Torque oscillations at frequency LF Torque oscillations at frequency LF*(1-s) (Milovan’s motor, J reduced to 20) Unrelated oscillation after reach full speed... this one dependent upon inertia etc see LC circuit in other PPT 0-speed 1*LF osc half-speed LF*(1-s) = 0.5*LF osc

q-d currents show same oscillation pattern at frequency LF. (1-s) q-d currents show same oscillation pattern at frequency LF*(1-s). (M’s motor, J reduced to 20)

a/b/c currents have dc offset a/b/c currents have dc offset. The offset has initial transient decay (red) followed by oscillation(blue) that varies at frequency of s*LF (M’s motor, J=104)

Zoom-in on oscillating dc component 0-0. 8 sec Zoom-in on oscillating dc component 0-0.8 sec. This is the frequency of the rotor currents when viewed in the rotor reference frame.

Rotor Iq current transformed to rotor ref frame shows it has a decaying LF-frequency component component plus a steady component at frequency of s*LFduring acceleration. Perhaps this is the cause of the s*LF oscillation of offset during acceleration... not sure

Locked rotor start - current shows no oscillation (as expected s Locked rotor start - current shows no oscillation (as expected s*LF remains 0)

Locked rotor start - torque oscillations remain constant frequency of LF (as expected LF*(1-s) remains LF). Also interesting that the oscillations die down much slower during LR start.

Krause’s equivalent circuit for transient analysis of SCIM Krause’s equivalent circuit for transient analysis of SCIM. I believe that current will circulate in the loop shown at frequency (w-wr) = LF*s w = ref frame radian speed * wr  (Poles/2)* 2*pi* Rotor Speed ** Te=1.5*P/2*Lm*(Iqs*Idr-Ids*Iqr) * Most convenient reference frame is synchronous i.e w = we = 2*pi*LF. In that case Vqs and Vds are dc. (With proper choice of supply phase Vds = 0) ** note wr is NOT 2*pi*RotorSpeed, instead it is adjusted by number of pole pairs so that it matches we when rotor speed is zero Lamba_qs = Lls*Iqs+Lm*(Iqs+Iqr) Lamba_ds = Lls*Ids+Lm*(Ids+Idr) Lamba_qr = Llr*Iqr+Lm*(Iqs+Iqr) Lamba_dr = Llr*Idr+Lm*(Ids+Idr)

Rough proof of claimed resonant frequency from previous slide Assume the rotor currents in sync frame oscillates at some unknown frequency wn Neglect resistive losses. Linear system -> neglect other currents flowing in magnetizing branch (response is sum of responses) We have loop currents Iqr*cos(wn*t+theta) and Iqd*cos(wn*t+phi) Let s be derivative operator. Let Ltot = Llr + Lm KVL around q axis rotor loop: Ltot * s*Iqr = - (we-wr)*LAdr = - (we-wr)*Ltot * Idr Cancel Ltot and solve for Iqr s*Iqr = - (we-wr)* Idr [1] KVL around d axis rotor loop: Ltot * s*Idr = (we-wr)*LAqr = (we-wr)*Ltot*Iqr Cancel Ltot and solve for Idr Idr = (we-wr)*Iqr/s [2] Plug Ird from [2] into [1] s*Iqr = - (we-wr)* (we-wr)*Iqr/s s^2*Iqr = -(we-wr)^2*Iqr d^2/dt^2(iqr(t) = -(we-wr)^2*iqr(t) This differential equation is satisfied by sinusoid of frequency (we-wr) we - wr = s*LF is a resonant frequency associated with circulating current in the path shown in the previous slide From equation [2] we observe Iqr =(we-wr)*Idr so we expect Iqr leads Idr by 90 degrees... that is exactly what we see in simulation (next slide). Stator currents also oscillate at the same frequency because they share the magnetizing branch so their flux linkage is affected. They go along for the ride.

Iqr leads Idr

Lamba_qs = Lls*Iqs+Lm*(Iqs+Iqr) Lamba_ds = Lls*Ids+Lm*(Ids+Idr) Circuilating loops shown will oscillate at frequency w-(w-wr) = wr = LF*(1-s). Proof could proceed identically to previous slide. Given the defined polarity of currents we expect equal/opposite stator androtor currents, which is what we see (next slide) w = ref frame radian speed * wr  (Poles/2)* 2*pi* Rotor Speed ** Te=1.5*P/2*Lm*(Iqs*Idr-Ids*Iqr) * Most convenient reference frame is synchronous i.e w = we = 2*pi*LF. In that case Vqs and Vds are dc. (With proper choice of supply phase Vds = 0) ** note wr is NOT 2*pi*RotorSpeed, instead it is adjusted by number of pole pairs so that it matches we when rotor speed is zero Lamba_qs = Lls*Iqs+Lm*(Iqs+Iqr) Lamba_ds = Lls*Ids+Lm*(Ids+Idr) Lamba_qr = Llr*Iqr+Lm*(Iqs+Iqr) Lamba_dr = Llr*Idr+Lm*(Ids+Idr)

Compare red-to-red and yellow to yellow Compare red-to-red and yellow to yellow. See that rotor and stator current oscillations have opposite polarity, consistent with circulating loop shown on previous slide (when we consider the defined polarity rotor and stator current both positive entering top of magnetizing branch). Similar behavior shown in slide 2 - that was J=20, this is J=104