Fast electric (voltage) transients (in the microseconds range), typical to atmospheric discharges and to voltage steep repetitive pulses of PWM static power converters with long power cables in variable speed motor/generator drives, require totally different models to describe properly the response of IMs to them.
Voltage strikes and restrikes produced during on/off switching operations of induction motors fed from standard power grid may cause severe dielectric voltage stresses on the stator induction machine (IM) windings, leading, eventually, to failure.
In industrial installations, high dielectric stresses may occur during second- and third-pole circuit breaker closure. The second- and the third-pole closure in electromagnetic power circuit breakers has been shown to occur within 0–700 μs .
For such situations, electrical machine modelling in the frequency range of a few KHz is required. Steep-fronted voltage waves with magnitudes up to 5 p.u. may occur at the machine terminals under certain circuit breaker operating conditions.
On the other hand, PWM voltage source inverters produce steep voltage pulses which are applied repeatedly to induction motor terminals in modern electric drives.
In insulated gate bipolar transistor (IGBT) inverters, the voltage switching rise times of 0.05–2 μs, in presence of long cables, have been shown to produce strong winding insulation stresses and premature motor bearing failures.
With short rise time IGBTs and power cables longer than a critical length lc, repetitive voltage pulse reflection may occur at motor terminals.
The reflection process depends on the parameters of the feeding cable between motor and inverter, the IGBT voltage pulse time tr, and the motor parameters.
The peak line-to-line terminal overvoltage (VpK) at the receiving end of an initially uncharged transmission line (power cable) subjected to a single PWM pulse with rise time tr  is
where the critical cable length lc corresponds to the situation when the reflected wave is fully developed; Vdc is the D.C. link voltage in the voltage source inverter, and Γm is the reflection coefficient (0 < Γm < 1).
Z0 is the power cable and Zm the induction motor surge impedance. The distributed nature of a long-cable L–C parameters favours voltage pulse reflection, besides inverter short rising time. Full reflection occurs along the power cable if the voltage pulses take longer than one-third of the voltage rising time to travel from converter to motor at speed u* ≈ 150–200 m/μs.
Figure 3.1 Critical power cable length lc.
The receiving (motor) end may get 3Vdc for cable lengths greater than lc when the transmission line (power cable) has initial trapped charges due to multiple PWM voltage pulses.
Inverter rise times of 0.1–0.2 μs lead to equivalent frequency in the MHz range.
Consequently, Super-high-frequency modelling of IMs involves frequencies in the range of 1 kHz to 10 MHz.
The effects of such fast voltage pulses on the machine windings include
The super-high frequency or surge impedance of the IM may be approached globally either when it is to be identified through direct tests or as a complex distributed parameter (capacitor, inductance, resistance) system, when identification from tests at the motor terminals is in fact not possible.
In such a case, either special tests are performed on a motor with added measurement points inside its electric (magnetic) circuit, or analytical or FEM methods are used to calculate the distributed parameters of the IM.
IM modelling for surge voltages may then be used to conceive methods to attenuate reflected waves and change their distribution within the motor so as to reduce insulation stress and bearing failures.
We will start with global (lumped) equivalent circuits and their estimation, and continue with distributed parameter equivalent circuits.
When PWM inverter-fed, the IM terminals experience three pulse voltage components.
Figure 3.2 PWM inverter with zero sequence impedance of the load (insulated neutral point).
Assume that the zero sequence impedance of IM is Z0.
The zero sequence voltage and current are then
Also, by definition,
and again from (3.3),
This is how the equivalent lumped impedance (Figure 3.2) for the common voltage evolved.
It should be noted that for the differential voltage mode, the currents flow between phases and thus no interference between the differential and common modes occurs.
To measure the lumped IM parameters for the three modes, terminal connections as shown in Figure 3.3 are made.
Figure 3.3 Line (Zm) neutral (Zon) and ground (common mode Zog).
Impedances shown in Figure 3.3, referred here as differential (Zm), zero sequence – neutral – (Zon), and common mode (Zog), can be measured directly by applying single-phase A.C. voltage of various frequencies. Both the amplitude and the phase angle are of interest.
Figure 3.4 Differential mode impedance (Zm). (After Ref. .)
