Induction Machine Drives: Dynamic / Static Characteristics

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The mathematical models of induction machines developed in the preceding sections will be applied here in the simulations and calculations of typical dynamic states and static characteristics of the drive. In the first stage the presentation will involve dynamic characteristics calculated for standardized electromechanical parameters of a drive. In the subsequent section the equations of motion will be reduced to the steady state and on this basis an equivalent circuit diagram of an induction motor will be derived together with static characteristics, typical parameters and graphical images for the characteristics.

Standardized Equations of Motion for Induction Motor Drive

Despite the common operating principle and a unique description in the form of a mathematical model, induction machines form a class that is considerably distinctive. The range of the rated powers varies from a fraction of a [kW] to the machines exceeding 10 [MW]. Concurrently, speed ratings resulting from the number of pole pairs applied in the construction, typically range for a machine from p=1 to p=6 pole pairs, and particular manufacturers offer machines with a higher number of pole pairs. Rated voltages applied to supply the primary windings also tend to very across the machines in accordance with the standardized series of voltages, while the majority of the motor run off a 230/400 [V] supply voltage or a high voltage of 6 [kV]. In addition, induction machines are differentiated by the structure of the windings of the secondary side (rotor) and in particular by the shape and profile of the cross-section of the bars in the squirrel-cage rotor. This part is responsible for the increase in the value of resistance R theta_r, which is the basic parameter which characterizes the mathematical model and for the fact that this parameter is considerably relative to the frequency of the currents in the cage's bars.

Squirrel-cage machines, which are distinguished by the tall and slender shape or particular profiles that tend to become thinner towards the air gap are characterized by resistance R theta_r, whose value increases for higher current frequencies in the cage occurring during motor's start-up. For high frequencies of the current in the cage the leakage reactance definitely dominates in the impedance of the bar in the cage rotor and the current is displaced towards the air gap, which brings a reduction of the active cross-section of the bar and increases resistance. --- the development of the mathematical models of a motor with constant resistance of rotor's windings R theta_r hence, they are relevant for slip-ring and single cage motors with a weak effect of current displacement. Their application with regard to squirrel-cage motors, for instance deep slot motors introduces a consider able error in particular in terms of the characteristics at the phase of start-up and small angular velocities of the rotor. In industrial practice, double-cage induction motors are applied in order to improve the start-up properties. In the upper cage such motors have bars with a smaller diameter, which for higher current frequencies have higher resistance levels. As a result, more complex mathematical models are necessary for their modeling.

In order to rationally perform the standardization of the parameters of various induction motors the equations of motion in flux coordinates will be inconsiderably transformed and expressed in the system a,ß, (?c = 0). It’s necessary to rescale the variables regarding the axial fluxes by dividing them by the rated voltage Usn:

In a similar manner, the equation of rotor's motion is transformed and it’s multi plied by p, hence standardized to the reference of one pole pair machine: ...

... is the electromechanical constant for an induction motor, while

... represent, respectively: moment of inertia, coefficient of viscous damping and load torque derived for the number of pole pairs p = 1.

Tb - is the break torque of the motor in a steady state ?e = p theta_r - is the angular velocity of the rotor expressed in terms of a motor with a single pole pair, called 'electrical angular speed'. This version of the equations of motion represents standardized equations for an induction motor for which the entire class of single-cage motors, not accounting for current displacement in the rotor's cage, are expressed in terms of a single synchronous velocity (for p = 1) regardless of the values of the rated voltage and the number of pole pairs. This is achieved as a result of dividing axial fluxes by the rated voltage Usn. In this model we have to do with the following independent parameters relative to the design of induction motors: ks, kr - coefficients of magnetic coupling of windings as, ar coefficients of damping (inverse of the time constants) cem electromechanical constant for a drive.

The above parameters are standardized, synthetic parameters of the mathematical model of an induction motor in its most simple version that does not account for the magnetic saturation of the core and current displacement in the rotor's cage.

