Most textbooks covering induction machines cover only the motor use-case scenario. When the machine is used as a motor, the slip factor is determined by the frequency of the stator current by the equation

\$ \Huge{s=f_s - \frac{\frac{p}{2} \cdot \frac{n_r}{60}}{f_s}} = \frac{n_s - n_r}{n_s} \$


\$ s \$ is the slip factor,

\$ f_s \$ is the stator frequency

\$ p \$ is the number of poles

\$ n_s \$ is the motor synchronous speed in rpm

\$ n_r \$ is the actual rotor speed in rpm

This all makes perfect sense when the machine is being operated as a motor and/or is connected directly to the transmission network which provides magnetizing current and a stable frequency to the stator windings.

My question: What happens if the machine is being operated as a generator and is on an island network? I understand that we need a source of reactive power -- provided by soft-start capacitor bank, for example -- but if the speed of the machine is variable, how in heaven's name is the slip factor determined? There's no alternating current magnetization, and hence no frequency on the stator winding. How is the frequency being set?

And how can I create a simple model of this that I can use in Simulink or in Scilab xcos?

I want to model a wind generating system on an island network, or a variable-speed system coupled to the transmission network via a rectifier and then a DC-DC converter. As usual I bit off more than I could chew and now I understand why everybody else does these models with permanent magnet synchronous machines: it's way easier.

Perhaps you can prove me wrong?

  • 2
    \$\begingroup\$ Do you mean "alternator on an island network"? If so, When you say "Island network" do you mean the alternator is the sole member of the network? If so (and this is NOT a specialist area of mine) I think you may have a "special case" and have to provide 'signal' / excitation to suit. You may have to assume that there is a remnant of permanent magnetisation present to start the system - or provide some sort of excitation. Ive seen people use 3 phase motors with capacitor arrangements to allow self excitation and use as an alternator. \$\endgroup\$
    – Russell McMahon
    Commented Sep 22, 2012 at 13:21
  • \$\begingroup\$ @RussellMcMahon: yes, that's basically what I mean. I get the part about a reactive power source for excitation, but that's normally done with a capacitor bank. As the bank discharges, the stator windings are magnetised, but this stator current doesn't alternate initially. The rotor turns through the stator's field and a current flows, which creates a magnetic field. Eventually, the induced field in the rotor will drive an alternating current in the stator... at least, that's my understanding of it. \$\endgroup\$ Commented Sep 22, 2012 at 14:08
  • \$\begingroup\$ @RussellMcMahon: my challenge is this: the frequency of the stator current is what determines the synchronous speed of the machine. The slip factor (see above) is a function of the synchronous speed and the actual rotor speed, except that, without an external ac source, there is no fixed frequency; it would have to be coming entirely from the induction generator. My goal is to model this mathematically so that I can simulate it, but this is a closed loop. \$\endgroup\$ Commented Sep 22, 2012 at 14:23
  • \$\begingroup\$ I just realised that I had written "motor" when I meant "generator". Sorry for the confusion. I've edited the original question. \$\endgroup\$ Commented Sep 22, 2012 at 15:42
  • 1
    \$\begingroup\$ For an 3 phase induction machine to work as a generator, you need to feed it first with an rotating field. Then wind will make the rotor turn faster than the rotating field in stator and only then slip can become negative and the machine will start to run in generator mode. Therefore, with too little wind, a windmill will run in motor mode and usually be turned off all together. \$\endgroup\$
    – jippie
    Commented Sep 22, 2012 at 18:30

5 Answers 5


So, just an induction motor employed as a generator? Yes there IS an ac magnetization on the stator winding. Spin the shaft, and a sine wave appears. An induction generator is an electromechanical sinewave oscillator. Small residual polarization of iron parts gets it started, and it builds up as a mechanically-driven RLC resonance between the capacitors and the generator inductance (but operating way off resonance, of course.)

In that case the "synchronous" speed would be the frequency of the AC signal measured at the stator coil (or at cap bank terminals,) same as when running in motor-mode. The slip is then taking place between this coil frequency versus generator rotor RPM. Just put the AC frequency in terms of RPM = 2/#poles x 60 x HzFreq

So, if a 4-pole induction motor (as a generator) with a particular capacitor value puts out 70Hz across the cap bank, the b-field inside the motor is rotating at 2/4*60*70 = 2100 RPM. If the actual shaft RPM is 2200, then slip factor is (2100 - 2200)/2200 = -0.045 I put it as negative slip, since it's opposite of the grid-driven slip of an induction motor.

I haven't messed with one of these beasts myself, so take this all with a grain of salt.

Classic diy page: QSL ham radio site


The concept of slip is a bit misleading for variable speed wind turbines. The type which uses an induction machine are called doubly fed induction generators (DFIGs) and their speeds are allowed to vary through the use of an AC-AC converter applied to their rotor windings. Because there is no actual induction in these machines, the term slip might be a bit odd to use, but the value for s can still be calculated, and is negative.

For these type of generators no capacitor banks are needed to supply reactive power, as the converters can be used to control the production of reactive power from the induction machine.

A DFIG must have an AC source to run, but this does not prohibit it from being run on an island system. Island systems are often AC systems.

I'm not really sure why permanent magnet synchronous machines would be easier to model. For wind turbine applications these machines also use power electronics, and their rotation speed is also variable through a wide range.

If you have the SimPowerSystems toolbox then these types of generators can be modelled in Simulink


but there are also free models which are available


Another free MATLAB tool which can be used for modelling power systems is


  • \$\begingroup\$ Your statement "Because there is no actual induction in these machines" doesn't make any sense to me; there is induction by definition, without induction the machine cannot develop stator voltage (in an island system). Also, DFIG are not the only induction machines used in windpower applications; there are others (the Danish concept comes to mind). Further, all the examples I have seen assume a reactive power source on the stator windings, so I don't see how the presence of an AC-AC inverter on the rotor will help here. \$\endgroup\$ Commented Nov 21, 2012 at 20:33
  • \$\begingroup\$ Also - even if there is an AC-AC inverter on the rotor windings: in a black-start condition, it needs to get its power from somewhere. \$\endgroup\$ Commented Nov 21, 2012 at 20:35
  • \$\begingroup\$ There is no induction induced in the rotor windings. The voltage there is directly applied from the network through the converter to the rotor windings. The stator is also connected to the network, and therefore has a voltage. By Danish concept I assume you mean a squirrel cage induction generator, but these are not variable speed turbines, which was what was asked for in the question. Yes, reactive power sources on the stator side are used for these turbines, but they do not have power electronics, so reactive compensation is required. \$\endgroup\$
    – Katt
    Commented Nov 22, 2012 at 15:58
  • \$\begingroup\$ But maybe I see that what the question was asking for is how should a variable speed wind turbine be used for black start, not actually island operation? \$\endgroup\$
    – Katt
    Commented Nov 22, 2012 at 16:42
  • \$\begingroup\$ Shouldn't "island operation" imply that it must also work in black-start conditions, since in either case the system has to be started somehow? I know this is almost never done in practice, but that is why I want to simulate it. This site shows a lab bench example: arthropodsystems.com/AsynchronousGenerator1/… I have since discovered this variable speed condition in an island is usually modeled with space vectors, which is something I am still learning and is apparently fairly challenging to do. \$\endgroup\$ Commented Dec 2, 2012 at 13:38

According to this wikipedia article, if the motor is used as a stand alone generator:frequency and voltage are complex function of machine parameters, capacitance used for excitation, and load value and type.


This link may provide information in regards to using an induction motor as a generator.
Motors as Generator
Another possibility would be to use a sine wave inverter for excitation of your generator.
Or use this method to excite the field windings of a DC motor/generator,
this should provide a constant frequency independent of rotor speed.
I hope this helps.


(This might help) The slip refers to the fact that the rotor physically lags the rotating flux in the stator. Approaching it on a physical basis what I did was to take a small single phase induction generator apart. A small permanent magnet is embedded in the rotor (nowhere near the size to actually affect the stator) and passes through 2 rotor coils which are in-line with the 2 rotor shoes. Taking the fact that the capacitor is in parallel with the stator there is therefore a resonant parallel circuit which reaches a resonant condition at a certain speed (preset by the value of C). There is at that speed a current magnification causing flux which by transformer action cuts the rotor. The rotor coils each have a diode to ensure that the current flows in one direction and the 2 poles are N and S. The rest is similar to the synchronous case.


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