Open Loop

FOC with Position Sensor Feedback

The design supports FOC sensor control where the motor position feeds back to form a closed loop with position and speed PI control. The design senses the motor position by incremental (quadrature) encoders.

FOC Sensorless

The design supports FOC sensorless control in which the design samples and uses both the motor phase voltages and currents as the feedback to the control loop.

The phase voltage calc block derives signals V_α and V_β, scaled and normalized with respect to the DC bus voltage (V_dclink). The DC bus voltage may drop during quick acceleration or rise during regeneration. If you do not expect the bus voltage to change much (e.g. large bus capacitance), you may use V_α and V_β generated from the inverse Clarke transform. The software function Phase_Volt_Calc_f() implements the phase voltage calculations using floating-point arithmetic.

The design integrates the speed estimator with the sliding mode observer (SMO) to allow a second order observer to calculate both estimated angle and estimated speed together. In FOC sensorless mode, the motor starts initially in open loop with a requested speed and switches to sensorless mode after a preset time to allow the SMO to settle. The software function SMO_Calc_f() implements the SMO calculations using floating-point arithmetic.

Sliding Mode Observer Theory

The observer, based on the electrical model, estimates the rotating back-EMF vector. The rotor position determines the direction of the back-EMF, which enables estimation of position. At constant speed, back-EMF components are sinusoidal in quadrature, as are error signals. Considering voltage V and current I in one motor phase with resistance Rs, inductance Ls. Net applied voltage V is at the motor terminal, and the centre of the motor is at 0 V.

${v=i{R}_s+\frac{di}{dt}}_{}{L}_s$

Solvinge the differential equation for i, assuming a constant v applied over sample time T, initial current i , and current at the end of T i +1, gives:

$i_{k+1}=i_k{e}^{-\left(\frac{R_s}{L_s}\right)T}+v_k\frac{1}{R_s}(1-e^{-\left(\frac{R_s}{L_s}\right)T})$

Normalizing by some V _max and I _max to create variables that are non-dimensional and always <1, gives:

$\left(\frac{i}{I_{max}}\right){}_{k+1}=\left(\frac{i}{I_{max}}\right){}_k{e}^{-\left(\frac{R_s}{L_s}\right)T}+\left(\frac{v}{V_{max}}\right)\frac{V_{max}}{I_{max}{R}_s}(1-e^{-\left(\frac{R_s}{L_s}\right)T})$

$\left(\frac{i}{I_{max}}\right){}_{k+1}=\left(\frac{i}{I_{max}}\right)Aparm+\left(\frac{v}{V_{max}}\right)Bparm$

$Aparm=e^{-\left(\frac{R_s}{L_s}\right)T}$

$Bparm=(1-Aparm)\frac{V_{max}}{I_{max}{R}_s}$

Aparm is non-dimensional, represents the electrical system dynamics. Ls/Rs is the electrical time constant, Ts is the discrete sample time. Aparm should be close to 1 if the sample time is much faster than the motor time constant.

Bparm uses V_max/(I_max*R_s) so is also nondimensional. (1-Aparm) is small.

Aparm and Bparm are constant and are calculated once during initialization of the software.

Angle Tracking Observer Theory

The angle tracking observer (part of SMO_Calc_f()) takes the estimated back-EMF vector as input and outputs an estimated rotor position (phi_SMO). The back-EMF estimate may be noisy so the observer filters it and converts it to position. The design provides two methods:

1. #if (TRACKER_ENABLE == 0), arctan method: A first order filter with gain Lpf_Gain is applied to both alpha and beta components of the back-EMF before using arctan to convert them to phi_SMO. The advantage is simplicity, but velocity must be estimated separately. The design does not use this method as the speed estimation may introduce lag.
2. Angle observer method: The design combines the back-EMF estimates with the last phi_SMO to calculate AObsError, which is proportional to speed*sin(actual-estimate). Assuming small errors and constant speed, AobsError=K*(θ_Elec - phi_SMO). Phi_SMO has units of fraction of 1rev (0..1 for 0..2π rad).

The back-EMF operates in the q-direction,, which is at right-angles to the d-direction (θelec_rad). The components of the back-EMF in the fixed alpha and beta directions are:

• vBemfAlpha_V = -dθmech_dt_rad_s*Ke_Vs_rad*sin(θelec_rad)
• vBemfBeta_V = dθmech_dt_rad_s*Ke_Vs_rad*cos(θelec_rad)
• AobsError is the back-EMF in the estimated d direction (phi_SMO), which is zero if phi_SMO is equal to θelec_rad
• AObsError = -vBemfAlpha_V*cos(phi_SMO)-vBemfBeta_V*sin(phi_SMO)

Substituting the Bemf equations into the AObsError equation:

AObsError = dθmech_dt_rad_s*Ke_Vs_rad*( sin(θelec_rad)*cos(phi_SMO)-cos(θelec_rad)*sin(phi_SMO) )

= dθmech_dt_rad_s*Ke_Vs_rad*sin(θelec_rad - phi_SMO)

Hence, if the BemfAlpha and BemfBeta values are correct, AObsError measures the angle estimation error (θelec_rad - phi_SMO).

For small error angles and at constant speed, assume AObsError = K*(θelec - phi_SMO)

AObsError = K(θ_k- φ_k)

The software creates a second-order filter from θ_elec to phi_SMO. Filter natural frequency Ω and damping coefficient ζ are design parameters tuned for the Tamagawa motors in the Tandem Motion-Power 48 V motor kit. The normalized scaling makes them suitable for other motors too with possible retuning. Carefully tune this non-linear observer because incorrect position estimates can make the FoC algorithm oscillate or become unstable by causing positive feedback.

SMO Parameters

The design derives various SMO parameters from the motor parameters for each motor type, such as resistance and inductance. Other SMO parameters, and default values, are:

Trapezoidal

The design supports trapezoidal control of BLDC motors using Hall sensor feedback on the Tandem Motion-Power 48 V Board. The software supports Duty Mode and Torque Mode, but the demonstration GUI only uses Velocity Mode. The software reconstructs the motor current from the individual phase current readings using the Hall encoder state to determine which phase current is relevant.