Zm has been determined by measuring the line voltage Vac during the PWM sequence with phase a, b together and c switched from the + A.C. bus to the − D.C. bus. The value of Δe is the transient peak voltage Vac above D.C. bus magnitude during trise, and ΔI is phase c peak transient phase current.
As shown in Figure 3.4, Zm decreases with increasing IM power. Also it has been found out that Zm varies from manufacturer to manufacturer, for given power, as much as 5 times. When using an RLC analyser (with phases a and b in parallel, connected in series with phase c), the same impedance has been measured by a frequency response test (Figure 3.5) .
Figure 3.5 Differential impedance Zm versus frequency (a) amplitude and (b) phase angle. (After Ref. .)
The phase angle of the differential impedance Zm approaches a positive maximum between 2 and 3 kHz (Figure 3.5b). This is due to lamination skin effect which reduces the iron core A.C. inductance.
At critical core frequency firon, the field penetration depth equals the lamination thickness dlam.
The relative iron permeability μrel is essentially determined by the fundamental magnetization current in the IM. Above firon eddy current shielding becomes important, and the iron core inductance starts to decrease until it approaches the wire self-inductance and stator air core leakage inductance at the resonance frequency fr = 25 kHz (for the 1 HP motor) and 55 kHz for the 100 HP motor.
Beyond fr, turn-to-turn and turn-to-ground capacitances of wire perimeter as well as coil-to-coil (and phase-to-phase) capacitances prevail such that the phase angle approaches now − 90° (pure capacitance) around 1 MHz.
So, as expected, with increasing frequency the motor differential impedance switches character from inductive to capacitive (Figure 3.5b). Zm is important in the computation of reflected wave voltage at motor terminals with the motor fed from a PWM inverter through a power (feeding) cable. Many simplified lumped equivalent circuits have been tried to model the experimentally obtained wide-band frequency response [4,5].
As very high-frequency phenomena are confined to the stator slots, due to the screening effect of rotor currents – and to stator end connection conductors, a rather simple line-to-line circuit motor model may be adopted to predict the line motor voltage surge currents.
Figure 3.6 Simplified differential mode IM equivalent circuit.
Chf and Rhf determine the model at high frequencies (above 10 kHz in general), whereas Llf and Rlf are responsible for lower frequency modelling. The identification of the model in Figure 3.6 from frequency response may be done through optimization (regression) methods.
A rather linear increase of Chf and a rather linear decrease of Rhf, Rlf, and Llf both with increasing power are shown in Table 3.1.
The high-frequency resistance Rhf is much smaller than the low-frequency resistance Rlf.
Figure 3.7 Frequency dependence of (a) Zon and (b) Zog. (After Ref. .)
Many lumped parameter circuits to fit results such as those shown in Figure 3.7 may be tried on.
Figure 3.8 (a) Simplified high-frequency phase circuit and (b) universal (low- and high-frequency) circuit per phase.
Harmonics copper losses are represented by Rsw, μLsl – stator first turn leakage inductance, Csw – inter-turn equivalent capacitor, Cft – total capacitance to ground, and Rframe – stator initial frame to ground damping resistance, which are all computable by regression methods from frequency response.
The impedances Zon and Zog (with the three phases in parallel) shown in Figure 3.8a are
The poles and zeros are
At low frequency (1–10 kHz), Zog is almost purely capacitive:
Taking f = 1 kHz and Zog (1 kHz) from the graph shown in Figure 3.7b, the capacitance Cg is determined.
With the pole frequency fp (Zon, Zog), corresponding to peak Zon value, the resistance Re is
This is a high-frequency inductance.
It is thus expected that at low frequency (1–10 kHz), fitting of Zon and Zog with the model will not be so good.
Typical values for a 1.1 kW four-pole, 50 Hz, 220/380 V motor are Cg = 0.25 nF, Ld(hf) = 28 mH, and Re = 17.5 kΩ.
For 55 kW, Cg = 2.17 nF, Ld(hf) = 0.186 mH, and Re = 0.295 kΩ.
On a logarithmic scale, the variation of Cg and Ld(hf) with motor power in kW is approximated  to
for 220/380 V, 50 Hz, 1–55 kW IMs.
