Animation is based on chapter 3 (p97) of the book Electrical Machines and Drives by Jan A. Melkebeek (ISBN 978-3-319-72729-5).[br][br]"The jumps of the mmf curve at each slot k correspond to the Dirac functions for the ampere-turns in the slot k."[br][br]Compare the mmf wave of this animation (with [math]q=25[/math]) with that of the following animations (where [math]q=+\infty[/math] and the mmf wave is the result of a stepwise linearly constant current layer): [url=https://www.geogebra.org/m/azhgwttv]https://www.geogebra.org/m/azhgwttv[/url] and [url=https://www.geogebra.org/m/tny9ykfg]https://www.geogebra.org/m/tny9ykfg[/url].[br][br][i]Please download the .ggb file and open with the Geogebra Classic 5 application (see [url=https://www.geogebra.org/download]https://www.geogebra.org/download[/url]) if the animation is too slow in your browser. The animation is designed to be opened on a 1080p screen and with "Make everything bigger" setting of Windows set to 100% ([url=https://support.microsoft.com/en-us/windows/change-the-size-of-text-in-windows-1d5830c3-eee3-8eaa-836b-abcc37d99b9a]Change the size of text in Windows - Microsoft Support[/url]).[/i][br][br][i][b]Any and all feedback is welcome and can be sent to [url=mailto:timon.dewispelaere@kuleuven.be]timon.dewispelaere@kuleuven.be[/url].[br][br][/b][/i]Please be patient while moving the sliders: geogebra is calculating!

In this animation the rotor (2) and stator (1) are fed by a controllable (in phase, frequency and amplitude) three phase current and the rotor speed follows from the slip equation: [math]N_p\Omega_m=\omega_1-\omega_2[/math] with [math]N_p[/math] the number of pole pairs ([math]N_p=1[/math] in the animation). Note also that for [math]t=0s[/math] phase belt 1U and 2U are in line when [math]\alpha_0=0°[/math] (with [math]\alpha_0[/math] the rotor angular displacement at [math]t=0s[/math]). [math]\frac{d\Omega_m}{dt}=0[/math] is assumed so that the motor torque [math]T_M=T_2=-T_1[/math] is always equal to the load torque [math]T_L[/math]: [math]T_M=T_L[/math] (i.e. only steady state conditions are animated). [br]Red and pink waveforms are the current densities [math]a_1\left(x,t\right)[/math] and [math]a_2\left(x,t\right)[/math] (A/m) of stator and rotor respectively. It is assumed that the phase conductors are spread very finely/thin over the phase width ([math]q=\infty[/math]), so that the current density is a constant over a phase width. [br]The black waveforms [math]f_1\left(x,t\right)[/math] (dashed line) and [math]f_2\left(x,t\right)[/math] (dash-dot line) are the accompanying mmfs (Aw) produced by the current densities [math]a_1\left(x,t\right)[/math] and [math]a_2\left(x,t\right)[/math] respectively (where [math]f_x=\int_{x_0}^xa_{ }\cdot dx[/math] and symmetry requirements allow to locate the neutral point where [math]f_x\left(x_0\right)=0[/math]). Please see also: [url=https://www.geogebra.org/m/azhgwttv]https://www.geogebra.org/m/azhgwttv[/url] and [url=https://www.geogebra.org/m/tny9ykfg]https://www.geogebra.org/m/tny9ykfg[/url].[br]The solid black line is the total mmf of rotor and stator, [math]f_{tot}\left(x,t\right)=f_1\left(x,t\right)+f_2\left(x,t\right)[/math]. [br]The torque resulting from a rotating fundamental field layer and rotating fundamental current layer can be calculated with [math]T=\frac{N_p\tau_p}{\pi}\cdot2\cdot N_p\int_{-\frac{\tau_p}{2}}^{\frac{\tau_p}{2}}b\left(x,t\right)\cdot a\left(x,t\right)\cdot l\cdot dx[/math]. In this animation the saturation of the magnetic circuit is neglected so that the air-gap induction in each point of the armature circumference follows directly from the local total mmf of rotor and stator: [math]b_{tot}\left(x,t\right)=\mu_0\frac{f_{tot}\left(x,t\right)}{\delta_x\left(x\right)}[/math] with [math]f_{tot}\left(x,t\right)=f_1\left(x,t\right)+f_2\left(x,t\right)[/math] and [math]\delta_x[/math] the air gap length. The torques [math]T_1[/math] and [math]T_2[/math] (produced by the fundamental functions) are given by [Nm]:[math]T_1=\frac{N_p\tau_p}{\pi}\cdot N_p\int_0^{2\tau_p}b_{tot}\left(x,t\right)\cdot