[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. The squirrel-cage (modelled as an equivalent sinusoïdally distributed three-phase winding) rotor is fitted with three imaginary current probes that measure the current in three rotor bars: [math]i_{RB,A}[/math], [math]i_{RB,B}[/math], [math]i_{RB,C}[/math].[br][br]The induced voltages (emfs) in the rotor are calculated using Faraday's law of induction: [math]e=-\frac{d\Phi}{dt}[/math] with [math]\Phi[/math] the coupled fluxes with the equivalent rotor phases. The rotor currents are calculated from the equivalent RL rotorcircuit for each phase: [math]e=L_r\frac{di}{dt}+R_ri[/math] solved for [math]i[/math]. The rotor resistance and inductance are assumed constant for all slip frequencies (no skin effect). The (changing) impressed stator currents together with the (changing) induced rotor currents form the (changing) resulting mmf wave [math]f_{tot}\left(x,t\right)[/math] (which is then used again to calculate the emfs in the equivalent rotor phases...).[br][br]The indirect method is used for field orientation in this animation: "[i]In the indirect methods for field orientation, synchronisation is obtained by means of the slip equation, which is indeed a necessary and sufficient condition for field orientation. From the desired q- and d-axis current components, the required slip frequency ([/i][math]\omega_s=\omega_2[/math][i]) is calculated with a slip calculator. This slip frequency is then combined with the measured rotor speed or angle to obtain the required stator frequency and phase. Variations of the torque producing current component [/i][math]I_{1q}[/math][i] cause a change of the torque proportional to this current variation (without any further transient); variations of the flux current component [/i][math]I_{1d}[/math][i] are accompanied with a transient determined by the field time constant (as the laws of physics prohibit any sudden changes of induction or flux).[/i]" [Electrical Machines and Drives - J. A. Melkebeek][br][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. For the stator 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/w2cvs3kd]https://www.geogebra.org/m/w2cvs3kd[/url], [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][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]. [br][br][b][br]Important note: [/b]Despite the best efforts of the author, it can be observed that the field orientation is [i][u]not perfect[/u][/i] in this animation:[br]- at lower values of [math]I_{1d}[/math], a variation of [math]I_{1q}[/math] does not anymore result in a perfectly proportional variation of the torque,[br]- a transient in the torque can be observed when [math]I_{1q}[/math] is quickly changed.[br][br][br][i]Please download the .ggb file and open with the Geogebra Classic 5 application (see [/i][url=https://www.geogebra.org/download]https://www.geogebra.org/download[/url][i]) 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% ([/i][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]Any and all feedback is welcome and can be sent to [url=mailto:timon.dewispelaere@kuleuven.be]timon.dewispelaere@kuleuven.be[/url].[/i][/justify]