Difference between revisions of "Motor Control"

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(Add Clarke and Park transforms)
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=DC Brushed Motor Control=
 
=DC Brushed Motor Control=
 
=Six-Step Control=
 
=Six-Step Control=
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=Field Oriented Control for Synchronous Machines=
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<div><h1>Field Oriented Control for Synchronous Machines</h1></div>Most FSAE grade motor controllers use field oriented control. This is due to the efficiency and dynamic performance of the controller and motor. Field oriented control relies on controlling the DQ-axis currents through applying a specific DQ-axis voltage to the motor. For a good explaination of field oriented control see the video below (though it focuses on sensorless field oriented control).<br data-attributes=""><br class="mw_emptyline_first"><br class="mw_emptyline"><a href="https://www.youtube.com/watch?v=cdiZUszYLiA&amp;t=11s" data-mce-href="https://www.youtube.com/watch?v=cdiZUszYLiA&amp;t=11s" title="[1]" data-mw-type="external_link" class="link external mw-external-link mceNonEditable" data-mw-wikitext="%5Bhttps://www.youtube.com/watch?v=cdiZUszYLiA&amp;t=11s%5D" contenteditable="false"><mwspan>[1]<div class="mceNonEditableOverlay"></div></mwspan></a><div><h2>Transformations</h2></div><div><h3>Clarke Transform</h3></div>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<span class="single_linebreak" title="single linebreak" contenteditable="false"><mwspan>¶</mwspan></span>of the magnetic field that the current will create. The real axis/the axis aligned with the phase A vector is called <span id="<@@@TAG155056@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>\alpha</math>" data-mw-type="tag" data-mw-id="155056" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
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Most high performance motor controllers use field oriented control. This is due to the improved efficiency and dynamic performance of the controller.
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  <semantics>
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==Transformations==
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    <mrow class="MJX-TeXAtom-ORD">
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===Clarke Transform===
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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
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        <mi>α<!-- α --></mi>
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of the magnetic field that the current will create. The real axis/the axis aligned with the phase A vector is called <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
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      </mstyle>
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<math>k = \frac{2}{3}</math> results in a vector with the same magnitude as the original 3-phase signals, and <math>k = \sqrt{\frac{2}{3}}</math> results in a power-invariant transform.
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    <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
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<math>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span>, while the imaginary axis/perpendicular axis is labeled <span id="<@@@TAG125345@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>\beta</math>" data-mw-type="tag" data-mw-id="125345" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Cbeta%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
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    <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span>. k is a scaling factor, where<br class="mw_emptyline"><span id="<@@@TAG104565@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>k = \frac{2}{3}</math>" data-mw-type="tag" data-mw-id="104565" data-mw-name="math" data-mw-wikitext="%3Cmath%3Ek%20=%20%5Cfrac%7B2%7D%7B3%7D%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {2}{3}}}">
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    <annotation encoding="application/x-tex">{\displaystyle k={\frac {2}{3}}}</annotation>
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  </semantics>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32f140d5d8c9832545636b79bf45e81478334ca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:6.308ex; height:5.176ex;" alt="{\displaystyle k={\frac {2}{3}}}"></span>
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span> results in a vector with the same magnitude as the original 3-phase signals, and <span id="<@@@TAG109486@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>k = \sqrt{\frac{2}{3}}</math>" data-mw-type="tag" data-mw-id="109486" data-mw-name="math" data-mw-wikitext="%3Cmath%3Ek%20=%20%5Csqrt%7B%5Cfrac%7B2%7D%7B3%7D%7D%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\sqrt {\frac {2}{3}}}}">
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    <annotation encoding="application/x-tex">{\displaystyle k={\sqrt {\frac {2}{3}}}}</annotation>
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  </semantics>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0412cac912c88cf59bc4156599e46640e80d09" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:8.632ex; height:6.176ex;" alt="{\displaystyle k={\sqrt {\frac {2}{3}}}}"></span>
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span> results in a power-invariant transform.<br class="mw_emptyline_first"><span id="<@@@TAG191952@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
 
\begin{bmatrix}
 
\begin{bmatrix}
 
x_\alpha \\
 
x_\alpha \\
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= k
 
= k
 
\begin{bmatrix}
 
\begin{bmatrix}
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1 & -\frac{1}{2} & -\frac{1}{2} \\
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1 &amp; -\frac{1}{2} &amp; -\frac{1}{2} \\
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0 & \frac{\sqrt 3}{2} & -\frac{\sqrt 3}{2}
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0 &amp; \frac{\sqrt 3}{2} &amp; -\frac{\sqrt 3}{2}
 
\end{bmatrix}
 
\end{bmatrix}
 
\begin{bmatrix}
 
\begin{bmatrix}
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x_C
 
x_C
 
\end{bmatrix}
 
\end{bmatrix}
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</math>
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</math>" data-mw-type="tag" data-mw-id="191952" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%5Cbegin%7Bbmatrix%7D%0Ax_%5Calpha%20%5C%5C%0Ax_%5Cbeta%0A%5Cend%7Bbmatrix%7D%0A=%20k%0A%5Cbegin%7Bbmatrix%7D%0A1%20&amp;%20-%5Cfrac%7B1%7D%7B2%7D%20&amp;%20-%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%0A0%20&amp;%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%20&amp;%20-%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0Ax_A%20%5C%5C%0Ax_B%20%5C%5C%0Ax_C%0A%5Cend%7Bbmatrix%7D%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}=k{\begin{bmatrix}1&amp;-{\frac {1}{2}}&amp;-{\frac {1}{2}}\\0&amp;{\frac {\sqrt {3}}{2}}&amp;-{\frac {\sqrt {3}}{2}}\end{bmatrix}}{\begin{bmatrix}x_{A}\\x_{B}\\x_{C}\end{bmatrix}}}">
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===Park Transform===
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  <semantics>
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The Park transform is used to transform quantities from being referenced to the stator to being referenced to the rotor. The <math>\alpha\beta</math>-frame is "stationary", as in the axes remain fixed while the current
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    <mrow class="MJX-TeXAtom-ORD">
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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.
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The transform itself is simply a rotation matrix, rotating the <math>\alpha\beta</math>-frame by the electrical angle of the rotor.
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}=k{\begin{bmatrix}1&amp;-{\frac {1}{2}}&amp;-{\frac {1}{2}}\\0&amp;{\frac {\sqrt {3}}{2}}&amp;-{\frac {\sqrt {3}}{2}}\end{bmatrix}}{\begin{bmatrix}x_{A}\\x_{B}\\x_{C}\end{bmatrix}}}</annotation>
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><div><h3>Park Transform</h3></div>The Park transform is used to transform quantities from being referenced to the stator to being referenced to the rotor. The <span id="<@@@TAG142237@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>\alpha\beta</math>" data-mw-type="tag" data-mw-id="142237" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%5Cbeta%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \beta }">
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span>-frame is "stationary", as in the axes remain fixed while the current<span class="single_linebreak" title="single linebreak" contenteditable="false"><mwspan>¶</mwspan></span>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.<span class="single_linebreak" title="single linebreak" contenteditable="false"><mwspan>¶</mwspan></span>The transform itself is simply a rotation matrix, rotating the <span id="<@@@TAG142237@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>\alpha\beta</math>" data-mw-type="tag" data-mw-id="142237" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%5Cbeta%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \beta }">
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</p></div><div class="mceNonEditableOverlay"></div></mwspan></span>-frame by the electrical angle of the rotor.<br class="mw_emptyline_first"><span id="<@@@TAG167346@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
 
\begin{bmatrix}
 
\begin{bmatrix}
 
x_d\\
 
x_d\\
Line 38: Line 236:
 
=
 
=
 
\begin{bmatrix}
 
\begin{bmatrix}
−
cos(\theta_r) & sin(\theta_r) \\
+
cos(\theta_r) &amp; sin(\theta_r) \\
−
-sin(\theta_r) & cos(\theta_r)
+
-sin(\theta_r) &amp; cos(\theta_r)
 
\end{bmatrix}
 
\end{bmatrix}
 
\begin{bmatrix}
 
\begin{bmatrix}
Line 45: Line 243:
 
x_\beta
 
x_\beta
 
\end{bmatrix}
 
\end{bmatrix}
−
</math>
+
</math>" data-mw-type="tag" data-mw-id="167346" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%5Cbegin%7Bbmatrix%7D%0Ax_d%5C%5C%0Ax_q%0A%5Cend%7Bbmatrix%7D%0A=%0A%5Cbegin%7Bbmatrix%7D%0Acos(%5Ctheta_r)%20&amp;%20sin(%5Ctheta_r)%20%5C%5C%0A-sin(%5Ctheta_r)%20&amp;%20cos(%5Ctheta_r)%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0Ax_%5Calpha%20%5C%5C%0Ax_%5Cbeta%0A%5Cend%7Bbmatrix%7D%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x_{d}\\x_{q}\end{bmatrix}}={\begin{bmatrix}cos(\theta _{r})&amp;sin(\theta _{r})\\-sin(\theta _{r})&amp;cos(\theta _{r})\end{bmatrix}}{\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}}">
−
==Current Regulation==
+
  <semantics>
−
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).  
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{d}\\x_{q}\end{bmatrix}}={\begin{bmatrix}cos(\theta _{r})&amp;sin(\theta _{r})\\-sin(\theta _{r})&amp;cos(\theta _{r})\end{bmatrix}}{\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3936dfef8cb177c2c3c1ae223fe87335c86a1dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:36.206ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}x_{d}\\x_{q}\end{bmatrix}}={\begin{bmatrix}cos(\theta _{r})&amp;sin(\theta _{r})\\-sin(\theta _{r})&amp;cos(\theta _{r})\end{bmatrix}}{\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><div><h2>Current Regulation</h2></div>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). <br class="mw_emptyline"><span id="<@@@TAG149363@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
 
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
 
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
−
</math>
+
</math>" data-mw-type="tag" data-mw-id="149363" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0AI_%7Bdq%7D%20=%20%5Cfrac%7BV_%7Bdq%7D%20-%20V_%7Bemf%7D%7D%7BR_s%20+%20sL_%7Bdq%7D%7D%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{dq}={\frac {V_{dq}-V_{emf}}{R_{s}+sL_{dq}}}}">
−
 
+
  <semantics>
−
 
+
    <mrow class="MJX-TeXAtom-ORD">
−
We can then write the equation for the system as follows:<br /><math> 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}})} </math>
+
      <mstyle scriptlevel="0" displaystyle="true">
−
 
+
        <msub>
−
 
+
          <mi>I</mi>
−
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:
+
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    <annotation encoding="application/x-tex">{\displaystyle I_{dq}={\frac {V_{dq}-V_{emf}}{R_{s}+sL_{dq}}}}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3a5976ef651c6e22e536a8166ce9194ab517a3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:17.55ex; height:6.176ex;" alt="{\displaystyle I_{dq}={\frac {V_{dq}-V_{emf}}{R_{s}+sL_{dq}}}}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><br class="mw_emptyline_first"><br class="mw_emptyline">We can then write the equation for the system as follows:<br data-attributes=""><span id="<@@@TAG144167@@@>" class="mceNonEditable wikimagic mw-tag" title="<math> 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}})} </math>" data-mw-type="tag" data-mw-id="144167" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%20G(s)%20=%20%5Cfrac%7BI_%7Bdq%7D(s)%7D%7BV_%7Bdq%7D(s)%20-%20V_%7BEMF,dq%7D%7D%20=%20%5Cfrac%7B1%7D%7BR_s%20+%20sL_%7Bdq%7D%7D%20=%20%5Cfrac%7B%5Cfrac%7B1%7D%7BL_%7Bdq%7D%7D%7D%7B(s%20+%20%5Cfrac%7BR_s%7D%7BL_%7Bdq%7D%7D)%7D%20%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}})}}}">
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          </mfrac>
 +
        </mrow>
 +
      </mstyle>
 +
    </mrow>
 +
    <annotation encoding="application/x-tex">{\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}}})}}}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe9d2f69dac0e8e27f742f225ddb717f169b038" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.338ex; width:52.649ex; height:9.509ex;" alt="{\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}}})}}}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><br class="mw_emptyline_first"><br class="mw_emptyline">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:<br class="mw_emptyline"><span id="<@@@TAG151981@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
 
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)
 
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)
  
  
−
</math>
+
</math>" data-mw-type="tag" data-mw-id="151981" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0AG_c(s)%20=%20%5Cfrac%7BV_%7Bdq%7D%20-%20V_%7BEMF,dq%7D%7D%7BI_%7Be,dq%7D(s)%7D%20=%20k_p%20+%20%5Cfrac%7Bk_i%7D%7Bs%7D%20=%20%5Cfrac%7B1%7D%7Bs%7D(k_i%20+%20sk_p)%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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})}">
−
 
+
  <semantics>
−
 
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    <mrow class="MJX-TeXAtom-ORD">
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More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>
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    <annotation encoding="application/x-tex">{\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})}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b50211a6364f8445185950d57919bd1f11915ea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:49.698ex; height:6.343ex;" alt="{\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})}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><br class="mw_emptyline_first"><br class="mw_emptyline">More explicitly, the desired output voltage at any given point in time can be written as:<br data-attributes=""><span id="<@@@TAG107039@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
  
  
Line 70: Line 657:
  
  
−
</math>
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</math>" data-mw-type="tag" data-mw-id="107039" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%0A%0AV_%7Bdq%7D%20=%20(I_%7Bdq,requested%7D%20-%20I_%7Bdq,measured%7D)%20%5Cfrac%7B1%7D%7Bs%7D(k_i%20+%20sk_p))%20+%20V_%7BEMF,dq%7D%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{dq}=(I_{dq,requested}-I_{dq,measured}){\frac {1}{s}}(k_{i}+sk_{p}))+V_{EMF,dq}}">
−
 
+
  <semantics>
−
 
+
    <mrow class="MJX-TeXAtom-ORD">
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The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>
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    <annotation encoding="application/x-tex">{\displaystyle V_{dq}=(I_{dq,requested}-I_{dq,measured}){\frac {1}{s}}(k_{i}+sk_{p}))+V_{EMF,dq}}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21901ad9dcaaa3b4618afdebaf27a680089a853d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:55.367ex; height:5.176ex;" alt="{\displaystyle V_{dq}=(I_{dq,requested}-I_{dq,measured}){\frac {1}{s}}(k_{i}+sk_{p}))+V_{EMF,dq}}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><br class="mw_emptyline_first"><br class="mw_emptyline">The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br data-attributes=""><span id="<@@@TAG103648@@@>" class="mceNonEditable wikimagic mw-tag" title="<math>
  
  
Line 79: Line 751:
  
  
−
</math>
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</math>" data-mw-type="tag" data-mw-id="103648" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%0A%0AG_%7Bopen%20loop%7D(s)%20=%20G(s)G_c(s)%20=%20%5Cfrac%7B1%7D%7Bs%7D%5Cfrac%7Bk_p%7D%7BL_%7Bdq%7D%7D%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan><div class="mw-parser-output"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{openloop}(s)=G(s)G_{c}(s)={\frac {1}{s}}{\frac {k_{p}}{L_{dq}}}}">
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==Torque Regulation==
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            <mi>o</mi>
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==Surface Mount Permanent Magnet Machines==
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==Interior Permanent Magnet Machines==
+
            <mi>e</mi>
−
==Position Feedback<br />==
+
            <mi>n</mi>
−
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.
+
            <mi>l</mi>
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            <mi>o</mi>
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===Resolvers & Encoders===
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==Space Vector PWM==
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    <annotation encoding="application/x-tex">{\displaystyle G_{openloop}(s)=G(s)G_{c}(s)={\frac {1}{s}}{\frac {k_{p}}{L_{dq}}}}</annotation>
 +
  </semantics>
 +
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc5d51749701c323f3d0daa405c72c74cf698d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:34.107ex; height:6.176ex;" alt="{\displaystyle G_{openloop}(s)=G(s)G_{c}(s)={\frac {1}{s}}{\frac {k_{p}}{L_{dq}}}}"></span>
 +
</p></div><div class="mceNonEditableOverlay"></div></mwspan></span><br class="mw_emptyline_first"><br class="mw_emptyline"><br class="mw_emptyline"><br class="mw_emptyline"><br class="mw_emptyline"><div><h2>Torque Regulation</h2></div><div><h2>Surface Mount Permanent Magnet Machines</h2></div><div><h2>Interior Permanent Magnet Machines</h2></div><div><h2>Position Feedback<br data-attributes=""></h2></div>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 entire operating range.<div><h3>Resolvers &amp; Encoders</h3></div><div><h4>Calibration</h4></div><div><h2>Space Vector PWM</h2></div>
  
 
=Field Oriented Control for Asynchronous Machines=
 
=Field Oriented Control for Asynchronous Machines=

Revision as of 20:00, 20 June 2022

DC Brushed Motor Control

Six-Step Control

Field Oriented Control for Synchronous Machines

Most FSAE grade motor controllers use field oriented control. This is due to the efficiency and dynamic performance of the controller and motor. Field oriented control relies on controlling the DQ-axis currents through applying a specific DQ-axis voltage to the motor. For a good explaination of field oriented control see the video below (though it focuses on sensorless field oriented control).


<a href="https://www.youtube.com/watch?v=cdiZUszYLiA&t=11s" data-mce-href="https://www.youtube.com/watch?v=cdiZUszYLiA&t=11s" title="[1]" data-mw-type="external_link" class="link external mw-external-link mceNonEditable" data-mw-wikitext="%5Bhttps://www.youtube.com/watch?v=cdiZUszYLiA&t=11s%5D" contenteditable="false"><mwspan>[1]

</mwspan></a>

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<mwspan>¶</mwspan>of the magnetic field that the current will create. The real axis/the axis aligned with the phase A vector is called <span id="<@@@TAG155056@@@>" class="mceNonEditable wikimagic mw-tag" title="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} " data-mw-type="tag" data-mw-id="155056" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">

</mwspan>, while the imaginary axis/perpendicular axis is labeled <span id="<@@@TAG125345@@@>" class="mceNonEditable wikimagic mw-tag" title="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} " data-mw-type="tag" data-mw-id="125345" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Cbeta%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }">

</mwspan>. k is a scaling factor, where
<span id="<@@@TAG104565@@@>" class="mceNonEditable wikimagic mw-tag" title="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}} " data-mw-type="tag" data-mw-id="104565" data-mw-name="math" data-mw-wikitext="%3Cmath%3Ek%20=%20%5Cfrac%7B2%7D%7B3%7D%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32f140d5d8c9832545636b79bf45e81478334ca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:6.308ex; height:5.176ex;" alt="{\displaystyle k={\frac {2}{3}}}">

</mwspan> results in a vector with the same magnitude as the original 3-phase signals, and <span id="<@@@TAG109486@@@>" class="mceNonEditable wikimagic mw-tag" title="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}}} " data-mw-type="tag" data-mw-id="109486" data-mw-name="math" data-mw-wikitext="%3Cmath%3Ek%20=%20%5Csqrt%7B%5Cfrac%7B2%7D%7B3%7D%7D%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0412cac912c88cf59bc4156599e46640e80d09" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:8.632ex; height:6.176ex;" alt="{\displaystyle k={\sqrt {\frac {2}{3}}}}">

