AC machines with three or more phases (n ≥ 3) operate utilizing the principle of rotating field,^{3} which is created by spatially shifting individual phases along the circumference of the machine by an angle that equals the phase shift in the multiphase system of voltages (currents), used to supply such a multiphase winding. Such machines are of either synchronous or induction type. All rotating fields in multiphase machines, caused by the fundamental harmonic of the supply, rotate at synchronous speed, governed with the stator winding frequency. When the rotor rotates at the same speed, the machine is of synchronous type. When the rotor rotates at a speed different from synchronous, the machine is called asynchronous or induction machine.
Principles of mathematical modeling of multiphase machines have been developed in the first half of the twentieth century [1–3]. These include a number of different mathematical transformations that replace original phase variables (voltages, currents, flux linkages) with some new fictitious variables, the principal aim being simplification of the system of dynamic equations that describes a multiphase ac machine. Matrices are customarily used in the process of the model transformation, typically in real form. A somewhat different and nowadays very popular approach, which utilizes space vectors and derives from Fortescue's symmetrical component (complex) transformation [1], was developed in [4]. Its principal advantage, when compared to the matrix method, is a more compact form of the resulting model (that is otherwise the same), which is also easier to relate to the physics of the machinery.
Following the extensive work, conducted in relation to multiphase machine modeling in the beginning of the last century, numerous textbooks have been published, which detail the model transformation procedures for induction and synchronous machines, as well as the applications of the models in analysis of ac machine transients [5–23]. The principles of multiphase machine modeling, model transformations, and resulting models for both induction and synchronous machine (including machines with an excitation winding, permanent magnet synchronous machines, and synchronous reluctance machines) are presented here in a compact and easy-to-follow manner. Although most of the industrial machines are with three phases, the general case of an n-phase machine is considered throughout, with subsequent discussion of the required particularization to different phase numbers.
Modeling of multiphase ac machines is customarily subject to a number of simplifying assumptions. In particular, it is assumed that all individual phase windings are identical and that the multiphase winding is symmetrical. This means that the spatial displacement between magnetic axes of any two consecutive phases is exactly equal to α = 2π/n electrical degrees.^{4} Further, the winding is distributed across the circumference of the stator (rotor) and is designed in such a way that the magneto-motive force (mmf) and, consequently, flux have a distribution around the air-gap, which can be regarded as sinusoidal. This means that all the spatial harmonics of the mmf, except for the fundamental, are neglected. Next, the impact of slotting of stator (rotor) is neglected, so that the air-gap is regarded as uniform in machines with circular cross section of both stator and rotor (induction machines and certain types of synchronous machines). If there is a winding on the rotor, which is of a squirrel-cage type (as the case is in the most frequently used induction machines and in certain synchronous machines), bars of such a rotor winding are distributed in such a manner that the mmf of this winding has the same pole pair number as the stator winding and the complete winding can be regarded as equivalent to a winding with the same number of phases as the stator winding.
Some further assumptions relate to the parameters of the machines. In particular, resistances of stator (rotor) windings are assumed constant (temperature-related variation and frequency-related variation due to skin effect are thus neglected). Leakage inductances are also assumed constant, so that any leakage flux saturation and frequency-related leakage inductance variation are ignored. Nonlinearity of the ferromagnetic material is neglected, so that the magnetizing characteristic is regarded as linear. Consequently, magnetizing (mutual) inductances are constant. Finally, losses in the ferromagnetic material due to hysteresis and eddy currents are neglected, as are any parasitic capacitances.
The assumptions listed in the preceding two paragraphs enable formulation of the mathematical model of a multiphase machine in terms of phase variables. Of particular importance is the assumption on sinusoidal mmf distribution, which, combined with the assumed linearity of the ferromagnetic material, leads to constant inductance coefficients within a multiphase (stator or rotor) winding in all machines with uniform air-gap. In machines with nonuniform air-gap, however, inductance coefficients within a multiphase winding are governed by a sum of a constant term and the second harmonic, which imposes certain restrictions in the process of the model transformation. Hence, a machine with uniform air-gap is selected for the discussions of the modeling procedure and subsequent model derivation. The machine is a multiphase induction machine, since obtained dynamic models can easily be accommodated to various types of synchronous machines. Motoring convention for positive power flow is utilized throughout, so that the positive direction for current is always from the supply source into the phase of the machine. The number of rotor bars (phases) is, for simplicity, taken as equal to the number of stator phases n.
Consider an n-phase induction machine. Let the phases of both stator and rotor be denoted with indices 1 to n, according to the spatial distribution of the windings, and let additional indices s and r identify the stator and the rotor, respectively. Schematic representation of the machine is shown in Figure 3.1, where magnetic axes of the stator winding are illustrated. The machine's phase windings are assumed to be connected in star, with a single isolated neutral point.
Since all the windings of the machine are of resistive-inductive nature, voltage equilibrium equation of any phase of either stator or rotor is of the same principal form, v = Ri + dψ/dt. Here, v, i, and ψ stand for instantaneous values of the terminal phase to neutral voltage, phase current, and phase flux linkage, respectively, while R is the phase winding resistance. Since there are n phases on both stator and rotor, the voltage equilibrium equations can be written in a compact matrix form, separately for stator and rotor, as
Figure 3.1 Schematic representation of an n-phase induction machine, showing magnetic axes of stator phases ((α = 2π/n)).
Connection between stator (rotor) phase flux linkages and stator/rotor currents can be given in a compact matrix form as
Here, for both stator and rotor, winding phase self-inductances are governed with L_{11} = L_{22} = … = L_{nn}, while for mutual inductances within the stator (rotor) winding L_{ij} = L_{ji} holds true, where i≠ j, i, j = 1 … n. For example, in a three-phase winding L_{12} = L_{13} = L_{21} = L_{31} = L_{23} = L_{32} = M cos 2π/3, since cos 2π/3 = cos 4π/3, so that there is a single value of all the mutual inductances within a winding. Also, L_{ii} = L_{l} + M, where L_{l} is the leakage inductance. However, taking as an example a five-phase winding, one has two different values of mutual inductances within a winding, L_{12} = L_{21} = L_{15} = L_{51} = L_{23} = L_{32} = L_{34} = L_{43} = L_{45} = L_{54} = M cos 2π/5 and L_{13} = L_{31} = L_{14} = L_{41} = L_{24} = L_{42} = L_{35} = L_{53} = L_{52} = L_{25} = M cos 2(2π/5). In general, given an n-phase winding, there will be, due to symmetry, (n−1)/2 different mutual inductance values within the winding.
