fswiki.us - User contributions [en]

Motor Control

2022-06-21T00:21:23Z

Etrain Troy: /* Torque Regulation */

[[Category:Electric Vehicle]]
=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 improved efficiency and dynamic performance of the controller and motor controller. This control scheme relies on controller the DQ-axis currents by applying a DQ-axis voltage.
==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 <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
<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.

<math>
\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}
</math>
===Park Transform===
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
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 <math>\alpha\beta</math>-frame by the electrical angle of the rotor.

<math>
\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}
</math>
==Current Regulation==
A PI controller is commonly used to regulate the 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).
<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>

==Torque Regulation==
There are a couple ways to calculation the required current to achieve a desired torque
===Zero D-Axis Current===
===Maximum Torque Per Amp (MTPA)===
===Thermal Limiting===
Losses in the motor and motor controller are a function of current. Therefore we need to reduce the current through the devices to limit the temperature of both the motor and controller.
===Torque Ramp===
Slowly ramp up torque (over a period of 10's or 100's of milliseconds) in order to prevent mechanical damage to drivetrain.

==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
==Position Feedback<br />==
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.
===Resolvers & Encoders===
====Calibration====
==Space Vector PWM==

=Field Oriented Control for Asynchronous Machines=

Motor Control

2022-06-21T00:15:19Z

Etrain Troy: /* Current Regulation */

[[Category:Electric Vehicle]]
=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 improved efficiency and dynamic performance of the controller and motor controller. This control scheme relies on controller the DQ-axis currents by applying a DQ-axis voltage.
==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 <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
<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.

<math>
\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}
</math>
===Park Transform===
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
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 <math>\alpha\beta</math>-frame by the electrical angle of the rotor.

<math>
\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}
</math>
==Current Regulation==
A PI controller is commonly used to regulate the 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).
<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>

==Torque Regulation==
==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
==Position Feedback<br />==
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.
===Resolvers & Encoders===
====Calibration====
==Space Vector PWM==

=Field Oriented Control for Asynchronous Machines=

Motor Control

2022-06-21T00:14:16Z

Etrain Troy: /* Field Oriented Control for Synchronous Machines */

[[Category:Electric Vehicle]]
=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 improved efficiency and dynamic performance of the controller and motor controller. This control scheme relies on controller the DQ-axis currents by applying a DQ-axis voltage.
==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 <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
<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.

<math>
\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}
</math>
===Park Transform===
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
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 <math>\alpha\beta</math>-frame by the electrical angle of the rotor.

<math>
\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}
</math>
==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).
<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>

==Torque Regulation==
==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
==Position Feedback<br />==
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.
===Resolvers & Encoders===
====Calibration====
==Space Vector PWM==

=Field Oriented Control for Asynchronous Machines=

Motor Control

2022-06-21T00:08:57Z

Etrain Troy: /* Resolvers & Encoders */

[[Category:Electric Vehicle]]
=DC Brushed Motor Control=
=Six-Step 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 <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
<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.

<math>
\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}
</math>
===Park Transform===
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
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 <math>\alpha\beta</math>-frame by the electrical angle of the rotor.

<math>
\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}
</math>
==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).
<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>

==Torque Regulation==

==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
==Position Feedback<br />==
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.

===Resolvers & Encoders===
====Calibration====

==Space Vector PWM==

=Field Oriented Control for Asynchronous Machines=

Motor Control

2022-06-21T00:08:29Z

Etrain Troy: Undo revision 2391 by Etrain Troy (talk)

[[Category:Electric Vehicle]]
=DC Brushed Motor Control=
=Six-Step 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 <math>\alpha</math>, while the imaginary axis/perpendicular axis is labeled <math>\beta</math>. k is a scaling factor, where
<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.

<math>
\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}
</math>
===Park Transform===
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
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 <math>\alpha\beta</math>-frame by the electrical angle of the rotor.

<math>
\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}
</math>
==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).
<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>

==Torque Regulation==

==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
==Position Feedback<br />==
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.

