I present a complete electrical and mechanical design study of a 60 kW, 800 V electrically excited synchronous motor intended for modern electric cars. The motivation comes from the continuing growth of the electric-vehicle market and the need to reduce dependence on rare-earth permanent magnets. In this article, I report my design procedure, finite-element validation, multi-objective optimization, and the electromagnetic vibration and acoustic analysis. The machine described here is an 8-pole, 48-slot, three-phase electrically excited synchronous motor with a rated speed of 6000 r/min and a maximum speed of 13000 r/min. The rated torque is 95.5 N·m, while the peak torque is 200 N·m, making the motor suitable for a compact electric car.
The role of the traction motor in an electric car is not limited to providing steady-state torque. The motor also defines the acceleration feel, hill-climbing capability, continuous cruising performance, and the acoustic comfort experienced by occupants. Permanent-magnet synchronous motors dominate current electric cars because of their high efficiency and high torque density. However, the price volatility of rare-earth materials and the environmental concerns associated with magnet production motivate the investigation of magnet-free traction motors. An electrically excited synchronous motor is attractive because the rotor field is produced by a copper winding fed with direct current. It avoids demagnetization risk, permits flexible field weakening, and can offer a more secure supply chain for the automotive industry.
1. Requirements of an Electric Car and Motor Matching
I first studied the vehicle dynamics of a representative electric car. The complete vehicle was assumed to have a curb mass of 1500 kg and a loaded mass of 1700 kg. The wheel rolling radius is 0.25 m, the frontal area is 2.38 m2, and the drag coefficient is 0.4. I used a rolling-resistance coefficient of 0.015, a driveline efficiency of 0.97, and a fixed gear ratio of 8. The required maximum speed of the electric car is 150 km/h. The acceleration requirement is from rest to 100 km/h in no more than 8 s, and the maximum climbing angle is 20° at a stable vehicle speed of at least 30 km/h. In addition, the motor should sustain the cruising speed of 70 km/h under rated operating conditions.
The force balance for a moving electric car can be expressed by the sum of rolling resistance, aerodynamic drag, grade resistance, and acceleration resistance. I write the total traction force \(F_t\) as
\[
F_t = m_0 g f_r \cos\alpha + \frac{C_D A v^2}{21.15} + m_0 g \sin\alpha + \delta m_0 \frac{\mathrm{d}v}{\mathrm{d}t},
\tag{1}
\]
where \(m_0\) is the loaded vehicle mass, \(g\) is the gravitational acceleration, \(f_r\) is the rolling-resistance coefficient, \(\alpha\) is the road angle, \(C_D\) is the aerodynamic drag coefficient, \(A\) is the frontal area, \(v\) is the vehicle speed in km/h, and \(\delta\) is the rotational-mass conversion factor. The factor 21.15 is appropriate when the speed is expressed in km/h. Multiplying the force balance by vehicle speed and including the driveline efficiency gives the traction power.
\[
P = \frac{v}{3600\eta_t}
\Bigl[
m_0 g f_r \cos\alpha
+ \frac{C_D A v^2}{21.15}
+ m_0 g \sin\alpha
+ \delta m_0 \frac{\mathrm{d}v}{\mathrm{d}t}
\Bigr].
\tag{2}
\]
Using the maximum-speed condition, the steady-grade condition, and the acceleration condition separately, I identified the three main power requirements for the electric car. The maximum-speed power is the largest when the vehicle is driven at 150 km/h on level ground. The grade-climbing power is evaluated for a continuous crawl at 30 km/h on a 20° grade. The acceleration power is evaluated as an averaged power over the 0–100 km/h interval. Table 1 summarizes the main parameters of the reference electric car.
| Vehicle parameter | Value |
| Vehicle mass | 1500 kg |
| Loaded vehicle mass | 1700 kg |
| Wheel rolling radius | 0.25 m |
| Frontal area | 2.38 m² |
| Aerodynamic drag coefficient | 0.4 |
| Rolling resistance coefficient | 0.015 |
| Rotational mass factor | 1.02 |
| Maximum speed | 150 km/h |
| 0–100 km/h acceleration time | 8 s |
| Maximum grade | 20° |
| Stable speed on maximum grade | 30 km/h |
| Continuous cruise speed | 70 km/h |
These calculations produced an interesting set of motor specifications. My peak-power calculation gave values close to 54 kW for maximum speed, 51 kW for grade climbing, and about 120 kW for acceleration. I therefore selected a peak output power of 130 kW to provide a margin over the most demanding acceleration condition. The rated power was set to 60 kW, corresponding to continuous operation of the electric car at the cruising condition. The rated motor speed was selected from the 70 km/h cruise speed and the driveline gear ratio.