Stator-to-rotor mutual inductance matrix of (3.3) contains time-varying coefficients. Time dependence is indirect, through the instantaneous rotor position variation, since the position of any rotor phase winding magnetic axis constantly changes with respect to any stator phase winding magnetic axis, due to rotor rotation. Let the instantaneous position of the rotor phase 1 magnetic axis with respect to the stator phase 1 magnetic axis be θ degrees (electrical). Electrical rotor speed of rotation and the rotor position are related through
Due to the assumption of sinusoidal mmf distribution, mutual inductances between stator and rotor phase windings can be described with only the first harmonic terms, so that
Note that in (3.6) one has cos (θ − (n − 1)α) ≡ cos (θ + α), cos (θ − (n − 2) α) ≡ cos (θ + 2α), etc.
Model (3.1) through (3.6) completely describes the electrical part of a multiphase induction machine. Since there is only one degree of freedom for rotor movement, the equation of mechanical motion is
The inductances of (3.6) are functions of electrical rotor position and, hence, according to (3.5), electrical rotor speed of rotation. The equation of mechanical motion (3.7) is, therefore, customarily given in terms of electrical speed of rotation ω, which is related to the mechanical angular speed of rotation through the number of magnetic pole pairs P, ω = Pω_{m}. Hence,
Equation of mechanical motion (3.7) is always of the same form, regardless of whether original variables or some new variables are used. Symbol T_{e} stands for the electromagnetic torque, developed by the machine. It in essence links the electromagnetic subsystem with the mechanical subsystem and is responsible for the electromechanical energy conversion. In general, electromagnetic torque is governed with
As stator and rotor winding inductance matrices, given with (3.4), do not contain rotor-position-dependent coefficients, Equation 3.8 reduces for smooth air-gap multiphase machines to
This means that, in machines with uniform air-gap, electromagnetic torque is solely created due to the interaction of the stator and rotor windings.
Any multiphase induction machine is completely described, in terms of phase variables (or, as it is said, in the original phase domain) with the mathematical model given with (3.1) through (3.8) (or (3.10) instead of (3.8)). The model is composed of a total of 2n + 1 first-order differential equations (3.1) and (3.7), where 2n differential equations are voltage equilibrium equations, while the (2n+1) th differential equation is the mechanical equilibrium equation. In addition, there are 2n + 1 algebraic equations (3.3) and (3.8). The first 2n algebraic equations provide correlation between flux linkages and currents of the machine, while the (2n+1) th algebraic equation is the torque equation. Finally, the model is completed with an integral equation (3.5), which relates instantaneous rotor electrical position with the angular speed of rotation.
Substitution of flux linkages (3.3) into voltage equilibrium equations (3.1) and electromagnetic torque (3.10) into the equation of mechanical motion (3.7) eliminates algebraic equations, so that the machine model contains 2n + 1 first-order differential equations in terms of winding currents, plus the integral equation (3.5). This is a system of nonlinear differential equations, with time-varying coefficients due to variable stator-to-rotor mutual inductances of (3.6). While solving this model directly, in terms of phase variables, is nowadays possible with the help of computers, this was not the case 100 years ago. Hence, a range of mathematical transformations of the basic phase-variable model has been developed, with the prime purpose of simplifying the model by the so-called change of variables. Model transformation is therefore considered next.
Before proceeding further, one important remark is due. Since stator and rotor variables and parameters in general apply to two different voltage levels, rotor winding is normally referred to the stator winding voltage level. This is in principle the same procedure that is customarily applied in conjunction with transformers, and it basically brings all the windings of the machine to the same voltage (and current) base. In all machines where the squirrel-cage rotor winding is used (induction machines and synchronous machines with damper winding), the actual values of rotor currents and rotor parameters cannot anyway be measured and, hence, this change of the rotor winding voltage level has no consequence on the subsequent model utilization since rotor voltages of (3.2) are by default equal to zero. However, if there is excitation at the rotor winding side, as the case may be with slip-ring induction machines (and as the case is with the field winding of the synchronous machines), in which case rotor winding voltages are not zero, it is important to have in mind that rotor voltages and currents (as well as parameters) will in what follows be values referred to the stator winding. No distinction is made here in terms of notation between original rotor winding variables and parameters, and corresponding values referred to the stator voltage level. As a matter of fact, it has already been implicitly assumed in the development of the model (3.1) through (3.10) that rotor winding has been referred to the stator winding.
Variables of an n-phase symmetrical induction machine can be viewed as belonging to an n-dimensional space. Since the stator winding is star connected and the neutral point is isolated, the effective number of the degrees of freedom is (n−1); this applies to the rotor winding also. The machine model in the original phase-variable form can be transformed using decoupling (Clarke's) transformation matrix, which replaces the original sets of n variables with new sets of n variables. This transformation decomposes the original n-dimensional vector space into n/2 two-dimensional subspaces (planes) if the phase number is an even number. If the phase number is an odd number, the original space is decomposed into (n−1)/2 planes plus one single-dimensional quantity. The main property of the transformation is that new two-dimensional subspaces are mutually perpendicular, so that there is no coupling between them. Further, in each two-dimensional subspace, there is a pair of quantities, positioned along two mutually perpendicular axes. This leads to significant simplification of the model, compared to the original one in phase-variable form, as demonstrated next.
Let the correlation between any set of original phase variables and a new set of variables be defined as
It is the same for both stator and rotor multiphase windings and, for an arbitrary phase number n, it can be given as
Here once more α = 2π/n. The coefficient in (3.12) in front of the matrix, $\sqrt{2/n}$
, is associated with the powers of the original machine and the new machine, obtained after transformation. Selection as in (3.12) keeps the total powers invariant under the transformation.^{5} Also, due to such a choice of the scaling factor, the transformation matrix satisfies the condition that [C]^{−1} = [C]^{t}, so that [f_{1,2,… n}] = [C]^{t} [f]_{αβ}.The first two rows in (3.12) define variables that will lead to fundamental flux and torque production (α−β components; stator-to-rotor coupling will appear only in the equations for α−β components). The last two rows define the two zero-sequence components and the last row of the transformation matrix (3.12) is omitted for all odd phase numbers n. In between, there are (n−4)/2 (or (n−3)/2 for n = odd) pairs of rows that define (n−4)/2 (or (n−3)/2 for n = odd) pairs of variables, termed further on x–y components. Upon application of (3.12) in conjunction with the phase-variable model (3.1) through (3.6) and (3.10), assuming without any loss of generality that the phase number n is an odd number and that rotor n-phase winding is short-circuited, one gets the following new model equations:
Per-phase equivalent circuit magnetizing inductance is introduced in (3.13) through (3.15) as L_{m} = (n/2) M and symbols L_{ls} and L_{lr} stand for leakage inductances of the stator and rotor windings, respectively. These are in essence the same parameters that appear in the well-known equivalent steady-state circuit of an induction machine and which can be obtained from standard no-load and locked rotor tests on the machine. Subscript + in designation of the zero-sequence component of (3.12) is omitted since there is a single such component when the phase number is an odd number.
Torque equation (3.15) shows that the torque is entirely developed due to the interaction of stator/rotor α–β current components and is independent of the value of x–y current components. This also follows from the α–β voltage equilibrium equations of both stator and rotor in (3.13) and (3.14), since these are the only axis component equations where coupling between stator and rotor remains to be present, through the rotor position angle θ. From rotor equations (3.14) it follows that, since the rotor winding is short-circuited and stator x–y components are decoupled from rotor x–y components, equations for rotor x–y components and the zero-sequence component equation can be omitted from further considerations.
The same applies to the stator zero-sequence component equation. Note that zero sequence is governed by the sum of all instantaneous phase quantities. Since winding is considered as star connected with isolated neutral, no zero-sequence current can flow in the stator winding (if the number of phases is even and such that n ≥ 6, the second zero-sequence 0_{−} current component can flow if the supply is such that is not zero). As far as the x–y stator current components are concerned, they will also be zero as long as the supply voltages upon application of the decoupling transformation do not yield nonzero stator voltage x–y components. Thus, under ideal symmetrical and balanced sinusoidal multiphase voltage supply, the total number of equations that has to be considered in the electromagnetic subsystem is only four differential equations (two pairs of α–β equations in (3.13) and (3.14)) instead of the 2n differential equations in the original phase-variable model.
As is obvious from (3.13) and (3.14), the basic form of the voltage equilibrium equations has not been changed by applying the decoupling transformation, and they are still governed with v = Ri + dψ/dt. However, by comparing the phase-variable model of the previous section with the relevant equations obtained after application of decoupling transformation, it is obvious that a considerable simplification has been achieved. Regardless of the actual phase number, one only needs to consider further four voltage equilibrium equations, instead of 2n, as long as the machine is supplied from a balanced symmetrical n-phase sinusoidal source. Torque equation (3.15) is also of a considerably simpler form than its counterpart in (3.10). Needless to say, Equations 3.5 and 3.7 do not change the form in the model transformation process. However, the problem of time-varying coefficients and nonlinearity of the system of differential equations has not been resolved.
New fictitious α–β and x–y stator and rotor windings are still firmly attached to the corresponding machine's member, meaning that stator windings are stationary, while rotor windings rotate together with the rotor. In order to get rid of the time-varying inductance terms in (3.13) through (3.15), it is necessary to perform one more transformation, usually called rotational transformation. This means that the fictitious machine's windings, obtained after application of the decoupling transformation, are now transformed once more into yet another set of fictitious windings. This time, however, the transformation for stator and rotor variables is not the same any more.
As stator-to-rotor coupling takes place only in α−β equations, rotational transformation is applied only to these two pairs of equations. Its form for an n-phase machine is identical as for a three-phase machine, since x–y component equations do not need to be transformed. The transformation is defined in such a way that the resulting new sets of stator and rotor windings, which will replace α−β windings, rotate at the same angular speed, so-called speed of the common reference frame. Thus, relative motion between stator and rotor windings gets eliminated, leading to a set of differential equations with constant coefficients. Since in an induction machine air-gap is uniform and all inductances within both stator and rotor multiphase winding in (3.4) are constants, selection of the speed of the common reference frame is arbitrary. In other words, any convenient speed can be selected. Let us call such an angular speed arbitrary speed of the common reference frame, ω_{a}. This speed defines instantaneous position of the d-axis of the common reference frame with respect to the stationary stator phase 1 axis,
The second axis of the common reference frame, which is perpendicular to the d-axis, is customarily labeled as q-axis. The correlation between variables obtained upon application of the decoupling transformation and new d−q variables is defined similarly to (3.11):
However, rotational transformation matrix [D] is now different for the stator and rotor variables:
As is evident from (3.19), rotational transformation is applied only to α−β equations, while x−y and zero-sequence equations do not change the form. The inverse relationship of (3.18), [f_{αβ}] = [D]^{−1} [f_{dq}], is again a simple expression since once more [D]^{−1} = [D]^{t}. An illustration of the various spatial angles in the cross section of the machine is shown in Figure 3.2.
When the decoupled model (3.13) through (3.15) of an n-phase induction machine with sinusoidal winding distribution is transformed using (3.18) and (3.19), the set of voltage equilibrium and flux linkage equations in the common reference frame for a machine with an odd number of phases is obtained in the following form:
Figure 3.2 Illustration of various angles used in the rotational transformation of an induction machine's model (1s and 1r denote magnetic axes of the first stator and rotor phases).
Since rotor winding is regarded as short-circuited, zero-sequence and x–y component equations of the rotor have been omitted from (3.20) and (3.21). If there is a need to consider these equations (as the case may be if the rotor winding has more than three phases and is supplied from a power electronic converter in a slip-ring machine), one only needs to add to the model (3.20) and (3.21) rotor x–y equations of (3.14), which are of identical form as in (3.20b) and (3.21b) and only index s needs to be replaced with index r.
Upon application of the rotational transformation torque expression (3.15) becomes
Model (3.20) through (3.22) fully describes a general n-phase induction machine, of any odd phase number. If the number of phases is even, it is only necessary to add the equations for the second zero-sequence component, which are of the identical form as in (3.20) and (3.21) for the first zero-sequence component. However, the complete model needs to be considered only if the supply of the machine contains components that give rise to the stator voltage x–y components. If the machine is considered to be supplied with a set of symmetrical balanced sinusoidal n-phase voltages (of equal rms value and phase shift of exactly 2π/n between any two consecutive voltages), then stator voltage x–y components are all zero, regardless of the phase number. This means that analysis of an n-phase machine can be conducted under these conditions by using only stator and rotor d − q pairs of equations, in exactly the same manner as for a three-phase machine.
A closer inspection of the d − q voltage equilibrium equations in (3.20a) of the stator and the rotor shows that, upon application of the rotational transformation, these equations are not of the same form as in phase domain (i.e., the form is not any more v = Ri + dψ/dt). The equations contain an additional term, a product of an angular speed and a corresponding flux linkage component. The reason for this is that, by means of rotational transformation, the speed of the windings has been changed. Instead of being zero and ω for the stator and the rotor, respectively, the speeds of new windings have been equalized and are now ω_{a}. The new additional terms account for this change and they represent rotational induced electromotive forces in fictitious d − q windings of the stator and the rotor.
A schematic representation of the fictitious machine that results upon application of the rotational transformation is shown in Figure 3.3. Assuming ideal symmetrical and balanced n-phase sinusoidal supply of the machine, the representation of the machine, regardless of the number of phases, is as in Figure 3.3. What this means is that an n-phase machine can be replaced with an equivalent two-phase machine for modeling purposes. Zero sequence is along a line perpendicular to the d − q plane (or, for even phase numbers, in a plane perpendicular to the d − q plane). If the supply is such that x–y stator voltage components are not zero, the representation of a machine with five or more phases has to include also x–y voltage and flux linkage equations. However, since the equivalent x–y windings are situated in the planes perpendicular to the one of Figure 3.3, simultaneous graphical representation of all the new windings is not possible any more.
Figure 3.3 Fictitious d − q windings of stator and rotor obtained using rotational transformation.
As can be seen from (3.21), time-varying inductance terms have been eliminated by means of rotational transformation. Hence, electromagnetic torque equation does not contain such time-varying terms either. The system of differential equations is now with constant coefficients. Further, if the speed of rotation is considered as constant, Equations 3.20 and 3.21 become linear differential equations, analysis of which can be done using, say, Laplace transform. This was just about the only technique available in the beginning of the last century, so that the model transformation has enabled initial analytical analyses of the transients to be conducted (albeit at a constant speed).
Electromagnetic torque equation (3.22) can be given in a number of alternative ways, by utilizing the correlations between d − q axis stator/rotor currents and d − q axis stator/rotor flux linkages of (3.21). Some alternative formulations of the electromagnetic torque are the following:
As noted already, angular speed of the common reference frame can be selected freely in an induction machine. However, some selections are more favorable than the others.
For simulation of transients of a mains-fed squirrel-cage induction machine, the most opportune common reference frame is the stationary reference frame, such that ω_{a} = 0, θ_{s} = 0, since then the stator variables actually involve only decoupling transformation. It should be noted that rotor variables are practically never of interest in squirrel-cage induction machines since they are immeasurable anyway. The other frequently used reference frame is the synchronous reference frame, in which the common d − q reference frame rotates at the angular speed equal to the angular frequency of the fundamental stator supply. Such a reference frame is very convenient for various analytical studies of, for example, inverter-supplied induction machines. The common reference frame fixed to the rotor (ω_{a} = ω) is only suitable if a slip-ring induction machine is under consideration, with a power electronic supply connected to the rotor winding.
A completely different selection of the angular speed of the common reference frame is utilized for the realization of high-performance induction motor drives with closed-loop control. Such control schemes are termed vector- or field-oriented control schemes, and the speed of the common reference frame is selected as speed of rotation of one of the rotating fields (stator, air-gap, or rotor) in the machine.
Since the relationship between original phase variables and variables obtained after decoupling transformation is governed by (3.11), while d − q variables are related to variables obtained after decoupling transformation through (3.18), it is possible to express the two individual transformations as a single matrix transformation that will relate phase variables 1,2,…,n with d − q variables. Let such a transformation matrix be denoted as [T]. From (3.11) and (3.18), one has [f_{dq}] = [D][C][f_{1,2,…,n}], so that [T] = [D][C]. Since the rotational transformation matrix is different for stator and rotor variables, the complete transformation matrix will also be different. Taking as an example a three-phase machine, the combined decoupling/rotational transformation matrix for stator variables will be
In the general n-phase case one has, instead of (3.24), the following:
Transformation matrices for the rotor are, in form, identical to those for the stator (3.24) and (3.25), and it is only necessary to replace the angle of transformation θ_{s} with θ_{r}.
When the model of the machine is used for simulation purposes, it is typically necessary to apply the appropriate transformation matrix in both directions. For the sake of example, consider a three-phase induction machine, supplied from a three-phase voltage source. Hence, stator phase voltages are known. Corresponding d − q axis voltage components are calculated using (3.24) for the selected reference frame:
These are the inputs of the d − q axis model, together with the disturbance, load torque. The model is solved for the electromagnetic torque, rotor speed, and stator d − q axis currents (rotor d − q currents are usually not of interest; however, they are obtained too). Since actual stator phase currents are of interest, then d − q axis stator current components have to be now transformed back into the phase domain, using inverse transformation:
Note that, due to assumed stator winding connection into star, with isolated neutral point, zero-sequence current cannot flow, and hence zero-sequence components are not considered.
Assuming that stator voltages are sinusoidal, balanced, and symmetrical, of rms value V, it is simple to show that the amplitude of d − q axis voltage components in (3.26) is, regardless of the selected reference frame, equal to $\sqrt{3}V$
. This is the consequence of the adopted power-invariant form of the transformation matrices. In general, for an n-phase machine, the amplitude is $\sqrt{n}V$ . In contrast to this, if the transformation is power-variant and keeps transformed power per-phase equal (coefficient in (3.25) is 2/n rather than $\sqrt{2/n}$ ), amplitudes of d − q axis components are equal to $\sqrt{2}V$ regardless of the phase number.Since, upon application of the decoupling transformation, one gets pairs of axis components in mutually perpendicular planes and these pairs are in mutually perpendicular axes as well, it is possible to consider all the planes as complex and define one axis component as a real part and the other axis component as an imaginary part of a complex number. Such complex numbers are known as space vectors and they differ considerably from phasors (complex representatives of sinusoidal quantities). To start with, space vectors can be used for both sinusoidal and nonsinusoidal supply. Second, space vectors describe a machine in both transient and steady-state operating conditions. In what follows, space vectors are denoted with underlined symbols.
Consider decoupling transformation matrix (3.12). As can be seen, each pair of rows contains sine and cosine functions of the same angles. Let a complex operator a be introduced as a = exp (jα) = cos α + j sin α, where once more α = 2π/n. Each pair of rows in (3.12) then defines one space vector, with odd rows determining the real parts and the even rows imaginary parts of the corresponding complex numbers, that is, space vectors. Let f stand once more for voltage, current, or flux linkage of either the stator or the rotor. Space vectors are then governed with
It is again assumed that the phase number is an odd number and neutral point is isolated, so that zero sequence cannot be excited. It is therefore not included here, but it in general remains to be governed with the corresponding penultimate row of the decoupling transformation matrix (3.12).
Since rotational transformation is applied only to α−β components, then only the corresponding α−β space vector will undergo a further transformation, governed with (3.19) in real form. Of course, the transformation is once more different for stator and rotor quantities. The stator and rotor voltage, current, and flux linkage space vectors are obtained in the common reference frame by rotating corresponding α−β space vector by an angle, which is for stator θ_{s} and for rotor θ_{r}. This is done by means of the vector rotator, exp (−jθ_{s}) for stator and exp (−jθ_{r}) for rotor variables. Hence, space vectors that will describe the machine in an arbitrary common reference frame are governed with
To form the induction machine's model in terms of space vectors, it is only necessary to combine d − q axis equations of the real model (3.20) and (3.21) as real and imaginary parts of the corresponding complex equations. Hence, the torque-producing part of the model is, regardless of the phase number, described with
Indices d − q, used in (3.29) to define space vectors, have been omitted in (3.30) and (3.31) for simplicity. In (3.30) and (3.31), space vectors are v_{s} = v_{ds} + jv_{qs}, i_{s} = i_{ds} + ji_{qs}, ψ_{s} = ψ_{ds} + jψ_{qs} and v_{r} = v_{dr} + jv_{qr}, i_{r} = i_{dr} + ji_{qr}, ψ_{r} = ψ_{dr} + jψ_{qr}. Torque equation (3.22) can be given, using space vectors, as
Model (3.30) and (3.31) is the dynamic model of an induction machine. Consider now steady-state operation with symmetrical balanced sinusoidal supply. Regardless of the selected common reference frame, model (3.30) and (3.31) under these conditions reduces to the well-known equivalent circuit of an induction machine, described with
Stator voltage space vector under symmetrical sinusoidal supply conditions is shown in Figure 3.5 for a three-phase machine. It travels around the circle of radius equal to $\sqrt{3}V$
. Instantaneous projections of the space vector onto α- and β-axis represent space vector real and imaginary parts, in accordance with the definition in (3.28). Upon application of the vector rotator of (3.29) with ${\theta}_{s}={\displaystyle \int {\omega}_{s}dt}={\omega}_{s}t$ the stator voltage space vector becomes aligned with the d-axis of the common rotating reference frame so that the q-component is zero. Since the d − q system of axes rotates, its position continuously changes; thus, the illustration in Figure 3.5a applies to one specific instant in time, when the angle is 45°. Since the machine is in steady state, the stator current space vector is in essence determined with the ratio of the stator voltage space vector and impedance. The angle that appears between the stator voltage and stator current space vectors is the power factor angle ϕ (Figure 3.5b). Speed of rotation of the stator current space vector is of course equal to the speed of the voltage space vector, but the radius of the circle along which the stator current space vector travels is different.Figure 3.4 Equivalent circuit of an induction machine for steady-state operation with sinusoidal supply in terms of space vectors.
Figure 3.5 Illustration of the stator voltage and current space vectors for symmetrical sinusoidal supply conditions. (a) Stator voltage space vector and (b) stator voltage and current space vectors.
If the machine has five or more phases and the stator supply is either not balanced/symmetrical, or it contains certain time harmonics that map into x−y stator voltage components, then it becomes necessary to use additional equivalent circuits, one per each x−y plane (i.e., only one for a five-phase machine, but two for a seven-phase machine, and so on). In principle, the form of equivalent circuits for x−y components is governed with (3.33). However, since x−y voltages may contain more than one frequency component, a separate equivalent circuit is needed for steady-state representation at each such frequency. Assuming, for the sake of illustration, that stator x−y voltages contain a single-frequency component, the equivalent circuit is as given in Figure 3.6.
Whether or not the stator winding x−y circuits are excited entirely depends on the properties of the stator winding supply. If the supply is a power electronic converter, which produces time harmonics in the output phase voltage, then some of these harmonics will map into each x–y plane. As an example, Table 3.1 shows harmonic mapping, characteristic for five-phase and seven-phase stator windings [24]. As can be seen, one particular time harmonic in each x–y plane for each phase number is shown in bold font. These are the time harmonics of the supply that can be used, in addition to the fundamental, to produce an average torque. The idea is to increase the torque density available from the machine, and this applies equally to both generating operation [25] and motoring operation [26]. However, for this to be possible, it is necessary that the stator winding is of the concentrated type, so that, in addition to the fundamental space harmonic, there exist the corresponding low-order space harmonics of the mmf. In simple terms, this means that the spatial distribution of the mmf is not regarded as sinusoidal any more; it is quasi-rectangular instead. Modeling of such machines is beyond the scope of this article. It suffices to say that, while the decoupling transformation matrix remains the same, rotational transformation changes the form. Also, the starting phase-variable model in this case has to take into account the existence of the low-order spatial harmonics through appropriate harmonic inductance terms. In the final model, d − q equations remain the same but electromagnetic torque equation and x–y circuit equations change.
Figure 3.6 Equivalent circuit, applicable to each frequency component of every x−y stator voltage space vector in machines with more than three phases.
Plane |
Five-Phase System |
Seven-Phase System |
---|---|---|
α − β |
10j ± 1 (1, 9, 11…) |
14j ± 1 (1, 13, 15…) |
x_{1} −y_{1} |
10j ± 3 (3, 7, 13…) |
14j ± 5 (5, 9, 19…) |
x_{2} −y_{2} |
n/a |
14k ± 3 (3, 11, 17…) |
Zero-sequence |
5(2j + 1) (5, 15…) |
7(2j + 1) (7, 21…) |
In high-power applications, it is more and more common that, instead of using three-phase machines, machines with multiple three-phase windings are used. The most common case is a six-phase machine. The stator winding is composed of two three-phase windings, which are spatially shifted by 30°. The outlay is shown schematically in Figure 3.7 for an induction machine. Since there are now two three-phase windings, phases are labeled as a, b, c, and indices 1 and 2 apply to the two three-phase windings (index s is omitted). As can be seen from Figure 3.7, this spatial shift leads to asymmetrical positioning of the stator phase magnetic axes in the cross section of the machine. Such a type of the multiphase machine is, therefore, usually termed asymmetrical machine, since spatial shift between any two consecutive phases is not equal any more and it is not governed by 2π/n. Instead, there is a shift between three-phase windings, equal to π/n. Furthermore, since the machine is based on three-phase windings and there are in general a of them, then the neutral points of each individual three-phase winding are kept isolated, so that there are a isolated neutral points.
Figure 3.7 Asymmetrical six-phase induction machine, illustrating magnetic axes of the stator phases.
Modeling principles, discussed so far, are valid for asymmetrical multiphase machines as well. As a matter of fact, final machine models in the common reference frame (3.20) through (3.22) and (3.30) through (3.33) remain to be valid, provided that decoupling transformation matrix (3.12) is adapted to the winding layout in Figure 3.7. In particular, [C] is, for an asymmetrical six-phase machine, given with [27]
Here, the first three terms in each row relate to the first three-phase winding, while the second three terms relate to the second three-phase winding, as indicated in the row above the transformation matrix. The form of the last two rows in (3.36) takes into account that neutral points of the two windings are isolated.
Provided that the asymmetrical six-phase machine's phase-variable model is decoupled using (3.36), rotational transformation matrices (3.19) remain the same and identical equations are obtained in the d − q common reference frame and in space vector form as for a symmetrical multiphase machine (of course, the complete transformation matrix of (3.25) has to be modified in accordance with (3.36)). One important note is however due in relation to the total number of x–y equation pairs. Use of a individual and isolated neutral points means that, upon transformation, there will be only (n−a) voltage equilibrium equations to consider, since zero-sequence current cannot flow in any of the three-phase windings. Since n = 3a, then the total number of equations is 2a. As the first pair is always for d − q components, then the resulting number of x–y voltage equation pairs is only (a−1). This comes down to one d − q pair and two x–y pairs for an asymmetrical nine-phase machine with three isolated neutral points. Had the neutral points been connected, there would have been three pairs of x–y equations.
Figure 3.8 An asymmetrical nine-phase stator winding structure.
As an example, consider an asymmetrical nine-phase machine, with disposition of stator phase magnetic axes shown in Figure 3.8. Stator phases of any of the three-phase windings are labeled again as a, b, c and additional index 1, 2, 3 denotes the particular three-phase winding. The angle between three-phase windings is α = π/n = 20°. The winding may have a single neutral point or three isolated neutral points. Decoupling transformation matrix for the asymmetrical nine-phase winding with a single neutral point is determined with (the ordering of terms in the rows of the transformation matrix now corresponds to the spatial ordering of phases in Figure 3.8, as indicated in the row above the transformation matrix):
Modeling principles, detailed in preceding sections for multiphase induction machines, apply in general equally to synchronous machines, since the stator winding of all synchronous machines is identical as for an induction machine, regardless of the number of phases. However, the rotor of synchronous machines differs considerably from the induction machine's rotor, both in terms of the winding disposition used and in terms of its construction. Moreover, synchronous machines are much more versatile than induction machines and come in a variety of configurations.
Most of synchronous machines have excitation on rotor, which can be provided either by permanent magnets or by a dc-supplied excitation (or field) winding. The exception is synchronous reluctance machine, where rotor is not equipped with either magnets or the excitation winding. Further, rotor of a synchronous machine may or may not carry a squirrel-cage short-circuited winding, depending on whether the machine is designed to operate from mains or from a power electronic supply with closed-loop speed (position) control. Finally, rotor may be of circular cross section, but it may also have a so-called salient-pole structure.
Two principal geometries of the rotor are illustrated in Figure 3.9. Only one phase (1s) of the stator multiphase winding is shown and it is illustrated schematically with its magnetic axis. The rotor is shown as having an excitation winding, which is supplied from a dc source and which produces rotor field. This field is stationary with respect to rotor and acts along the d-axis. But, since rotor rotates at synchronous speed, the rotor field rotates at synchronous speed in the air-gap as well. In both types of synchronous machines, which are normally used for electric power generation and high-power motoring applications, rotor will either physically have a squirrel-cage winding (salient-pole rotor; not shown in Figure 3.9) or will behave as though there is a squirrel-cage winding (cylindrical rotor structure).
If permanent magnets are used instead of the excitation winding, then they may be either fixed along the circumference of a cylindrical rotor (surface-mounted permanent magnet synchronous machine, often abbreviated as SPMSM) or they may be embedded (or inset) into the rotor (interior PMSM or IPMSM). If the machine is designed for variable-speed operation with closed-loop control, the rotor will not have any windings. If the machine is aimed at line operation, then the rotor will have to have a squirrel-cage winding (recall that a synchronous motor develops torque at synchronous speed only; hence, if supplied from mains, it cannot start unless there is a squirrel-cage winding that will provide asynchronous torque at nonsynchronous speeds of rotation).
Figure 3.9 Basic structures of synchronous machines with (a) cylindrical rotor and (b) with a salient-pole rotor.
Transformations discussed in Sections 3.3 and 3.4 and given with (3.12) and (3.19) remain to be valid in exactly the same form for synchronous machine. However, what is different and therefore impacts considerably on the transformation procedure is the fact that the air-gap in a synchronous machine is not uniform any more. This is obvious for the salient-pole structure of Figure 3.9, but also applies to the cylindrical rotor structure, since the excitation winding occupies only a portion of the rotor circumference, so that the effective air-gap is the lowest in the d-axis and is the highest in the axis perpendicular to d-axis (i.e., q-axis). Nonuniform air-gap length means that the magnetic reluctance, seen by stator phase windings, continuously changes as rotor rotates. Note, however, that as far as the inductances of rotor windings are concerned, the situation is identical as for induction machines, since stator cross section is circular (and the same as in induction machines). Thus, rotor winding inductances will all be constant, as the case was in an induction machine.
As far as permanent magnet synchronous machines are concerned, in terms of magnetic behavior IMPSM corresponds to the salient-pole structure (since permeability of permanent magnets is very close to the permeability of the air, thus causing considerably higher magnetic reluctance in the rotor area where magnets are embedded, compared to the rotor area where there is only ferromagnetic material). On the other hand, SPMSMs behave similar to the machines with cylindrical rotor structure. Since magnets are effectively increasing the air-gap length and are placed uniformly on the rotor surface, the difference between the magnetic reluctance in SPMSMs along d- and q-axis is very small and is usually neglected.
Magnetic reluctance, seen by stator phase windings, varies continuously as the rotor rotates. It changes between two extreme values, the minimum one along d-axis and the maximum one along q-axis. Hence, one can define two corresponding extreme stator phase winding self-inductances, L_{sd} and L_{sq}. Assuming again that the spatial distribution of the mmf is sinusoidal, it can be shown that the stator phase 1 inductance is now governed with
As can be seen from (3.39), self-inductance is a constant position-independent quantity if and only if the inductances along d- and q-axis are the same, which applies only if the air-gap is perfectly uniform. When there is a variation in the air-gap, the self-inductance contains the second harmonic of a continuously changing value as the rotor rotates. Self-inductance of (3.39) will during each revolution of the rotor take the maximum and minimum values (L_{sd} and L_{sq}) twice. Similar considerations also apply to mutual inductances within the multiphase stator winding, which will now also contain the second harmonic in addition to a constant value. Hence, in synchronous machines, all elements of the stator inductance matrix (3.4a) contain rotor-position-dependent terms, which is a very different situation when compared to an induction machine. Dependence of stator inductance matrix terms on rotor position also means that the electromagnetic torque of the machine (3.8) does not reduce any more to the form given in (3.10), since there is an additional term,
The first torque component in (3.40) is again the consequence of the interaction of the stator and rotor windings (fundamental torque component) and it exists in all synchronous machines with excitation on rotor (using either permanent magnets or an excitation winding). The second component is, however, purely produced due to the variable air-gap and is called reluctance torque component. In synchronous reluctance machines, where there is no excitation on rotor, this torque component is the only one available if squirrel-cage rotor winding does not exist.
The consequence of the rotor-position-dependent inductances of the stator winding on modeling procedure is that any synchronous machine can be described with a set of differential equations with constant coefficients if and only if one selects the common reference frame as firmly fixed to the rotor. Hence, d-axis of the common reference frame is selected as the axis along which the rotor field winding (or permanent magnets) produces flux. Thus, in (3.19) one now has θ_{s} ≡ θ, which simultaneously means that θ_{r} ≡ 0. Such transformation matrix is often called Park's transformation in literature. In simple terms, this means that rotational transformation is applied only to the stator fictitious windings, obtained after decoupling transformation. The machine is therefore modeled in the rotor reference frame. If the machine runs at synchronous speed, this coincides with the synchronous reference frame. However, in a more general case and especially in motoring applications, one needs to have in mind that fixing the reference frame to the rotor means that the transformation angle for stator variables has to be continuously recalculated using (3.5), where speed of rotation is a variable governed by (3.7).
As noted at the end of Section 3.2, it is important to observe that in synchronous machines with field winding there are separate voltage levels at the stator and field winding. It is assumed further on that the field winding (and the squirrel-cage winding, if it exists) has been referred already to the stator winding voltage level. Models are further given separately for synchronous machines with excitation winding and permanent magnet synchronous machines. Only the torque-producing part of the model is given, which is the same for all machines with three or more phases on the stator and in essence comes down to rearranging appropriately Equations 3.20a, 3.21a, and 3.22 of the induction machine model. If the machine has more than three phases, the models given further on need to be complemented with the x–y voltage and flux equations of the stator winding, (3.20b) and (3.21b). These remain to be given with identical expressions as for an induction machine and are therefore not repeated further on.
Stator voltage equilibrium equations (3.20a) are in principle identical as for an induction machine, except that now ω_{a} = ω. Rotor short-circuited winding (damper winding) voltage equations are also the same as in (3.20a) with the last term set to zero, since ω_{a} = ω. Hence,
Resistances of the rotor damper winding along d- and q-axis are not necessarily the same, and this is taken into account in (3.41b). Zero-sequence voltage equation of the stator winding is the same as in (3.13) and is not repeated. Voltage equilibrium equation of the excitation winding, identified with index f(which has not undergone any transformation, except for the voltage level referral to stator voltage level) is of the same form as for damper windings, except that the voltage is not zero:
Flux linkage equations of various windings, however, now involve two different values of the magnetizing inductance, L_{md} and L_{mq}, which is the consequence of the uneven air-gap. These inductances are related with the corresponding phase mutual inductance terms through L_{md} = (n/2) M_{d} and L_{mq} = (n/2) M_{q}. Hence, flux linkages along d- and q-axis are
It should be noted that in certain cases damper winding of the rotor is modeled with one equivalent d-axis winding (as in (3.41b) and (3.42)) but with two equivalent q-axis windings. In such a case, one more voltage equilibrium equation and one more flux equation are needed for the q-axis. Their form is identical as for the q-axis damper winding in (3.41b) and (3.42), but the parameters (resistance and leakage inductance) are in general different.
Electromagnetic torque equation (3.40) upon transformation reduces in the rotor reference frame to a simple form,
The form of (3.43b) can be re-arranged so that the fundamental torque component is separated from the reluctance torque component,
Mechanical equation of motion of (3.7) is of course the same as for an induction machine. Relationship between original stator phase variables and transformed stator d − q axis quantities is in the general case and in the three-phase case governed with (3.25) and (3.24), respectively, where ${\theta}_{s}\equiv \theta ={\displaystyle \int \omega dt}$
.Since in permanent magnet synchronous machines field winding does not exist, the field winding equations ((3.41c) and the last of (3.42)) are omitted from the model. It is also observed that the permanent magnet flux ψ_{m} now replaces term L_{md}i_{f} in the flux linkage equations of the d-axis. If the machine has a damper winding, it can again be represented with an equivalent dr–qr winding. Hence, voltage, flux, and torque equations of a permanent magnet machine can be given as
In torque equation (3.46), the first and the third component are the synchronous torques produced by the interaction of the stator and the rotor and due to uneven magnetic reluctance, respectively, while the second component is the asynchronous torque (the same conclusions apply to (3.43c), valid for a synchronous machine with a field winding). This component exists only when the speed is not synchronous, since at synchronous speed there is no electromagnetic induction in the short-circuited damper windings.
Model (3.44) through (3.46) describes an IPMSM. If the machine is not equipped with a damper winding, as the case will be in machines designed for variable-speed operation with power electronic supply, it is only necessary to remove from the model (3.44) through (3.46) all variables associated with the rotor winding. This comes down to omission of (3.44b) and (3.45b) and setting of rotor d − q currents in (3.45a) and (3.46) to zero.
If the magnets are surface-mounted, it is usually assumed that the machine is with uniform air-gap, so that L_{md} = L_{mq} = L_{m}. This makes magnetizing inductances along the two axes equal in (3.45) and (3.46) and, consequently, eliminates the reluctance component in the torque equation (3.46). Thus, for a SPMSM without damper winding, one gets an extremely simple model, which consist of the following equations:
The electrical part of the model (3.47) and (3.48) is usually written with eliminated stator d − q axis flux linkages, as
With regard to the correlation between stator phase and transformed variables, the same remarks apply as given in conjunction with a synchronous machine with excitation winding.
This type of synchronous machine does not have any excitation on rotor. Depending on whether the machine is designed for line operation or for power electronic supply, the rotor may or may not have the squirrel-cage winding. To get the model of this type of synchronous machine, it is only necessary to remove from the IPMSM model terms related to the permanent magnet flux linkage. Hence, from (3.44) through (3.46), one now gets
Figure 3.10 Equivalent dynamic d − q circuits of permanent magnet synchronous machines (Xd = ω Ld, Xq = ω Lq, Em = ωψm).
If the machine does not have squirrel-cage winding on rotor, rotor voltage equations (3.52b) and rotor flux linkage equations (3.53b) are omitted. Hence, the stator voltage equations and the electromagnetic torque in such a machine take an extremely simple form,
The form of the d − q axis equivalent circuits is the same as in Figure 3.10, provided that the electromotive force term ωψ_{m} is set to zero.
As noted already, permanent magnet machines and synchronous reluctance machines without rotor damper (squirrel-cage) windings are exclusively used in conjunction with power electronic supply and closed-loop control, which requires information on the instantaneous rotor position.
A basic review of the modeling procedure, as applied in conjunction with multiphase ac machines with sinusoidal mmf distribution around the air-gap, has been provided. The material has been presented in a systematic way so that not only three-phase but also machines with any phase number are covered. All the types of ac machinery that operate on the basis of the rotating field have been encompassed. This includes both induction and synchronous machines of various designs. Given modeling procedure and the models in developed form are valid under the simplifying assumptions introduced in Section 3.2. In this context a couple of remarks seem appropriate.
In a number of cases the assumptions of constant machine parameters represent physically unjustifiable simplifications. This is sometimes due to the machine construction and sometimes due to the transient phenomenon under consideration. For example, frequency-dependent variation of parameters (resistance and leakage inductance) is of importance in rotor windings of squirrel-cage induction machines, which are often designed with deep-bar winding, or there may even exist physically two separate cage windings. In both cases the accuracy of the model is significantly improved if the rotor is represented as having two (rather than one) squirrel-cage windings. In terms of the final model, this comes down to expanding the Equations 3.20 through 3.22) (or 3.30 through 3.32) so that the representation contains voltage equilibrium and flux equations for two rotor windings (note that this also affects the torque equation (3.22)). For more detailed discussion the reader is referred to [22].
Assumption of constant stator leakage inductance is usually accurate enough. The exception are the investigations related to starting, reversing, re-closing, and similar transients of mains-fed induction machines, where the stator current may typically reach values of five to seven times the stator rated current. Means for accounting for stator leakage flux saturation in the d − q axis models have been developed, and such modified models require knowledge of the stator leakage flux magnetizing curve, which can be obtained from locked rotor test.
The iron losses are of magnetic nature, and accounting for them in the d − q axis models can only ever be approximate. The usual procedure is the same as in the steady-state equivalent circuit phasor representation. An equivalent iron loss resistance can be added in parallel to the magnetizing branch in the circuit of Figure 3.4. This of course requires expansion of the model with additional equations and an appropriate modification of the torque equation. It should be noted that such a representation of iron losses can only ever relatively accurately represent the phenomenon if the machine is supplied from a sinusoidal source.
By far the most frequently inadequate assumption is the one related to the linearity of the magnetizing characteristic, which has made the magnetizing (mutual) inductance (or inductances in synchronous machines) constant. This applies to both induction and synchronous machines. There are even situations where this assumption essentially means that a certain operating condition cannot be simulated at all; for example, self-excitation of a stand-alone squirrel-cage induction generator. It is for this reason that huge amount of work has been devoted during the last 30 years or so to the ways in which main flux saturation can be incorporated into the d–q axis models of induction and synchronous machines. Numerous improved machine models, which account for magnetizing flux saturation (and therefore utilize the magnetizing characteristic of the machine), are nowadays available. Some methods are discussed in references [14,20,21]. In principle, the machine model always becomes considerably more complicated than the case is when saturation of the main flux is neglected.
Finally, resistances of all windings change with operating temperature. Since temperature does not exist as a variable in the d − q models, this variation cannot be accounted for unless the d − q model is coupled with an appropriate thermal model of the machine.