Wound-rotor induction generators (WRIGs) have been built for power per unit as high as 400 MW/unit in pump storage power plants, to as low as 4.0 MW/unit in wind power plants. Diesel engine or gas-turbine-driven WRIGs for standby or autonomous operation up to 20–40 MW may also be feasible to reduce fuel consumption and pollution for variable loads.
Wound-rotor induction generators (WRIGs) have been built for power per unit as high as 400 MW/unit in pump storage power plants, to as low as 4.0 MW/unit in wind power plants. Diesel engine or gas-turbine-driven WRIGs for standby or autonomous operation up to 20–40 MW may also be feasible to reduce fuel consumption and pollution for variable loads.
Below 1.5–2 MW/unit, the use of WRIGs is not justifiable in terms of cost versus performance when compared with full-power rating converter synchronous or cage rotor induction generator systems.
The stator-rated voltage increases for power up to 18–20 kV (line voltage, root mean squared [RMS]) at 400 mega volt ampere (MVA). Because of limitations with voltage, for low-cost power converters, the rotor-rated (maximum) voltage occurring at maximum slip currently is about 3.5–4.2 kV (line voltage, RMS) for direct current (DC) voltage link alternating current (AC)–AC pulse-width modulated (PWM) converters with integrated gate-controlled thyristors (IGCTs).
Higher voltage levels are being tested and will be available soon for industrial use, based on multiple-level DC voltage link AC–AC converters made of insulated power cells in series and other high voltage technologies.
Thus far, for the 400 MW WRIGs, the rated rotor current may be in the order of 6500 A; and thus, for S_{max} = ±0.1, approximately, it would mean 3.6 kV line voltage (RMS) in the rotor. A transformer is necessary to match the 3.6 kV static power converter to the rotor with the 18 kV power source for the stator. The rotor voltage V_{r} is as follows:
For |S_{max}| = 0.1, V_{r} = 3.6 ⋅ 10^{3} V (per phase), ${V}_{s}=18\cdot {10}^{-3}\text{/}\sqrt{3}\text{\hspace{0.17em}}\text{V}$
(phase): K_{rs} = 2/1. Therefore, the equivalent turn ratio is decisive in the design. In this case, however, a transformer is required to connect the DC voltage link AC–AC converter to the 18 kV local power grid.For powers in the 1.5–4 MW range, low stator voltages are feasible (690 V line voltage, RMS). The same voltage may be chosen as the maximum rotor voltage V_{r}, at maximum slip.
For S_{max} = ±0.25, ${V}_{s}=690\text{/}\sqrt{3}\text{\hspace{0.17em}}\text{V}$
, V_{r} = V_{s}, K_{rs} = 1/|S_{max}| = 4. In this case, the rotor currents are significantly lower than the stator currents. A transformer to match the rotor voltage to the stator voltage is not required.Finally, for WRIGs in the 3–10 MW range to be driven by diesel engines, 3000 (3600) rpm, or gas turbines, stator and rotor voltages in the 3.5–4.2 kV range are feasible. The transformer is again avoided.
Once the stator and rotor rated voltages are calculated and fixed, further design can proceed smoothly. Electromagnetic (emf) and thermomechanical designs are the two types of design. In what follows, we will touch on mainly the emf design.
Even for the emf design, we should distinguish three main operation modes:
The motor mode is required in applications such as pump storage power plants or even with microhydro or wind turbine prime movers.
The emf design implies a machine model, analytical, numerical, or mixed; one or more objective functions; and an optimization method with a computer program to execute it.
The optimization criteria may include the following:
Deterministic, stochastic, and evolutionary optimization methods have been applied to electric machine design [1] (see also Chapter 10). Whatever the optimization method, it is useful to have sound geometrical parameters for the preliminary (or general) design with regard to performance when starting the optimization design process. This is the reason why the general emf design is primarily discussed in what follows. Among the operation modes listed above, the generator at the power grid is the most frequently used, and, thus, it is the one of interest here.
High- and low-voltage stators and rotors are considered to cover the entire power range of WRIGs (from 1 to 400 MW).
