1. Overview Part1 - Analysis of base case (Resonant frequency ~ 6hz, excitation frequency ~ 6hz). Sensitivity analysis of eigenvalue. Proposed intuitive RCL circuit to predict eigenvalue behavior. Transfer functions and simulations for base case. Part 2 - Base case system (6hz resonant frequency), but change excitation to 9hz. Simulation results agree with part 1 transfer function. Part 3 - New system (discard flywheel, increase voltage) creates resonant frequency of 9hz. Excite the new system at 6hz. Includes transfer functions and simualtios Part 4 - Review of current waveforms. The base case chosen is about 50% higher than posted current waveforms. Re-simulation with lower torque recreates the posted waveforms well. Resonant frequencies don’t change and oscillation magnitudes are scaleable

2. Part 1 - Base Case: 100hp 1800 rpm motor parameters and operating point (for linearization) Rs = 0.043934 Xls = 0.335347 Xm = 8.433329 Xlr = 0.338430 Rr = 0.025981 J = 1.8589 (acts as monolithic rigid inertia at the frequencies of interest) Operating Point –LoadLevel = 25% - multiplier for base load –Voltage = 1 - multiplier for base voltage. Base quantities –Base voltage = 460vac –Base load = 100hp –Base torque ~ 100 N-m

3. Sensitivity analysis of “rotor” eigenvalue based on 10% change in each parameter. Found that rotor torsional resonant freq is proportional to voltage, inversely proportional to square root of inertia. If we add together both leakage reactances, we find resonant frequency is inversely proportional to the sqrt of sum of leakage reactances. Rotor resistance has a big impact on damping (but not on resonant frequency). Magnetizing reactance shows very little contribution (probably current through large magnetizing reactance cannot change fast enough to participate). The next two slides present a crude attempt to create an intuitive RLC circuit that roughly recreates this behavior based on assumptions: 1 - assume magnetizing inductance is so large that magneitizing current is constant during oscilaltion at the frequencies of interest (confirmed by simulation) 2 - assume d axis parameters are constant and do not oscillate (approximately confirmed by simulation). As shown, we replace the d-axis rotor speed voltage source with Ceq =-J/Lambda_dr^2. It creates in the q-axis circuit a series RLC circuit which includes L=total leakage reactance, C = as above, R = Rs+Rr We know the resonant frequency of a lightly-damped RLC circuit is roughly = 1/sqrt(LC). This matches many of features above (F~J^-0.5, F~Xleakage^-0.5, f~V since Lamba_dr is roughly proportional to V). This explains the resonant frequency behavior very well, but does not do as well on the damping. For example does not explain why Rr contributes to damping and Rs does not. Also does not explain the leakage inductance contribution to damping.

4. Induction motor transient eq ckt. Fig 4.5-1 of Krauss marked to show cage rotor and applied voltage in phase with q axis (d axis applied voltage is 0). 0-axis circuit ommitted under assumption that applied voltages sum to 0. 0 0 0 Note: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)

5. Now add an oscillating condition related to load torque and slip. LM is very large inductance. Assume the frequency of oscillaton is high enough that there is no significant change in Im (but not so high as to preclude oscillation of leakage currents). The current in magnetizing branch resembles a constant (DC) current source at a value corresponding to SS before start of the oscillation. Since Lambda_qr is small, the oscillation does not show up much in the d-axis rotor circuit. The d-axis circuit remains roughly steady so we consider Lambda_dr as a constant. The speed voltage source in the dq circuit (call it Vceq) is then Vceq = Lambda_dr*wslip. Telec-Tmech = -J*d/dt(wsync-wslip). Can simplify thru linearization etc to Telec = -J*d/dt(wslip). Substitute Telec ~ - Lamdba_dr*Iqs. - Lamdba_dr*Iqs=J*d/dt(wslip). Iqs = - J*d/dt(wslip)/Lambda_dr [Equation 1]. But since Vceq=Lambda_dr*wslip we know d/dt(wslip) = dVceq/dt/Lambda_dr. Substitute this into Equation 1 to give Iqs = J*dVc/dt/Lambda_dr^2. Acts like capacitance with Ceq = J/Lambda_dr^2 0 0 0 Idm0~ Vqs/[wsync*(Lls+Lm)] Ceq dc Iqm0~0

