Appendix

Reference sheet for variables used for derivations, found here.

$a_{chord}$ : Chording angle (electrical rad)

$a_{skew}$ : Skew angle (electrical rad)

$\alpha$ : Temperature coefficient of resistance ($1/^{\circ}C$)

$\alpha_i$ : Pole arc coefficient

$A(n_g)$ : Strand cross sectional area ($mm^2$)

$A_{wind}$ : Total winding area ($mm^2$)

$A_{slot}$ : Slot area on stator tooth ($mm^2$)

$A_{total}$ : Effective total copper cross-section area for multiple strands of a certain AWG ($mm^2$)

$A_{pole}$ : Effective pole span area (mm^2)

$AWG_{equiv}$ : Equivalent American Wire Gauge rating, derived from equivalent diameter for a bundle

$B$ : Magnetic flux density ($T$, or $Wb/m^2$)

$B_r$ : Remanent flux density, dependent on NdFeB grade. ($T$ or $Wb/m^2$)

$B_g$ : Airgap flux density ($T$ or $Wb/m^2$)

$C_{SD}$ : Multiplier, 1 for delta config or $\sqrt{3}$ for star.

$C_w$ : Winding buildup correction (mm)

$d_{equiv}$ : Equivalent diameter for multiple strands of a certain AWG (mm)

$d(n_g)$ : Strand diameter for gauge n ($mm$)

$\frac{d\lambda}{d\theta}$ : Change in flux linkage with rotor angle ($Wb/rad$)

$E$ : Back-EMF ($V$)

$E_{max}$ : Peak back-EMF ($V$)

$f$ : Copper fill factor to available slot area

$g$ : Airgap length - the very small space between rotor and stator (mm)

$g’$ : Airgap length, corrected via Carter coefficients

$\gamma$ : Electrical angle between two slots (rad)

$I_{coil}$ : Coil current ($A$), ($I_{coil} = I_{phase}$ in series, $I_{coil} = \frac{I_{phase}}{N_{TPP}}$ for parallel)

$I_{line}$ : Line-to-line current ($A$)

$I_{phase}$ : Phase current, measured line-to-neutral ($A$)

$I_max$ : Peak current ($A$)

$I_{total}$ : Combined sum of phase currents ($A$)

$K$ : Proportional motor constant, for scaling KV.

$K_c, K_{cs}, K_{cs}$ : Carter’s coefficient for stator and rotor, a multiplier for increasing effective air gap length

$K_e$ : Motor b-EMF constant ($V/rad/s$)

$K_d$ : Distribution factor

$K_p$ : Pitch factor for distributed windings (=1 for concentrated)

$K_{sigma} : Slot leakage coefficient

$K_{skew}$ : Skew factor for PM magnets

$K_t$ : Motor torque constant ($N*m/A$)

$KV$ : Speed constant of a motor ($RPM/V$)

$K_w$ : Winding factor

$L_{stack}$ : Stack length or thickness of stator (mm)

$L_M$ : Magnet thickness (mm)

$\lambda$ : Flux linkage, function of Kw and flux ($Wb$)

$MTL$ : Mean Turn Length ($mm$)

$\mu_r$ : Magnetic recoil permeability, ($\approx1.05$ for NdFeB magnets). Dimensionless slope of the linear portion of the demagnetization curve.

$n_g$ : Copper AWG gauge

$n_s$ : Strand count

$n_t$ : Turn count

$N_{poles}$ : Number of poles

$N_{TPP}$ : Teeth per phase

$\omega$ : Angular velocity of the rotor ($rad/s$)

$P$ : Power ($W$)

$\phi$ : Magnetic flux ($Wb$, or $V*s$)

$\phi_{Pole}$ : Flux per pole (Wb)

$q$ : Slots per pole per phase

$R_{per_m}$ : Resistance per meter ($\Omega/m$)

$R_{coil}$ : Coil resistance ($\Omega$)

$R_{line}$ : Line-to-line resistance ($\Omega$) ($R_{line} = 2*R_{phase}$ for star, $R_{line} = \frac{2}{3} * R_{phase}$ for delta)

$R_{phase}$ : Phase Resistance, or line-to-neutral ($\Omega$)

$r_{gap}$ : Radial moment arm of the motor, distance from the center of the motor to the middle of the air gap (mm)

$\rho$ : Resistivity ($\Omega*m$)

$\sigma$ : Slot pitch to airgap ratio

$T$ : Motor operating temperature ($^{\circ}C$)

$t_s$ : Slot pitch (electrical rad)

$\tau$ : Torque ($N*m$)

$\theta$ : Rotor angle ($rad$)

$\theta_{elec}$ : Electrical angle ($rad$)

$V_{line}$ : Line-to-line voltage ($V$) ($V_{line} = \sqrt{3}*V_{phase}$ for star, $V_{line} = V_{phase}$ for delta)

$V_{phase}$ : Line-to-neutral voltage, also known as phase voltage ($V$)