Optimization Techniques for Cogging Torque Reduction and Thermal Characterization in Brushless DC Motor

Abstract

This paper presents soft computing-based optimization techniques for the cogging torque reduction and thermal characterization by finite element analysis in a permanent magnet brushless DC motor (BLDC). Stator and rotor structure of BLDC motor are optimized to reduce the cogging torque, noise, and vibration by using the design parameters namely: length of magnet, length of air gap and opening in the stator slot which are selected by performing variance-based sensitivity analysis. The proposed method is suitable in the preliminary design phase of the motor to determine the optimal structure to improve the efficiency. The comparison of results obtained using firefly algorithm , ant colony optimization algorithm and Bat algorithm indicate that Firefly-based optimization algorithm is capable of giving improved design parameter output. Cogging torque is created due to the interaction of magnets in the rotor and the stator slot of the motor. Thorough thermal analysis is also conceded out to investigate the thermal characteristics at dissimilar portions of the motor namely: stator core, stator slot, rotor core and permanent magnet at different operating environments in the continuous operation mode. Thermal investigation is required for the various high speed e-vehicle applications. The usefulness of the designed machine simulation is compared with the results obtained from hardware analysis. The outcomes attained from software simulation studies are validated through experimental hardware setup.

Appendix

To predict the open-circuit magneto static field of electric motors, analytical magnetic circuit approach is needed. The magnetic flux $\Phi_{m}$ is the flux actually linking with the magnet, which is equal to the summation of air gap flux and leakage flux. The leakage flux is an essential quantity for predicting the average flux density in the air gap. The magnetic circuit analysis takes the leakage flux into account.

The assumptions for magnetic modeling are as follows:

• (1) The magnetic field intensity in the stator winding is negligible.

• (2) Infinite permeability in stator and rotor core.

• (3) There is no magnetic saturation in the steel region.

Let,

$\Phi_{r}$ as the flux source in weber,

$\Phi_{g}$ as the air gap flux in weber,

$\Phi_{m}$ as the flux leaving the magnet in weber,

$\Phi_{{{\text{mm}}}}$ as the magnet to magnet leakage flux in weber,

$\Phi_{{{\text{mr}}}}$ as the magnet to rotor leakage flux in weber,

$R_{r}$ as the reluctance of the rotor in AT/weber,

$R_{s}$ as the reluctance of the stator back irons in AT/weber,

$R_{g}$ as the reluctance of the air gap in AT/weber,

$R_{{{\text{mo}}}}$ as the reluctance of the magnet in AT/weber,

$R_{{{\text{mm}}}}$ as the reluctance from magnet to magnet, and.

$R_{{{\text{mr}}}}$ as the reluctance from magnet to rotor in AT/weber.

Based on assumption 1 listed above, the values of $R_{s}$ and $R_{r}$ can be ignored. The $R_{m}$ can be expressed as given in Eq. (2).

$$R_{m} = \frac{{R_{{{\text{mo}}}} }}{1 + 2\eta + 4\lambda }$$

(2)

where

$$\eta = R_{{{\text{mo}}}} /R_{{{\text{mr}}}}$$

(3)

and

$$\lambda = R_{{{\text{mo}}}} /R_{{{\text{mm}}}}$$

(4)

Also, the air gap flux is expressed as given in Eq. (5)

$$\Phi_{g} = \frac{{R_{m} \Phi_{r} }}{{R_{m} + R_{g} }} = \frac{{R_{{{\text{mo}}}} }}{{R_{{{\text{mo}}}} + R_{g} \left( {1 + 2\eta + 4\lambda } \right)}}\Phi_{r}$$

(5)

The flux leaving the magnet is expressed as given in Eq. (6)

$$\Phi_{m} = \frac{{R_{{{\text{mo}}}} + R_{g} \left( {2\eta + 4\lambda } \right)}}{{R_{{{\text{mo}}}} + R_{g} \left( {1 + 2\eta + 4\lambda } \right)}}\Phi_{r}$$

(6)

In addition, the flux source of the magnet $\Phi_{r}$ can be written as given in Eq. (7)

$$\Phi_{r} = A_{m} B_{r}$$

(7)

where $A_{m}$ is the flux passing area and.

$B_{r }$ represents the residential flux density of the magnet.