They cover each individual motor and make it possible to calculate its dynamic and static characteristics. Other parameters used in the modeling of an induction motor, for example coefficient of windings' leakage s are relative to the ones presented in or are involved in them already (for instance number of pole pairs p). The presented method of standardization of equations and parameters is based on. On the basis of the calculations of parameters conducted on of data gained from industrial catalogues from several meaningful manufacturers for a few dozen of squirrel-cage motors with various power ratings and rated voltages, the table found below has been developed.

==== Standardized parameters of typical induction squirrel-cage motors

It contains standardized parameters for a wide range of squirrel-cage induction motors with basic design and a small influence of current displacement in the rotor's cage. Beside the basic parameters it contains relative parameters, which are encountered in several versions of the mathematical model. The data give mean values of the parameters for the group of examined motors with the number of pole pairs p = 1…5. The research that follows serves for the purpose of setting an example and illustration and is based on four motors from the group of 60 motors that were used for the preparation of Table 1. The selected motors are representative of the groups of small, medium and large power ones, respectively. Their parameters are presented below; one can make an effort to compare them with the parameters.

Typical Dynamic States of an Induction Machine Drive - Examples of Trajectories of Motion

Equations of motion for an induction machine drive presented for various coordinate systems have 5 degrees of freedom: four for electric variables in axes u,v of the transformed system of electric variables of a machine and one degree of freedom for the mechanical variable in the form of the angle of rotation of the rotor. It’s possible to have to do with more than one coordinate of the mechanical motion, for instance under the assumption of a flexible drive shaft, where the angles of shaft rotation on the side of the motor and on the side of the drive are different and this difference corresponds to the torsion angle of the shaft. The particular dynamic states formally constitute distinct initial conditions for a system of ordinary differential equations that form the mathematical model of a drive. The number of these states can be infinite for various initial conditions; however, from the practical point of view a few of them are encountered most frequently and hence they deserve a more in-depth analysis here. The typical dynamic states for an induction machine drive include: start-up from standstill, motor start-up under non-zero rotational speed, change of a load, reversal - i.e. change in the direction of rotation and electrical braking. In the first order we will discuss dynamic states, which can be easily and effectively solved using models with transformed variables of the stator and rotor. The presentation will cover, respectively: start-up from a standstill, start-up under angular speed different from zero (repeated start up and reversal) and drive regime of operation under cyclic variable load. The following stage will include the presentation of dynamic states, which can be conveniently calculated on the basis of models in which single side of a motor is untransformed. Such cases include the issue of a soft-start and DC braking of an induction motor.

Start-Up during Direct Connection to Network

A computer simulation of this dynamic state is performed for zero initial conditions. It’s possible to conduct calculations by application of various versions of the mathematical model in a transformed coordinate system. It’s beneficial to apply the model in current coordinates in a,ß axes or in flux coordinates. Both systems are in the standard form and the application of the a,ß system enables one to achieve the natural frequency of voltages in the transformed sys tem. The calculations of the current curves require the application of a reverse transformation Ts T under the assumption that ?c = 0, is0 = 0. For the case of the model in (with flux coordinates, the transformation of Ts T , which leads to transformation of currents in the axial a,ß system has to be preceded by the trans formation of to convert flux variables into axial currents. As a result, we obtain:

The examples of trajectories of the motion presented in this section apply a model in the flux coordinates, while phase currents are obtained using. The calculations of the trajectories of start-up from a stall was conducted for three induction motors with respective, small, medium and large power, whose parameters are given, respectively. === presents the curves of the phase current, electromagnetic torque and angular speed for an unloaded small power motor whose moment of inertia on the shaft is J = 3Js, where Js de notes the moment of inertia of the motor's rotor. The waveforms of the same type are presented === for medium and large power motors. In +-+-+-8 for the medium power motor the trajectory of electromagnetic torque, i.e. the relation of the torque and angular speed is additionally presented.

