Motor Control

Field Oriented Control for Synchronous Machines

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).

$I_{dq}={\frac {V_{dq}-V_{emf}}{R_{s}+sL_{dq}}}$

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}) (k_p + \frac{k_i}{s} = \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}} }