Based on Eqs. (5)–(7), the average flux densities within the air gap and the magnet can be obtained as follows.

$$B_{{g,{\text{ave}}}} = \frac{{A_{m} /A_{g} }}{{1 + \left( {R_{g} /R_{{{\text{mo}}}} } \right)\left( {1 + 2\eta + 4\lambda } \right)}}B_{r}$$

(8)

and

$$B_{m} = \frac{{1 + \left( {R_{g} /R_{{{\text{mo}}}} } \right)\left( {2\eta + 4\lambda } \right)}}{{1 + \left( {R_{g} /R_{{{\text{mo}}}} } \right)\left( {1 + 2\eta + 4\lambda } \right)}} .$$

(9)

where $A_{m} /A_{g}$ is the ratio of the magnetic flux passing area of the magnet to that of the air gap.

Based on geometric and magnetic relationships, the following expressions are obtained:

$$A_{m} = \tau_{m} L$$

(10)

$$A_{g} = \left( {\tau_{m} + \tau_{f} } \right)L$$

(11)

$$R_{{{\text{mo}}}} = \frac{{l_{m} }}{{\mu_{o} \mu_{r} A_{m} }}$$

(12)

$$R_{g} = \frac{{g_{e} }}{{\mu_{o} \left( {\tau_{m} + 2g_{e} } \right)L}}$$

(13)

$$g_{e} = k_{c} g$$

(14)

where $L$ is the stack length of the stator in mm, $\tau_{m}$ is the magnet width in mm, $\tau_{f}$ is the length of the two adjacent magnets, $l_{m}$ is the magnet height in mm, $\mu_{o}$ is the permeability of free space, $\mu_{r}$ is the relative permeability of the magnet, $g$ is the air gap length in mm, $g_{e}$ is the effective air gap length in mm, and $k_{c}$ is the carter’s co-efficient. In particular, the fringing effect is taken into consideration by adding the length 2 $g_{e}$ to $\tau_{m}$ for calculating $R_{g}$ as expressed in Eq. (13).

The magnet-to-rotor as well as magnet-to-magnet reluctances can be further obtained by calculating their permeances using corresponding circular-arc and straight-line permanence models. Since, the reluctances are reciprocal to the permeance, thus can be represented as given in Eqs. (2–16).

$$R_{{{\text{mr}}}} = \frac{1}{{P_{{{\text{mr}}}} }}$$

(15)

$$R_{{{\text{mm}}}} = \frac{1}{{P_{{{\text{mm}}}} }}$$

(16)

The fringing permeance $P_{{{\text{mr}}}}$ is an infinite sum of differential width permeances, each of length $l_{m} + \pi x$, i.e.

$$P_{{{\text{mr}}}} = \sum \frac{{\mu_{o} {\text{d}}A}}{l} = \sum \frac{{\mu_{o} L}}{{l_{m} + \pi x}}{\text{d}}x$$

(17)

After simplification, the $P_{{{\text{mr}}}}$ can be expressed as given in Eq. (17).

$$P_{mr} = \frac{{\mu_{o} L}}{\pi }\ln \left[ {1 + \frac{{\pi \min \left( {g_{e} ,\tau_{f} /2} \right)}}{{l_{m} }}} \right]$$

(18)

where ${\text{d}}A = L{\text{d}}x$ is the cross-sectional area of each differential permeance.

$P_{{{\text{mm}}}}$ can be expressed as given in Eq. (19).

$$P_{{{\text{mm}}}} = \frac{{\mu_{0} L}}{\pi }\ln \left( {1 + \frac{{\pi g_{e} }}{{\tau_{f} }}} \right)$$

(19)

Based on Eqs. (10), (12), (15), (18) and (19), Eqs. (3) and (4) can be rewritten as given in Eqs. (20 and 21).

$$\eta = \frac{{l_{m} }}{{\mu_{r} \pi \tau_{m} }}\ln \left( {1 + \frac{{\pi \min \left( {g_{e} ,\tau_{f} /2} \right)}}{{l_{m} }}} \right)$$

(20)

$$\lambda = \frac{{l_{m} }}{{\mu_{r} \pi \tau_{m} }}\ln \left( {1 + \frac{{\pi g_{e} }}{{\tau_{f} }}} \right)$$

(21)

The magnetic flux density can be expressed as given in Eq. (22).

$$B_{g,ave} = \frac{{B_{r} }}{{1 + \frac{{\tau_{f} }}{{\tau_{m} }} + \mu_{r} \frac{{g_{e} }}{{l_{m} }}\frac{{\tau_{m} + \tau_{f} }}{{\tau_{m} + 2g_{e} }}\left( {1 + 2\eta + 4\lambda } \right)}}$$