+-+-+-7 a) phase current b) electromagnetic torque c) angular velocity during free acceleration after direct connecting to the supply network, for the small power motor. Motor is unloaded and J = 3Js

a) phase current b) angular velocity c) electromagnetic torque time history; d) torque trajectory, during free acceleration after direct connecting to the supply network, for the medium power motor, while Tl = 0 and J = 2Js; a) phase current; b) electromagnetic torque; c) angular velocity, during free acceleration after direct connecting to the supply network, for the high power motor, while Tl = 0 and J = 2Js

One can note the considerable oscillatory changes of electromagnetic torque with a large initial value (4…6 Tn) and a frequency similar to the network voltage during the direct connection of the induction motor to the supply network. This results from the occurrence of an aperiodic component of the magnetic flux generated by the stator's windings in association with slowly increasing flux of the rotor's windings. The oscillatory state of the torque occurs until the instant when the two fluxes reach a steady state during the rotation over a circular trajectory.

+-+-+- Magnetic flux vector trajectory in the air gap of induction motor during free acceleration after direct connection to the network: a) ?s b) theta_r for the small power motor c) ?s d) theta_r for the high power motor.

The presentation of fluxes during the start-up of the small and large power motors. This figure refers to the start-up curves. The presented torque waveforms during direct connection to the network pose a hazard to the mechanical parts of the drive such as the shaft, clutch as well as the very device that is connected. For this reason the direct connection is more and more frequently replaced with the methods of soft-start, which are more widely discussed in the further part of this subsection.

+-+-+- Reconnection of the small power induction motor at the synchronous speed and zero current initial conditions: a) stator current b) electromagnetic torque c) angular speed; d) stator flux; e) rotor flux trajectory.

+-+-+- Reconnection of the high power induction motor at the synchronous speed and zero current initial conditions: a) stator current b) angular speed c) electromagnetic torque; d) torque-speed trajectory e) stator flux f) rotor flux trajectory

Reconnection of an Induction Motor

Reconnection is the term which denotes the dynamic state encountered during engaging a motor during coasting i.e. for a non-zero angular speed. One can note the difference between a reconnection: resulting from a breakdown in power supply for 0.3...1 [s], when the magnetic field in the motor from the weakening current of the cage does not decay completely and a reconnection after a breakdown of power supply of over 1 [s], when the magnetic field decays completely. For fast reconnections one has to take into account non-zero initial conditions for electric variables on the rotor's windings since the formed electromagnetic torque is then considerably dependent on the phase of the voltage connected to the stator's terminals. Just as in the case of synchronizing a synchronous machine with the network it’s possible to undertake a reconnection in accordance with the phase, whose characteristics include low values of the connection currents and electromagnetic torque. In contrast, in the most adverse case, such reconnection can occur in the circumstances of the opposition between the phases of network voltages and the one on the motor's terminals. In the latter case we have to do with a large connection current, which is hazardous for the drive due to a torque surge. For this reason it’s best to avoid fast and direct reconnections of an induction motor into the network. For large power motors the duration period of the hazardous reconnection lasts for about 0.8-1.0 [s], while for small and medium size ones the breakdown the time is 0.5 [s]. The reconnection of the motor after the period of the voltage breakdown over 1 [s] could be considered as the connection from the zero initial conditions of electric variables. The curve of the current and torque after such a reconnection is relative to the angular speed theta_r(0) after which the reconnection has occurred; however, it does not exceed the values that are present during the direct connection of a motor during standstill.

The curves of the currents, electromagnetic torque and angular speed and magnetic fluxes in the stator and rotor after connection to the network for synchronous speed are presented for small and large power motors, respectively.

+-+-+- Reversal of the medium power induction motor drive (J = 2Js): a) stator current b) rotor flux c) relative rotor speed d) torque-speed trajectory

Drive Reversal

The term reversal denotes turning on a drive under a speed that is reverse to the direction of the rotation resulting from the sequence of phases of the supply network after turning on. A reversal may be associated with the needs of a technology or may form a type of braking resulting from a counter current. In this case the reversal should be discontinued when the rotational speed of the drive is close to zero and before the drive starts its rotation in the opposite direction. The dynamic curves of the current and electromagnetic torque in the initial phase of the reversal are similar to the values of the curves for these values during start-up under direct turning on. The calculations of the trajectories of the drive motion apply zero initial conditions for currents (fluxes), by assuming an adequately long interval in the supply (1 [s] or more) and an angular speed similar to the synchronous speed but with a negative sign. Examples of curves during the reversal of a middle power induction motor are presented.