a_1\left(x,t\right)\cdot l\cdot dx=\frac{N_p\tau_p}{\pi}\cdot N_p\cdot l\cdot\frac{\mu_0}{\delta_x}\int_0^{2\tau_p}f_{tot}\left(x,t\right)\cdot a_1\left(x,t\right)\cdot dx=k\cdot\int_0^{2\tau_p}f_{tot}\left(x,t\right)\cdot a_1\left(x,t\right)\cdot dx[/math] and [math]T_2=\frac{N_p\tau_p}{\pi}\cdot N_p\int_0^{2\tau_p}b_{tot}\left(x,t\right)\cdot a_2\left(x,t\right)\cdot l\cdot dx=\frac{N_p\tau_p}{\pi}\cdot N_p\cdot l\cdot\frac{\mu_0}{\delta_x}\int_0^{2\tau_p}f_{tot}\left(x,t\right)\cdot a_2\left(x,t\right)\cdot dx=k\cdot\int_0^{2\tau_p}f_{tot}\left(x,t\right)\cdot a_2\left(x,t\right)\cdot dx[/math] with [math]k=\frac{N_p\tau_p}{\pi}\cdot N_p\cdot l\cdot\frac{\mu_0}{\delta_x}[/math] some (machine) constant. In the animation the torques are then given in relation to the maximum attainable torque in the animation [pu].[br][br]Field orientation can be achieved when:[br][list][*]the fundamental parts of [math]a_1(x,t)[/math] and [math]f_2(x,t)[/math] are in phase, or[/*][*]the fundamental parts of [math]a_2(x,t)[/math] and [math]f_1(x,t)[/math] are in phase[/*][/list]Indeed, for field orientation the torque of the rotor [math]T_2[/math] or stator [math]T_1[/math] (action = reaction) is maximum for some given stator and rotor current magnitudes.[br][br]Animation and equations are based on the book Electrical Machines and Drives by Jan A. Melkebeek (ISBN 978-3-319-72729-5).[br][br][i][i]Please download the .ggb file and open with the Geogebra Classic 5 application (see [url=https://www.geogebra.org/download]https://www.geogebra.org/download[/url]) if the animation is too slow in your browser. The animation is designed to be opened on a 1080p screen and with "Make everything bigger" setting of Windows set to 100% ([url=https://support.microsoft.com/en-us/windows/change-the-size-of-text-in-windows-1d5830c3-eee3-8eaa-836b-abcc37d99b9a]Change the size of text in Windows - Microsoft Support[/url]).[/i][br][br][/i][b][i]Any and all feedback is welcome and can be sent to timon.dewispelaere@kuleuven.be.[/i][/b]

[justify]In this animation the stator (three phase symmetrical winding, phase width [math]S=\frac{1}{3}\tau_p[/math], terminals 1U, 1V and 1W) is fed by a three phase current and the rotor (smooth rotor, distributed winding, phase width [math]S=\frac{2}{3}\tau_p[/math], terminals + and -) is fed by a DC current. The machine is field oriented at all times which allows the rotor speed to be dynamically controlled by controlling the rotor torque (through the torque producing component of the stator current [math]I_{1,q}[/math]). The instantaneous stator currents ([math]i_U[/math], [math]i_V[/math] and [math]i_W[/math]) phase and amplitude follow from: (1) the rotor angle, (2) setting the torque producing component [math]I_{1,q}[/math] of the stator current, and (3) setting the flux producing component [math]I_{1,d}[/math] of the stator current. [math]\frac{d\Omega_m}{dt}J_{ }=T_M[/math] is assumed (load torque [math]T_L[/math] is zero) so that the rotor accelerates/decelerates according to the applied electromechanical torque [math]T_M=T_2=-T_1[/math] (i.e. dynamic conditions are animated). Equations of motion were implemented as shown in [url=https://www.youtube.com/watch?v=oGWvkP9O86o]'Geogebra Physics Simulation - Tutorial' by Kornél Rokolya (Youtube)[/url]. [/justify]See also https://www.geogebra.org/m/kj4bu8by for further explanations.[br][br][i]Please download the .ggb file and open with the Geogebra Classic 5 application (see [url=https://www.geogebra.org/download]https://www.geogebra.org/download[/url]) if the animation is too slow in your browser. The animation is designed to be opened on a 1080p screen and with "Make everything bigger" setting of Windows set to 100% ([url=https://support.microsoft.com/en-us/windows/change-the-size-of-text-in-windows-1d5830c3-eee3-8eaa-836b-abcc37d99b9a]Change the size of text in Windows - Microsoft Support[/url]).[br][br][/i][b][i]Any and all feedback is welcome and can be sent to [url=mailto:timon.dewispelaere@kuleuven.be]timon.dewispelaere@kuleuven.be[/url].[/i][/b]