6. Torque Transfer Function TelectOsc/TmechOsc - based on linearization about the base case. Resonant frequency ~ 6hz. As expected at low freq, the ratio is 1 (motor load responds in same amount as load), and at high freq ratio appraoches 0 (motor doesn’t see load osc due to i nertia). At resonant peak frequency, the torque variation is amplified by a factor close to 3

7. Torque transfer function Zoom-in. Peak roughly matches 6.4hz eigenvalue (as expected-both come from same linearized model). For 6hz excitation (as simulated), expect ratio appro 2.65. Note we expect ratio ~ 1 at 9 hz excitation.

8. Torque transfer function phase. In phase far below resonance falling to out-of- phase far above resonance. This matches SDOF mechanical system if we consider Te to play the role of the spring force.

9. Speed transfer function: SpeedOscFraction/TloadOsc. Torque oscillation of base (full load) would cause speed oscillation of 8% (!) at resonance.

10. Speed transfer function phase.

11. Simulation: unloaded start followed by application of pulsing load Tload=100(1+sin(2*pi*6* )N-m. (100N-m=25% of full load torque). Initial osc confirms resonant freq ~ 6.4hz. Motor torque oscillates 550Npk/pk which is approx 2.65 times as much as load (same as predicted by transfer function). Speed oscillation is approx +-0.6hz = 2% of speed. This matches prediction based on transfer function. (Expect 8% variation at full load and 2% variatio at 25% load). Phases also obey transfer function if we remember to invert Tload 3 osc/0.46 sec ~ 6.4hz

12. Part 2 - Change excitation frequency to 9hz (keep base system which has nat freq ~ 6hz). Result is that motor torque oscillation magnitude is approx same as load torque oscillation magnitude (ratio about 1 as predicted by transfer functionanalysis).

13. Longer view of the 9hz scenario confirms it does reach a quasi-steady state pattern with constant amplitudes of motor torque.

14. Part 3 - manipulate system parameters toward 9hz resonant frequency. Keep 6hz excitation frequency Part 3 System: Same as base case EXCEPT:  V=1.1  J = 1.16 (corresponds to inertia of motor alone... base case was 1.8 corresponding to inertial of motor plus flywheel)

15. Torque transfer function for “part 3 system” (No Flywheel, V=1.1) has resonant frequency at 9hz.

16. Zoom-in on “part 3 system” torque transfer function shows torque amplification at 6hz is about 1.7

17. Speed transfer function for “part 3 system”

18. Simulation of part 3 system excited at 6hz. Torque amplification is 1.7 which matches transfer function prediction.. Note eigenvalue analysis suggests resonant freq has moved to 9hz. Initial consideration of symmetry might lead us to believe that a 9hz nat-freq system excited at 6hz would have a similar amplification factor as a 6hz nat-freq system excited at 9z (torque amplification =1). This is not t he case for several reasons : 1- the transfer function drops off slower to the left than to the right 2- Fractional change from 6 to 9 is larger than fractional ecrease from 9 to 6, 3- part 3 system has lower damping factor than base system which gives higher amplification around resonance (the resonant peak of part 3 system reaches 4 whereas resonance peak of base system was 2.7) 3 osc/0.32 sec ~ 9hz

19. Part 4 - examine currents Overview - base case with 25% constant load and 25% oscillating load has higher current than recorded. If we apply 15% constant load and 15% oscillating load the current waveform matches much more closely. The resonant frequencies and amplifications do not change. Torque and current oscillations change (from base case) by a factor of roughly 0.6 = 15/25

20. Base Case - Current. Note that the simulation time-step size is based on the slower-varying d-q simulation. This looks choppy when converted to (60hz) abc variables but is more than adequate to capture the slower- varying (6hz) transient.

21. Measured Current. About 60% as high as the base case.

22. Base case except 15% (vs 25%) load. Matches recorded current magnitude very closely. Current and torque results are roughly scaled by a factor of 60%. Note the eigenvalue and damping have been shown to be insensitive to load, so conclusions (transfer function, eigenvalues0) are not affected.

23. Base case except 15% load (vs 25% load). Simulation plot of torques and speeds. Very similar to base case except excitation and oscillation is scaled by 60%.