+-+-+- The cyclic load torque of the drive

+-+-+- Induction motor drive characteristics under periodic load changes: fl = 3[Hz], Tlav

= Tn, Tlmax = 4.2 Tn: a) phase current b) rotational speed c) torque time history d) torque speed trajectory

Cyclic Load of an Induction Motor

Load on a motor may contain a variable term. In this case steady operation state, understood as fixed point, is not achieved by a drive on its characteristics. In contrast, the drive operates in a closed trajectory when the operating regime becomes steady. For high inertia of the drive and a relatively small variable term of the load torque the trajectory of the motion is close to a fixed point. In the opposite case the trajectory of the drive's motion forms a curve that considerably diverges from static characteristic. The trajectory is relative to the value of the variable term, frequency of the load variation and moment of inertia relative to the motor's shaft.

The examples of the drive regime of operation for a large component of variable load are presented. +-+-+-14 that precedes them presents the stepwise variable load torque acting on the induction machine's shaft presents the dynamic curve for the mean load equal to the rated torque in the cycle of the load, in which for tm = 0.2tl the load torque is equal to Tlmax = 4.2Tn, and in the remaining part of the cycle Tlmin = 0.2Tn with the frequency of the torque variation fl =3 [Hz]. The trajectory of the electromagnetic torque in respect to angular speed forms a closed curve with the shape of an eight. …the cases of the identical load on a drive but for frequency fl =6 [Hz] and fl =15 [Hz]. As a consequence, there is a considerable reduction and limitation of the trajectory loop. +-+-+- a variable load with the frequency of fl =3 [Hz] and fl =2 [Hz] at the boundary of drive break.

+-+-+-Delay soft-start of the small power induction motor (J = 3Js): a) phase current b) angular speed c) electromagnetic torque d) stator flux trajectory e) and rotor flux trajectory

Soft-Start of an Induction Motor for Non-simultaneous Connection of Stator's Windings to the Network

+-+-+- Induction motor electromagnetic torque under periodic load changes for various frequency values: a) fl = 6[Hz] b) fl = 15[Hz]

+-+-+- Induction motor performance under periodic load changes close to the break torque loading: a) fl = 3[Hz], Tlmax = 5.2 Tn, Tlav = 1.87 Tn, b) fl = 2[Hz], Tlmax = 3.5 Tn, Tlav = 1.85 Tn

+-+-+- Delay soft-start of the medium power induction motor (J = 2Js): a) electromagnetic torque b) stator flux trajectory c) rotor flux trajectory

+-+-+-Delay soft-start of the high power induction motor (J = Js): a) phase current b) electromagnetic torque c) stator flux trajectory d) and rotor flux trajectory.

As it was mentioned earlier, the direct connection of an induction motor to the supply network results in the high value of an oscillatory component of electromagnetic torque during start-up. Besides, there is considerable value of the start-up current. This is well for small, medium and high power motors. The oscillations of the electromagnetic torque can be considerably limited and, hence, it’s virtually possible eliminate their effect as a result of the application of synchronized connection of phase windings in the network. In the first stage, two clamps of the stator's windings are connected with a suitable synchronization with the network and in the second stage the third clamp is connected with an adequate phase delay. A computer simulation of the examples that illustrate this issue can be conveniently conducted by use of a model of an induction motor with untransformed electric variables of the stator's windings. This model is presented in equations, while the notation used for stator's windings. Under the assumption that the voltage of the supply network Us12 is given in the form of the function ...the soft-start of the motor follows for the phase angle:

...while the connection of the remaining, third, clamp follows with a phase delay:

The values of coefficients _1, _2 are determined as a result of the calculations involving simulations for selected squirrel-cage motors; their extreme values are relevant with regard to motors from a small to large power. The instances of such connections, result in a virtual lack of aperiodic component in the generated magnetic flux _s, theta_r. This issue has been illustrated using examples based on computer simulations for motors from small to large power, and the obtained results. One can easily note the smooth curve of the current without the aperiodic component and small oscillations of the electromagnetic torque at the initial stage of the start-up. This comes as a consequence of the curve of trajectories of magnetic fluxes presented in the figures. This type of start-up, that is associated with the need to apply power semiconductor switches, has considerably more advantages than direct connection, and this can be concluded from a comparison between the above illustrations and results presented for the same motors.

DC Braking of an Induction Motor

Braking using direct current involves DC supply to the suitably connected stator's windings in such a way that enables the potentially high constant magnetic flux in which the rotor is put in motion. The current produced by rotor windings as a con sequence of induction combines with the magnetic field thus producing braking torque, which approaches idle run for a DC supply, i.e. the condition when the rotor is stalled. --- two typical layouts from among the list of the possible connections between the stator's windings for braking.

The modeling is based on equations for an induction motor for untransformed currents of stator's windings. For a three-phase system of connections, we directly apply equations by assuming that:

For two-phase power supply during braking the following constraints are applicable:

+-+-+- Connection of induction motor stator windings for the DC three-phase breaking and the two-phase breaking

+-+-+- 3-phase DC breaking of the small power induction motor with J = 5Js, iDC = 2In.:

a) stator current is1 b) stator current is2 c) magnetizing current m i d) MMF trajectory imu / imv e) electromagnetic torque f) rotational speed

Consequently, the stator windings connected in a star have a single degree of freedom, for which we assume the variable, is_1. Using and after elimination of the latter of equations, we obtain the model for this type of braking:

On the basis of the obtained versions of the mathematical model, simulations were conducted for braking of a small and large power motors for a braking current, which in the steady state is equal to iDC = 2In. The characteristic wave forms are presented.

+-+-+- 2-phase DC breaking of the small power induction motor with J = 5Js, iDC = 2In.:

a) stator current is1 b) magnetizing current im c) MMF trajectory imu / imv d) electromagnetic torque e) rotational speed

+-+-+- 3-phase DC breaking of the high power induction motor with J = 1.5Js, iDC = 2In:

a) stator current is1 b) stator current is2 c) magnetizing current im d) MMF trajectory imu / imv

e) electromagnetic torque f) rotational speed

+-+-+- 2-phase DC breaking of the high power induction motor with J = 1.5Js, iDC = 2In:

a) stator current is1 b) magnetizing current im

c) MMF trajectory imu / imv d) electromagnetic torque e) rotational speed

The closer familiarity with the results of calculations for DC breaking leads to the following general conclusions:

- 2-phase braking is considerably more effective than 3-phase braking with direct current; however, its characteristics include oscillations of torque and speed in the final phase of braking. This results from the lack of damping of the clamped circuits in the windings in phase 2 and 3

- magnetizing current during braking is quite small and is definitely smaller than the magnetizing current during symmetric motor regime. After the rotor is stalled the magnetizing current reaches the value of iDC. Hence, the saturation of the magnetic circuit over the entire range of speeds during braking is similar to characteristics of motor regime of operation and the applied models with constant parameters remain in the same precision range as during motor regime. This concerns the supply of the motor with the direct voltage, for which current iDC does not exceed several times the rated current.

- considerable differences are absent from the dynamic curves of braking for small and large power motors. Smaller motors tend to brake more dynamically, in accordance with the larger value of the electromechanical constant cm.