To improve the low-frequency (between 1 and 10 kHz) fitting between the equivalent circuit and the measured frequency response, an additional low-frequency Re, Le branch may be added in parallel to Ld shown in Figure 3.8.
The addition of an eddy current resistance representing the motor frame Rframe (Figure 3.8) may also improve the equivalent circuit precision.
Test results with triangular voltage pulses and with a PWM converter and short cable have proven that the rather simple high-frequency phase equivalent circuit shown in Figure 3.8 is reliable.
More elaborated equivalent circuits for both differential and common voltage modes may be adopted for better precision . For their identification however, regression methods have to be used, when the skin effect branch shown in Figure 3.8 is to be considered at least when Zog is identified. Both amplitude and phase of frequency response up to 1 MHz are used for identification by regression methods .
Alternatively, the d-q model is placed in parallel, and thus, a general equivalent circuit acceptable for digital simulations at any frequency is obtained. This way, a universal low, wide frequency IM model is obtained (Figure 3.8b) .
The lumped equivalent circuits for high frequency presented here are to be identified from frequency response tests. Their configuration retains a large degree of approximation. They serve only to assess the impact of differential and common mode voltage surges at the induction motor terminals.
The voltage surge and its distribution within the IM will be addressed next.
When power grid is connected, IMs undergo surge-connection or atmospherical surge voltage pulses. When PWM voltage is surge-fed, IMs get trains of steep front voltage pulses. Their distribution, in the first few microseconds, along the winding coils, is not uniform.
Higher voltage stresses in the first 1–2 terminal coils and their first turns occur.
This nonuniform initial voltage distribution is due to the presence of stray capacitors between turns (coils) and the stator frame.
The complete distributed circuit parameters should contain individual turn-to-turn and turn-to-ground capacitances, self-turn, turn-to-turn and coil-to-coil inductances and self-turn, and eddy current resistances.
Some of these parameters may be measured through frequency response tests only if the machine is tapped adequately for the purpose. This operation may be practical for a special prototype to check design computed values of such distributed parameters.
The computation process is extremely complex, even rather impossible, in terms of turn-to-turn parameters in random wound coil windings.
Even via 3D-FEM, the complete set of distributed circuit parameters valid from 1 kHz to 1 MHz is not yet feasible.
However, at high frequencies (in the MHz range), corresponding to switching times in the order of tens of a microsecond, the magnetic core acts as a flux screen and thus most of the magnetic flux will be contained in air, as leakage flux.
The high-frequency eddy currents induced in the rotor core will confine the flux within the stator. In fact, the magnetic flux will be nonzero in the stator slot and in the stator coil end connections zone.
Let us suppose that the end connection resistances, inductances and capacitances between various turns and to the frame can be calculated separately. The skin effect is less important in this zone, and the capacitances between the end turns and the frame may be neglected, as their distance to the frame is notably larger in comparison with conductors in slots which are much closer to the slot walls.
So, in fact, the FEM may be applied within a single stator slot, conductor by conductor (Figure 3.9).
Figure 3.9 Flux distribution in a slot for eddy current FE analysis.
The magnetic and electrostatic field is zero outside the stator slot perimeter Γ and nonzero inside it.
The turn in the middle of the slot has a lower turn-to-ground capacitance than turns situated closer to the slot wall.
The conductors around a conductor in the middle of the slot act as an eddy current screen between turn and ground (slot wall).
In a random wound coil, positions of the first and of the last turns (nc) are not known and thus may differ in different slots.
Computation of the inductance and resistance slot matrixes is performed separately within an eddy current FEM package .
The first turn current is set to 1 A at a given high frequency, whereas the current is zero in all other conductors. The mutual inductances between the active turn and the others are also calculated.
The computation process is repeated for each of the slot conductors as active.
With I = 1 A (peak value), the self-inductance L and resistance R of the conductor are
The output of the eddy current FE analysis per slot is an nc × nc impedance matrix.
The magnetic flux lines for nc = 55 turns/coil (slot), , from 1 to 10 MHz, with one central active conductor (Ipeak = 1 A) (Figure 3.9, ) show that the flux lines are indeed contained within the slot volume.
