10.4: Solid Iron Rotor Bodies
- Page ID
- 57425
Solid steel rotor electric machines (SSRM) can be made to operate with very high surface speeds and are thus suitable for use in high RPM situations. They resemble, in form and function, hysteresis machines. However, asynchronous operation will produce higher power output because it takes advantage of higher flux density. We consider here the interactions to be expected from solid iron rotor bodies. The equivalent circuits can be placed in parallel (harmonic-by-harmonic) with the equivalent circuits for the squirrel cage, if there is also a cage in the machine.
To estimate the rotor parameters \(\ R_{2 s}\) and \(\ X_{2 s}\), we assume that important field quantities in the machine are sinusoidally distributed in time and space, so that radial flux density is:
\[\ B_{r}=\operatorname{Re}\left(\underline{B}_{r} e^{j(\omega t-p \phi)}\right)\label{127} \]
and, similarly, axially directed rotor surface current is:
\[\ K_{z}=\operatorname{Re}\left(\underline{K}_{z} e^{j(\omega t-p \phi)}\right)\label{128} \]
Now, since by Faraday’s law:
\[\ \nabla \times \bar{E}=-\frac{\partial \bar{B}}{\partial t}\label{129} \]
we have, in this machine geometry:
\[\ \frac{1}{R} \frac{\partial}{\partial \phi} E_{z}=-\frac{\partial B_{r}}{\partial t}\label{130} \]
The transformation between rotor and stator coordinates is:
\[\ \phi^{\prime}=\phi-\omega_{m} t\label{131} \]
where \(\ \omega_{m}\) is rotor speed. Then:
\[\ p \omega_{m}=\omega-\omega_{r}=\omega(1-s)\label{132} \]
and
Now, axial electric field is, in the frame of the rotor, just:
\[\ E_{z}=\operatorname{Re}\left(\underline{E_{z}} e^{j(\omega t-p \phi)}\right)\label{133} \]
\[\ =\operatorname{Re}\left(\underline{E}_{z} e^{j\left(\omega_{r} t-p \phi^{\prime}\right)}\right)\label{134} \]
and
\[\ \underline{E}_{z}=\frac{\omega_{r} R}{p} \underline{B}_{r}\label{135} \]
Of course electric field in the rotor frame is related to rotor surface current by:
\[\ \underline{E}_{z}=\underline{Z}_{s} \underline{K}_{z}\label{136} \]
Now these quantities can be related to the stator by noting that air-gap voltage is related to radial flux density by:
\[\ \underline{B}_{r}=\frac{p}{2 l N_{a} k_{1} R \omega} \underline{V}_{a g}\label{137} \]
The stator-equivalent rotor current is:
\[\ \underline{I}_{2}=\frac{\pi}{3} \frac{R}{N_{a} k_{a}} \underline{K}_{z}\label{138} \]
Then we can find stator referred, rotor equivalent impedance to be:
\[\ \underline{Z}_{2}=\frac{\underline{V}_{a g}}{\underline{I}_{2}}=\frac{3}{2} \frac{4}{\pi} \frac{l}{R} N_{a}^{2} k_{a}^{2} \frac{\omega}{\omega_{r}} \frac{\underline{\underline{E}_{z}}}{\underline{K}_{z}}\label{139} \]
Now, if rotor surface impedance can be expressed as:
\[\ \underline{Z}_{s}=R_{s}+j \omega_{r} L_{s}\label{140} \]
then
\[\ \underline{Z}_{2}=\frac{R_{2}}{s}+j X_{2}\label{141} \]
where
\[\ R_{2}=\frac{3}{2} \frac{4}{\pi} \frac{l}{R} N_{a}^{2} k_{1}^{2} R_{s}\label{142} \]
\[\ X_{2}=\frac{3}{2} \frac{4}{\pi} \frac{l}{R} N_{a}^{2} k_{1}^{2} X_{s}\label{143} \]
Now, to find the rotor surface impedance, we make use of a nonlinear eddy-current model proposed by Agarwal. First we define an equivalent penetration depth (similar to a skin depth):