The design methodology that follows covers only the essentials. A comprehensive design methodology is beyond the scope of this text.
The rated line voltage of the stator (RMS) (Y) V_{SN} = 0.38(0.46), 0.69, 4.2(6), …, 18 kV. The rated stator frequency f_{1} = 50 (60) Hz. The rated ideal speed n_{1N} = f_{1}/p_{1} = 3000 (3600), 1500 (1800). The maximum (ideal) speed n_{max} = (f_{1}/p_{1})(1 + |S_{max}|). The maximum slip S_{max} = ±0.05, ±0.1, ±0.2, ±0.25. The rated stator power (at unity power factor) S_{1Ns} = 2 MW. The rated rotor power (at unity power factor) S_{Nr} = S_{Ns}|S_{max}| = 0.5 MW. The rated (maximum) rotor line voltage (Y) is V_{RN} ≤ V_{SN}. As already discussed, the stator power may reach up to 350 MW (or more) with the rotor delivering maximum power (at maximum speed) of up to 40–50 MW and voltage ${V}_{N}^{r}$
≤ 4.2(6) kVHere we will consider the following for our discussion: total 2.5 MW at 690 V, 50 Hz, V_{RN} = V_{SN} = 690 V, S_{max} = ±0.25, and rated ideal speed n_{1N} = 50/p_{1} = 1500 rpm (p_{1} = 2).
The emf design factors are as follows:
Two main design concepts need to be considered when calculating the stator interior diameter D_{is}: the output coefficient design concept (C_{e}—Esson coefficient) and the shear rotor stress (f_{xt}) [1]. Here, we will make use of the shear rotor stress concept with f_{xt} = 1.5–8 N/cm^{2}.
The shear rotor stress increases with torque. First, the emf torque has to be estimated, noting that at 2.5 MW, 2p_{1} = 4 poles, the expected rated efficiency η_{N} > 0.95.
The total emf power S_{gN} at maximum speed and power is as follows:
The corresponding emf torque T_{e} is
This torque is at stator rated power and ideal synchronous speed (S = 0). The stator interior diameter, D_{is}, based on the shear rotor stress concept, f_{xt}, is
with l_{i} equal to the stack length. The stack length ratio λ = 0.2–1.5, in general. Smaller values correspond to a larger number of poles. With λ = 1.0 and f_{xt} = 6 N/cm^{2} (aiming at high torque density), the stator internal diameter, D_{is}, is as follows:
At this power level, a unistack stator is used, together with axial air cooling.
The external stator diameter D_{out} based on the maximum air gap flux density per given magnetomotive force (mmf) is approximated in Table 3.1 [1]. Therefore, from Table 3.1, D_{out} is as follows:
The rated stator current at unity power factor in the stator I_{SN} is
The air gap flux density B_{g1} = 0.75 T (the fundamental value). On the other hand, the air gap emf E_{S} per phase is as follows:
2p_{1} |
2 |
4 |
6 |
8 |
≥10 |
D_{out}/D_{is} |
1.65–1.69 |
1.46–1.49 |
1.37–1.40 |
1.27–1.30 |
1.24–1.20 |
where τ is the pole pitch:
K_{W1} is the total winding factor:
where
q_{1} is the number of slot/pole/phase in the stator
y/τ is the stator coil span/pole pitch ratio
As the stator current is rather large, we are inclined to use a_{1} = 2 current paths in parallel in the stator. With two current paths in parallel, full symmetry of the windings with respect to stator slots may be provided. For the time being, let us adopt K_{W1} = 0.910. Consequently, from Equation 3.9, the number of turns per current path W_{1a} is as follows:
Adopting q_{1} = 5 slots/pole/phase, a two-layer winding with two conductors per coil, n_{c1} = 2,
The final number of turns/coil is n_{c1} = 2, q_{1} = 5, with two symmetrical current paths in parallel. The North and South poles pairs constitute the paths in parallel.
Note that the division of a turn into elementary conductors in parallel with some transposition to reduce skin effects will be discussed later in this chapter.