===Resolvers & Encoders===
==Space Vector PWM==

=Field Oriented Control for Asynchronous Machines=

Motor Control

2022-06-21T00:00:28Z

Etrain Troy: /* Field Oriented Control for Synchronous Machines */

[[Category:Electric Vehicle]]
=DC Brushed Motor Control=
=Six-Step Control=
<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&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]<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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle scriptlevel="0" displaystyle="true">
<mi>α</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</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>
</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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle scriptlevel="0" displaystyle="true">
<mi>β</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</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>
</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}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle scriptlevel="0" displaystyle="true">
<mi>k</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k={\frac {2}{3}}}</annotation>
</semantics>
</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>
</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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</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}
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}
</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&%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><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&-{\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}}}">
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<annotation encoding="application/x-tex">{\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}}}</annotation>
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</math></span><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}}}"></span>
</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 }">
<semantics>
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<mstyle scriptlevel="0" displaystyle="true">
<mi>α</mi>
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<annotation encoding="application/x-tex">{\displaystyle \alpha \beta }</annotation>
</semantics>
</math></span><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 }"></span>
</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 }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle scriptlevel="0" displaystyle="true">
<mi>α</mi>
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<annotation encoding="application/x-tex">{\displaystyle \alpha \beta }</annotation>
</semantics>
</math></span><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 }"></span>
</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}
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}
</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&%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><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})&sin(\theta _{r})\\-sin(\theta _{r})&cos(\theta _{r})\end{bmatrix}}{\begin{bmatrix}x_{\alpha }\\x_{\beta }\end{bmatrix}}}">
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<annotation encoding="application/x-tex">{\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}}}</annotation>
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</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})&sin(\theta _{r})\\-sin(\theta _{r})&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}}
</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>
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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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<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)

</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})}">
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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>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</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}}">
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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>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</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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<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 & Encoders</h3></div><div><h4>Calibration</h4></div><div><h2>Space Vector PWM</h2></div>

=Field Oriented Control for Asynchronous Machines=

Electric Motors

2022-06-20T23:44:57Z

Etrain Troy: /* Common FSAE Motors */

To propel an electric vehicle, electricity must be converted to mechanical force. Electric motors use current flowing through wires to create magnetic fields that interact with other magnetic fields in the motor to create motion.
=Motor Types=

==DC Motors==
DC motors are electro-mechanical machines which take DC inputs to generate torque. These machines include Brushed DC, Brushless DC machines (BLDC), and homopolar machines among others. While some use mechanical commutators like brushes, some DC machines commutate electrically based on the rotor position.
===Brushed DC Motor===
Brushed DC Motors are the conventional hobbyist motor and come in two main varieties: Permanent Magnet Based (PMDC) and Field-Winding Based (FWDC). They use mechanical brushes to commutate the the flow of current through the rotor to generate a torque. Because of this, they are easy to control, and do not require a position sensor to track the rotation of the motor. For the PMDC machine, the flux generated by the rotor interacts with the permanent magnet flux to generate torque. For the FWDC machine, the permanent magnets are replaced by another set of coils which generate the stator flux. This allows the machines to operate at high speeds using flux weakening techniques.

For FSAE applications, Brushed DC machines can reduce system complexity; however, at higher power levels, these machines have low efficiencies and must have regular maintenance of brushes. In addition, these machines have relatively low torque and power density which can increase system mass and cause packaging issues. For these reasons, they are not commonly used in FSAE powertrains.
===Brushless DC Motor===
Brushless DC motors (BLDC) are another form of DC machines that commutate electrically (i.e. without brushes). BLDC machines typically consist of a rotor with permanent magnets and a stator which is wound such that the EMF has a trapezoidal shape. While the structure shares many similarities with the AC Synchronous machines, the distinction is in the control methodology and corresponding winding structure.

Rather than using a high resolution control algorithm like Field Oriented Control, BLDC motors use a heavily discretized, six-step control method. This is implemented by commanding a DC voltage (averaged through PWM) through two of the motor phases based on the rotor position. This is implemented by Hall Effect sensors embedded in the machine which determine which of the six possible phases to regulate current in.

In addition to the Hall Effect sensor used for commutation, BLDC motors often use an [[Rotary Position Sensor#Encoder|encoder]] to track the rotational position and velocity of the motor for control purposes.

BLDC motors are advantageous for their high torque density (higher than SMPM machines); however, the six-step method generates a large amount of torque ripple which reduces efficiency. BLDC machines also cannot flux weaken with conventional techniques, so problems can arise at higher speeds if voltage limited.
==AC Synchronous Motors==
AC Synchronous Machines a class of machines characterized by the fact that their rotor flux is synchronous with the rotor angle. These category includes surface mount (SMPM), synchronous reluctance (synRel), and interior permanent magnet (IPM) machines among others. These machines use PM or mutual torque along with reluctance torque to generate torque over a wide speed range. AC Synchronous machines also have sinusoidal EMFs due to their winding distribution. These machines typically use high performance control aglorithms like Field Oriented Control which require a high resolution position sensors like a [[Rotary Position Sensor#Resolver|resolver]].

===Surface Mount Permanent Magnet Machines===
Surface mount permanent magnet machines (SMPM or SMPMSM) are the most straight-forward type of synchronous machines. The AC stator consists of three phase wired in either a WYE or Delta configuration and usually has a sinusoidal winding distribution to achieve a sinusoidal Back-EMF. The rotor contains permanent magnets which are fixed on the surface of the rotor iron.