The eddy current field solver calculates A and Φ in the field equation:
A(x,y) – the magnetic potential (Wb/m)
Φ(x,y) – the electric scalar potential (V)
μr – the magnetic permeability
ω – the angular frequency
σ – the electric conductivity
εr – the dielectric permittivity.
The capacitance matrix may be calculated by an electrostatic FEM package.
This time, each conductor is set to 1 V (D.C.) source, whereas the others are set to 0 V. The slot walls are defined as a zero potential boundary. The electrostatic field simulator solves now for the electric potential Φ(x,y).
where ρ(x,y) is the electric charge density.
The result is an nc × nc capacitance matrix per slot which contains the turn-to-turn and turn-to-slot wall capacitance of all conductors in slot. The computation process is done nc times with always a different single conductor as the 1 V (D.C.) source.
The matrix terms Cij are
where Welij is the electric field energy associated with the electric flux lines that connect charges on conductor “i” (active) and “j” (passive).
As electrostatic analysis is performed, the dependence of capacitance on frequency is neglected.
A complete analysis of a motor with 24, 36, 48, … stator slots with each slot represented by nc × nc (e.g. 55 × 55) matrixes would be hardly practical.
Figure 3.10 Equivalent circuit of the line-end coil.
To consider extreme possibilities, the line-end coil and the last five turns of the first coil are also simulated turn-by-turn. The rest of the turns are simulated by lumped parameters. All the other coils per phase are simulated by lumped parameters. Only the diagonal terms in the impedance and capacitance matrixes are nonzero. Saber-simulated voltage drops across the line-end coil and in its first three turns are shown in Figure 3.11a and b for a 750 V voltage pulse with trise = 1 μs .
Figure 3.11 Voltage drop versus time for trise = 1 μs: (a) line-end coil (after Ref. ). (b) Turns 1–3 of line-end coil (after Ref. ).
The voltage drop along the line-end coil was 280 V for trise = 0.2 μs and only 80 V for trise = 1 μs. Notice that there are six coils per phase. Feeder cable tends to lead to 1.2–1.6 kV voltage amplitude by wave reflection and thus dangerously high electric stress may occur within the line-end coil of each phase, especially for IGBTs with trise < 0.5 μs, and random wound machines.
Thorough frequency response measurements with a tapped winding IM have been performed to measure turn–turn and turn-to-ground distributed parameters .
Further on, the response of windings to PWM input voltages (rise time: 0.24 μs) with short and long cables have been obtained directly and calculated through the distributed electric circuit with measured parameters. Rather satisfactory but not very good agreement has been obtained .
It was found that the line-end coil (or coil 01) takes up 52% and the next one (coil 02) takes also a good part of input peak voltage (42%) . This is in contrast to FE analysis results which tend to predict a lower stress on the second coil .
As expected, long power cables tend to produce higher voltage surges at motor terminals. Consequently, the voltage drop peaks along the line-end coil and its first turn are up to two to three times higher. The voltage distribution of PWM voltage surges along the winding coils, especially along its line-end first 2 coils and the line-end first 3 turns thus obtained, is useful to winding insulation design.
Also preformed coils seem more adequate than random coils.
It is recommended that in power grid-fed IMs, the timing between the first-, second-, and third- pole power switch closure be from 0 to 700 μs. Consequently, even if the commutation voltage surge reaches 5 p.u. , in contrast to 2–3 p.u. for PWM inverters, the voltage drops along the first two coils and their first turns are not necessarily higher because the trise of commutation voltage surges is much larger than 0.2–1 μs. As we already mentioned, the second main effect of voltage surges on the IM is the bearing early failures with PWM inverters. Explaining this phenomenon requires, however, special lumped parameter circuits.
Rotor eccentricity, homopolar flux effects, or electrostatic discharge are known causes of bearing (shaft) A.C. currents in power grid-fed IMs [12,13]. The high-frequency common mode large voltage pulse at IM terminals, when fed from PWM inverters, has been suspected to further increase bearing failure. Examination of bearing failures in PWM inverter-fed IM drives indicates fluting, induced by electrical discharge machining (EDM). Fluting is characterized by pits or transverse grooves in the bearing race which lead to premature bearing wear.
When riding the rotor, the lubricant in the bearing behaves as a capacitance. The common mode voltage may charge the shaft to a voltage that exceeds the lubricant’s dielectric field rigidity believed to be around 15 Vpeak/μm. With an average oil film thickness of 0.2–2 μm, a threshold shaft voltage of 3–30 Vpeak is sufficient to trigger EDM.
As described in Section 3.1, a PWM inverter produces zero sequence besides positive and negative sequence voltages.
These voltages reach the motor terminals through power cables, online reactors, or common mode chokes . These impedances include common mode components as well.
Figure 3.12 Three-phase PWM inverter plus IM model for bearing currents.
The common mode voltage, originating from the zero sequence PWM inverter source, is distributed between stator and rotor neutral and ground (frame):
With the feeding cable represented by a series/parallel impedance Zs, Zp, the common mode voltage equivalent circuit may be extracted from Figure 3.12, as shown in Figure 3.13. R0 and L0 are the zero sequence impedances of IM to the inverter voltages. Calculating Csf, Csr, Crf, Rb, Cb, and Zl is still a formidable task.
Figure 3.13 Common mode lumped equivalent circuit.
Consequently, experimental investigation has been performed to somehow segregate the various couplings performed by Csf, Csr, and Crf.
The physical construction to the scope implies adding an insulated bearing support sleeve to the stator for both bearings. Also brushes are mounted on the shaft to measure Vrg.
Figure 3.14 The test motor.
In region A in Figure 3.15, the shaft voltage Vrg charges to about 20 Vpk. At the end of region A, Vsng jumps to a higher level causing a pulse in Vrg. In that moment, the oil film breaks down at 35 Vpk and a 3 Apk bearing current pulse is produced. At high temperatures, when oil film thickness is further reduced, the breakdown voltage (Vrg) pulse may be as low as 6–10 V.
Figure 3.15 Bearing breakdown parameters. (After Ref. .)
Region B is without bearing current. Here, the bearing is charged and discharged without current.
Region C shows the rotor and bearing (Vrg and Vsng) charging to a lower voltage level. No EDM occurs this time. Vrg = 0 with Vsng high means that contact asperities are shorting Cb.
The shaft voltage Vrg, measured between the rotor brush and the ground, is a strong indicator of EDM potentiality. Test results in Figure 3.15  show the shaft voltage Vrg, the bearing current Ib, and the stator neutral to ground voltage Vsng.
An indicator of shaft voltage is the bearing voltage ratio (BVR):
With insulated bearings, neglecting the bearing current (if the rotor brush circuits are open and the ground of the motor is connected to the inverter frame), the ground current IG refers to stator winding to stator frame capacitance Csf (Figure 3.16a). With an insulated bearing, but with both rotor brushes connected to the inverter frame, the measured current IAB is related to stator winding to rotor coupling (Csr) (Figure 3.16b).
Figure 3.16 Capacitive coupling modes: (a) stator winding to stator, (b) stator winding to rotor, and (c) uninsulated bearing current Ib. (After Ref. .)
In contrast, short-circuiting the bearing insulation sleeve (G’) allows the measurement of initial (uninsulated) bearing current Ib (Figure 3.16c).
Reducing the shaft voltage Vrg to <1 to 1.5 Vpk is apparently enough to avoid EDM and thus eliminate bearing premature failure.
To do so, bearing currents should be reduced or their path be bypassed by larger capacitance path, that is, increasing Csf or decreasing Csr.
Three main practical procedures have evolved  so far:
Figure 3.17 Conductive shields to bypass bearing currents (Csr is reduced).
Various degrees of shaft voltage attenuation rates (from 50% to 100%) have been achieved with such methods depending on the relative area of the shields. Shaft voltages close to NEMA (National Electrical Manufacturers Association) specifications have been obtained.
The conductive shields do not affect notably the machine temperatures.
Besides the EDM discharge bearing current, a kind of circulating bearing current that flows axially through the stator frame has been identified . Essentially, a net high-frequency axial flux is produced by the difference in stator coil end currents due to capacitance current leaks along the stack length between conductors in slots and the magnetic core. However, the relative value of this circulating current component proves to be small in comparison with EDM discharge current.