\r\n\tThe transcriptional factor Nrf2 is a regulator of cytoprotection - for example, of antioxidant enzymes, pro-inflammatory cytokines, apoptosis, detoxification, proliferative processes, etc. At present, it is unknown how Nrf2 is regulated, both in physiological and pathological conditions.Therefore, it is essential to understand the mechanism by which Nrf2 is regulated. This book aims to give the scientific community the opportunity to disseminate their research and readers to deepen or expand their understanding in this field of knowledge. Potential authors are encouraged to expand the list of topics as best suits them in order to present the new scientific advances on Nrf2 and its regulation in oxidative stress.
A proper study of the induction machine operation, especially when it comes to transients and unbalanced duties, requires effective mathematical models above all. The mathematical model of an electric machine represents all the equations that describe the relationships between electromagnetic torque and the main electrical and mechanical quantities.
The theory of electrical machines, and particularly of induction machine, has mathematical models with distributed parameters and with concentrated parameters respectively. The first mentioned models start with the cognition of the magnetic field of the machine components. Their most important advantages consist in the high generality degree and accuracy. However, two major disadvantages have to be mentioned. On one hand, the computing time is rather high, which somehow discountenance their use for the real-time control. On the other hand, the distributed parameters models do not take into consideration the influence of the temperature variation or mechanical processing upon the material properties, which can vary up to 25% in comparison to the initial state. Moreover, particular constructive details (for example slots or air-gap dimensions), which essentially affects the parameters evaluation, cannot be always realized from technological point of view.
The mathematical models with concentrated parameters are the most popular and consequently employed both in scientific literature and practice. The equations stand on resistances and inductances, which can be used further for defining magnetic fluxes, electromagnetic torque, and et.al. These models offer results, which are globally acceptable but cannot detect important information concerning local effects (Ahmad, 2010; Chiasson, 2005; Krause et al., 2002; Ong, 1998; Sul, 2011).
The family of mathematical models with concentrated parameters comprises different approaches but two of them are more popular: the phase coordinate model and the orthogonal (dq) model (Ahmad, 2010; Bose, 2006; Chiasson, 2005; De Doncker et al., 2011; Krause et al., 2002; Marino et al., 2010; Ong, 1998; Sul, 2011; Wach, 2011).
The first category works with the real machine. The equations include, among other parameters, the mutual stator-rotor inductances with variable values according to the rotor position. As consequence, the model becomes non-linear and complicates the study of dynamic processes (Bose, 2006; Marino et al., 2010; Wach, 2011).
The orthogonal (dq) model has begun with Park’s theory nine decades ago. These models use parameters that are often independent to rotor position. The result is a significant simplification of the calculus, which became more convenient with the defining of the space phasor concept (Boldea & Tutelea, 2010; Marino et al., 2010; Sul, 2011).
Starting with the ″classic″ theory we deduce in this contribution a mathematical model that exclude the presence of the currents and angular velocity in voltage equations and uses total fluxes alone. Based on this approach, we take into discussion two control strategies of induction motor by principle of constant total flux of the stator and rotor, respectively.
The most consistent part of this work is dedicated to the study of unbalanced duties generated by supply asymmetries. It is presented a comparative analysis, which confronts a balanced duty with two unbalanced duties of different unbalance degrees. The study uses as working tool the Matlab-Simulink environment and provides variation characteristics of the electric, magnetic and mechanical quantities under transient operation.
The structure of the analyzed induction machine contains: 3 identical phase windings placed on the stator in an 120 electric degrees angle of phase difference configuration; 3 identical phase windings placed on the rotor with a similar difference of phase; a constant air-gap (close slots in an ideal approach); an unsaturated (linear) magnetic circuit that allow to each winding to be characterized by a main and a leakage inductance. Each phase winding has Ws turns on stator and WR turns on rotor and a harmonic distribution. All inductances are considered constant. The schematic view of the machine is presented in Fig. 1a.