\[\ \delta=\sqrt{\frac{2 H_{m}}{\omega_{r} \sigma B_{0}}}\label{144} \]
where \(\ \sigma\) is rotor surface material volume conductivity, \(\ B_{0}\), ”saturation flux density” is taken to be 75 % of actual saturation flux density and
\[\ H_{m}=\left|\underline{K}_{z}\right|=\frac{3}{\pi} \frac{N_{a} k_{a}}{R}\left|\underline{I}_{2}\right|\label{145} \]
Then rotor surface resistivity and surface reactance are:
\[\ R_{s}=\frac{16}{3 \pi} \frac{1}{\sigma \delta}\label{146} \]
\[\ X_{s}=.5 R_{s}\label{147} \]
Note that the rotor elements \(\ X_{2}\) and \(\ R_{2}\) depend on rotor current \(\ I_{2}\), so the problem is nonlinear. We find, however, that a simple iterative solution can be used. First we make a guess for \(\ R_{2}\) and find currents. Then we use those currents to calculate \(\ R_{2}\) and solve again for current. This procedure is repeated until convergence, and the problem seems to converge within just a few steps.
Aside from the necessity to iterate to find rotor elements, standard network techniques can be used to find currents, power input to the motor and power output from the motor, torque, etc.
Solution
Not all of the equivalent circuit elements are known as we start the solution. To start, we assume a value for \(\ R_{2}\), possibly some fraction of \(\ X_{m}\), but the value chosen doesn not seem to matter much. The rotor reactance \(\ X_{2}\) is just a fraction of \(\ R_{2}\). Then, we proceed to compute an “air-gap” impedance, just the impedance looking into the parallel combination of magnetizing and rotor branches:
\[\ Z_{g}=j X_{m} \|\left(j X_{2}+\frac{R_{2}}{s}\right)\label{148} \]
(Note that, for a generator, slip s is negative).
A total impedance is then
\[\ Z_{t}=j X_{1}+R_{1}+Z_{g}\label{149} \]
and terminal current is
\[\ I_{t}=\frac{V_{t}}{Z_{t}}\label{150} \]
Rotor current is just:
\[\ I_{2}=I_{t} \frac{j X_{m}}{j X_{2}+\frac{R_{2}}{s}}\label{151} \]
Now it is necessary to iteratively correct rotor impedance. This is done by estimating flux density at the surface of the rotor using (145), then getting a rotor surface impedance using (146) and using that and (143 to estimate a new value for \(\ R_{2}\). Then we start again with (148). The process “drops through” this point when the new and old estimates for \(\ R_{2}\) agree to some criterion.
Harmonic Losses in Solid Steel
If the rotor of the machine is constructed of solid steel, there will be eddy currents induced on the rotor surface by the higher-order space harmonics of stator current. These will produce magnetic fields and losses. This calculation assumes the rotor surface is linear and smooth and can be characterized by a conductivity and relative permeability. In this discussion we include two space harmonics (positive and negative going). In practice it may be necessary to carry four (or even more) harmonics, including both ‘belt’ and ‘zigzag’ order harmonics.
Terminal current produces magnetic field in the air-gap for each of the space harmonic orders, and each of these magnetic fields induces rotor currents of the same harmonic order.
The “magnetizing” reactances for the two harmonic orders, really the two components of the zigzag leakage, are:
\[\ X_{z p}=X_{m} \frac{k_{p}^{2}}{N_{p}^{2} k_{1}^{2}}\label{152} \]
\[\ X_{z n}=X_{m} \frac{k_{n}^{2}}{N_{n}^{2} k_{1}^{2}}\label{153} \]
where \(\ N_{p}\) and \(\ N_{n}\) are the positive and negative going harmonic orders: For ‘belt’ harmonics these orders are 7 and 5. For ‘zigzag’ they are:
\[\ N_{p}=\frac{N_{s}+p}{p}\label{154} \]
\[\ N_{n}=\frac{N_{s}-p}{p}\label{155} \]
Now, there will be a current on the surface of the rotor at each harmonic order, and following 67, the equivalent rotor element current is:
\[\ \underline{I}_{2 p}=\frac{\pi}{3} \frac{R}{N_{a} k_{p}} \underline{K}_{p}\label{156} \]
\[\ \underline{I}_{2 n}=\frac{\pi}{3} \frac{R}{N_{a} k_{n}} \underline{K}_{n}\label{157} \]
These currents flow in response to the magnetic field in the air-gap which in turn produces an axial electric field. Viewed from the rotor this electric field is:
\[\ \underline{E}_{p}=s_{p} \omega R \underline{B}_{p}\label{158} \]
\[\ \underline{E}_{n}=s_{n} \omega R \underline{B}_{n}\label{159} \]
where the slip for each of the harmonic orders is:
\[\ s_{p}=1-N_{p}(1-s)\label{160} \]
\[\ s_{n}=1+N_{p}(1-s)\label{161} \]
and then the surface currents that flow in the surface of the rotor are:
\[\ \underline{K}_{p}=\frac{E_{p}}{Z_{s p}}\label{162} \]
\[\ \underline{K}_{n}=\frac{\underline{E}_{n}}{Z_{s n}}\label{163} \]
where \(\ Z_{s p}\) and \(\ Z_{s n}\) are the surface impedances at positive and negative harmonic slip frequencies, respectively. Assuming a linear surface, these are, approximately:
\[\ Z_{s}=\frac{1+j}{\sigma \delta}\label{164} \]
where \(\ \sigma\) is material restivity and the skin depth is
\[\ \delta=\sqrt{\frac{2}{\omega_{s} \mu \sigma}}\label{165} \]
and \(\ \omega_{s}\) is the frequency of the given harmonic from the rotor surface. We can postulate that the appropriate value of \(\ \mu\) to use is the same as that estimated in the nonlinear calculation of the space fundamental, but this requires empirical confirmation.
The voltage induced in the stator by each of these space harmonic magnetic fluxes is:
\[\ V_{p}=\frac{2 N_{a} k_{p} l R \omega}{N_{p} p} \underline{B}_{p}\label{166} \]
\[\ V_{n}=\frac{2 N_{a} k_{n} l R \omega}{N_{n} p} \underline{B}_{n}\label{167} \]
Then the equivalent circuit impedance of the rotor is just:
\[\ Z_{2 p}=\frac{V_{p}}{I_{p}}=\frac{3}{2} \frac{4}{\pi} \frac{N_{a}^{2} k_{p}^{2} l}{N_{p} R} \frac{Z_{s p}}{s_{p}}\label{168} \]
\[\ Z_{2 n}=\frac{V_{n}}{I_{n}}=\frac{3}{2} \frac{4}{\pi} \frac{N_{a}^{2} k_{n}^{2} l}{N_{n} R} \frac{Z_{s n}}{s_{n}}\label{169} \]
The equivalent rotor circuit elements are now:
\[\ R_{2 p}=\frac{3}{2} \frac{4}{\pi} \frac{N_{a}^{2} k_{p}^{2} l}{N_{p} R} \frac{1}{\sigma \delta_{p}}\label{170} \]
\[\ R_{2 n}=\frac{3}{2} \frac{4}{\pi} \frac{N_{a}^{2} k_{n}^{2} l}{N_{n} R} \frac{1}{\sigma \delta_{n}}\label{171} \]
\[\ X_{2 p}=\frac{1}{2} R_{2 p}\label{172} \]
\[\ X_{2 n}=\frac{1}{2} R_{2 n}\label{173} \]
Stray Losses
So far in this document, we have outlined the major elements of torque production and consequently of machine performance. We have also discussed, in some cases, briefly, the major sources of loss in induction machines. Using what has been outlined in this document will give a reasonable impression of how an induction machine works. We have also discussed some of the stray load losses: those which can be (relatively) easily accounted for in an equivalent circuit description of the machine. But there are other losses which will occur and which are harder to estimate. We do not claim to do a particularly accurate job of estimating these losses, and fortunately they do not normally turn out to be very large. To be accounted for here are:
- No-load losses in rotor teeth because of stator slot opening modulation of fundamental flux density,
- Load losses in the rotor teeth because of stator zigzag mmf, and
- No-load losses in the solid rotor body (if it exists) due to stator slot opening modulation of fundamental flux density.