(22)

and

$$B_{m} = \frac{{\left( {1 + \frac{{2g_{e} }}{{\tau_{m} }}} \right)\frac{1}{{\mu_{r} }}\frac{{l_{m} }}{{g_{e} }} + 2\eta + 4\lambda }}{{\left( {1 + \frac{{2g_{e} }}{{\tau_{m} }}} \right)\frac{1}{{\mu_{r} }}\frac{{l_{m} }}{{g_{e} }} + 1 + 2\eta + 4\lambda }}B_{r}$$

(23)

The leakage coefficient ($f_{{{\text{LKG}}}}$) is defined as the ratio of air gap flux to magnet flux and it is analytically expressed as given in Eq. (24).

$$f_{{{\text{LKG}}}} = \frac{{\Phi_{g} }}{{\Phi_{m} }} = \frac{1}{{1 + \frac{{\mu_{r} g_{e} \tau_{m} }}{{l_{m} \left( {\tau_{m} + 2g_{e} } \right)}}\left( {2\eta + 4\lambda } \right) }}$$

(24)

The average electromagnetic torque is expresses as given in Eq. (25).

$$T_{{{\text{ave}}}} = mk_{l} k_{w} N_{{{\text{tol}}}} B_{{g,{\text{ave}}}} R_{{{\text{ro}}}} Li_{{\text{ph }}}$$

(25)

where $k_{w}$ is the winding factor, $N_{{{\text{tol}}}}$ is the total conductors per phase, $R_{{{\text{ro}}}}$ is the outside rotor radius in meter, $m$ is the phases of conduction, $k_{l }$ is the correction factor due to losses, $i_{{{\text{ph}}}}$ is the phase current in amperes.

The cogging torque equation is expressed as given in Eq. (26)

$$T_{{{\text{cog}}}} = \left( {\frac{{{\text{d}}\omega_{\alpha } }}{{{\text{d}}\omega_{\theta } }}} \right)$$

(26)

where

$$\omega_{\alpha } = \frac{1}{{2\mu_{0} }}\int {B^{2} \left( \theta \right){\text{d}}V}$$

(27)

$$\begin{aligned} B\left( \theta \right) = & \;\;\mathop \sum \limits_{n = 1,3,5, \ldots }^{\infty } \frac{{2B_{r} \beta }}{{\mu_{r1} }} \times \frac{{\sin \left( {n\pi \beta /2} \right)}}{{\left( {n\pi \beta /2} \right)}} \\ & \;\; \times \frac{np}{{\left( {np} \right)^{2} - 1}}\left[ {\left( {\frac{{r_{g} }}{{r_{s} }}} \right)^{np - 1} \left( {\frac{{r_{r} }}{{r_{s} }}} \right)^{np + 1} + \left( {\frac{{r_{r} }}{{r_{g} }}} \right)^{np + 1} } \right] \\ & \;\; \times \left\{ {\frac{{\left( {np - 1} \right) + 2\left( {{\raise0.7ex\hbox{${r_{y} }$} \!\mathord{\left/ {\vphantom {{r_{y} } {r_{r} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{r} }$}}} \right)^{np + 1} - \left( {np + 1} \right)\left( {{\raise0.7ex\hbox{${r_{y} }$} \!\mathord{\left/ {\vphantom {{r_{y} } {r_{r} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{r} }$}}} \right)^{{2{\text{np}}}} }}{{\left( {{\raise0.7ex\hbox{${\left( {\mu_{r1} + 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {\mu_{r1} + 1} \right)} {\left( {\mu_{r1} } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {\mu_{r1} } \right)}$}}} \right)\left[ {1 - \left( {r_{y} /r_{s} } \right)^{{2{\text{np}}}} } \right] - \left( {\left( {\mu_{r1} - 1} \right)/\mu_{r1} } \right)\left[ {\left( {r_{r} /r_{s} } \right)^{{2{\text{np}}}} - \left( {r_{y} /r_{r} } \right)^{{2{\text{np}}}} } \right]}}} \right\} \\ \end{aligned}$$

(28)

$\begin{gathered} np \ne 1 \hfill \\ r_{s} = r_{r} + l_{g} + l_{w} \hfill \\ r_{y} = r_{r} - l_{m} \hfill \\ r_{g} = r_{r} + l_{g} \hfill \\ \end{gathered}$