Dynamic Permanent Magnet Synchronous Motor

Notes:

• FOC is working, however a solver similar to Matlab's ODE15s has to replace the current 4th order Runge-Kutta. ode15s is a variable-step, variable-order (VSVO) solver based on the numerical differentiation formulas (NDFs) of orders 1 to 5. Optionally, it can use the backward differentiation formulas (BDFs, also known as Gear's method) that are usually less efficient. Like ode113, ode15s is a multistep solver. Use ode15s if ode45 fails or is very inefficient, and you suspect that the problem is stiff, or when solving a differential-algebraic equation (DAE)

• It has been verified that a similar behaviour occurs in matlab with ode4 (4th Order Runge-Kutta) solver in Matlab/Simulink

Dynamic driving equations

$$ \frac{d}{dt}i_q = \frac{u_q}{L_q} - i_q\frac{R_s}{L_q} - \frac{\omega_e L_d i_d}{L_q} - \frac{\omega_e \lambda_m}{L_q} $$

$$ \frac{d}{dt}i_d = \frac{u_d}{L_d} - i_d\frac{R_s}{L_d} + \frac{\omega_e L_q i_q}{L_d} $$

$$ T_{em} = \frac{3}{2}\cdot\frac{p}{2}\left[\lambda_mi_q + (L_d - L_q)i_qi_d\right] $$

$$ \alpha_m = \frac{1}{J}(T_{em} - T_{load} - \omega_m b_v) $$

Where:

Symbol:		Description:	Value:	Unit:
$\lambda_m$	-	Flux linkage	$\sim$	$Wb$
$\omega_e$	-	Electrical angular velocity	$\sim$	$rad/s$
$\omega_m$	-	Mechanical angular velocity	$\sim$	$rad/s$
$\alpha_m$	-	Mechanical angular acceleration	$\sim$	$rad/s^2$
$i_q$	-	q-axis current	$\sim$	$A$
$i_d$	-	d-axis current	$\sim$	$A$
$u_q$	-	q-axis voltage	$\sim$	$V$
$u_d$	-	d-axis voltage	$\sim$	$V$
$T_{em}$	-	Electromagnetic torque	$\sim$	$Nm$
$T_{load}$	-	External load torque	$\sim$	$Nm$
$L_q$	-	Quadrature inductance	$\sim$	$H$
$L_d$	-	Direct inductance	$\sim$	$H$
$R_s$	-	Phase resistance	$\sim$	$\Omega$
$b_v$	-	Viscous damping	$\sim$	$Nms/rad$
$J$	-	Motor inertia	$\sim$	$kg/m^2$

Aux equations

$$ \omega_e = \frac{p}{2}\omega_m $$

$$ k_t = \frac{15\sqrt{3}}{\pi K_V}$$

$$ \lambda_m = \frac{2k_t}{3p_p} = \frac{2\cdot15\sqrt{3}}{3\pi} \frac{1}{p_pK_V} $$

Symbol:		Description:	Value:	Unit:
$\lambda_m$	-	Flux linkage	$\sim$	$Wb$
$\omega_e$	-	Electrical angular velocity	$\sim$	$rad/s$
$\omega_m$	-	Mechanical angular velocity	$\sim$	$rad/s$
$p_p$	-	Motor pole pairs	$\sim$	$-$
$k_t$	-	Torque constant	$\sim$	$Nm/A$
$K_V$	-	Speed Constant	$\sim$	$rad/(sV)$