The voltage equations that describe the 3+3 circuits are:
In a matrix form, the equations become:
The quantities in brackets represent the matrices of voltages, currents, resistances and total flux linkages for the stator and rotor. Obviously, the total fluxes include both main and mutual components. Further, we define the self-phase inductances, which have a leakage and a main component: Ljj=Lσs+Lhs for stator and LJJ=LΣR+LHR for rotor. The mutual inductances of two phases placed on the same part (stator or rotor) have negative values, which are equal to half of the maximum mutual inductances and with the main self-phase component: Mjk=Ljk=Lhj=Lhk. The expressions in matrix form are:
where u denotes the angle of 1200 (or 2π/3 rad).
The analysis of the induction machine usually reduces the rotor circuit to the stator one. This operation requires the alteration of the rotor quantities with the coefficient k=Ws/WR by complying with the conservation rules. The new values are:
where the reluctances of the flux paths have been used. The new matrices, with rotor quantities denoted with lowercase letters are:
By virtue of these transformations, the voltage equations become:
By using the notations:
and after the separation of the currents derivatives, (8) can be written under operational form as follows:
Besides (10), the equations concerning mechanical quantities must be added. To this end, the electromagnetic torque has to be calculated. To this effect, we start from the coenergy expression, , of the 6 circuits (3 are placed on stator and the other 3 on rotor) and we take into consideration that the leakage fluxes, which are independent of rotation angle of the rotor, do not generate electromagnetic torque, that is:
The magnetic energy of the stator and the rotor does not depend on the rotation angle and consequently, for the electromagnetic torque calculus nothing but the last term of (11) is used. One obtains:
The equation of torque equilibrium can now be written under operational form as:
where ωR represents the rotational pulsatance (or rotational pulsation).
The simulation of the induction machine operation in Matlab-Simulink environment on the basis of the above equations system is rather complicated. Moreover, since all equations depend on the angular speed than the precision of the results could be questionable mainly for the study of rapid transients. Consequently, the use of other variables is understandable. Further, we shall use the total fluxes of the windings (3 motionless windings on stator and other rotating 3 windings on rotor).
It is well known that the total fluxes have a self-component and a mutual one. Taking into consideration the rules of reducing the rotor circuit to the stator one, the matrix of inductances can be written as follows:
Now, the equation system (8) can be written shortly as:
By using the multiplication with the reciprocal matrix:
than (15) becomes:
This is an expression that connects the voltages to the total fluxes with no currents involvement. Now, practically the reciprocal matrix must be found. To this effect, we suppose that the reciprocal matrix has a similar form with the direct matrix. If we use the condition: than through term by term identification is obtained:
where the following notations have been used:
Further, the matrix product is calculated:, which is used in (17). After a convenient grouping, the system becomes:
For the calculation of the electromagnetic torque we can use the principle of energy conservation or the expression of stored magnetic energy. The expression of the electromagnetic torque corresponding to a multipolar machine (p is the number of pole pairs) can be written in a matrix form as follows:
To demonstrate the validity of (21), one uses the expression of the matrix, (18), in order to calculate its derivative:
where the following notation has been used:
This expression defines the permeance of a three-phase machine for the mathematical model in total fluxes.
Observation: One can use the general expression of the electromagnetic torque where the direct and reciprocal matrices of the inductances (which link the currents with the fluxes) should be replaced, that is:
A more convenient expression that depends on sinθR and cosθR, leads to the electromagnetic torque equation in fluxes alone:
Ultimately, by getting together the equations of the 6 electric circuits and the movement equations we obtain an 8 equation system, which can be written under operational form:
This equation system, (26-1)-(26-8) allows the study of any operation duty of the three-phase induction machine: steady state or transients under balanced or unbalanced condition, with simple or double feeding.
Generally, the symmetrical three-phase squirrel cage induction machine has the stator windings connected to a supply system, which provides variable voltages according to certain laws but have the same pulsation. Practically, this is the case with 4 wires connection, 3 phases and the neutral. The sum of the phase currents gives the current along neutral and the homopolar component can be immediately defined. The analysis of such a machine can use the symmetric components theory. This is the case of the machine with two unbalances as concerns the supply. The study can be done either using the equation system (26-1...8) or on the basis of symmetric components theory with three distinct mathematical models for each component (positive sequence, negative sequence and homopolar).
