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]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. [br][br][/i][b][i]Any and all feedback is welcome and can be sent to timon.dewispelaere@kuleuven.be.[/i][/b]