[[File:Cross-section-of-SPM-motor.png|right|middle|thumb|Four pole SMPM Machine Cross Section <ref name="Wang"> Wang, Yawei & Bianchi, N. & Bolognani, Silverio & Alberti, Luigi. (2017). Synchronous motors for traction applications. 1-8. 10.23919/EETA.2017.7993210. </ref> ]]

Because the permeability of the magnets is close to air, the ideal SMPM has no rotor saliency and therefore no reluctance torque. The resultant torque expression is dependent solely on the current orthogonal to the rotor flux. I.E The only current that produces torque in the machine is that which generates a magnetic field in the quadrature axis in the rotor reference frame. The resultant torque expression is below.

<math>

\tau = \frac{3Np}{4}\Lambda_{PM}I_q^r

</math>

Where...

<math>Np</math> = Number of Motor Poles

<math>\Lambda_{PM}</math> = Permanent Magnet Flux Linkage

<math>I_q^r</math> = Quadrature Current in rotor reference frame

For FSAE, SMPMs are good choice because of their high torque density and efficiency. At low speeds SMPM machines can also achieve unity power factor which is ideal when the vehicle is power limited. Using Field oriented control, SMPM's can flux-weaken as well to achieve higher speeds. Depending on the inductance of the machine, can achieve constant power after a certain speed. One problem with the SMPM design is that at high speeds the magnets experience high amounts of tensile stress. Because of non-idealities in the magnetic material, this stress worsens the performance of the material and inhibits high speed operation. This is not usually a problem for FSAE vehicles however.

===Synchronous Reluctance Machines===
Rather than using permanent magnets, Synchronous Reluctance (SynRel) Machines rely on the magnetic saliency of the rotor to generate a reluctance torque. The rotors of these machines have complex flux barriers cut out of the rotor lamination to achieve these salient poles. These machines are not as common as there SMPM counterparts, but they allow easy control of the flux in the machine. Because they do not contain permanent magnets, they are often thought of as the synchronous analogy to induction machines. Furthermore, a variation of SynRel machines, Switched Reluctance Machines, are often thought of as BLDC Machines are to SMPM machines.

[[File:Syn rel reference frames.png|right|middle|thumb|Four Pole PM Assisted Synchronous Reluctance Machine <ref name="SynRelIMG"> Bianchi, N. & Bolognani, Silverio & Bon, Diego & Pre, Michele. (2009). Rotor Flux-Barrier Design for Torque Ripple Reduction in Synchronous Reluctance and PM-Assisted Synchronous Reluctance Motors. Industry Applications, IEEE Transactions on. 45. 921 - 928. 10.1109/TIA.2009.2018960. </ref>]]
<math>

\tau = \frac{3Np}{4}(L_d - L_q)I_d^rI_q^r

</math>

Where...

<math>Np</math> = Number of Motor Poles

<math>L_q</math> = Quadrature Axis Inductance in rotor reference frame

<math>L_d</math> = Direct Axis Inductance in rotor reference frame

<math>I_q^r</math> = Quadrature Axis Current in rotor reference frame

<math>I_d^r</math> = Direct Axis Current in rotor reference frame

As previously stated, SynRel machines have intricate flux barriers that generate salient poles in the rotor. While this allows for easy control of machine's flux levels, the flux barriers can cause large mechanical stress concentrations which can cause problems for high torque applications. With that said, these machines can still achieve high levels of efficiency and have started to become more common place. Specifically, variations like the switched reluctance machine have been used in some electric vehicle applications despite their noise and torque ripple issues.

===Interior Permanent Magnet Machines===
Interior Permanent Magnet (IPM or IPMSM) Machines are newer type of synchronous machine that embed the rotor magnets in the rotor iron. There are several different types of IPM machines including inset, embedded, spoke, etc. Because the magnets which have a low incremental permeability are embedded in the highly permeable rotor iron, IPM machines a substantial rotor saliency. IPM machines, therefore, can use both permanent magnet and reluctance torque.

[[File:IPM cross section.png|right|middle|thumb|Four Pole IPM Cross Section <ref name="IPM_Machine">Sim, Hyun-Woo & Lee, June-Seok & Lee, Kyo-Beum. (2014). On-line Parameter Estimation of Interior Permanent Magnet Synchronous Motor using an Extended Kalman Filter. Journal of Electrical Engineering and Technology. 9. 10.5370/JEET.2014.9.2.600. </ref>]]
<math> \tau = \frac{3Np}{4}((L_d - L_q)I_d^r+\Lambda_{PM})I_q^r </math>

Where...