The vast majority of electric drives uses however the 3 wires connection (no neutral). Consequently, there is no homopolar current component, the homopolar fluxes are zero as well and the sum of the 3 phase total fluxes is null. This is an asymmetric condition with single unbalance, which can be studied by using the direct and inverse sequence components when the transformation from 3 to 2 axes is mandatory. This approach practically replaces the three-phase machine with unbalanced supply with two symmetric three-phase machines. One of them produces the positive torque and the other provides the negative torque. The resultant torque comes out through superposition of the effects.
The operation of the machine with 2 unbalances can be analyzed by considering certain expressions for the instantaneous values of the stator and rotor quantities (voltages, total fluxes and currents eventually, which can be transformed from (a, b, c) to (α, β, 0) reference frames in accordance with the following procedure :
We define the following notations:
By using these notations in (17) and after convenient groupings we obtain:
Typical for the cage machine or even for the wound rotor after the starting rheostat is short-circuited is the fact that the rotor voltages become zero. The equations of the six circuits get different as a result of certain convenient math operations. (29-2) and (29-3) are multiplied by (-1/2) and afterwards added to (29-1); (29-3) is subtracted from (29-2); (29-1), (29-2) and (29-3) are added together. We obtain three equations that describe the stator. Similarly, (29-4), (29-5) and (29-6) are used for the rotor equations. The new equation system is:
Further, the movement equation has to be attached. It is necessary to establish the detailed expression of the electromagnetic torque in fluxes alone starting with (25) and using convenient transformations:
Ultimately, the 8 equation system under operational form is:
These equations allow the study of three-phase induction machine for any duty. It has to be mentioned that the electromagnetic torque expression has no homopolar components of the total fluxes.
For the study of the single unbalance condition is necessary to consider expressions of the instantaneous values of the stator and rotor quantities (voltages, total fluxes and eventually currents in a,b,c reference frame) whose sum is null. The real quantities can be transformed to (d,q) reference frame (Simion et al., 2011). By using the notations (28-1), (28-2), (28-3) and (28-4) then after convenient grouping we obtain (Simion, 2010):\n\t\t\t\t\n\t\t\t\t
Further, the movement equation (31) must be attached. The operational form of the equation system (4 electric circuits and 2 movement equations) is:
The equation sets (33-1...4) and (34-1...6) prove that a three-phase induction machine connected to the supply system by 3 wires can be studied similarly to a two-phase machine (two-phase mathematical model). Its parameters can be deduced by linear transformations of the original parameters including the supply voltages (Fig. 2a).
The windings of two-phase model are denoted with (αs, βs) and (αr, βr) in order to trace a correspondence with the real two-phase machine, whose subscripts are (as, bs) and (ar, br) respectively. We shall use the subscripts xs and ys for the quantities that corresponds to the three-phase machine but transformed in its two-phase model. This is a rightful assumption since (αs, βs) axes are collinear with (x, y) axes, which are commonly used in analytic geometry. Further, new notations (35) for the flux linkages of the right member of the equations (33-1...4) will be defined by following the next rules:
projection sums corresponding to rotor flux linkages from (αr, βr) axes along the two stator axes (denoted with x and y that is ψxr, ψyr) when they refer to the flux linkages from the right member of the first two equations, Fig. 2b.
projection sums corresponding to stator flux linkages from (αs, βs) axes along the two rotor axes (denoted with X and Y that is ψXS, ψYS) when they refer to the flux linkages from the last two equations, Fig. 2c.
Some aspects have to be pointed out. When the machine operates under motoring duty, the pulsation of the stator flux linkages from (αs, βs) axes is equal to ωs. Since the rotational pulsation is ωR then the pulsation of the rotor quantities from (αr, βr) axes is equal to ωr=sωs=ωs\n\t\t\t\t\t–ωR. The pulsation of the rotor quantities projected along the stator axes with the subscripts xr and yr is equal to ωs. The pulsation of the stator quantities projected along the rotor axes with the subscripts XS and YS is equal to ωr. The equations (33-1...4) become:
The first two equations join the quantities with the pulsation ωs and the other two, the quantities with the pulsation ωr\n\t\t\t\t\t= sωs. The expression of the magnetic torque, in total fluxes and rotor position angle becomes:
or a second equivalent expression:
which shows the ″total symmetry″ of the two-phase model of the three-phase machine regarding both stator and rotor. The equations of the four circuits together with the movement equation (37) under operational form give:
This last equation system allows the study of transients under single unbalance condition. It is similar with the frequently used equations (Park) but contains as variables only total fluxes and the rotation angle. There are no currents or angular speed in the voltage equations.
For the steady state analysis of the symmetric three-phase induction machine, one can define the simplified space phasor of the stator flux, which is collinear to the total flux of the (αs) axis and has a times higher modulus. In a similar way can be obtained the space phasors of the stator voltages and rotor fluxes and the system equation (39-1...6) that describe the steady state becomes:
When the speed regulation of the cage induction machine is employed by means of voltage and/or frequency variation then the simultaneous control of the two total flux space vectors is difficult. As consequence, new strategies more convenient can be chosen. To this effect, we shall deduce expressions of the electromagnetic torque that include only one of the total flux space vectors either from stator or rotor.
One of the methods used for the control of induction machine consists in the operation with constant stator total flux space vector. From (40), the rotor total flux space vector is:
where θ is the angle between stator and rotor total flux space vectors. This angle has the meaning of an internal angle of the machine.
The expression of the magnetic torque that depends with the stator total flux space vector becomes:
Assuming the ideal hypothesis of maintaining constant the stator flux, for example equal to the no-load value, then the pull-out torque, Temax, corresponds to sin2θ = 1 that is:
Now an observation can be formulated. Let us suppose an ideal static converter that operates with a U/f=constant=k1 strategy. For low supply frequencies, the pull-out torque decreases in value since the denominator increases with the pulsatance decrease, ωs (Fig. 3). Within certain limits at low frequencies, an increase of the supply voltage is necessary in order to maintain the pull-out torque value. In other words, U/f = k2, and k2>k1.\n\t\t\t\t
A proper control of the induction machine requires a strategy based on U/f = variable. More precisely, for low frequency values it is necessary to increase the supply voltage with respect to the values that result from U/f = const. strategy. At a pinch, when the frequency becomes zero, the supply voltage must have a value capable to compensate the voltage drops upon the equivalent resistance of the windings. Lately, the modern static converters can be parameterized on the basis of the catalog parameters of the induction machine or on the basis of some laboratory tests results.
From (40) we can deduce:\n\t\t\t\t
if the term νtt was neglected. By inspecting the square root term, which is variable with the slip (and load as well), we can point out the following observations.
Constant maintaining of the stator flux for low pulsations (that is low angular velocity values including start-up) can be obtained with a significant increase of the supply voltage. The ″additional″ increasing of the voltage depends proportionally on the load value. Analytically, this fact is caused by the predominance of the term G against F, (45). From the viewpoint of physical phenomena, a higher voltage in case of severe start-up or low frequency operation is necessary for the compensation of the leakage fluxes after which the stator flux must keep its prescribed value.
Constant maintaining of the stator flux for high pulsations (that is angular speeds close or even over the rated value) requires an insignificant rise of the supply voltage. The U/f ratio is close to its rated value (rated values of U and f) especially for low load torque values. However, a certain increase of the voltage is required proportionally with the load degree. Analytically, this fact is now caused by the predominance of the term F against G, (45).
In conclusion, the resultant stator flux remain constant for U/f =constant=k1 strategy if the load torque is small. For high loads (especially if the operation is close to the pull-out point), the maintaining of the stator flux requires an increase of the U/f ratio, which means a significant rise of the voltage and current.
If the machine parameters are established, then a variation rule of the supply voltage can be settled in order to have a constant stator flux (equal, for example, to its no-load value) both for frequency and load variation.