</mwspan> results in a power-invariant transform.
<span id="<@@@TAG191952@@@>" class="mceNonEditable wikimagic mw-tag" title="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 &amp; -\frac{1}{2} &amp; -\frac{1}{2} \\ 0 &amp; \frac{\sqrt 3}{2} &amp; -\frac{\sqrt 3}{2} \end{bmatrix} \begin{bmatrix} x_A \\ x_B \\ x_C \end{bmatrix} } " data-mw-type="tag" data-mw-id="191952" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%5Cbegin%7Bbmatrix%7D%0Ax_%5Calpha%20%5C%5C%0Ax_%5Cbeta%0A%5Cend%7Bbmatrix%7D%0A=%20k%0A%5Cbegin%7Bbmatrix%7D%0A1%20&%20-%5Cfrac%7B1%7D%7B2%7D%20&%20-%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%0A0%20&%20%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%20&%20-%5Cfrac%7B%5Csqrt%203%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0Ax_A%20%5C%5C%0Ax_B%20%5C%5C%0Ax_C%0A%5Cend%7Bbmatrix%7D%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4000c82275c067f5f6aafcbfd1792306c269e58" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.005ex; width:34.754ex; height:9.176ex;" alt="{\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}}}">

</mwspan>

Park Transform

The Park transform is used to transform quantities from being referenced to the stator to being referenced to the rotor. The <span id="<@@@TAG142237@@@>" class="mceNonEditable wikimagic mw-tag" title="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} " data-mw-type="tag" data-mw-id="142237" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%5Cbeta%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efb0e5523f52275f3193b0dfd9a92ad5b76830c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle \alpha \beta }">

</mwspan>-frame is "stationary", as in the axes remain fixed while the current<mwspan>¶</mwspan>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.<mwspan>¶</mwspan>The transform itself is simply a rotation matrix, rotating the <span id="<@@@TAG142237@@@>" class="mceNonEditable wikimagic mw-tag" title="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} " data-mw-type="tag" data-mw-id="142237" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%5Calpha%5Cbeta%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efb0e5523f52275f3193b0dfd9a92ad5b76830c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle \alpha \beta }">

</mwspan>-frame by the electrical angle of the rotor.
<span id="<@@@TAG167346@@@>" class="mceNonEditable wikimagic mw-tag" title="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) &amp; sin(\theta_r) \\ -sin(\theta_r) &amp; cos(\theta_r) \end{bmatrix} \begin{bmatrix} x_\alpha \\ x_\beta \end{bmatrix} } " data-mw-type="tag" data-mw-id="167346" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%5Cbegin%7Bbmatrix%7D%0Ax_d%5C%5C%0Ax_q%0A%5Cend%7Bbmatrix%7D%0A=%0A%5Cbegin%7Bbmatrix%7D%0Acos(%5Ctheta_r)%20&%20sin(%5Ctheta_r)%20%5C%5C%0A-sin(%5Ctheta_r)%20&%20cos(%5Ctheta_r)%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0Ax_%5Calpha%20%5C%5C%0Ax_%5Cbeta%0A%5Cend%7Bbmatrix%7D%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3936dfef8cb177c2c3c1ae223fe87335c86a1dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:36.206ex; height:6.176ex;" alt="{\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}}}">

</mwspan>

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).
<span id="<@@@TAG149363@@@>" class="mceNonEditable wikimagic mw-tag" title="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}} } " data-mw-type="tag" data-mw-id="149363" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0AI_%7Bdq%7D%20=%20%5Cfrac%7BV_%7Bdq%7D%20-%20V_%7Bemf%7D%7D%7BR_s%20+%20sL_%7Bdq%7D%7D%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3a5976ef651c6e22e536a8166ce9194ab517a3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:17.55ex; height:6.176ex;" alt="{\displaystyle I_{dq}={\frac {V_{dq}-V_{emf}}{R_{s}+sL_{dq}}}}">

</mwspan>

We can then write the equation for the system as follows:
<span id="<@@@TAG144167@@@>" class="mceNonEditable wikimagic mw-tag" title="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}})} } " data-mw-type="tag" data-mw-id="144167" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%20G(s)%20=%20%5Cfrac%7BI_%7Bdq%7D(s)%7D%7BV_%7Bdq%7D(s)%20-%20V_%7BEMF,dq%7D%7D%20=%20%5Cfrac%7B1%7D%7BR_s%20+%20sL_%7Bdq%7D%7D%20=%20%5Cfrac%7B%5Cfrac%7B1%7D%7BL_%7Bdq%7D%7D%7D%7B(s%20+%20%5Cfrac%7BR_s%7D%7BL_%7Bdq%7D%7D)%7D%20%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe9d2f69dac0e8e27f742f225ddb717f169b038" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.338ex; width:52.649ex; height:9.509ex;" alt="{\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}}})}}}">

</mwspan>

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:
<span id="<@@@TAG151981@@@>" class="mceNonEditable wikimagic mw-tag" title="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) } " data-mw-type="tag" data-mw-id="151981" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0AG_c(s)%20=%20%5Cfrac%7BV_%7Bdq%7D%20-%20V_%7BEMF,dq%7D%7D%7BI_%7Be,dq%7D(s)%7D%20=%20k_p%20+%20%5Cfrac%7Bk_i%7D%7Bs%7D%20=%20%5Cfrac%7B1%7D%7Bs%7D(k_i%20+%20sk_p)%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b50211a6364f8445185950d57919bd1f11915ea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:49.698ex; height:6.343ex;" alt="{\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})}">

</mwspan>

More explicitly, the desired output voltage at any given point in time can be written as:
<span id="<@@@TAG107039@@@>" class="mceNonEditable wikimagic mw-tag" title="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} } " data-mw-type="tag" data-mw-id="107039" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%0A%0AV_%7Bdq%7D%20=%20(I_%7Bdq,requested%7D%20-%20I_%7Bdq,measured%7D)%20%5Cfrac%7B1%7D%7Bs%7D(k_i%20+%20sk_p))%20+%20V_%7BEMF,dq%7D%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21901ad9dcaaa3b4618afdebaf27a680089a853d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:55.367ex; height:5.176ex;" alt="{\displaystyle V_{dq}=(I_{dq,requested}-I_{dq,measured}){\frac {1}{s}}(k_{i}+sk_{p}))+V_{EMF,dq}}">

</mwspan>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:
<span id="<@@@TAG103648@@@>" class="mceNonEditable wikimagic mw-tag" title="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}} } " data-mw-type="tag" data-mw-id="103648" data-mw-name="math" data-mw-wikitext="%3Cmath%3E%0A%0A%0AG_%7Bopen%20loop%7D(s)%20=%20G(s)G_c(s)%20=%20%5Cfrac%7B1%7D%7Bs%7D%5Cfrac%7Bk_p%7D%7BL_%7Bdq%7D%7D%0A%0A%0A%3C/math%3E" contenteditable="false"><mwspan>

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc5d51749701c323f3d0daa405c72c74cf698d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:34.107ex; height:6.176ex;" alt="{\displaystyle G_{openloop}(s)=G(s)G_{c}(s)={\frac {1}{s}}{\frac {k_{p}}{L_{dq}}}}">

</mwspan>




Torque Regulation

Surface Mount Permanent Magnet Machines

Interior Permanent Magnet Machines

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 entire operating range.

Resolvers & Encoders

Calibration

Space Vector PWM

Field Oriented Control for Asynchronous Machines

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