Motor Control

Field Oriented Control for Synchronous Machines

Most high performance motor controllers use field oriented control. This is due to the improved efficiency and dynamic performance of the controller.

Transformations

Clarke Transform

The Clarke transform is used to convert the measurements of the phase currents or voltages to a rectangular 2d space vector, which represents both a complex phasor representing the AC current in the phases, and a vector with the same direction of the magnetic field that the current will create. The real axis/the axis aligned with the phase A vector is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} , while the imaginary axis/perpendicular axis is labeled Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} . k is a scaling factor, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \frac{2}{3}} results in a vector with the same magnitude as the original 3-phase signals, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \sqrt{\frac{2}{3}}} results in a power-invariant transform.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x_\alpha \\ x_\beta \end{bmatrix} = k \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt 3}{2} & -\frac{\sqrt 3}{2} \end{bmatrix} \begin{bmatrix} x_A \\ x_B \\ x_C \end{bmatrix} }

Park Transform

The Park transform is used to transform quantities from being referenced to the stator to being referenced to the rotor. The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\beta} -frame is "stationary", as in the axes remain fixed while the current and voltage vector spins around with the rotor. in the DQ-frame, because it is referenced so the D axis always is aligned with a north pole on the rotor, constant current results in constant torque regardless of rotation. The transform itself is simply a rotation matrix, rotating the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\beta} -frame by the electrical angle of the rotor.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x_d\\ x_q \end{bmatrix} = \begin{bmatrix} cos(\theta_r) & sin(\theta_r) \\ -sin(\theta_r) & cos(\theta_r) \end{bmatrix} \begin{bmatrix} x_\alpha \\ x_\beta \end{bmatrix} }

Current Regulation

A PI controller is commonly used to regulate the phase currents. This is explained by starting with a simple motor model in the DQ-frame (the motor is represented by a voltage source, resistance, and inductance). Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}} }

We can then write the equation for the system as follows:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(s) = \frac{I_{dq}(s)}{V_{dq}(s) - V_{EMF,dq}} = \frac{1}{R_s + sL_{dq}} = \frac{\frac{1}{L_{dq}}}{(s + \frac{R_s}{L_{dq}})} }

The pole of the system can be cancelled out by a an appropriately tuned PI controller. The equation for the entire current controller can then be written as follows: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_c(s) = \frac{V_{dq} - V_{EMF,dq}}{I_{e,dq}(s)} = k_p + \frac{k_i}{s} = \frac{1}{s}(k_i + sk_p) }

More explicitly, the desired output voltage at any given point in time can be written as:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq} }

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}} }

Position Feedback

Field oriented control relies on having the magnetic angle of the motor for the park transform. This can either be directly measured through an encoder or resolver (sensored control), or estimated/calculated from either the back EMF or motor impedance (sensorless control). One of the main limitations of sensorless control is that the angle is only easily estimated at either high or low speeds (depending on the method). A sensor ensures reliable angle feedback across the operating range.