Reduction of a Mathematical Model to an Equivalent Circuit Diagram

A dynamic system, such as electric drive described with ordinary differential equations for given initial conditions and input functions, is characterized with a specific trajectory of the motion. This trajectory represents the history of all variables in a system. The steady state of such a system occurs when the trajectory is represented by a fixed point, that is ... or by a periodic function with the period of T, when ...For an electric drive this occurs when variables in a system forming the vector of generalized coordinates q are either constant functions or periodically variable ones. In a induction motor drive we can assume in an idealized way that the steady state occurs when the angular speed is constant, i.e. const r r = = O ? _ and the electric currents which supply the windings are periodic functions with the period in conformity with the voltages enforcing the flow of the currents.

One can note that the history of both the supply voltages and the resulting cur rents is relative to the transformation of the co-ordinates of the system, as presented in the models of the motor in a,ß, d,q or x,y axes. In a x,y system rotating with the speed ?c = ?s = p?f the symmetric system of sinusoidal voltages supplying phase windings as a result of transformation is reduced to constant voltages. In such coordinate system the steady state literally means a fixed point on the trajectories of all variables. The situation will be different for a steady state in the case of asymmetry of the supply voltages or cyclically variable load torque. In such a case the steady state will be characterized by periodically variable waveforms of electric currents and angular speed, while in the speed waveform the constant component will form the predominant element. The acquaintance with steady states is relevant for the design and exploitation of a drive since it provides information regarding its operating conditions and, hence, forms the basis for the development of strategies regarding methods of drive control. The familiarity with the steady states makes it possible to determine the characteristics of the drive, i.e. functional relations between variables that form the sets of constant points on a trajectory and ones that are time invariable. For the reasons given here the steady state of the induction motor drive can be conveniently described in axial coordinates x,y. Therefore, we will take as the starting point the transformed equations in current coordinates, which after the substitution ?c = ?s gives:... - is the slip of the rotor speed in relation to the rotating magnetic field. We assume that the steady state forms the fixed point of the trajectory ... hence, it denotes the constant angular speed theta_r = const and the constant slip s = const. This condition is possible due to the constant values of currents isxy, irxy, and, as a result, the constant electromagnetic torque Te. This requires the constant sup ply voltages after the transformation of x,y, which take the following form in accordance with:

This makes it possible to develop an equivalent circuit for an induction motor in the steady state as a result of merging equations in the form of a two port, using a common magnetizing reactance term jXm. The equivalent circuit in the form, beside the voltage and current relations presented in every two port, also realizes in an undisturbed manner the energetic relations occurring in the steady state. This comes as a result of the application of orthogonal trans formations that preserve scalar product and quadratic forms in the transformation of equations.

+-+-+- Equivalent circuit of an induction motor for the steady state

+-+-+-Equivalent circuit of induction motor with physical interpretation of electric power components

In this circuit we have to do with a resistance term Rr /s, which realizes in the energetic sense both Joule's losses in the rotor windings and the mechanical out put of the drive transferred via the machine's shaft as the product of torque Te and the angular speed of the shaft theta_r. Hence the resistance term can be divided into two terms: Rr, Rr(1-s)/s, which realize the losses of the power in the stator's windings and mechanical power Pm. The following components of the electric power are encountered in the equivalent diagram: input power Joule's losses in stator windings ir gap field power Joule's losses in rotor windings mechanical power

The energy balance for a 3-phase machine is preserved due to the fact that ..., hence, the transformed power is three times higher than the power of a single phase. In the analysis of the expression for the mechanical power output of an induction motor drive we can distinguish the following areas of operation:

  • - motor regime
  • - generating regime
  • - braking regime
  • - stall of the motor
  • - idle run

From the expression for the mechanical power we can calculate the motor's torque in the steady state:

The equivalent circuit can additionally be useful in the calculation of the stator and motor currents:

It would be valuable to present the currents in the standardized parameters since as a consequence of such presentation it’s possible to depart from the particular design of an induction motor. The standardized parameters assume values in the ranges presented. In this case the relations take the form:

Currents s r I, I , represent symbolic values of stator and rotor currents for steady state sine curves. The electromagnetic torque in the steady state can be derived from relation

...- is a blocked-rotor reactance. The electromagnetic torque can also be presented using standardized parameters, and takes this form:

The expressions, representing electromagnetic torque relative to supply voltages and motor parameters are frequently subjected to certain simplifications in order to simplify the analysis of these expressions. The basic procedure applies disregarding of the resistance of the stator winding Rs and, subsequently, as in some or all terms of this expression. A detailed analysis of this type of simplification will be conducted later on during the determination of the characteristics of the drive regime.