Fig. 4 presents (for a machine with predetermined parameters: supply voltage with the amplitude of 490 V (Uas=346.5V); Rs=Rr=2; Lhs=0,09; Lσs= Lσr=0,01; J=0,05; p=2; kz=0,02; ω1=314,1 (SI units)) the variation of the resultant stator voltage with the pulsatance (in per unit description) for three constant slip values. The variation is a straight line for reduced loads and has a certain inflection for low frequency values (a few Hz). For under-load operation, a significant increase of the voltage with the frequency is necessary. This fact is more visible at high slip values, close to pull-out value (in our example the pull-out slip is of 0,33).
The variation rule based on UsR=f(ωs) strategy (applied to the upper curve from Fig. 4) provide an operation of the motor within a large range of angular speeds (from start-up to rated point) under a developed torque, whose value is close to the pull-out one. Obviously, the input current is rather high (4-5 I1N) and has to be reduced. Practically, the operation points must be placed within the upper and the lower curves, Fig. 4. It is also easy to notice that the operation with higher frequency values than the rated one does not generally require an increase of the supply voltage but the developed torque is lower and lower. In this case, the output power keeps the rated value.
Usually, the electric drives that demand high value starting torque use constant rotor total flux space vector strategy. The stator total flux space vector can be written from (41) as:
and the expression of the electromagnetic torque on the basis of rotor flux alone becomes:
Assuming the ideal hypothesis of maintaining constant the rotor flux, for example equal to the no-load value, then the electromagnetic torque expression is:
where the voltage and pulsation is supposed to have rated values. Taking into discussion a machine with predetermined parameters (supply voltage with the amplitude of 490 V (Uas=346.5V); Rs=Rr=2; Lhs=0,09; Lσs= Lσr=0,01; J=0,05; p=2; kz=0,02; ω1=314,1 (SI units)) then the expression of the mechanical characteristic is:
which is a straight line, A1 in Fig. 5. The two intersection points with the axes correspond to synchronism (Te=0, ΩR=ωs/2=157) and start-up (Te=995 Nm, ΩR=0) respectively.
The pull-out torque is extremely high and acts at start-up. This behavior is caused by the hypothesis of maintaing constant the rotor flux at a value that corresponds to no-load operation (when the rotor reaction is null) no matter the load is. The compensation of the magnetic reaction of the rotor under load is hypothetical possible through an unreasonable increase of the supply voltage. Practically, the pull-out torque is much lower.
Another unreasonable possibility is the maintaining of the rotor flux to a value that corresponds to start-up (s = 1) and the supply voltage has its rated value. In this case the expression of the mechanical characteristic is (50) and the intersection points with the axes (line A2, Fig. 5) correspond to synchronism (Te=0, ΩR=ωs/2=157) and start-up (Te=78 Nm, ΩR=0) respectively.
The supply of the stator winding with constant voltage and rated pulsation determines a variation of the resultant rotor flux within the short-circuit value (ΨrRk=0,5Wb) and the synchronism value (ΨrR0=1,78Wb). The operation points lie between the two lines, A1 and A2, on a position that depends on the load torque. When the supply pulsation is two times smaller (and the voltage itself is two times smaller as well) and the resultant rotor flux is maintained constant to the value ΨrR0=1,78Wb, then the mechanical characteristic is described by the straight line B1, which is parallel to the line A1. Similarly, for ΨrRk=0,5Wb, the mechanical characteristic become the line B2, which is parallel to A2.
When the applied voltage and pulsation are two times smaller regarding the rated values then the operation points lie between B1 and B2 since the rotor flux varies within ΨrRk=0,5Wb (short-circuit) and ΨrR0=1,78Wb (synchronism).
The control based on constant rotor flux strategy ensures parallel mechanical characteristics. This is an important advantage since the induction machine behaves like shunt D.C. motor. A second aspect is also favorable in the behavior under this strategy. The mechanical characteristic has no sector of unstable operation as the usual induction machine has.
The modification of the flux value (generally with decrease) leads to a different slope of the characteristics, which means a significant decrease of the torque for a certain angular speed.
The question is ″what variation rule of UsR/ωs must be used in order to have constant rotor flux″? The expression of the modulus of the resultant rotor flux can be written as: