5. Stator Winding Resistance Analyzer

This analyzer calculates stator winding resistance and coil length, considering effects of end windings.

5.1. Model Background

This analyzer implements the coil length and resistance calculation approach provided in the following reference:

1. Bianchi, S. Bolognani, and P. Frare, “Design Criteria for High-Efficiency SPM Synchronous Motors,” in IEEE Transactions on Energy Conversion, June 2006.

The calculation approach is summarized below. The following equation is used to calculate stator winding resistance:

\[\begin{split}R_\text{wdg} &= \frac{z_Q z_C l_\text{coil}}{\sigma_\text{cond} A_\text{cond}}\\\end{split}\]

where \(z_Q\) is the number of turns per coil, \(z_C\) is the number of series coils per phase, \(l_\text{coil}\) is the average length of a single coil loop, \(\sigma_\text{cond}\) is the conductor conductivity, and \(A_\text{cond}\) is the cross-section area.

Conductor area is found as:

\[A_\text{cond} = \frac{K_\text{Cu}A_\text{slot}}{n_\text{layers}z_Q}\]

where \(A_\text{slot}\) is the stator slot area, \(K_\text{Cu}\) is the slot fill factor, and \(n_\text{layers}\) is the number of layers (\(n_\text{layers}=1\) for a single-layer winding and \(n_\text{layers} = 2\) for a double-layer winding).

To calculate \(l_\text{coil}\), the following coil shape is assumed:

This is the view when one side of the stator is cut along the axial direction and unrolled. Using this diagram, \(l_\text{coil}\) is calculated as

\[\begin{split}l_\text{coil} &= 2(l_\text{st} + l_\text{end wdg})\\\end{split}\]

where \(l_\text{st}\) is the length of a coil along the axial length of a motor and \(l_\text{end wdg} = l_1 + 2l_2\) is the end winding length.

The segments with length \(l_1\) represent the parts of the end winding that appear only in distributed windings, which are calculated as

\[\begin{split}l_1 &= \tau_u K_\text{ov} (y-1)\\\end{split}\]

where \(y\) is a coil pitch represented in a number of slots and \(K_\text{ov}\) is a coil overlength factor. For a concentrated winding, \(y = 1\), resulting in \(l_1 = 0\).

The segments with length \(l_2\) represent the bent parts of a coil and are calculated as 1/4th of the circle circumference. The diameter of this circle is estimated as the average of the stator tooth width \(w_\text{st}\) and the length between adjacent slots \(\tau_u\).

\[\begin{split}l_2 &= \frac{1}{4} \pi \frac{\tau_u + w_\text{st}}{2}\\\end{split}\]

Here, \(\tau_u\) is calculated as the length of an arc (when a stator is in its normal cylindrical shape) at the median depth of a slot:

\[\begin{split}\tau_u &= \frac{2 \pi}{Q} (r_\text{si} + d_\text{sp} + \frac{d_\text{st}}{2})\\\end{split}\]

where \(Q\) is the number of stator slots, \(r_\text{si}\) is the stator inner radius, \(d_\text{sp}\) is the stator pole thickness, and \(d_\text{st}\) is the stator tooth depth (see here or the figure below).

5.2. Input from User

The user is required to provide the following parameters:

Input to stator winding resistance problem
Arguments	Description	Units
r_si	Stator inner radius	m
d_sp	Stator pole thickness	m
d_st	Stator tooth depth	m
w_st	Stator tooth width	m
l_st	Stack length	m
Q	Number of slots
y	Slot pitch (in number of slots)
z_Q	Number of turns per coil
z_C	Number of series coils per phase
Kcu	Slot fill factor
Kov	Winding overlength factor
sigma_cond	Conductor conductivity	S/m
slot_area	Stator slot area	m^2
n_layers	Number of layers (single or double)

For the definition of dimensions, please refer to the figure below:

5.3. Output to User

The stator winding resistance analyzer returns a dictionary that has the following parameters:

Output of stator winding resistance analyzer
Arguments	Description	Units
l_coil	Length of a single loop of a coil	meters
l_ew	Length of an end winding	meters
R_coil	Resistance of a coil	Ohms
R_ew	Resistance of an end winding	Ohms
R_wdg	Resistance of a phase winding	Ohms

Here, the total phase winding resistance R_wdg is the product of R_coil and the number of coils per phase z_C.

Example code using the stator winding resistance analyzer is provided below:

import numpy as np
from eMach.mach_eval.analyzers.electromagnetic.stator_wdg_res import (
    StatorWindingResistanceProblem,
    StatorWindingResistanceAnalyzer
    )

# define problem and analyzer
res_prob = StatorWindingResistanceProblem(
    r_si=34.45/1000,
    d_sp=3.95/1000,
    d_st=20.75/1000,
    w_st=5.38/1000,
    l_st=50/1000,
    Q=24,
    y=9,
    z_Q=16,
    z_C=4,
    Kcu=0.5,
    Kov=1.8,
    sigma_cond=5.7773*1e7,
    slot_area=251*1e-6,
    n_layers=2,
    )
res_analyzer = StatorWindingResistanceAnalyzer()

# analyze the problem
results = res_analyzer.analyze(res_prob)

The output of the code is the dictionary with the following key-value pairs:

Results of stator winding resistance analyzer
Arguments	Value	Units
l_coil	0.496	meters
l_ew	0.198	meters
R_coil	0.035	Ohms
R_ew	0.014	Ohms
R_wdg	0.14	Ohms