Note that these losses have a somewhat different character from the other miscellaneous losses we compute. They show up as drag on the rotor, so we subtract their power from the mechanical output of the machine. The first and third of these are, of course, very closely related so we take them first.
The stator slot openings ‘modulate’ the space fundamental magnetic flux density. We may estimate a slot opening angle (relative to the slot pitch):
\(\ \theta_{D}=\frac{2 \pi w_{d} N_{s}}{2 \pi r}=\frac{w_{d} N_{s}}{r}\)
Then the amplitude of the magnetic field disturbance is:
\(\ B_{H}=B_{r 1} \frac{2}{\pi} \sin \frac{\theta_{D}}{2}\)
In fact, this flux disturbance is really in the form of two traveling waves, one going forward and one backward with respect to the stator at a velocity of \(\ \omega / N_{s}\). Since operating slip is relatively small, the two variations will have just about the same frequency as viewed from the rotor, so it seems reasonable to lump them together. The frequency is:
\(\ \omega_{H}=\omega \frac{N_{s}}{p}\)
Now, for laminated rotors this magnetic field modulation will affect the tips of rotor teeth. We assume (perhaps arbitrarily) that the loss due to this magnetic field modulation can be estimated from ordinary steel data (as we estimated core loss above) and that only the rotor teeth, not any of the rotor body, are affected. The method to be used is straightforward and follows almost exactly what was done for core loss, with modification only of the frequency and field amplitude.
For solid steel rotors the story is only a little different. The magnetic field will produce an axial electric field:
\(\ \underline{E}_{z}=R \frac{\omega}{p} B_{H}\)
and that, in turn, will drive a surface current
\(\ \underline{K}_{z}=\frac{\underline{E}_{z}}{\underline{Z}_{s}}\)
Now, what is important is the magnitude of the surface current, and since \(\ \left|\underline{Z}_{s}\right|=\sqrt{1+.5^{2}} R_{s} \approx1.118R_s\), we can simply use rotor resistance. The nonlinear surface penetration depth is:
\(\ \delta=\sqrt{\frac{2 B_{0}}{\omega_{H} \sigma\left|\underline{K}_{z}\right|}}\)
A brief iterative substitution, re-calculating \(\ \delta\) and then \(\ \left|\underline{K}_{z}\right|\) quickly yields consistent values for \(\ \delta\) and \(\ R_{s}\). Then the full-voltage dissipation is:
\(\ P_{r s}=2 \pi R l \frac{\left|\underline{K}_{z}\right|^{2}}{\sigma \delta}\)
and an equivalent resistance is:
\(\ R_{r s}=\frac{3\left|V_{a}\right|^{2}}{P_{r s}}\)
Finally, the zigzag order current harmonics in the stator will produce magnetic fields in the air gap which will drive magnetic losses in the teeth of the rotor. Note that this is a bit different from the modulation of the space fundamental produced by the stator slot openings (although the harmonic order will be the same, the spatial orientation will be different and will vary with load current). The magnetic flux in the air-gap is most easily related to the equivalent circuit voltage on the \(\ n^{t h}\) harmonic:
\(\ B_{n}=\frac{n p v_{n}}{2 l R N_{a} k_{n} \text {omega }}\)
This magnetic field variation will be substantial only for the zigzag order harmonics: the belt harmonics will be essentially shorted out by the rotor cage and those losses calculated within the equivalent circuit. The frequency seen by the rotor is that of the space harmonics, already calculated, and the loss can be estimated in the same way as core loss, although as we have pointed out it appears as a ‘drag’ on the rotor.