus, as the frequency increases, the resistance increases and the inductance decreases.
Before analyzing any electromagnetic device in highfrequency, a specific frequency range must be defined depending on the application standards, as shown in this International Special Committee on Radio Interference (CISPR) guideline ^{[16]}[74]. EMI conducted standards cover the frequency range from 150 kHz to 30 MHz, whereas the radiated standards cover the frequency range of 30 MHz to 100 MHz. However, in most cases, it is difficult to obtain models to achieve good accuracy in the whole range of frequencies, especially beyond 10 MHz ^{[17][18][19][20][21]}[17,20,28,39,51]. Moreover, in ^{[22]}[75], it is stated that the model of a transformer is not accurate beyond 10 MHz due to the current coupling between Differentialmode (DM) and Commonmode (CM) paths.
There are 4 main phenomena that must betaken into account when modelling electric machines in high frequency:
This surface distribution of the current increases the resistance and decreases the inductance due to the flux distribution inside the conductor. The skin effect is a local effect in each conductor, independent from the neighboring ones, so it can be analyzed simplifying the problem to a single conductor ^{[23][24]}[76,77]. As it depends on the frequency and geometry, it can be mitigated using a smaller conductor diameter than the skin depth.
In the case of the laminated iron core, the inductance decreases less than in the bulk core since the magnetic flux is not completely pushed out of the laminated core. On the other hand, the resistance increases more in the laminated core, as the surface area where Eddy current flows is larger and the proximity effect between sheets increases the eddy currents in the core. It should be noted that the nonlinearity of the magnetic saturation of iron core is neglected for those simulations, as well as the displacement current term, which is negligible at least below 1 GHz ^{[30]}[55].
The main parasitic capacitances involved in an electrical motor are shown in Figure 4. Cwr is the windingtorotor capacitance, Cb is the bearing capacitance,Cws is the windingtostator capacitance, Csr is the statortorotor capacitance,Cwh is the windingtohousing capacitance and Cwsh is the windingtoshaft capacitance.
In the winding, different capacitances appear. The interturn capacitance and the capacitance between phases that can be neglected if there is a single layer winding in the slot ^{[27]}[58]. The complete commonmode (CM) paths are identified in ^{[14]}[73]. To calculate the interturn capacitances, it is essential to consider the geometry, material and positions of the conductors inside the slot, as well as the thickness and the material of the insulation ^{[32][33]}[43,54].
There are two different ways to analyze an electrical machine in high frequency. The first way is to determine the value of the Lumped Parameter Model (LPM) parameters of the motor by experimental measurements. This method is fairly the most used in the literature [12–36]. It is a practical method that achieves accurate EMC simulation of motor drives. Nevertheless, the manufactured motor is necessary for the measurements, so it is not valid for the design stage before prototyping.
In the second way, an equivalent circuit model or LPM is made so that each circuit constant is calculated with the design parameters of the motor either analytically [11,37–50], by Finite Element Method (FEM) [28,39,51–70] or even with hybrid methods [71–73]. With FEMbased electromagnetic field analysis, the inductance, resistance and capacitance of each turn of winding can be obtained, which are difficult to measure experimentally [55].
Regarding the FEM analysis, the detailed geometry of an electric machine can be accurately modeled obtaining precise resistance and inductance values, taking into account the skin and proximity effects, as well as eddy current losses in the stator and the capacitive couplings. The mesh of the geometry is essential to obtain accurate results in FEM. When working in high frequency, the skin depth of the materials must be finely meshed, both in the conductors to consider skin and proximity effects, and in the core, as the Eddy currents flow along the skin depth of the iron sheets. Thus, the mesh size must be thinner than the skin depth. However, this meshing requirement leads to a higher time consumption and computational load. Thus, an equilibrium must be found between the model complexity and the accuracy. For the active length of the machine, 2D simulations are usually used, even if the resistivity of the sheets must be calibrated to take into account the lamination effects, usually using 3D models. It must be remarked that not all iron sheets comprising the magnetic core can be modeled in 3D with a mesh size thinner than the skin depth, otherwise, the model will take ages to solve. Commonly, 2 or 3 sheets are simulated and the results are extrapolated to the whole length. Even if the 3D FEM is the most complete simulation tool, it is the most computationally demanding too, so it is usually only used for the endwinding calculation, as it is the only way to calculate it accurately. Once a 3D FEM simulation is done, some authors propose different coefficients to take into account this end winding, by using just 2D FEM for successive simulations. With respect to the capacitance calculation in electrostatic FEM, the same applied to magnetic calculation is applied, needing 3D models for the end winding, at least for the winding torotor capacitance calculation, where it has a big impact. In this case, a 2 mesh layer must be defined between the conductors for turntoturn capacitance accuracy. Gauss Law method is recommended for the calculation of the capacitance matrix, as it requires fewer simulations.
In the case of analytical tools, commonly they require less computation time, as some assumptions and simplifications are considered for each specific case. The skin effect can be easily calculated analytically reducing the effective crosssection area of the conductors. Nonetheless, the computation of the proximity effect is not so simple due to the non uniform distribution of the magnetic field in the slot. Bessel functions have been used to calculate the AC resistance of the winding, leading to 12% errors at 50 MHz. An analytical method is found to calculate the Joule losses taking into account the proximity losses, but an uniform conductor distribution is assumed in a rectangular slot, so it may not be applicable to all cases, and it is only tested until 1.5 kHz so it may need further development for using it in EMC analysis of electric machines. With respect to the capacitance values, geometric simplifications are done to simplify the calculations to plate capacitors in the case of the windingtorotor capacitance or cylindrical ones for the statortorotor capacitance, and even if they converge to the same order of magnitude, they are not accurate enough. A novel approach is proposed for the determination of the windingtorotor capacitance using the method of image charges, validating the results with FEM. In the turntoturn capacitance, some methods can reach acceptable results. For the turntoturn capacitance calculations, the method must be chosen depending on the specific disposition of the winding, otherwise, large errors can appear.
Concerning measurementbased tools, rather good accuracy can be reached by adjusting the model behavior in the whole frequency range. Nevertheless, it is important to underline that this approach is not suitable for the design stage of the motor, as the prototype must be already built to perform the experimental impedance measurements, and then obtain the parameter values by curve fitting. For example, this approach should be suitable for analyzing the behavior of the overall electric drive in simulation, considering the highfrequency models of the motor, the inverter and the EMC filter, but not for predicting the behavior of the electrical machine during the design process. Inside the measuredbased tools, there are two different ways of obtaining the electrical circuit model, by looking for the physical meaning of each parameter and relating it to the impedance curve, or just by parameter fitting procedures, obtaining even negative values. This model can obtain excellent accuracy in the whole frequency range.
In the previous section, three different analysis tools are described for highfrequency electrical machine analysis. From that analysis, some electrical parameters are obtained (L, R, C), to introduce them in different models. The main models are classified based on their topology, parameter extraction methods and their main characteristics.
Concerning their complexity or size, the models can be classified into two categories. Distributed Parameter Models (DPM) and Lumped Parameter Models (LPM). DPM tend to be more accurate, but they may not be integrated with other system components, as they need intensive computation ^{[20][21][25][34][27][15][35][36]}[39,51,52,56,58,61,62,86].
By contrast, LPM is more practical, as it can be introduced in a complete electrical drive model, and its parameters can be obtained from simple impedance measurements ^{[37][17][18][38][39][40][29][41][42][43][12][44][45][33][30][14]}[13,17,20,21,22,23,24,26,27,31,32,35,37,54,55,73]. Some authors develop high accuracy DPM models and simplify it to LPM with matrix reduction methods ^{[25]}[52] or by grouping the RL parameters ^{[27]}[58]. A review of the different models is made in Table 1.
Model 
Frequency Range 
Model Type 
Parameter Extraction 
Interturn Effects 
Bearing Model 
Rotor Model 
Integration in Drive 
Simulation Domain 
Iron Loss 
[32] 
1 k–13 M 

Measured 
x 
x 

x 
Freq & Time 
Implicit 
[17] 
10 k–10 M 

Measured 
x 
x 

x 
Freq & Time 
Implicit 
[20] 
10 k–10 M 

Measured 



x 
Freq & Time 
R Parallel 
[23] 
10 k–10 M 

Measured 
x 



Frequency 
Implicit 
[31] 
100 k–500 M 

Measured 




Frequency 
Implicit 
[12] 
10 k–10 M 

Measured 
x 

x 
x 
Freq & Time 
Implicit 
[13] 
100–100 M 

Measured 
x 



Freq & Time 
R 
[21] 
150 k–10 M 
LPM 
Measured 
x 

x 
x 
Freq & Time 
R 
[33] 
100–30 M 
Fixed 
Measured 
x 

x 
x 
Freq & Time 
R 
[24] 
100–10 M 
Segments 
Measured 
x 



Frequency 
RLRC 
[15] 
100 k–100 M 

Measured 




Freq & Time 
Implicit 
[35] 
10 k–1 M 

Measured 
x 



Freq & Time 
Implicit 
[36] 
100–10 M 

Measured 
x 



Freq & Time 
R Parallel 
[37] 
10–10 M 

Analytic 
x 
x 
x 
x 
Freq & Time 
R Parallel 
[73] 
30–5M 

FEM 

x 
x 

Time 
RL 
[54] 
100–100 M 

FEM 
x 
x 
x 
x 
Freq & Time 
Implicit 
[27] 
1 k–10 M 
f(F) 
Measured 



x 
Freq & Time 
Implicit 
[22] 
10 k–3 M 
Segments 
Measured 

x 


Freq & Time 
Implicit 
[62] 
20–4 M 

FEM 
x 



Frequency 
Implicit 
[86] 
10 k 

FEM 
x 

x 
x 
Time 
Implicit 
[58] 
100–1 M 

FEM 
x 

x 
x 
Frequency 
R Parallel 
[61] 
0–100 k 
DPM 
FEM 
x 

x 

Time 
Implicit 
[39] 
10–10 M 

FEM 
x 



Freq & Time 
Implicit 
[51] 
1 k–10 M 

FEM 
x 



Freq & Time 
Implicit 
[55] 
10 k–20 M 

FEM 
x 



Frequency 
Implicit 
[70] 
10 k–1 M 

FEM 
x 



Frequency 
Implicit 
With respect to the simulation domain, some models are working in the frequency domain, for example, to obtain the CM and DM impedance versus frequency. However, to simulate the overvoltages and currents, the time domain is needed. In this domain, the frequency dependency of the parameters is usually considered using lumped parameter circuits, as varying the value for each frequency may not be practical. Typically, parallel RL branches are used to reproduce skin and proximity effects in the resistance and inductance values, where each branch represents a frequency range ^{[39][41][19]}[22,26,28].
Finally, depending on the objective of the simulation, the~developed model may highlight or neglect some parts of the machine. On~the basis, all models are RLC circuits, with~different number of segments and different physical meanings, but~in origin, all refer to winding self and mutual inductances, resistance and parasitic coupling capacitances. The~interturn effects in the winding are essential to obtain accurate results, whereas the bearing capacitance is just included when the bearing currents are analyzed. The~iron losses produced by Eddy currents in the stator sometimes are included as a resistor in parallel with the winding, whereas other times the losses are just implicit in the circuit~values.
Concerning the rotor, its influence may be only significant when analyzing bearing currents, shaft voltages, or~terminal overvoltage. The~rotor position is also important in the lowfrequency range for salient pole permanent magnet machines as the inductance changes with the rotor position. Hence, if~a full frequency range (0 Hz30 MHz) model is required, the~rotor should be~considered.
Once the machine EMI behavior is analyzed by developing different models shown in the previous section, the influence of different aspects must be evaluated using those models. Different factors may affect the EMC behavior, such as design parameters, manufacturing materials, and fabrication processes and tolerances. To go to the detail of the design parameters and tolerances, normally FEM analysis is used as the main option. the following table is a review of the different design parameters.
Parameter 
Impact 
Optimum 
Parallel Circuits 
First resonance frequency 
\( f_{rSeries}=\frac{1}{2}f_{rParallel} \) 
Y∆ Connection 
First resonance frequency in CM Impedance DM Impedance Amplitude 
(∆ higher than Y) (Y higher than ∆)

Conductor placement 
Cwr & insulation stress Bearing & CM Currents Shaft Voltage Copper Losses 
Furthest from rotor Middle of the slot Strands aligned with flux lines

Winding Topology 
Windingtoground capacitance 
Circumferential winding 
Impregnation level of conductors 
Stray Capacitance 
Low (Affects thermal) 
Slot Shape 
C_{wr}, Shaft voltage 
 
Electrodes in slot wedge 
C_{wr} & insulation stress Bearing & CM Currents Shaft Voltages 
High diameter Nearest from rotor >1 electrodes together 
Airgap size 
C_{rs} 
Minimum 
Shielding 
Induced voltage 
Depends on frequency 
Endwinding shielding 
C_{wr} 
Faraday’s cage 
To get a more in depth knowledge on the field, read the fulll article.