\[
n = \frac{v i_g}{0.377 r}
\tag{3}
\]
where \(i_g\) is the total gear ratio and \(r\) is the wheel rolling radius. This gives approximately 5942 r/min at 70 km/h, so I set the rated speed to 6000 r/min. The maximum speed of the motor was chosen as 13000 r/min to meet high-speed cruising requirements. The rated torque is obtained from the power-speed relationship as
\[
T_e = 9550 \frac{P_e}{n_e}.
\tag{4}
\]
The peak torque is determined by the maximum grade demand:
\[
T_{\max} \geq \frac{F_t r}{i_g \eta_t}.
\tag{5}
\]
The resulting peak torque is not lower than 192.4 N·m, so I selected a nominal peak torque of 200 N·m. Because of the charging and on-board system advantages of high-voltage architectures, I chose an 800 V direct-current bus for the electric car. With the same power, an 800 V bus reduces the motor current compared with a 400 V bus, thus reducing copper losses and allowing faster charging.
| Motor specification | Value |
| Motor type | Electrically excited synchronous motor |
| Rated power | 60 kW |
| Rated voltage | 800 V |
| Rated speed | 6000 r/min |
| Maximum speed | 13000 r/min |
| Rated torque | 95.5 N·m |
| Peak torque | 200 N·m |
| DC-bus voltage | 800 V |
2. Main Dimensions and Structural Selection
Once the global performance requirements were known, I proceeded with the electromagnetic sizing of the motor. The classical motor-size equation relates the stator bore volume to the calculated electromagnetic power, magnetic loading, electric loading, and speed. I expressed this relation in the form
\[
D_1^2 L_{\mathrm{ef}} = \frac{6.1 P’}{\alpha_p K_{\mathrm{Nm}} K_{\mathrm{dp}} A B_\delta n},
\tag{6}
\]
where \(D_1\) is the stator bore diameter, \(L_{\mathrm{ef}}\) is the effective iron length, \(\alpha_p\) is the calculated pole-arc coefficient, \(K_{\mathrm{Nm}}\) is the air-gap field-waveform coefficient, \(K_{\mathrm{dp}}\) is the armature winding factor, \(A\) is the electric loading, \(B_\delta\) is the air-gap flux density, and \(P’\) is the electromagnetic power. For an electric car traction motor, the acceleration requirement demands a relatively short magnetic path with a low axial length to minimize the volume of the drive unit.
I selected a stator bore diameter of 175 mm and an effective axial length of 179.9 mm. The stator outer diameter is 255 mm and the rotor outer diameter is 172.6 mm. The rotor inner diameter is 50 mm. The rated air-gap length begins at 1.2 mm. The rotor is built with salient poles, and the pole faces are eccentric to shape the air-gap field toward a sinusoidal distribution. The rotor field winding is wound around the pole bodies and is supplied by direct current through a brushless excitation system in the final vehicle integration. Table 3 shows the main structural dimensions of the machine.
| Motor dimension | Value |
| Stator outer diameter | 255 mm |
| Stator bore diameter | 175 mm |
| Rotor outer diameter | 172.6 mm |
| Rotor inner diameter | 50 mm |
| Effective axial length | 179.9 mm |
| Number of poles | 8 |
| Number of stator slots | 48 |
| Initial air-gap length | 1.2 mm |
| Initial stator tooth width | 6.5 mm |
| Initial rotor pole width | 33 mm |
| Initial rotor pole-shoe width | 10 mm |
The choice of an 8-pole, 48-slot combination is important for balanced three-phase winding operation and for reducing the amplitude of low-order electromagnetic force waves. I used a double-layer distributed winding with two parallel branches. The three-phase stator winding is star-connected. For the electromagnetic steel, I chose DW310-35 non-oriented silicon steel because it offers low specific iron loss under the switching frequencies encountered in electric-car traction inverters.