<math>Np</math> = Number of Motor Poles

<math>\Lambda_{PM}</math> = Permanent Magnet Flux Linkage

<math>L_q</math> = Quadrature Axis Inductance in rotor reference frame

<math>L_d</math> = Direct Axis Inductance in rotor reference frame

<math>I_q^r</math> = Quadrature Axis Current in rotor reference frame

<math>I_d^r</math> = Direct Axis Current in rotor reference frame

IPM machines are advantageous for multiple reasons. First, because the magnets are embedded in the rotor, the magnets see predominately compressive forces rather than tensile in SMPM designs at high speeds. Because high performance magnets can withstand higher compressive forces than tensile, IPM motors can typically run at higher speeds than SMPM machines. Second, because of IPM's saliency and large inductance's, these machines can easily flux weaken and achieve constant power over a wide speeds range.

==AC Asynchronous Motors==
Asynchronous machines are a class of electric motors which are characterized by the fact that their rotor flux is not necessarily synchronous with the rotor itself. The most common and pervasive type of AC asynchronous machines in the induction machine. There are several different configurations of induction motors (wound rotor, squirrel cage rotor, slip-ring, etc); however, they all have similar core operating fundamentals. That is, each machine generates torque by inducing currents on the rotor, thus generating a rotor flux density which interacts with the stator flux.

[[File:Rotterdam_Ahoy_Europort_2011.JPG|right|middle|thumb|Induction Machine <ref name="Induction Machine">Wikipedia contributors. (2020, May 17). Induction motor. In Wikipedia, The Free Encyclopedia. Retrieved 16:56, May 20, 2020, from https://en.wikipedia.org/w/index.php?title=Induction_motor&oldid=957240578 </ref> ]]

The torque analysis for an induction machines is more complex than that of a synchronous machine as there are many synchronous reference frames to choose from (e.g. Stator Current Frame, Stator Flux Reference Frame, Rotor Flux reference frame, etc.). For control purposes, namely Direct Field Oriented control, the machine dynamics are represented in the rotor flux reference frame. Additionally, the induction machines must regulate both the stator currents and flux levels in the rotor.

<math> \tau = \frac{3Np}{4}\frac{M}{L_r}|\lambda_r|i_{sq}^{\lambda_r} </math>

Where...

<math>Np</math> = Number of Motor Poles

<math>\lambda_{r}</math> = Rotor flux linkage

<math>M</math> = Mutual Inductance

<math>L_r</math> = Rotor Inductance

<math>i_{sq}^{\lambda_r}</math> = Stator Quadrature Current in rotor flux linkage reference frame

While Induction motors are widely used in industrial applications because of their low cost (no permanent magnets) and self starting features, induction machines are not often used in electric vehicles. This is because they are often not as torque dense as their PMSM counterparts and for efficiency purposes due to added rotor losses due to the induced rotor currents. In addition, even with Direct Field Oriented control methods, induction machines are not as responsive as synchronous machines.

=Common FSAE Motors=
When selecting a motor or motors for a car, one must consider the mass, size, power, efficiency, available controllers, and ease of integration. The most common motors in FSAE are those from Emrax, AMK, and Plettenberg.

=References=
<references />

[[Category:Electric Vehicle]]

Motor Control

2020-06-03T01:24:41Z

Etrain Troy: Added more sections to FOC

Motor Control

2020-06-03T01:04:15Z

Etrain Troy: Wrote some stuff about current regulation

[[Category:Electric Vehicle]]
=DC Brushed Motor Control=
=Six-Step Control=
=Field Oriented Control for Synchronous Machines=
==Transformations==
===Clarke Transform===
===Park Transform===
==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).

<math>
I_{dq} = \frac{V_{dq} - V_{emf}}{R_s + sL_{dq}}
</math>

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>

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:
<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)

</math>

More explicitly, the desired output voltage at any given point in time can be written as:<br /><math>

V_{dq} = (I_{dq,requested} - I_{dq,measured}) (k_p + \frac{k_i}{s} = \frac{1}{s}(k_i + sk_p)) + V_{EMF,dq}

</math>

The back EMF is estimated using motor parameters and rotational speed. Also note that the entire system simplifies as follows:<br /><math>

G_{open loop}(s) = G(s)G_c(s) = \frac{1}{s}\frac{k_p}{L_{dq}}

</math>
==Surface Mount Permanent Magnet Machines==
==Interior Permanent Magnet Machines==
=Field Oriented Control for Asynchronous Machines=

Motor Control

2020-06-03T00:51:04Z

Etrain Troy: