# Analysis of Permanent Magnet Motors in High Frequency--A Review
## 3 Analysis Tools
### Analytic Methods
#### 3.2.2 Analysis of Coil Conductors  
At high frequencies, the windings are affected by skin and proximity effect as explained in Section 2.1. The first effect can be easily calculated with the well-known Equation (6), where \(L\) is the length of the winding, \(r\) is the radius of the conductor and \(\delta\) is the skin depth explained in (1).  
\[R_{skin}=\frac{\rho L}{A_{eff}}=\frac{\rho L}{\pi r^{2}-\pi(r-\delta)^{2}} \tag{6}\]  
For example, in [40] a single coil of electrical machine is analyzed and the model shows the same trend than the measurements. However, when a group of coils is analyzed, the resistance shoots up due to proximity effect that is not considered in the model, reducing the accuracy of the model.  
The proximity effect is not so easy to estimate analytically. The most used method to calculate proximity effect in the literature is [89] and it is applied in transformers. [PERSON] proposes a method to calculate the AC resistance of round conductor windings using the exact analytical equations for round conductors, obtaining more accurate results. Nevertheless, some assumptions are still present because they are 1-D flux density models, not considering the effect of leakage flux crossing the slot opening.  
In [50] a 2-D analytical model is presented to predict the proximity losses in p.m. machines stator slots, including the effects of slot opening. It is based on the solution of the Laplacian equation in the rectangular coordinate system of the slots in the machine, applying the boundary conditions of the magnetic field intensity. It makes the following assumptions:  
* Bundle-level proximity losses are neglected by transposing conductors.
* The slots have a rectangular shape.
* The conductors have a uniform distribution in the slot.  
The model is validated with FEM results for single- and double-layer windings, obtaining accurate results up to 1.5 kHz.  
In [38] Bessel functions are used to calculate the AC resistance of a shielded multi-conductor, obtaining good results in low frequency but increasing the error up to 12% in high frequency compared to FEM results. This method is also compared with FEM results in [39] for a machine winding obtaining the same error.  
In addition, the method proposed in [37] uses the Institute of Electrical and Electronics Engineers (IEEE) standard Induction Machine (IM) circuit [90] obtaining the parameters from the manufacturer data sheet. Then, some add-ons are proposed to account for the high-frequency effects. The inductance of the first turns of the winding is added, as it influences the high-frequency anti-resonance point. A stator turn-to-turn damping resistance is defined also to account for the skin and proximity effect of the wire. There is not perfect matching of DM impedance because the stator leakage inductance skin effect is not taken into account.  
Summarizing the relevant cases obtained with analytic methods in Table 3, it can be seen that the accuracy is lower than in FEM methods, and their frequency range is smaller. One method is only found to obtain proximity effect losses in electric machines directly [50], as the dumping resistance inserted in [37] accounts for skin and proximity effects in the winding and high-frequency iron losses in the machine.