[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 slip-ring rotor (three phase symmetrical winding, phase width [math]S=\frac{1}{3}\tau_p[/math], terminals 2U, 2V and 2W) is short circuited and blocked ([math]\Omega_m=0[/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 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 (changing) impressed stator currents together with the (changing) induced rotor currents form the (changing) resulting mmf field [math]f_{tot}\left(x,t\right)[/math] (which is then used again to calculate the emfs in the 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. 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: [br][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][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 (very) 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]