3. Finite-Element Electromagnetic Performance
Before optimizing the motor, I built a two-dimensional finite-element model of one eighth of the machine. The modeled sector includes one pole pair with six stator slots. This sector model preserves all spatial harmonics that can affect the motor because the field solution is periodic after one pole pitch. The mesh in the air-gap region and in the stator teeth is refined to capture the saturation and slotting effects accurately.
I first simulated the no-load condition. The armature current was set to zero, while the rotor field winding was supplied with the constant direct current required to produce the rated air-gap field. The resulting magnetic field lines pass from the rotor poles through the air gap into the stator teeth and stator yoke. The maximum flux density appears in the rotor pole body and reaches about 1.7 T. The stator yoke flux density remains below about 1.5 T, which is acceptable for the selected electrical steel. The no-load air-gap flux density has a maximum value close to 0.96 T, and its waveform is closer to a sine wave than that of a conventional uniform-air-gap salient machine because of the eccentric pole-face shaping.
The no-load back-electromotive-force waveform was calculated at 6000 r/min. The three phase back-electromotive-force waveforms are balanced and almost sinusoidal. The peak no-load line-to-neutral back-electromotive force is around 393 V. Some fifth- and eleventh-order harmonics remain because of slotting and local saturation, but they are manageable from the viewpoint of inverter current control.
The initial design was then tested at rated load. I applied three balanced sinusoidal currents to the armature winding, with the phase angle aligned to produce maximum torque per ampere. A constant current is applied to the rotor field winding. The instantaneous torque waveform reaches a steady value after an initial transient. The average torque is 95.99 N·m, which slightly exceeds the required rated value of 95.5 N·m. The torque ripple is defined as
\[
V_T = \frac{T_{\max}-T_{\min}}{T_{\max}+T_{\min}}\times 100\%.
\tag{7}
\]
For the initial geometry, the torque ripple is 6.59%. The cogging torque, calculated by evaluating the stored magnetic energy at different rotor positions while the armature is unexcited, has a peak value of 3.43 N·m. This value is close to 3.6% of the rated torque, which is too high for smooth operation of an electric car. Therefore, optimization is necessary.
I also evaluated the loss balance at rated load. The stator iron loss is about 1.1 kW. The armature copper loss is about 0.25 kW because the 800 V bus reduces the rated phase current. The rotor field copper loss is approximately 1.55 kW, owing to the relatively high direct current and the smaller conductor cross-section available in the rotor. The total loss at the rated operating point is approximately 2.9 kW, giving an estimated efficiency of 95.4% under the assumed ideal sinusoidal current excitation. The initial performance is summarized in Table 4.
| Electromagnetic performance | Initial value |
| Average torque at rated load | 95.99 N·m |
| Torque ripple at rated load | 6.59% |
| Cogging-torque peak | 3.43 N·m |
| Stator iron loss | 1.1 kW |
| Armature copper loss | 0.25 kW |
| Field copper loss | 1.55 kW |
| Estimated efficiency | 95.4% |
4. Taguchi Sensitivity Analysis and Multi-Objective Optimization
Because the electromagnetic performance is linked to many geometric parameters, I first applied the Taguchi method to identify the most influential design parameters. The Taguchi method is based on orthogonal arrays. It reduces the number of finite-element simulations compared with full factorial scanning. For five variables with four levels, full factorial scanning would require 45 = 1024 simulations. The Taguchi L16 orthogonal array, on the other hand, requires only 16 carefully chosen combinations.
I selected the air-gap length \(\delta\), the rotor pole-shoe height \(B_{m2}\), the stator tooth width \(B_t\), the rotor pole width \(B_m\), and the rotor pole-shoe width \(B_{m1}\) as Taguchi factors. These parameters directly influence the reluctance of the magnetic path, the flux concentration, and the interaction between the stator teeth and rotor poles. Table 5 lists the symbols and levels used in the Taguchi investigation.
| Factor | Description | Range (mm) | Level 1 | Level 2 | Level 3 | Level 4 |
| \(\delta\) | Air-gap length | 1.00–1.21 | 1.00 | 1.07 | 1.14 | 1.21 |
| \(B_{m2}\) | Rotor pole-shoe height | 8.8–10.0 | 8.8 | 9.2 | 9.6 | 10.0 |
| \(B_t\) | Stator tooth width | 6.2–7.1 | 6.2 | 6.5 | 6.8 | 7.1 |
| \(B_m\) | Rotor pole width | 31.5–34.5 | 31.5 | 32.5 | 33.5 | 34.5 |
| \(B_{m1}\) | Rotor pole-shoe width | 9.5–11.0 | 9.5 | 10.0 | 10.5 | 11.0 |
For each simulation I recorded the average torque \(T_{\mathrm{avg}}\), the torque ripple \(V_T\), and the cogging-torque peak \(T_{\mathrm{cog}}\). I then performed a mean-value analysis. For each factor and each level, I calculated the average response over the four experiments containing that level. The variance of a factor \(A\) is estimated as
\[
S_A = \frac{1}{4}\sum_{j=1}^{4}\bigl(\bar{y}_{A,j}-\bar{y}_{\mathrm{all}}\bigr)^2,
\tag{8}
\]
where \(\bar{y}_{A,j}\) is the average output when factor \(A\) is at level \(j\), and \(\bar{y}_{\mathrm{all}}\) is the grand mean over all 16 simulations. Table 6 shows the influence percentages of the five factors on the three objective functions.
| Objective | \(\delta\) | \(B_{m2}\) | \(B_t\) | \(B_m\) | \(B_{m1}\) |
| Average torque | 78.18% | 2.00% | 8.07% | 11.22% | 0.52% |
| Torque ripple | 34.59% | 1.60% | 39.73% | 5.33% | 18.75% |
| Cogging torque | 22.96% | 9.71% | 12.36% | 21.85% | 33.11% |
The variance analysis shows that \(B_{m2}\), the rotor pole-shoe height, has only a small influence on the average torque and torque ripple. I therefore removed \(B_{m2}\) from the next-stage optimization and retained the air-gap length, stator tooth width, rotor pole width, and rotor pole-shoe width as active variables. This decision reduced the dimensionality of the response-surface model and accelerated the optimization process.
5. Response-Surface Model and NSGA-II Optimization
To find a truly optimized geometry rather than merely selecting the best among a few discrete levels, I replaced the discrete Taguchi factor levels with a continuous response-surface model. The central composite design method was used because it gives a very good balance between experimental effort and model accuracy. The four retained factors were varied around their initial values. The central point is the baseline geometry, and the axial points are placed at a distance of \(\pm \alpha\) from the center. For four factors, the value of \(\alpha\) is
\[
\alpha = 2^{k/4} = 2,
\tag{9}
\]
since \(k=4\). Table 7 gives the actual and coded values used in the central composite design.
| Coded level | \(\delta\) (mm) | \(B_t\) (mm) | \(B_m\) (mm) | \(B_{m1}\) (mm) |
| \(-\alpha\) | 0.90 | 6.20 | 31.50 | 9.50 |
| \(-1\) | 1.00 | 6.50 | 32.50 | 10.00 |
| 0 | 1.10 | 6.80 | 33.50 | 10.50 |
| \(+1\) | 1.20 | 7.10 | 34.50 | 11.00 |
| \(+\alpha\) | 1.30 | 7.40 | 35.50 | 11.50 |
The central composite design contains 30 sample points. I created a complete quadratic response model of the form
\[
Y = \beta_0
+ \sum_{i=1}^{4}\beta_i x_i
+ \sum_{i=1}^{4}\beta_{ii}x_i^2
+ \sum_{i<j}\beta_{ij}x_i +=""
where \(Y\) represents the response, \(x_i\) are the coded geometric variables, and \(\varepsilon\) is the residual error. I used least-squares regression to estimate the coefficients. The model quality was assessed by the coefficient of determination \(R^2\), the adjusted coefficient \(R^2_{\mathrm{adj}}\), and the predicted coefficient \(R^2_{\mathrm{pred}}\). Table 8 presents these indicators for the three objective functions.
| Objective | \(R^2\) | \(R^2_{\mathrm{adj}}\) | \(R^2_{\mathrm{pred}}\) |
| Average torque | 0.9995 | 0.9991 | 0.9973 |
| Torque ripple | 0.9830 | 0.9672 | 0.9022 |
| Cogging torque | 0.9947 | 0.9898 | 0.9695 |
All \(R^2\) values are above 0.9. The predicted coefficients also agree sufficiently well with the adjusted coefficients, confirming that the response-surface model is accurate enough to guide the global search.
For multi-objective optimization, I used the non-dominated sorting genetic algorithm NSGA-II. This algorithm is especially useful for Pareto optimization because it combines fast non-dominated sorting, crowding distance, and an elitism strategy. I set the population size to 200, the number of generations to 200, the crossover probability to 80%, and the mutation probability to 5%. The three objective functions were defined as maximizing average torque, minimizing torque ripple, and minimizing cogging torque. Because the algorithm normally minimizes all objectives, I used the negative of the average torque in the objective vector.
\[
\begin{cases}
\min \bigl[-T_{\mathrm{avg}}(x),\; V_T(x),\; T_{\mathrm{cog}}(x)\bigr]^T \\
x_{L} \leq x \leq x_{U}
\end{cases}
\tag{11}
\]
The algorithm produced a Pareto front in the three-dimensional objective space. I selected a balanced compromise point that gives a very small cogging torque without sacrificing the average-torque improvement. The selected objective values predicted by the response-surface model were 104.27 N·m for the average torque, 6.01% for the torque ripple, and 0.74 N·m for the cogging torque.
The optimized variables obtained from NSGA-II are listed in Table 9. The most noticeable change is a reduction of the air-gap length from 1.2 to 1.02 mm. The stator tooth width increases from 6.5 to 7.10 mm. The rotor pole width and pole-shoe width are also adjusted slightly. After obtaining these geometric changes, I rebuilt the finite-element mesh and carried out the final electromagnetic verification.
| Geometric parameter | Initial value | Optimized value |
| Air-gap length \(\delta\) | 1.20 mm | 1.02 mm |
| Stator tooth width \(B_t\) | 6.50 mm | 7.10 mm |
| Rotor pole width \(B_m\) | 33.00 mm | 33.55 mm |
| Rotor pole-shoe width \(B_{m1}\) | 10.00 mm | 10.64 mm |
The finite-element verification of the optimized motor gives an average torque of 104.75 N·m. This is 9.13% higher than the initial value of 95.99 N·m. The torque ripple decreases to 6.06%, which is 8.04% lower than the initial value. The cogging torque decreases to 0.72 N·m, corresponding to a reduction of 79.01%. The predicted response-surface values are very close to these finite-element results, proving the usefulness of the surrogate-model approach. Table 10 compares the initial motor, the Taguchi-optimized motor, and the final response-surface and genetic-algorithm optimized motor.
| Objective | Initial | Taguchi | Response-surface + NSGA-II |
| Average torque (N·m) | 95.99 | 102.46 | 104.75 |
| Torque ripple (%) | 6.59 | 6.27 | 6.06 |
| Cogging torque (N·m) | 3.43 | 0.92 | 0.72 |
The final design also improves the peak-torque capability of the electric car. In the peak-torque operating condition, the optimized motor delivers an average torque of about 227.65 N·m, instead of 208.82 N·m in the initial design. The peak-condition torque ripple is reduced from 10.14% to 9.50%. The high peak torque is important for hill starts and emergency overtaking of an electric car.
6. Radial Electromagnetic Force Analysis
After the electromagnetic optimization, I analyzed the vibration and noise of the electrically excited synchronous motor. The vibration of a traction motor is mainly caused by radial electromagnetic forces acting on the stator teeth. These forces are time-varying in magnitude and spatially distributed around the air gap. If the spatial order of a force wave is low, the stator deforms more easily and radiates more noise. Therefore, the spatial and time harmonic orders must be identified carefully.
In an electrically excited synchronous motor, the air-gap magnetic field is produced by both the rotor field winding and the armature winding. I represent the rotor magnetomotive force and the stator magnetomotive force as sums of rotating harmonic waves. The total flux density can be approximated by
\[
B_r(\theta,t) = \bigl[F_R(\theta,t) + F_S(\theta,t)\bigr]\Lambda(\theta,t),
\tag{12}
\]
where \(F_R\) is the rotor field magnetomotive force, \(F_S\) is the armature-reaction magnetomotive force, and \(\Lambda\) is the air-gap permeance. The tangential component of the flux density is much smaller than the radial component, so the radial stress can be expressed with Maxwell’s stress tensor as
\[
p_r(\theta,t) \simeq \frac{B_r^2(\theta,t)}{2\mu_0}.
\tag{13}
\]
The product of two sinusoidal MMF harmonics produces radial force waves whose spatial orders and time frequencies are determined by the sum and difference of the original harmonic orders. In mathematical terms,
\[
\cos(m_1\theta – \omega_1 t)\cos(m_2\theta – \omega_2 t)
=
\frac{1}{2}\cos\bigl[(m_1\pm m_2)\theta – (\omega_1\pm\omega_2)t\bigr].
\tag{14}
\]
For an 8-pole, 48-slot machine, the pole pair number is \(p=4\). The stator slot number is \(Z=48\). The spatial order of any force component can therefore be expressed as
\[
r = A(2p) \pm B Z = 4A \pm 48B,
\tag{15}
\]
where \(A\) and \(B\) are integers. This expression yields low-order force waves with spatial orders 0 and 8. The zero-order force wave is spatially uniform around the stator and is known to be a strong source of electromagnetic noise in electric-car motors. The time frequencies of all meaningful radial force waves are even multiples of the electrical frequency because the MMF harmonic orders combine in pairs. Therefore, the main time harmonics are \(2f\), \(4f\), \(6f\), etc., where \(f\) is the motor fundamental frequency.
I computed the air-gap field in the time-stepping finite-element model. The radial air-gap flux density has a peak value of about 1.2 T and clearly shows the effect of the six stator slots in each pole pair. The tangential air-gap flux density is much smaller, which confirms that the tangential force component can be neglected in the first-order calculation. I then evaluated the radial force density on every stator tooth and performed a two-dimensional Fourier transformation in both space and time. The two-dimensional spectrum shows dominant components at different space and time orders. The 0-th spatial order appears at time frequencies of 0, \(6f\), and \(12f\), while the 8-th spatial order appears mainly at \(2f\), \(8f\), and \(14f\). The 12f component at zero spatial order is especially important for this motor.
7. Whole-Machine Modal Analysis
The dynamic response of the motor structure is strongly influenced by its modal properties. I simplified the three-dimensional model by representing the laminated stator core as a continuous solid with equivalent elastic properties. The cast aluminum housing was included together with the stator core. The end caps were omitted because their contribution to radial breathing and low-order radial modes is small. The contact between the stator core and the housing was treated as fully fixed. The modal equation used in the finite-element calculation is
\[
[M]\{\ddot{x}\} + [K]\{x\} = 0,
\tag{16}
\]
where \([M]\) is the mass matrix and \([K]\) is the stiffness matrix. The modal analysis solves the eigenvalue problem
\[
\det\bigl([K]-\omega_i^2[M]\bigr) = 0.
\tag{17}
\]
Table 11 summarizes the natural frequencies of the first important radial modes obtained from the modal simulation. These modes describe the deformation pattern of the stator and housing assembly in the transverse plane.
| Modal order \(m\) | Natural frequency (Hz) |
| 0 | 5782 |
| 2 | 1117.4 |
| 3 | 2937.5 |
| 4 | 5125.7 |
| 5 | 7365.3 |
| 6 | 9580.2 |
| 7 | 11511 |
| 8 | 12321 |
The 0-th modal order, also called the breathing mode, is particularly relevant because it radiates sound efficiently. The 2nd-order mode has a lower natural frequency of approximately 1117 Hz, while higher-order modes appear at increasingly higher frequencies. The natural frequencies are important because an electromagnetic force wave can produce high vibration when its time frequency is close to one of these structural natural frequencies.
8. Electromagnetic Vibration Response
I performed a harmonic-response analysis by importing the radial electromagnetic force distribution from Maxwell’s equations to the structural model. The forces were applied on the stator tooth surfaces as equivalent distributed pressures. The structural damping was assumed to be small. The radial force harmonic at 4800 Hz is one of the strongest components because the motor operates at a fundamental frequency of
\[
f = \frac{p n}{60} = \frac{4 \times 6000}{60} = 400\ \mathrm{Hz}.
\tag{18}
\]
For the 8-pole motor, the 12th time multiple is \(12 \times 400 = 4800\) Hz. The radial electromagnetic force at this frequency has zero spatial order. It therefore excites a nearly uniform radial breathing deformation of the stator. The harmonic-response calculation shows that the highest vibration acceleration on the housing occurs at 4800 Hz. This result is consistent with the spatial and temporal decomposition of the electromagnetic force. The vibration displacement mode shape at 4800 Hz is almost circular, with the largest displacement at the tooth tips and a smaller displacement at the stator back iron because the load is applied at the teeth.
I also examined the vibration response over a wider frequency range above and below the rated speed. The most visible acceleration peaks appear at the even multiples of the electrical frequency, confirming the analytical conclusion that the radial force harmonics occur only at even time orders for this motor topology. Odd multiples of the electrical frequency do not produce notable vibration peaks. The low-order force waves dominate the response, while high-order spatial waves are strongly attenuated because structural deformation decreases with the fourth power of the spatial order.
9. Noise Radiation Simulation
I used the vibration acceleration of the outer housing as the boundary condition for an acoustic radiation model. The motor was placed in an acoustic air domain extending one meter from the housing surface. The acoustic finite-element model computes the sound pressure level in the surrounding air. The A-weighted sound pressure level is used to emulate the frequency response of the human ear. For a single-frequency source, the sound pressure level is defined as
\[
L_p = 20\log_{10}\frac{p}{p_0},
\tag{19}
\]
where \(p_0\) is the reference sound pressure of 20 μPa. The total A-weighted level over a given spectrum is obtained by logarithmically summing the contributions of all frequency components:
\[
L_{\mathrm{eq}} = 10\log_{10}\sum_{i}10^{L_{p,i}/10}.
\tag{20}
\]
The acoustic simulation shows that the most important noise peak occurs at 4800 Hz, corresponding to the same 12th time harmonic that dominates the vibration response. At one meter from the machine housing, the calculated A-weighted sound pressure level reaches about 67.86 dBA at this frequency. The spatial directivity is almost uniform because the exciting force has zero spatial order. Near the housing surface, the sound pressure level at 4800 Hz reaches about 78.6 dB, while at 2400 Hz it is lower. The noise distribution spreads outward like a spherical wave from the motor surface.
In addition to the rated-speed analysis, I performed a speed sweep to evaluate the noise of the electric-car motor over the whole operating range. I used the equivalent radiated sound power as a scalar noise metric. The waterfall diagram shows diagonal noise ridges that track the even multiples of the electrical frequency. The maximum equivalent radiated sound power is 83.216 dB and occurs at 7000 r/min and 5600 Hz. At this speed, the 12th time harmonic of the fundamental electrical frequency is
\[
12 \times \frac{4 \times 7000}{60} = 5600\ \mathrm{Hz}.
\tag{21}
\]
This frequency is very close to the calculated 0-th order natural frequency of 5782 Hz. Therefore, the 0-th order 12th harmonic force wave excites a near-resonant breathing mode of the motor structure. This result clearly demonstrates that the vibration and noise behavior of the 8-pole, 48-slot electrically excited synchronous motor is governed by the spatial-zero-order, 12th-time-harmonic radial force component.
10. Concluding Remarks
I have presented a systematic approach for the design, optimization, and noise analysis of an electrically excited synchronous motor for electric cars. The motor specification was derived from the dynamic equations of a representative electric car, including maximum speed, grade ability, acceleration, and continuous cruise requirements. A rated power of 60 kW at 800 V, a rated speed of 6000 r/min, and a peak torque of 200 N·m define the reference machine.
The initial finite-element analysis showed that the motor met the average torque target but produced a cogging torque that was too high. Taguchi sensitivity analysis revealed the most important geometric parameters for average torque, torque ripple, and cogging torque. The response-surface model created with central-composite sampling was then optimized by the NSGA-II genetic algorithm. The final design increased the average torque by 9.13%, reduced the torque ripple by 8.04%, and reduced the cogging torque by 79.01%. The simulated torque and acoustic performance after optimization confirm that the combined response-surface and genetic-algorithm method is both accurate and efficient.
The vibration analysis was based on the radial electromagnetic force spectrum. I demonstrated analytically and numerically that the force waves in an 8-pole, 48-slot electrically excited synchronous motor have spatial orders that are multiples of four, with the zero-order and eight-order components being the most critical. Time harmonics appear only at even multiples of the fundamental frequency. The 12th time harmonic of the zero spatial order is the dominant source of radiated noise across the speed range of the electric car. The maximum equivalent radiated sound power occurs at 7000 r/min when the 12th harmonic approaches the breathing-mode natural frequency. This work provides a useful reference for future low-noise electrically excited synchronous motor designs for electric cars.