• Induction machines undergo transients when the amplitude and frequency of electric variables and (or) the speed vary in time.

• Direct starting, after turn-off, sudden mechanical loading, sudden shortcircuit , reconnection after a short supply fault, behavior during short intervals of supply voltage reduction, performance when PWM converterfed, are all typical examples of IM transients.

• The investigation of transients may be approached directly by circuit models or by coupled FEM circuit models of different degrees of complexity.

• The phase coordinate model is the “natural” circuit model, but its stator– rotor mutual inductances depend on rotor position. The order of the model is 8 for a single three-phase winding in the rotor. The symmetrical rotor cage may be replaced by one, two, n, three-phase windings to cater for skin (frequency) effects.

• Solving such a model for transients requires a notable computation effort on a contemporary PC. Dedicated numerical software (such as Matlab) has quite a few numerical methods to solve systems of nonlinear equations.

• To eliminate the parameter (inductance) dependence on rotor position, the space phasor (complex variable) or d–q model is used. The order of the system (in d–q, real, variables) is not reduced, but its solution is much easier to obtain through numerical methods.

• In complex variables, with zero homopolar components, only two electrical and one mechanical equations remain for a single cage rotor.

With stator and rotor flux space phasors Ψs and Ψr as variables and constant speed, the machine model exhibits only two complex eigen values. Their expressions depend on the speed of the reference system.

• As expected, below breakdown torque speed, the real part of the complex eigen values tends to be positive suggesting unstable operation.

• The model has two transient electrical time constants τs’, τr’, one of the stator and one of the rotor. So, for voltage supply, the stator and rotor flux transients are similar. Not so for current supply (only the rotor equation is used) when the rotor flux transients are slow (marked by the rotor time constant τr = Lr/Rr).

• Including magnetic saturation in the complex variable model is easy for main flux path if the reference system is oriented to main flux space phasor

clip_image002Ψm = Ψm , as Ψm = Lm(im)im.

• The leakage saturation may be accounted for separately by considering the leakage inductance dependence on the respective (stator or rotor) current.

• The standard equivalent circuit may be ammended to reflect the leakage and main inductances, Lsl, Lrl, Lm, dependence on respective currents is, ir, im. However, as both the stator and rotor cores experience a.c. fields, the transient values of these inductances should be used to get better results.

• For large stator (and rotor) currents such as in high performance drives, the main and leakage flux contribute to the saturation of teeth tops. In such cases, the airgap inductance Lg is separated as constant and one stator and one rotor total core inductance Lsi, Lri, variable with respective currents, are responsible for the magnetic saturation influence in the machine.

• Further on, the core loss may be added to the complex variable model by orthogonal short-circuited windings in the stator (and rotor). The ones in the rotor may be alternatively used as a second rotor cage.

• Core losses influence slightly the efficiency torque and power factor during steady-state. Only in the first few miliseconds the core loss “leakage” time constant influences the IM transients behavior. The slight detuning in field orientation controlled drives, due to core loss, is to be considered mainly when precise (sub 1% error) torque control is required.

• Reduced order models of IMs are used in the study of power system steadystate or transients, as the number of motors is large.

clip_image003Neglecting the stator transients (stator leakage time constant τs’) is the obvious choice in obtaining a third order model (with Ψr and ωr as state variables). It has to be used only in synchronous coordinates (where steadystate means dc). However, as the supply frequency torque oscillations are eliminated, such a simplification is not to be used in calculating starting transients.

• The supply frequency oscillations in torque during starting are such that the average transient torque is close to the steady-state torque. This is why calculating the starting time of a motor via the steady-state circuit produces reasonable results.

• Considering leakage saturation in investigating starting transients is paramount in calculating correctly the peak torque and current values.

• Large induction machines are coupled to plants through a kind of elastic coupling which, together with the inertia of motor and load, may lead to torsional frequencies which are equal to those of transient torque components. Large torsional torques occur at the load shaft in such cases. Avoiding such situations is the practical choice.

• The sudden short-circuit at the terminals of large power IMs represents an important liability for not-so-strong local power grids. The torque peaks are not, however, severely larger than those occuring during starting. The sudden short-circuit test may be used to determine the IM parameters in real saturation and frequency effects conditions.

• More severe transients than direct starting, reconnection on residual voltage, or sudden short-circuit have been found. The most severe case so far occurs when the primary of transformer feeding an IM is turned off for a short interval (tens of milliseconds), very soon after direct starting. The secondary of the transformer (now with primary on no-load) introduces a large stator time constant which keeps alive the d.c. decaying stator current components. This way, very large torque peaks occur. They vary from 26 – 40 p.u. in a 7.5 HP to 12 to 14 p.u. in a 500 HP machine.

• To treat unsymmetrical stator voltages (connections) so typical with PWM converter fed drives, the abc–dq model seems the right choice. Fault conditions may be treated rather easily as the line voltages are the inputs to the system (stator coordinates).

• For steady-state stability in power systems, first order models are recommended. The obvious choice of neglecting both stator and rotor electrical transients in synchronous coordinates is not necessarily the best one. For low power IMs, the first order system thus obtained produces acceptably good results in predicting the mechanical (speed) transients. The subtransient voltage (rotor flux) E’ and its angle δ represent the fast variables.

• For large power motors, a modified first order model is better. This new model reflects the subtransient voltage E’ (rotor flux, in fact) transients,

with speed ωr and angle δ as the fast transients calculated from algebraic equations.

• In industry, a number of IMs of various power levels are connected to a local power grid through a power bus that contains a series reactance and a parallel capacitor (to increase power factor).

Connection and disconnection transients are very important in designing the local power grid. Complete d–q models, with saturation included, proved to be necessary to simulate residual voltage (turn-off) of a few IMs when some of them act as motors and some as generators until the mechanical and magnetic energy in the system die down rather abruptly. The parallel capacitor delays the residual voltage attenuation, as expected.

• Series capacitors are used in some power grids to reduce the voltage sags during large motor starting. In such cases, subsynchronous resonance (SSR) conditions might occur.

• In such a situation, high current and torque peaks occur at certain speed. The maximum value of power bus plus stator resistance Re for which SSR occurs, shows how to avoid SSR. A better solution (in terms of losses) to avoid SSR is to put a resistance in parallel with the series capacitor. In general the critical resistance Rad (in parallel) for which SSR might occur increases with the series capacitance and so does the SSR frequency.

• Rotor bar and end ring segment faults occur frequently. To investigate them, a detailed modeling of the rotor cage loops is required. Detailed circuit models to this end are available. In general, such fault introduces mild torque and speed pulsations and (1-2S)f1 pulsations in the stator current. Information like this may be instrumental in IM diagnosis and monitoring. Interbar currents tend to attenuate the occurring asymmetry.

• Finite element coupled circuit models have been developed in the last 10 years to deal with the IM transients. Still when skin and saturation effects, skewing and the rotor motion, and the IM structural details are all considered, the computation time is still prohibitive. World-wide aggressive R & D FEM efforts should render such complete (3D) models feasible in the near future (at least for prototype design refinements with thermal and mechanical models linked together).

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