The stator slot pitch is now computable:
The stator winding factor K_{W1} may now be recalculated if only the y/τ ratio is fixed: y/τ = 12/15, very close to the optimum value, to reduce to almost zero the fifth-order stator mmf space harmonic:
This value is very close to the adopted one, and thus, n_{c1} = 2 and q_{1} = 5, a_{1} = 2 hold. The number of slots N_{S} is as follows:
As expected, the conditions to full symmetry, N_{S}/m_{1}q_{1} = 60/(3·2) = 10 = integer, 2p_{1}/a_{1} = 2·2/2 = integer are fulfilled.
The stator conductor cross-section A_{cos} is as follows:
With the design current density j_{co1} = 6.5 A/mm^{2}, rather typical for air cooling:
Open slots in the stator are adopted, but magnetic wedges are used to reduce the air gap flux density pulsations due to slot openings.
In general, the slot width W_{S}/τ_{S} = 0.45–0.55. Let us adopt W_{S}/τ_{S} = 0.5. The slot width W_{S} is
There are two coils (four turns in our case) per slot (Figure 3.1).
As we deal with a low-voltage stator (690 V, line voltage RMS), the total slot filling factor, with rectangular cross-sectional conductors, may be safely considered as K_{fills} ≈ 0.55.
The useful (above wedge) slot area A_{sn} is
The rectangular slot useful height h_{su} is straightforward:
The slot aspect ratio h_{su}/W_{s} = 70.315/13.33 = 5.275 is still acceptable. The wedge height is about h_{sw} ≈ 3 mm. Adopting a magnetic wedge leads to the apparent reduction of slot opening from W_{s}= 13.33 mm to about W_{s} = 4 mm, as detailed later in this chapter.
Figure 3.1 Stator slotting geometry.
The air gap g is
With a flux density B_{cs} = 1.55 T in the stator back iron, the back core stator height h_{cs} (Figure 3.1) may be calculated as follows:
The magnetically required stator outer diameter D_{outm} is
This is roughly equal to the value calculated from Table 3.1 (Equation 3.6).
The stator core outer diameter may be increased by the double diameter of axial channels for ventilation, which might be added to augment the external cooling by air flowing through the fins of the cast iron frame.
As already inferred, the rectangular axial channels, placed in the upper part of stator teeth (Figure 3.1), can help improve the machine cooling once the ventilator of the shaft is able to flow part of the air through these stator axial channels.
The division of a stator conductor (turn) with a cross-section of 128.8 mm^{2} is similar to the case of synchronous generator design in terms of transposition to limit the skin effects. Let us consider four elementary conductors in parallel. Their cross-sectional area A_{cos} _{e} is as follows:
Considering only a_{ce} = 12 mm, out of the 13.33 mm slot width available for the elementary stator conductor, the height of the elementary conductor h_{ce} is as follows:
Without transposition and neglecting coil chording the skin effect coefficient, with m_{e} = 16 layers (elementary conductors) in slot, is as follows [1]:
Finally, ξ = βh_{ce} = 0.6964 ⋅ 10^{−2} ⋅ 2.68 ⋅ 10^{−3} = 0.1866. With,
From Equation 3.27, K_{Rme} is
The existence of four elementary conductors (strands) in parallel leads also to circulating currents. Their effect may be translated into an additional skin effect coefficient K_{rad} [1]:
where
With l_{i}/l_{turn} ≈ 0.4, n_{cn} = 2, h_{ce} = 2.68 · 10^{−3} m, and β = 0.6964 · 10^{−2} m^{−1} (from Equation 3.28), K_{rad} is as follows:
The total skin effect factor K_{Rll} is
It seems that at least for this design, no transposition of the four elementary conductors in parallel is required, as the total skin effect winding losses add only 3.717% to the fundamental winding losses.
The stator winding is characterized by the following:
With t_{1} equal to the largest common divisor of N_{s} and p_{1} = 2, there are N_{s}/t_{1} = 60/2 = 30 distinct slot emfs. Their star picture is shown in Figure 3.2. They are distributed to phases based on 120° phase shifting after choosing N_{s}/2m_{1} = 60/(2 · 3) = 10 arrows for phase a and positive direction (Figure 3.3a and b).
Figure 3.2 Slot allocation to phases for NS = 60 slots, 2p1 = 4 poles, and m = 3 phases.
Figure 3.3 (a) Half of coils of stator phases A and (b) their connection to form two current paths in parallel.
The rotor design is based on the maximum speed (negative slip)/power delivered, P_{RN}, at the corresponding voltage V_{RN} = V_{SN}. Besides, the WRIG is designed here for unity power factor in the stator. Therefore, all the reactive power is provided through the rotor. Consequently, the rotor also provides for the magnetization current in the machine.
For V_{RN} = V_{SN} at S_{max}, the turn ratio between rotor and stator K_{rs} is obtained:
Now the stator rated current reduced to the rotor ${{I}^{\prime}}_{SN}$
is as follows:The rated magnetization current depends on the machine power, the number of poles, and so forth. At this point in the design process, we can assign ${{I}^{\prime}}_{m}$
(the rotor-reduced magnetization current) a per unit (P.U.) value with respect to I_{SN}:Let us consider here K_{m} = 0.30. Later on in the design, K_{m} will be calculated, and then adjustments will be made.
Therefore, the rotor current at maximum slip and rated rotor and stator delivered powers is as follows:
The rotor power factor cos φ_{2N} is
It should be noted that the oversizing of the inverter to produce unity power factor (at rated power) in the stator is not very important.
Note that, generally, the reactive power delivered by the stator Q_{1} is requested from the rotor circuit as SQ_{1}.
If massive reactive power delivery from the stator is required, and it is decided to be provided from the rotor-side bidirectional converter, the latter and the rotor windings have to be sized for the scope.
When the source-side power factor in the converter is unity, the whole reactive power delivered by the rotor-side converter is “produced” by the DC link capacitor, which needs to be sized for the scope.
Roughly, with cos φ_{2} = 0.707, the converter has to be oversized at 150%, while the machine may deliver almost 80% of reactive power through the stator (ideally 100%, but a part is used for machine magnetization).
Adopting V_{RN} = V_{SN} at S_{max} eliminates the need for a transformer between the bidirectional converter and the local power grid at V_{SN}.
Once we have the rotor-to-stator turns ratio and the rated rotor current, the designing of the rotor becomes straightforward.
The equivalent number of rotor turns, W_{2}K_{W2} (single current path), is as follows:
The rotor number of slots N_{R} should differ from the stator one, N_{S}, but they should not be too different from each other.
As the number of slots per pole and phase in the stator q_{1} = 5, for an integer q_{2}, we may choose q_{2} = 4 or 6. We choose q_{2} = 4. Therefore, the number of rotor slots N_{R} is as follows:
With a coil span of Y_{R}/τ = 10/12, the rotor winding factor (no skewing) becomes
From Equation 3.39, with Equation 3.41, the number of rotor turns per phase W_{2} is
The number of coils per phase is N_{R}/m_{1} = 48/3 = 16. With W_{2} = 80 turns/phase; it follows that each coil will have n_{c2} turns:
Therefore, there are ten turns per slot in two layers and one current path only in the rotor.
Adopting a design current density j_{con} = 10 A/mm^{2} (special attention to rotor cooling is needed) and, again, a slot filling factor K_{fill} = 0.55, the copper conductor cross-section A_{cor} and the slot useful area A_{slotUR} are as follows:
The rotor slot pitch τ_{R} is
Assuming that the rectangular slot occupies 45% of rotor slot pitch, the slot width W_{R} is
The useful height h_{RU} (Figure 3.4) is, thus,
This is an acceptable (Equation 3.48) value, as h_{SU}/W_{R} < 4.
Figure 3.4 Rotor slotting.
Open slots have been adopted, but with magnetic wedges (μ_{r} = 4–5), the actual slot opening is reduced from W_{R} = 15.2 mm to about 3.5 mm, which should be reasonable (in the sense of limiting the Carter coefficient and surface and flux pulsation space harmonics core losses).
We need to verify the maximum flux density, B_{tRmax}, in the rotor teeth at the bottom of the slot:
with
Though the maximum rotor tooth flux density is rather large (2.2 T), it should be acceptable, because it influences a short path length.
The rotor back iron radial depth h_{cr} is as follows:
The value of rotor back iron flux density B_{cr} of 1.6 T (larger than in the stator back iron) was adopted, as the length of the back iron flux lines is smaller in the rotor with respect to the stator.
To see how much of it is left for the shaft diameter, let us calculate the magnetic back iron inner diameter D_{ir}:
This inner rotor core diameter may be reduced by 20 mm (or more) to allow for axial channels—10 mm (or more) in diameter—for axial cooling, and thus, 0.267 m (or slightly less) are left of the shaft. It should be enough for the purpose, as the stack length is l_{i} = D_{is} = 0.52 m.
The rotor winding design is straightforward, with q_{2} = 4, 2p_{1} = 4 and a single current path. The cross-section of the conductor is 43.20 mm^{2}, and thus, even a single rectangular conductor with the width b_{cr} W_{R} ≈ 13 mm and its height h_{cr} = A_{cor}/b_{cr} = 43.40/13 = 3.338 mm, will do.
As the maximum frequency in the rotor, ${f}_{q\mathrm{max}}={f}_{1}^{*}\text{|}{S}_{\mathrm{max}}\text{|}=50\cdot 0.25=12.5\text{\hspace{0.17em}}\text{Hz}$
, the skin effect will be even smaller than in the stator (which has a similar elementary conductor size); and thus, no transposition seems to be necessary.The winding end connections have to be tightened properly against centrifugal and electrodynamic forces by adequate nonmagnetic bandages.
The electrical rotor design also contains the slip-rings and brush system design and the shaft and bearings design. These are beyond our scope here. Though debatable, the use of magnetic wedges on the rotor also seems doable, as the maximum peripheral speed is smaller than 50 m/s.
The rated magnetization current was previously assigned a value (30% of I_{SN}). By now, we have the complete geometry of stator and rotor slots cores, and the magnetization mmf may be considered. Let us consider it as produced from the rotor, though it would be the same if computed from the stator.
We start with the given air gap flux density B_{g1} = 0.75 T and assume that the magnetic wedge relative permeability μ_{RS} = 3 in the stator and μ_{RR} = 5 in the rotor. Ampere’s law along the half of the Γ contour (Figure 3.5), for the main flux, yields the following:
where
The air gap mmf ${F}_{A{A}^{\prime}}$
is as follows:K_{C} is the Carter coefficient:
The equivalent slot openings, with magnetic wedges,${W}_{{S}^{\prime}}$
and ${W}_{{R}^{\prime}}$ , are as follows:Figure 3.5 Main flux path.
where
Finally,
This is a small value that will result, however, in smaller surface and flux pulsation core losses. The small effective slot opening, ${W}_{{S}^{\prime}}$
and ${W}_{{R}^{\prime}}$ will lead to larger slot leakage inductance contributions. This, in turn, will reduce the short circuit currents.The air gap mmf (from Equation 3.54) is as follows: ${F}_{A{A}^{\prime}}=1.612\cdot {10}^{-3}\cdot 1.126\frac{0.75}{1.256\cdot {10}^{-6}}=1083.86\text{\hspace{0.17em}}\text{A}\text{\hspace{0.17em}}\text{turns}$
The stator and rotor teeth mmf should take into consideration the trapezoidal shape of the teeth and the axial cooling channels in the stator teeth.
We might suppose that, in the stator, due to axial channels, the tooth width is constant and equal to its value at the air gap W_{ts} = τ_{S} − W_{S} = 26.6 − 13.2 = 13.4 mm. Therefore, the flux density in the stator tooth B_{ts} is
B (T) |
0.05 |
0.1 |
0.15 |
0.2 |
0.25 |
0.3 |
0.35 |
0.4 |
0.45 |
0.5 |
H (A/m) |
22.8 |
35 |
45 |
49 |
57 |
65 |
70 |
76 |
83 |
90 |
B (T) |
0.55 |
0.6 |
0.65 |
0.7 |
0.75 |
0.8 |
0.85 |
0.9 |
0.95 |
1 |
H (A/m) |
98 |
106 |
115 |
124 |
135 |
148 |
162 |
177 |
198 |
220 |
B (T) |
1.05 |
1.1 |
1.15 |
1.2 |
1.25 |
1.3 |
1.35 |
1.4 |
1.45 |
1.5 |
H (A/m) |
237 |
273 |
310 |
356 |
417 |
482 |
585 |
760 |
1,050 |
1,340 |
B (T) |
1.55 |
1.6 |
1.65 |
1.7 |
1.75 |
1.8 |
1.85 |
1.9 |
1.95 |
2.0 |
H (A/m) |
1760 |
2460 |
3460 |
4800 |
6160 |
8270 |
11,170 |
15,220 |
22,000 |
34,000 |
From the magnetization curve of silicon steel (3.5% silicon, 0.5 mm thickness, at 50 Hz; Table 3.2), H_{ts} = 1290 A/m by linear interpolation.
Therefore, the stator tooth mmf F_{AB} is
The stator back iron flux density was already chosen: B_{cs} = 1.5 T.
H_{cs} (from Table 3.2) is H_{cs} = 1340 A/m. The average magnetic path length l_{csw} is as follows:
Therefore, the stator back iron mmf F_{BC} is
In the rotor teeth, we need to obtain first an average flux density by using the top, middle, and bottom tooth values B_{trt}, B_{tRm}, and B_{tRb}:
The average value of B_{tR} is as follows:
From Table 3.2, H_{tR} = 5334 A/m. The rotor tooth mmf ${F}_{{A}^{\prime}{B}^{\prime}}$
is, thus,Finally, for the rotor back iron flux density B_{CR} = 1.6 T (already chosen), H_{cr} = 2460 A/m, with the average path length l_{CRav} as follows:
The rotor back iron mmf ${F}_{{B}^{\prime}{C}^{\prime}}$
,The total magnetization mmf per pole F_{m} is as follows (Equation 3.55):
It should be noted that the total stator and rotor back iron mmfs are not much different from each other. However, the stator teeth are less saturated than the other iron sections of the core. The uniform saturation of iron is a guarantee that sinusoidal air gap tooth and back iron flux densities distributions are secured. Now, from Equation 3.55, the no-load (magnetization) rotor current I_{R0} might be calculated:
The ratio of I_{R0} to ${I}_{S{N}^{\prime}}$
(Equation 3.35; stator current reduced to rotor), K_{m}, is, in fact,The initial value assigned to K_{m} was 0.3, so the final value is smaller, leaving room for more saturation in the stator teeth by increasing the size of axial teeth cooling channels (Figure 3.1). Alternatively, the air gap may be increased up to 3 mm if so needed for mechanical reasons.
The total iron saturation factor K_{s} is
Therefore, the iron adds to the air gap mmf 79.568% more. This will reduce the magnetization reactance X_{m} accordingly.
The main WRIG parameters are the magnetization reactance X_{m}, the stator and rotor resistances R_{s} and R_{r}, and leakage reactances X_{sl} and X_{rl}, reduced to the stator. The magnetization reactance expression is straightforward [1]:
The base reactance ${X}_{b}={V}_{SN}\text{/}{I}_{SN}=690\text{/(}\sqrt{3}\cdot 1675.46\text{)}=0.238\text{\hspace{0.17em}}\Omega $
. As expected, x_{m} = X_{m}/X_{b} = 2.5254/0.238 = 10.610 = I_{SN}/I_{R0} from Equation 3.67.Figure 3.6 Stator coil end connection geometry.
The stator and rotor resistances and leakage reactances strongly depend on the end connection geometry (Figure 3.6). The end connection l_{fs} on one machine side, for the stator winding coils, is
The stator resistance R_{s} per phase has the following standard formula (skin effect is negligible as shown earlier):
The stator leakage reactance X_{sl} is written as follows:
where
For the case in point [1],
The differential geometrical permeance coefficient λ_{ds} [1] may be calculated as follows:
The coefficient σ_{ds} is the ratio between the differential and magnetizing inductance and depends on q_{1} and chording ratio β. For q_{1} = 5 and β = 0.8 from Figure 3.7 [1], σ_{ds} = 0.0042.
Finally,
The term λ_{ds} generally includes the zigzag leakage flux.
Finally, from Equation 3.73,
As can be seen, L_{sl}/L_{m} = 0.257 · 10^{−3}/(8.0428 · 10^{−3}) = 0.03195.
Figure 3.7 Differential leakage coefficient σd.
X_{sl} = ω_{1}L_{sl} = 2π · 50 · 0.257 · 10^{−3} = 0.0807 Ω. The rotor end connection length l_{fr} may be computed in a similar way as in Equation 3.71:
The rotor resistance expression is straightforward:
Equation 3.73 also holds for rotor leakage inductance and reactance:
with σ_{dR} = 0.0062 (q_{2} = 4, β_{R} = 5/6) from Figure 3.7:
From Equation 3.79, ${L}_{Rl}^{r}$
isNoting that ${R}_{R}^{r}$
, ${L}_{Rl}^{r}$ , and ${X}_{Rl}^{r}$ are values obtained prior to stator reduction, we may now reduce them to stator values with K_{RS} = 4 (turns ratio):Let us now add here the R_{S}, L_{sl}, X_{sl}, L_{m}, and X_{m} to have them all together: R_{s} = 0.429 · 10^{−2} Ω, L_{sl} = 0.257 · 10^{−3} H, X_{sl} = 0.0807 Ω, L_{m} = 8.0428 · 10^{−3} H, and X_{m} = 2.5254 Ω.
The rotor resistance reduced to the stator is larger than the stator resistance, mainly due to notably larger rated current density (from 6.5 to 10 A/mm^{2}), despite having shorter end connections.
Because of smaller q_{2} than q_{1}, the differential leakage coefficient is larger in the rotor, which finally leads to a slightly larger rotor leakage reactance than in the stator.
The equivalent circuit may now be used to compute the power flow through the machine for generating or motoring, once the value of slip S and rotor voltage amplitude and phase are set. We leave this to the interested reader. In what follows, the design methodology, however, explores the machine losses to determine the efficiency.
The electrical losses are made of the following:
As the skin effect was shown small, it will be considered only by the correction coefficient K_{R} = 1.037, already calculated in Equation 3.33. For the rotor, the skin effect is neglected, as the maximum frequency S_{max}f_{1} = 0.25f_{1}. In any case, it is smaller than in the stator, because the rotor slots are not as deep as the stator slots:
The slip-ring and brush losses p_{sr} are easy to calculate if the voltage drop along them is given, say V_{SR} ≈1 V. Consequently,
To calculate the stator fundamental core losses, the stator teeth and back iron weights G_{ts} and G_{cs} are needed:
The fundamental core losses in the stator, considering the mechanical machining influence by fudge factors such as K_{t} = 1.6–1.8 and K_{y} = 1.3–1.4, are as follows [1]:
With B_{ts} = 1.488 T, B_{cs} = 1.5 T, f_{1} = 50 Hz, and p_{10/50} = 3 W/kg (losses at 1 T and 50 Hz), ${P}_{irons}=3\cdot {\left(\frac{50}{50}\right)}^{1.3}[1.6\cdot {(1.488)}^{2}\cdot 313.8+1.3\cdot {(1.5)}^{2}\cdot 582]=8442\text{\hspace{0.17em}}\text{W}=8.442\text{\hspace{0.17em}}\text{kW}$
The rotor fundamental core losses may be calculated in a similar manner, but with f_{1} replaced by Sf_{1}, and introducing the corresponding weights and flux densities.
As the S_{max} = 0.25, even if the lower rotor core weights will be compensated for by the larger flux densities, the rotor fundamental iron losses would be as follows:
The surface and pulsation additional core losses, known as strayload losses, are dependent on the slot opening/air gap ratio in the stator and in the rotor [1]. In our design, magnetic wedges reduce the slot-openings-to-air gap ratio to 4.066/1.612 and 3.04/1.612; thus, the surface additional core losses are reduced. For the same reason, the stator and rotor Carter coefficients K_{c1} and K_{c2} are small (K_{c1}·K_{c2} = 1.126); and therefore, the flux-pulsation additional core losses are also reduced.
Consequently, all additional losses are most probably well within the 0.5% standard value (for detailed calculations, see Reference 1, Chapter 11) of stator rated power:
Thus, the total electrical losses ∑p_{e} are
Neglecting the mechanical losses, the “electrical” efficiency η_{e} is
This is not a very large value, but it was obtained with the machine size reduction in mind.
It is now possible to redo the whole design with smaller f_{xt} (rotor shear stress), and lower current densities to finally increase efficiency for larger size. The length of the stack l_{i} is also a key parameter to design improvements. Once the above, or similar, design methodology is computerized, then various optimization techniques may be used, based on objective functions of interest, to end up with a satisfactory design (see Reference 1, Chapter 10).
Finite element analysis (FEA) verifications of the local magnetic saturation, core losses, and torque production should be instrumental in validating optimal designs based on even advanced analytical nonlinear models of the machine.
Mechanical and thermal designs are also required, and FEA may play a key role here, but this is beyond the scope of our discussion [2,3]. In addition, uncompensated magnetic radial forces have to be checked, as they tend to be larger in WRIG (due to the absence of rotor cage damping effects) [4].
The experimental investigation of WRIGs at the manufacturer’s or user’s sites is an indispensable tool to validate machine performance.
There are international (and national) standards that deal with the testing of general use induction machines with cage or wound rotors (International Electrotechnical Commission [IEC]-34, National Electrical Manufacturers Association [NEMA] MG1-1994 for large induction machines).
Temperature, losses, efficiency, unbalanced operation, overload capability, dielectric properties of insulation schemes, noise, surge responses, and transients (short circuit) responses are all standardized.
We avoid a description of such tests [5] here, as space would be prohibitive, and the reader could read the standards above (and others) by himself. Therefore, only a short discussion, for guidance, will be presented here.
Testing is performed for performance assessment (losses, efficiency, endurance, noise) or for parameter estimation. The availability of rotor currents for measurements greatly facilitates the testing of WRIGs for performance and for machine parameters. On the other hand, the presence of the bidirectional power flow static converter connected to the rotor circuits, through slip-rings and brushes, poses new problems in terms of current and flux time harmonics and losses.
The IGBT static converters introduce reduced current time harmonics, but measurements are still needed to complement digital simulation results.
In terms of parameters, the rotor and stator are characterized by single circuits with resistances and leakage inductances, besides the magnetization inductance. The latter depends heavily on the level of air gap flux, whereas the leakage inductances decrease slightly with their respective currents.
As for WRIGs, the stator voltage and frequency stay rather constant, and the stator flux ${\overline{\Psi}}_{s}$
varies only a little. The air gap flux ${\overline{\Psi}}_{m}$varies a little with load for unity stator power factor:
The magnetic saturation level of the main flux path does not vary much with load. Therefore, L_{m} does not vary much with load. In addition, unless I_{s}/I_{sn} > 2, the leakage inductances, even with magnetic wedges on slot tops, do not vary notably with respective currents. Therefore, parameters estimation for dealing with the fundamental behavior is greatly simplified in comparison with cage rotor induction motors with rotor skin effect.
On the other hand, the presence of time harmonics, due to the static converter of partial ratings, requires the investigation of these effects by estimating adequate machine parameters of WRIG with respect to them. Online data acquisition of stator and rotor currents and voltages is required for the scope.
The adaptation of tests intended for cage rotor induction machines to WRIGs is rather straightforward; thus, the following may all be performed for WRIGs with even better precision, because the rotor parameters and currents are directly measurable (for details see Reference 1, Chapter 22):