Static Characteristics of an Induction Motor

Static characteristics concern the steady state of a drive and give in an analytic or graphic form the functional relations between the parameters characterizing motor regimes. Typical static characteristics can for instance indicate the relations between electromagnetic torque, current and the capacity of a motor or between efficiency and the slip, voltage supply and the power output of the drive and the like.

One can note that static characteristics constitute a set of constant points along a trajectory {} q ? i q for selected variables of the state qi or their functions, that illustrate the values that are interesting from the point of view of the specific regimes of a machine, for example electromagnetic torque Te. Static characteristics collect the end points of trajectories for which the system reaches a steady state. They don’t provide information regarding the transfer from a specific point on the characteristics to another one, how much time it will take and whether it’s attainable.

Hence, in static characteristics we don’t have to do with such parameters as moment of inertia J, and the electromagnetic torque Te and load torque Tl are equal since the drive is in the state of equilibrium, i.e. it does not accelerate or brake. For example, very relevant characteristics are presented using functions. They illustrate the electromagnetic torque for an induction machine depending on a number of parameters. A typical task involves the study of the relation between the characteristics of the electromagnetic torque and the slip Te(s)

for constant remaining parameters, since it informs of the driving capabilities of the motor in the steady state. The relation between machine's torque and slip Te(s) gives the maximum of this function for two slip values called break torque slip or pull-out slip.

... or in standardized parameters ...

The root term ? in formulae is the factor for correction of the value of the break torque slip as a result of the of stator windings resistance Rs influence.

Since leakage coefficient is 03 . 0 08 . 0 … = s

… the following inequality is fulfilled ...

In addition, these relations are inversely proportional to the square root of the frequency of the supply source. Two degrees of simplification that are applicable in the development of static characteristics of an induction motor result from the presented estimates. The first of them is not very far-reaching and involves disregarding of resistance Rs in the terms denoting torque and break torque ...

... slip, in which this effect is smaller in accordance with the estimation in. In this case we obtain:

The most extensive simplification concerns the case when the resistance of the stator windings is completely disregarded, i.e. Rs = 0. In this case we obtain: +-+-+- presents static characteristics of the motor's torque in the function of the slip for a small power induction motor for the three examined variants of simplification regarding resistance Rs. One can note the small difference between the curve for the torque marked with solid line (i.e. the one presenting relations with out simplifications, and dotted line (i.e. the one presenting the result of calculations on the basis of formulae involving the first degree of simplification). However, when the resistance of stator windings is totally disregarded (Rs = 0) in accordance with formulae, the error in the characteristics of torque Te is considerable, as the relative involvement of resistance Rs in the stall impedance of small power motor is meaningful.

+-+-+- Torque-slip characteristics for the small power induction motor illustrating simplifications concerning stator resistance Rs: _____ Rs taken into consideration completely, according to; into consideration taken only the most significant component containing Rs; ---- Rs totally disregarded

It’s noteworthy that for Rs = 0 the characteristic of motor torque becomes an odd function of the slip s, so it’s symmetrical in relation to the point of the idle run s = 0. Accounting for resistance Rs torque waveform on the side of the motor regime (s>0) is considerably smaller in terms of absolute values than for the case of generating regime, i.e. for s<0. In addition, on the side of the generator regime the effect of the first degree of simplification accounting for resistance Rs is more clearly discernible than for the case of the motor regime, which can be simply interpreted by analyzing relations. The presented effect of the resistance of stator windings on the characteristics of the torque increases along with the reduction of the pulsation of the supply voltage ?s and becomes very high for small frequencies. This subject will be covered in more detail later. This effect when graphically presented in the range of the supply frequencies 1 < fs = 50 [Hz].

+-+-+- Characteristic of the break-torque slip sb versus pulsation of the supply voltage ?s for the small power motor.

The formulae for the break-torque slip and motor torque accounting for simplifications concerning the resistance can be additionally presented in formulae containing standardized parameters. The equivalent of the formulae takes the form:

Concurrently, formulae are replaced with the form which disregards resistance Rs, by introducing as = 0:

Formula constitutes the basic rule applicable for adjusting the RMS value of sinusoidal supply voltage Us of the motor to the frequency of this voltage fs in such a manner, that guarantees a constant break-torque value of Tb. Hence, the relation takes the form:

During the course of action that follows in the discussion of frequency based control of motor's rotational speed it will become evident that this rule is completely insufficient within the range of small supply frequencies. This is so due to the rising share of the resistance Rs in the impedance of the motor stall along with the decrease in the frequency of supply. The relation denoting the break-torque with out simplifications, in which resistance Rs is not disregarded, is much more complex than the one in. The greater complexity of the relation results from the substitution of the break-torque slip sb in the expression denoting the electromagnetic torque of the motor. As a result we obtain: Under the simplifying assumption that Rs = 0, we have as = 0 and ? = 1 and, as a consequence, break-torque expression is reduced to this form. The relation in is applied to indicate the effect of resistance Rs on the break torque Tb more clearly. The following show voltage frequency relations required to provide constant value of nominal break-torque Tbn in the function of stator voltage pulsation _s. For the motor regime of operation the required voltage is clearly higher than for the generator regime. From +-+-+- we can also see that smaller motors, within low frequency range, require much higher supply voltages than large motors to sustain the nominal level of Tbn. A close inspection indicates that for higher pulsations ?s the differences between motors disappear, but still there is constant discrepancy between the symmetrical 'ideal' V-line for as = 0 and the curves, for which stator resistance Rs was accounted for. For the motor operation the required voltages are higher while for generator operation they are lower in comparison to the 'ideal' V line. One might say that the actual V-line for which resistance Rs is included is shifted in the direction of lower pulsations ?s in respect to the 'ideal' V-line for which Rs is completely ignored.

+-+-+-30 Voltage-pulsation curves indicating the a stator voltage level required to sustain a nominal break-torque Tbn while _s pulsation changes. The curves are presented for different induction motors with as = 18.8, 5.4, 1.7, 1.2, 0.0 : a) for full range of stator voltage pulsation _s, b) range of _s limited to low values.

Subsequently, the characteristics of the motors in the function of the slip in two versions: completely accounting for parameter Rs - smaller characteristic in each pair, and the one totally disregarding resistance, i.e. for Rs = 0 - with the above presented characteristic. For nominal value of ?s = 2pfs, the distinctive difference between the two versions take place for the small power motor.

+-+-+- Torque-slip curves (relative values) for the three induction motors: small, medium and high power. The effect of Rs = 0 simplification is illustrated for fs = 50 [Hz] Subsequently, the characteristics of stator current for the three motors accounting for resistance Rs. The relation is applied in this case, which comes as a consequence of:

When the resistance of stator windings is disregarded (as = 0), the relation which defines the current in the stator windings takes a considerably more succinct form, which is additionally easy to verify for the two extreme motor states, i.e. for s = 0 and s = 8.

Self reactance of the stator windings Xs is encountered in a multitude of relations concerning induction motors. The value of this parameter can be easily determined from calculations or manufacturers' data for idle run. From the equivalent diagram of the motor it results that ....

+-+-+- Stator current-slip curves (relative values) for the three exemplary motors according to …presented in relative values, for fs = 50 [Hz] ... where ... denote voltage, current, impedance and phase angle for the idle run of the motor. If the phase angle during idle run is not familiar, it’s possible to use assessment relevant for the rated frequency: Rs << Xs and calculate in an approximated way.

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