Control system and method for a rotating electromagnetic machine

[0002] 1. Field of the Invention

[0003] The invention relates generally to the control of electromagnetic rotating machines such as permanent magnet, switched reluctance and hybrid machines thereof, and more particularly, to adaptive, smooth torque control of such machines.

[0004] 2. Description of Related Art

[0005] Many electromagnetic machines in general, and electric motors employing permanent magnets in particular, exhibit torque irregularities as the rotor rotates with respect to the stator, the coils of which are typically energized with a sinusoidal waveform. Such irregularities are referred to as “torque ripple.” These torque irregularities may be caused by the physical construction of a given machine. For example, they may result from the use of bearings to support the rotor. In addition, because of the electromagnetic characteristics of machines that employ magnets, the rotor tends to prefer certain angular positions with respect to the stator. Torque irregularities resulting from the electromagnetic characteristics of an electromagnetic machine are commonly known as “cogging” irregularities and the resultant non-uniform rotation of the rotor or non-uniform torque output is known as “cogging.” Cogging is either current independent or current dependent. The first component is noted when the machine is spun unenergized. The second component is present when current flows—the cogging grows as the magnitude of the stator currents increases.

[0006] In rotating electromagnetic machines employing permanent magnets, cogging most often results from the physical construction of the machine. Irregularities due to the magnets can result, for example, from the magnets being incorrectly placed upon or in the rotor, or from some irregularity about how the magnets are energized. Moreover, the utilization of rotors having discrete north and south outer poles results in a circumferential distribution of magnetic flux about the rotor circumference that is not smooth, but choppy. Additionally, the stators commonly used with such machines are formed in such a way that the magnetic fluxes generated by the stator windings provide a flux distribution about the stator circumference that is not smooth. The combination of such rotors and stators, and the accompanying non-smooth flux distributions, produces undesired irregularities in the torque output of such machines. Rotor output non-uniformities may also be produced by back emf harmonics produced in certain machines.

[0007] Obtaining smooth torque is further complicated by other factors. For instance, manufacturing variances between motors makes it difficult, if not impossible, to apply a common solution to a group of motors. Such manufacturing variance includes the placing or misplacing of magnets upon the rotor (if surface mounted), variance introduced by the magnetizing process itself and irregularities in the stator coil windings. Other causes of variance include instances when the magnets are damaged or chipped. Further, variations exist even within individual motors. For example, variations typically exist between a motor's phases and over the motor's full mechanical cycle. Moreover, motor behavior changes over time as the motor ages.

[0008] The phase windings of certain types of electromagnetic machines are energized at least in part as a function of the instantaneous rotor position. Accordingly, such machines often use a rotor position sensor that provides an output indicating rotor position relative to the stator. A controller uses this information to produce control signals that are used to energize and de-energize the phase windings. Errors in the measurement of the angular position of the rotor also contribute to torque ripple.

[0009] For many motor applications a slight non-uniformity in the rotation of the rotor caused by torque irregularities is of little or no consequence. For example, in large motors driving large loads, slight variations in the output torque will not significantly affect the rotor speed and any slight variations in rotor speed will not significantly affect the system being driven by the machine. This assumes that the torque variation as the machine turns is small compared to the load. In other applications, where the rotation of the rotor or the torque output of the motor must be precisely controlled or uniform, such non-uniformity is not acceptable. For example, in servomotors used in electric power steering systems and is in disk drives, the rotational output of the rotor or the torque output of the motor must be smooth and without significant variation.

[0010] Prior art approaches to reducing the undesirable consequences of torque irregularities in electromagnetic machines have focused on relatively complex rotor or stator constructions designed to eliminate the physical characteristics of the machines that would otherwise give rise to the irregularities. While the prior art machine construction approaches can result in reduction of torque irregularities, the approaches require the design and construction of complex rotor and stator components, such complex components are typically difficult to design, difficult to manufacturer, and much more costly to produce than are conventionally constructed components. Moreover, many of the physical changes required by such prior art solutions result in a significant reduction in the efficiency or other performance parameters of the resulting machines over that expected of comparable conventional machines. Thus, many of the prior art attempts to reduce torque irregularities do so at the cost of machine performance.

[0011] Attempts to reduce torque ripple that focus on motor control schemes, rather than motor construction, have also been undertaken. For example, various learning or iterative schemes, based on either experimental procedures or well known physical relationships concerning motor voltages, currents and angular positions have been attempted. These attempted solutions often make assumptions concerning the behavior of the motor, such as motor flux being described by a linear relationship, or considering the effect of mutual flux insignificant. Still further, prior solutions to torque ripple typically ignore the effect of motor sensitivity to inaccuracies in angular measurement. To increase accuracy in position measurement, the use of sophisticated position sensors has been attempted, but this increases the machine's complexity and cost.

[0012] Thus, a need exists for a control system that addresses the shortcomings of the prior art.

[0013] In one aspect of the present invention, a system for controlling a rotating electromagnetic machine is presented. The rotating machine, such as a permanent magnet motor or switched reluctance motor, or some hybrid of the two, includes a stator having a plurality of phase windings and a rotor that rotates relative to the stator. A drive is connected to the phase windings for energizing the windings. The control system includes an estimator connectable to the machine for receiving signals representing the phase winding voltage, current and rotor position. The estimator outputs parameter estimations for an electrical model of the machine based on the received voltage, current and rotor position. The electrical model is a mathematical model that describes electrical behavior of the machine as seen at the motor terminals.

[0014] A torque model receives the parameter estimations from the estimator. The torque model is developed via a mathematical transform of the electrical model, and describes torque characteristics of the machine. Using the parameters received from the estimator, the torque model estimates torque for associated rotor position-phase current combinations. A controller has input terminals for receiving a torque demand signal and the rotor position signal. The controller outputs a control signal to the drive in response to the torque demand and rotor position signals and the torque model. In certain embodiments, a solver uses the torque model to generate energization current profiles according to desired machine behavior, such as smooth torque and/or minimal sensitivity to errors in rotor position measurement. It is to be noted in particular that the solver can be so defined that the solution possesses particular properties. It may be desired to deal with only the most significant components of cogging or torque ripple, a result of motor drive cost considerations. Such a solution can be achieved.

[0015] Some parameters of the torque model are unobservable via information immediately available from the machine terminals. For example, in machines employing permanent magnets, it is not mathematically obvious how changes in the machine current and voltage, as the rotor spins, indicate or measure the interaction of the machine's magnets with themselves. In accordance with further aspects of the present invention, a method of determining a non-load dependent cogging torque is provided. The rotor is spun unloaded at a predetermined angular velocity and the motor terminal voltage and current are measured. Rotor positions associated with the voltage and current measurements are determined, and a first mathematical model is developed based on the measured voltage and rotor position to describe electrical behavior of the machine. The first mathematical model is mathematically transformed to develop a second mathematical model to describe torque characteristics of the machine. The windings are then energized such that the rotor holds a predetermined position against the cogging torque, and the cogging torque is calculated for the predetermined position via the second mathematical model.

[0016] Other objects and advantages of the invention will become apparent upon reading the following detailed description and upon reference to the drawings in which:

[0017] FIG. 1 is a block diagram of an exemplary rotating electromagnetic machine system in accordance with embodiments of the present invention;

[0018] FIG. 2 is a block diagram illustrating an electromagnetic machine control system in accordance with embodiments of the present invention;

[0019] FIG. 3 is a chart illustrating an exemplary integration path for evaluating an integral giving coenergy used in embodiments of the invention;

[0020] FIG. 4 illustrates current profiles generated for a smooth torque solution in accordance with the present invention; and

[0021] FIG. 5 illustrates torque ripple for a machine controlled in accordance with aspects of the present invention.

[0022] While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

[0023] Illustrative embodiments of the invention are described below. In the interest of clarity, not all features of an actual implementation are described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the developers' specific goals, such as compliance with system-related and business-related constraints, which will vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure.

[0024] Turing to the drawings and, in particular, to FIG. 1, a system 10 constructed according to certain teachings of this disclosure is illustrated. Among other things, the illustrated system 10 actively controls the electric power supplied to an electromagnetic machine such that the negative consequences of torque irregularities that would otherwise be produced by the machine are reduced or eliminated.

[0025] The system 10 includes an electromagnetic machine 12 and a drive 14 that provides electric power to the electromagnetic machine 12. The machine 12 shown in FIG. 1 may comprise, for example, a permanent magnet motor, a switched reluctance motor, or a hybrid motor (permanent magnet and switched reluctance combination). The machine 12 is of conventional construction that includes a rotating component (a rotor 12a) and a stationary component (a stator 12b). Wound about the stator are a number of energizable phase windings 12cwhich may be energized through the application of electric power to motor terminals 15, 16, 17.

[0026] The drive 14 is coupled to provide electric power to terminals 15, 16 and 17 of the machine 12. The drive 14 receives control inputs from a control system 13, which is coupled to receive feedback from the machine 12 in terms of rotor position information 18 and energization feedback 19. Other feedback information may be provided to the controller 13. While the drive 14 is illustrated in exemplary form as providing three power terminals to the machine 12, it should be understood that more or fewer power terminals may be provided to accommodate motors or machines with greater than three phases, less than three phases or if various types of inverters (e.g., with neutral connections) are used.

[0027] The energization feedback 19 provides an indication of the operational characteristics of the machine 12 and may, for example, include feedback concerning the currents flowing in the stator windings and/or the voltages at the terminals 15, 16 and 17. The position and energization parameters may be detected through conventional detectors such as standard rotor position detectors and standard current/voltage sensors. Alternative embodiments are envisioned in which the rotor position and feedback parameters are not detected directly but are calculated or estimated through known techniques. For example, embodiments are envisioned where only the terminal voltages are known or sensed along with the currents flowing through the stator windings of the machine 12 and the sensed current and voltage values are used to derive rotor position information.

[0028] The control system 13 also receives input command signals 11 that correspond to a desired output parameter of machine 12 such as rotor speed, output torque, etc. As described in more detail below, the drive 14 controls the application of electric power to machine 12 in response to the control system 13 in such a manner that the difference between the input command signal 11 and the corresponding output of the machine 12 is minimized. In certain embodiments, the control system 13 also actively controls application of power to the machine 12 as a function of rotor position in such a manner to achieve a desired behavior of the machine 12 meeting one or more criteria in categories including, for example, torque ripple, cogging torque, angular sensitivity, harmonic content, etc. The use of the control system 13 to actively achieve desired machine behavior, as opposed to attempting to achieve such behavior through complex rotor or stator constructions, results in a better performing system in that, for example, conventional, low cost machines and machine construction techniques may be used.

[0029] An electromagnetic machine system 100 in accordance with an exemplary embodiment of the present invention is shown in FIG. 2. The machine system 100 includes a control system 13, which may be implemented by an appropriately programmed digital controller, such as a digital signal processor (DSP), microcontroller or microprocessor.

[0030] The control system 13 includes an input terminal 11 that receives, for example, a signal representing the torque demanded of the motor 12. Torque is a function of current and angle; hence, for any particular rotor angle there is a set of appropriate currents that will produce the desired torque. Based on the rotor angle and the required torque, appropriate current values are sent to the drive 14, which in turn provides the necessary voltage to the motor 12 to meet the current demand.

[0031] Rotor position feedback 18 and energization feedback 19, such as the motor terminal voltage and current, are provided to an estimator 30. In accordance with mathematical “good practice,” the voltage and current values may be normalized—the measured values are divided by the maximum expected value. The estimator 30 calculates motor parameters such as angular speed and the time derivatives of the phase currents. These values are used to derive and update a first mathematical model that describes the electrical behavior of the motor 12. The structure of the electrical model is such that it can accurately represent electrical machine characteristics such as resistance, back electromagnetic force (“BEMF”), self and mutual inductance, cogging, etc., depending on the type of machine 12 employed.

[0032] The parameters calculated by the estimator 30 are passed to a torque model 32 of the motor 12. The torque model 32 is developed by mathematically transforming the electrical model in an appropriate manner, dictated by the electromagnetic physics of the motor 12, into a second model that describes the torque characteristics of the machine 12. Since the electrical model coefficients flow naturally into the torque model 32, by constructing an accurate electrical model of the motor 12, the torque characteristics of the motor 12 are also known. For example, the torque model may describe the torque produced for any combination of phase current and rotor position in the normal operating envelope of the machine. Thus, using the values calculated by the estimator 30, an estimate of motor torque can be calculated for any current-angle combination.

[0033] The torque model 32 is interrogated by a solver 34, which calculates the required currents, or solution curves, according to some desired motor behavior and the known behavior of the motor 12 (as regards current and angle). Thus, the controller 36 provides the appropriate current for a given rotor angular position to achieve the desired output torque 40 or other output parameter, and further, to achieve the output parameter in accordance with the desired machine behavior. For example, the desired machine behavior may include the operating characteristics of the motor 12 meeting one or more criteria in categories including cogging torque, torque ripple, angular sensitivity of the solution to angular error and harmonic content of the solution curves. In the particular system 100 shown in FIG. 2, the solver 34 output is stored explicitly as a lookup table accessible by the controller 36. The torque demand 11 and rotor angle is applied to the lookup table to determine the appropriate phase current value to be applied to the phase windings via the driver 14. In other embodiments, the output of the solver 34 is in an analytic form, derived by fitting a function to the calculated numerical values.

[0034] Since the electrical model used by the estimator 30 is algebraic in nature, the estimator 30 can be allowed to run for some time period operating upon not necessarily sequential data before a new set of parameter estimates are released in to the torque model 32. At this point the solver 34 can then recalculate the necessary lookup tables 36. Once fully calculated, the new lookup tables can then replace those tables currently in use. Many of these operations can be background tasked; that is, they can occur as and when computational resources are available. This is one of the advantages of an algebraic motor model—it is not intricately wrapped up in the time variable.

[0035] Torque can be estimated via coenergy or field energy, though calculating torque via coenergy results in simpler expressions. Thus, the estimate of output torque 40 can be calculated using only feedback available from the machine terminals—such as the terminal voltage and current and the rotor position—to estimate the parameters of resistance and flux linkage. The following disclosure is generally provided in terms of a three phase hybrid motor, though the model form can be generalized into different types of rotating machines having any number of phases by one skilled in the art having the benefit of this disclosure.

[0036] It is common in many applications to utilize what is known as a balanced three phase feed. In such systems, when a three-phase motor is used, the sum of the three phase currents will equal zero. Hence, the αβ-Frame of Reference (FoR) can be used. If balanced feed is not used, it is necessary to use the abc-FoR. The αβ-FoR is considered first.

[0037] The electrical model may have the form of a product of polynomial expressions in current and angle. Typically, those for angle will involve trigonometric functions. The current polynomials may also be orthogonal and may be one of any number of suitable polynomial types. For more complex machines, an orthogonal model form may be appropriate. With the first model structure disclosed herein, it is assumed that flux linkage models are expressions that are products of polynomial terms involving phase currents and trigonometric polynomials of mechanical angle. Models using orthogonal functions are discussed further later in this specification. Generally, the following nomenclature is used in this disclosure:

[0038] φ is the phase index, ranges over defined set of numbers {1,2,3, . . . } or equivalent letters {a,b,c, . . . }

[0039] a,b,c are phase names, equivalent to 1,2,3 when numerically referenced

[0040] α,β,0 are αβ Frame of Reference (FoR) labels

[0041] λ_φ is the phase φ flux linkage

[0042] p,P . . . q,Q . . . r,R . . . n,N are summation indexes and maximum index values

[0043] sin( ), cos( ) are trigonometric functions

[0044] g_φpqrn, h_φpqrnare model parameters

[0045] i_α,i_β,i_Care variables representing αβ frame of reference currents

[0046] I_α,I_β are maximum values of αβ-frame of reference currents encountered in the coenergy integral

[0047] i_a,i_b,i_care variables representing abc-frame of reference currents

[0048] I_a,I_b,I_care maximum values of abc-FoR currents encountered in the coenergy integral

[0049] i_fis the current flow associated with the fictious rotor circuit modelling the presence of a magnet

[0050] v_φ is the phase φ voltage

[0051] R_φα,R_φβ,R_φαβ are resistance values associated with phase φ $\frac{d}{dy}x$

[0052] is the differential of x with respect to y

[0053] θ is the rotor angle

[0054] ω is the rotor angular velocity

[0055] t is time t

[0056] ω_cis coenergy

[0057] dx is x infintesimal

[0058] ∫ f(x) dx is the integral of f(x) with respect to x $\frac{\partial }{\partial x}f\left(x,y,\dots \right)$

[0059] is the partial derivative of function f(.) with respect to x

[0060] D₁, . . . ,D₆are components of the coenergy integral, along defined path

[0061] T is motor or true torque

[0062] o(xⁿ) is the remainder associated with n'th order and higher terms of the Jacobian matrix

[0063] J_ijis the ij'th entry of the Jacobian

[0064] F_i(x₁, . . . ,x_M) is the i'th function of variables x₁, . . . ,x_Mδx is delta x

[0065] δx_newis the new value of delta x, or change in x

[0066] x_new, x_oldare new and old values of x calculated during Newton iterative process

[0067] T_tvis estimated torque, directly from the terminal variables

[0068] T_cogis torque which cannot be calculated directly using terminal variables

[0069] S_ais solution sensitivity $I=\left(\begin{array}{c}{i}_{\alpha }\left(\theta \left(1\right)\right)\\ i_{\alpha }\left(\theta \left(2\right)\right)\\ \cdots \\ i_{\alpha }\left(\theta \left(N\right)\right)\\ i_{\beta }\left(\theta \left(1\right)\right)\\ i_{\beta }\left(\theta \left(2\right)\right)\\ \cdots \\ i_{\beta }\left(\theta \left(N\right)\right)\end{array}\right)$

[0070] is the vector of αβ frame of reference current values across the set of discrete angle values

[0071] I(n+1),I(n) are the (n+1)'th and n'th iterated current vectors

[0072] ΔI(n) is the calculated change in current vector at n'th interval

[0073] φ_Tk=(0 . . . 0 T(θ(k),i_α(k),i_β(k)) 0 . . . 0) is the k'th torque vector

[0074] φ_Sk=(0 . . . 0 S(θ(k),i_α(k),i_β(k)) 0 . . . 0) is the k'th sensitivity vector $A=\left(\begin{array}{c}{\phi }_{\mathrm{T1}}\\ \cdots \\ {\phi }_{\mathrm{TN}}\end{array}\right)$

[0075] are stacked torque vectors $B=\left(\begin{array}{c}{\phi }_{\mathrm{S1}}\\ \cdots \\ {\phi }_{\mathrm{SN}}\end{array}\right)$

[0076] are stacked current vectors

[0077] Assuming that the model structure for each machine phase (φ) is identical, the general form of the flux model using the αβ-FoR is: $\begin{array}{cc}{\lambda }_{\phi }=\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)& \left(1\right)\end{array}$

[0078] Such a model allows for a non-linear relationship between phase current and flux as well as mutual effects between any two or more phases. As noted above, for the purposes of the present disclosure it is assumed that model structure is invariant with respect to phase, although this need not be so. Contiguous powers of polynomial current and angle harmonic need not be used, as is the case in Equation (1). For example, consider the following: $\begin{array}{cc}{\lambda }_{\phi }=\sum _{p=p_1}^{p_S}{i}_{\alpha }^p·\sum _{q=q_1}^{q_T}{i}_{\beta }^q·\sum _{r=r_1}^{r_U}{i}_f^r·\sum _{n=n_1}^{n_V}\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)& \left(2\right)\end{array}$

[0079] where the indexing sets:

[0080] p=(p₁,p₂, . . . ,p_S.) r=(r₁,r₂, . . . ,r_U.)

[0081] q=(q₁,p₂, . . . ,q_T.) n=(n₁,n₂, . . . ,n_V.)

[0082] need not contain contiguous integers. In fact, most practical applications will have this form.

[0083] Relatively simple models of the form presented in Equation (2) that are sufficiently accurate can be obtained. Model structure can be allowed to vary between phases if so desired. This variation upon defining model structure has a significant impact upon the computational complexity of the associated algorithms. Some model components will be present as a result of manufacturing variance and would not be suggested by a theoretical consideration of the motor design. Further, model complexity can vary greatly between motors of different design. For example, a permanent magnet motor design with the express intent of reducing cogging, typically through the use of skew, may only require a very simple model to accurately predict torque. It is generally desirable to avoid models that are over or under determined.

[0084] The abc-FoR currents can be transformed into αβ-FoR currents using the following transform: $\begin{array}{cc}\left(\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ i_0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 1\\ \frac{-1}{2}& \frac{-1}{2}·\sqrt{3}& 1\\ \frac{-1}{2}& \frac{1}{2}·\sqrt{3}& 1\end{array}\right)·\left(\begin{array}{c}{i}_{\alpha }\\ i_b\\ i_c\end{array}\right)& \left(3\right)\end{array}$

[0085] Under the balanced feed assumption, the third phase current is zero. It is known that phase voltage (v_φ) is defined by $\begin{array}{cc}{v}_{\phi }=i_{\phi }·R_{\phi }+\frac{d}{dt}{\lambda }_{\phi }& \left(4\right)\end{array}$

[0086] where R_φ is the phase resistance. Thus, using the αβ-FoR: $\begin{array}{cc}{v}_{\phi }=R_{\phi }+i_{\alpha }·R_{\mathrm{\phi \alpha }}+i_{\beta }·R_{\mathrm{\phi \beta }}+i_{\alpha }·i_{\beta }·R_{\mathrm{\phi \alpha }\beta }+\frac{d}{dt}{\lambda }_{\phi }& \left(5\right)\end{array}$

[0087] It should be noted that there are more resistance terms in Equation (5) then is necessary from the perspective of how electric circuits operate. Such additional terms allow for the presence of test data offsets and the like to be directly compensated for; otherwise the estimator will set redundant terms to zero.

[0088] It is also known that angular velocity (ω) is defined by: $\begin{array}{cc}\omega =\frac{d}{dt}\theta & \left(6\right)\end{array}$

[0089] From Equations (1) and (4): $\begin{array}{cc}\frac{d}{dt}{\lambda }_{\phi }=\sum _{p=1}^P\left({\mathrm{pi}}_{\alpha }^{p-1}·\frac{d}{dt}{i}_{\alpha }\right)·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=1}^Q\left({\mathrm{qi}}_{\beta }^{q-1}·\frac{d}{dt}{i}_{\beta }\right)·\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=1}^R\left({\mathrm{ri}}_f^{r-1}·\frac{d}{dt}{i}_f\right)·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=1}^N\omega ·n·\left(g_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)-h_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)\right)& \left(7\right)\end{array}$

[0090] The imaginary rotor current (i_f) is nominally constant and its time derivative is zero. Hence, from Equations (5) and (7): $\begin{array}{cc}{v}_{\phi }=R_{\phi }+i_{\alpha }·R_{\mathrm{\phi \alpha }}+i_{\beta }·R_{\mathrm{\phi \beta }}+i_{\alpha }·i_{\beta }·R_{\mathrm{\phi \alpha \beta }\cdots }+\sum _{p=1}^P\left({\mathrm{pi}}_{\alpha }^{p-1}·\frac{d}{dt}{i}_{\alpha }\right)·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=1}^Q\left({\mathrm{qi}}_{\beta }^{q-1}·\frac{d}{dt}{i}_{\beta }\right)·\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=1}^N\omega ·n·\left(g_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)-h_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)\right)& \left(8\right)\end{array}$

[0091] In embodiments employing a switched reluctance machine, there is no imaginary rotor current state as there are no rotor magnets with which this state is associated, hence: $\begin{array}{cc}{i}_f\equiv 0\mathrm{and}\frac{d}{dt}{i}_f\equiv 0& \left(9\right)\end{array}$

[0092] Therefore, in the case of a switched reluctance machine, indexing variable (r) associated with the imaginary rotor phase may be removed from Equation (8): $\begin{array}{cc}{v}_{\phi }=R_{\phi }+i_{\alpha }·R_{\mathrm{\phi \alpha }}+i_{\beta }·R_{\mathrm{\phi \beta }}+i_{\alpha }·i_{\beta }·R_{\mathrm{\phi \alpha \beta }\cdots }+\sum _{p=1}^P\left({\mathrm{pi}}_{\alpha }^{p-1}·\frac{d}{dt}{i}_{\alpha }\right)·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=0}^N\left(g_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=1}^Q\left({\mathrm{qi}}_{\beta }^{q-1}·\frac{d}{dt}{i}_{\beta }\right)·\sum _{n=0}^N\left(g_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+h_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)\right)\cdots +\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=1}^N\omega ·n·\left(g_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)-h_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)\right)& \left(10\right)\end{array}$

[0093] It would be a routine undertaking for one skilled in the art having the benefit of this disclosure to apply a similar process of simplification to the specific case of an SR motor, for example.

[0094] There are several techniques available for calculating the coefficients in the model given test data. As noted above, the electrical model is algebraic, which allows the use of any of a number of parameter estimation techniques, such as least squares methods or grammian matrix methods. Least squares-based parameter estimators find the model coefficients that minimize the square of the difference between the observed data and the output from the model. Recursive least square parameter estimators are used in particular embodiments of the invention. The recursive least squares parameter estimators are most suitable for actual production systems. They operate in such a manner that they can produce an improved estimate with each new sample of data. That is, they are not restricted in their operation to complete sets of test data.

[0095] A further refinement of the recursive least squares parameter estimation technique involves the use of a “forgetting factor,” which operates in the following manner. As more and more data is captured, the effect that the old data has upon the calculation of the new data is reduced. In this way, only the most recent data will have a significant effect in the parameter estimation process. This forgetting factor operates over any appropriate time interval—for example, minutes, hours or days—depending on design considerations. Many of the variables considered do not vary significantly over time. Some, however, such as phase resistance, vary over time and with respect to other factors such as the machine temperature. This added refinement allows the control system to be tuned to any particular machine, and also allows adaptation to changes in that machine. This typically occurs as the machine ages.

[0096] Various data collection schemes may be used for parameter estimation. For example, one data collection technique requires constant phase current, which usually only occurs in controlled data collection situations. Another involves varying current, typical of practical applications.

[0097] In the constant phase current data collection scheme, $\frac{d}{dt}{i}_{\alpha }=\frac{d}{dt}{i}_{\beta }=0$

[0098] Under these conditions, Equation (8) reduces to: $\begin{array}{cc}{v}_{\phi }=R_{\phi }+i_{\alpha }·R_{\mathrm{\phi \alpha }}+i_{\beta }·R_{\mathrm{\phi \beta }}+i_{\alpha }·i_{\beta }·R_{\mathrm{\phi \alpha \beta }\cdots }+\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{r=0}^R{i}_f^r·\sum _{n=1}^N\omega ·n·\left(g_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)-h_{\phi \mathrm{pqrn}}·\sin \left(n·\theta \right)\right)& \left(11\right)\end{array}$

[0099] For notational convenience and to reflect the unobservability of the i_fterm, the following identities are defined as a result of considering the last two Σ terms in Equation (11): $\begin{array}{cc}\sum _{r=0}^R{i}_f^r·g_{\phi \mathrm{pqrn}}=G_{\phi \mathrm{pqn}}\mathrm{for}\mathrm{all}\phi ,p\mathrm{and}q& \left(12\right)\\ \sum _{r=0}^R{i}_f^r·h_{\phi \mathrm{pqrn}}=H_{\phi \mathrm{pqn}}\mathrm{for}\mathrm{all}\phi ,p\mathrm{and}q& \left(13\right)\end{array}$

[0100] To retain consistency with Equation (10) for the case in which phase current varies, the harmonic terms (n) in Equation (11) are not collected into the H and G terms defined by Equations (12) and (13). Substituting Equations (12) and (13) into Equation (11) results in: $\begin{array}{cc}{v}_{\phi }=R_{\phi }+i_{\alpha }·R_{\mathrm{\phi \alpha }}+i_{\beta }·R_{\mathrm{\phi \beta }}+i_{\alpha }·i_{\beta }·R_{\mathrm{\phi \alpha \beta }\cdots }+\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=1}^N\omega ·n·\left(G_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)-H_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)\right)& \left(14\right)\end{array}$

[0101] Division throughout Equation (14) by ω yields: $\begin{array}{cc}\frac{v_{\phi }}{\omega }=\frac{1}{\omega }·R_{\phi }+\frac{i_{\alpha }}{\omega }·R_{\mathrm{\phi \alpha }}+\frac{i_{\beta }}{\omega }·R_{\mathrm{\phi \beta }}+\frac{i_{\alpha }·i_{\beta }}{\omega }·R_{\mathrm{\phi \alpha \beta }\cdots }+\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=1}^{N}n·\left(G_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)-H_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)\right)& \left(15\right)\end{array}$

[0102] Using Equation (14), the motor parameters can be estimated in the case of constant phase current.

[0103] As noted above, practical motor applications involve varying current. Equation (8) describes how the flux model coefficients, first presented in Equation (1), propagate through three separate paths when calculating phase voltage in the variable current case. The first path, as previously encountered, passes through those expressions explicitly involving ω. The other two paths involve expressions with time derivatives of the currents. As before, ω is divided throughout and substitutions as indicated in Equations (12) and (13) are made, yielding: $\begin{array}{cc}\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{v_{\phi }}{\omega }=\frac{1}{\omega }·R_{\phi }+\frac{i_{\alpha }}{\omega }·R_{\phi \alpha }+\frac{i_{\beta }}{\omega }·R_{\phi \beta }+\frac{i_{\alpha }·i_{\beta }}{\omega }·R_{\phi \alpha \beta }\dots +\\ \left[\frac{1}{\omega }·\left(\frac{d}{dt}{i}_{\alpha }\right)\right]·\sum _{p=1}^P{\mathrm{pi}}_{\alpha }^{p-1}·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=0}^N\left(G_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+H_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\dots +\\ \left[\frac{1}{\omega }·\left(\frac{d}{dt}{i}_{\beta }\right)\right]·\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=1}^Q{\mathrm{qi}}_{\beta }^{q-1}·\sum _{n=0}^N\left(G_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+H_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)\right)\dots +\\ \sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=1}^{N}n·\left(G_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)-H_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)\right)\end{array}\\ \mathrm{Note}\mathrm{that}:\end{array}\\ \frac{1}{\omega }·\left(\frac{d}{dt}i\right)=\frac{1}{\left(\frac{d}{dt}\theta \right)}·\frac{d}{dt}i=\left(\frac{d}{d\theta }t\right)·\left(\frac{d}{dt}i\right)=\frac{d}{d\theta }i\end{array}& \left(16\right)\end{array}$

[0104] This allows a simple re-working of Equation (16) as follows: $\begin{array}{c}\frac{v_{\phi }}{\omega }=\frac{1}{\omega }·R_{\phi }+\frac{i_{\alpha }}{\omega }·R_{\phi \alpha }+\frac{i_{\beta }}{\omega }·R_{\phi \beta }+\frac{i_{\alpha }·i_{\beta }}{\omega }·R_{\phi \alpha \beta }\dots +\\ \left(\frac{d}{d\theta }{i}_{\alpha }\right)·\sum _{p=1}^P{\mathrm{pi}}_{\alpha }^{p-1}·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=0}^N\left(G_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+H_{\phi \mathrm{pqrn}}·\cos \left(n·\theta \right)\right)\dots +\\ \left(\frac{d}{d\theta }{i}_{\beta }\right)·\sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=1}^Q{\mathrm{qi}}_{\beta }^{q-1}·\sum _{n=0}^N\left(G_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)+H_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)\right)\dots +\\ \sum _{p=0}^P{i}_{\alpha }^p·\sum _{q=0}^Q{i}_{\beta }^q·\sum _{n=1}^{N}n·\left(G_{\phi \mathrm{pqn}}·\cos \left(n·\theta \right)-H_{\phi \mathrm{pqn}}·\sin \left(n·\theta \right)\right)\end{array}$

[0105] The advantage of this re-formulation of Equation (16) is that two possible sources of impulsive noise in the equation, and the problems they can cause with a parameter estimator, have been removed. If the driving current forms are assumed known, then their derivatives with respect to angle can be directly calculated. If the current forms are defined in an analytic form, such as from a BEMF model, then a closed expression exists for the derivative else a numerical estimate can be obtained. In particular, if at the (k−1) and k intervals the angle and desired currents are θ(k−1), θ(k), i_α(k−1) and i_α(k) then: $\frac{d}{d\theta }{i}_{\alpha }=\frac{i_{\alpha }\left(k\right)-i_{\alpha }\left(k-1\right)}{\theta \left(k\right)-\theta \left(k-1\right)}$

[0106] This approach implicitly assumes the system exhibits good current following properties.

[0107] Since resistance and flux linkage can be estimated via the electrical model, a torque model relating current and angle to torque generated can be derived through a series of standard operations. Very generally, these operations include integrating the flux linkage from zero current to the present value of current—coenergy, and differentiating this expression with respect to shaft angle. The result of these operations is an expression for torque.

[0108] Coenergy (ω_c) is defined by: $\begin{array}{cc}{\omega }_c=\int \sum _{\phi =1}^{N_{\phi }}{\lambda }_{\phi }d{i}_{\phi }& \left(17\right)\end{array}$

[0109] where N is the number of stator phases. Thus, for a three-phase switched reluctance motor, N_φ=3 (for the a, b, and c stator phases), while for a three phase PM motor N_φ=4 (for the three stator phases and the imaginary f rotor phase). Torque is found via coenergy by: $\begin{array}{cc}T=\frac{\partial }{\partial \theta }{\omega }_c& \left(18\right)\end{array}$

[0110] The relationship between the abc-FoR and the αβ FoR is given by: $\left(\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ i_0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 1\\ \frac{-1}{2}& \frac{-1}{2}·\sqrt{3}& 1\\ \frac{-1}{2}& \frac{1}{2}·\sqrt{3}& 1\end{array}\right)·\left(\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right)$

[0111] The derivatives of the abc FoR currents are related to the αβ-FoR currents by: $\begin{array}{cc}d{\lambda }_a\left(i_{\alpha },i_{\beta },i_f,\theta \right)=\frac{\partial }{\partial i_{\alpha }}{\lambda }_a·{\mathrm{di}}_{\alpha }+\frac{\partial }{\partial i_{\beta }}{\lambda }_a·{\mathrm{di}}_{\beta }+\frac{\partial }{\partial i_f}{\lambda }_a·{\mathrm{di}}_f& \left(19\right)\\ d{\lambda }_b\left(i_{\alpha },i_{\beta },i_f,\theta \right)=\frac{\partial }{\partial i_{\alpha }}{\lambda }_b·{\mathrm{di}}_{\alpha }+\frac{\partial }{\partial i_{\beta }}{\lambda }_b·{\mathrm{di}}_{\beta }+\frac{\partial }{\partial i_f}{\lambda }_b·{\mathrm{di}}_f& \left(20\right)\\ d{\lambda }_c\left(i_{\alpha },i_{\beta },i_f,\theta \right)=\frac{\partial }{\partial i_{\alpha }}{\lambda }_c·{\mathrm{di}}_{\alpha }+\frac{\partial }{\partial i_{\beta }}{\lambda }_c·{\mathrm{di}}_{\beta }+\frac{\partial }{\partial i_f}{\lambda }_c·{\mathrm{di}}_f& \left(21\right)\\ d{\lambda }_f\left(i_{\alpha },i_{\beta },i_f,\theta \right)=\frac{\partial }{\partial i_{\alpha }}{\lambda }_f·{\mathrm{di}}_{\alpha }+\frac{\partial }{\partial i_{\beta }}{\lambda }_f·{\mathrm{di}}_{\beta }+\frac{\partial }{\partial i_f}{\lambda }_f·{\mathrm{di}}_f& \left(22\right)\end{array}$

[0112] and

[0113] di_a=di_α  (23) $\begin{array}{cc}{\mathrm{di}}_b=\frac{-1}{2}·{\mathrm{di}}_{\alpha }-\frac{\sqrt{3}}{2}·{\mathrm{di}}_{\beta }& \left(24\right)\\ {\mathrm{di}}_c=\frac{-1}{2}·{\mathrm{di}}_{\alpha }+\frac{\sqrt{3}}{2}·{\mathrm{di}}_{\beta }& \left(25\right)\end{array}$

[0114] di_f=di_f  (25)

[0115] Since a switched reluctance machine has no permanent magnets, in the case of a switched reluctance machine Equation (26) is as follows:

[0116] di_f≡0

[0117] The expression for coenergy is thus written in the αβ-FoR: $\begin{array}{cc}{\omega }_c=\int {\lambda }_{a}d{i}_{\alpha }-\frac{1}{2}·\int {\lambda }_{b}d{i}_{\alpha }-\frac{\sqrt{3}}{2}·\int {\lambda }_{b}d{i}_{\beta }-\frac{1}{2}·\int {\lambda }_{c}d{i}_{\alpha }+\frac{\sqrt{3}}{2}·\int {\lambda }_{c}d{i}_{\beta }+\int {\lambda }_{f}d{i}_f& \left(27\right)\end{array}$

[0118] For convenience, the individual integral components of this expression are set equal to D₁, D₂, D₃, D₄, D₅and D₆, respectively: $\begin{array}{cc}{\omega }_c=D_1-\frac{1}{2}·D_2-\frac{\sqrt{3}}{2}·D_3-\frac{1}{2}{D}_4+\frac{\sqrt{3}}{2}·D_5+D_6& \left(28\right)\end{array}$

[0119] Again, in the particular case of a switched reluctance machine:

[0120] D₆≡0

[0121] For a switched reluctance machine, Equations (27) and (28) therefore reduce to: $\begin{array}{cc}{\omega }_c=\int {\lambda }_{a}d{i}_{\alpha }-\frac{1}{2}·\int {\lambda }_{b}d{i}_{\alpha }-\frac{\sqrt{3}}{2}·\int {\lambda }_{b}d{i}_{\beta }-\frac{1}{2}·\int {\lambda }_{c}d{i}_{\alpha }+\frac{\sqrt{3}}{2}·\int {\lambda }_{c}d{i}_{\beta }& \left(29\right)\\ {\omega }_c=D_1-\frac{1}{2}·D_2-\frac{\sqrt{3}}{2}·D_3-\frac{1}{2}{D}_4+\frac{\sqrt{3}}{2}·D_5& \left(30\right)\end{array}$

[0122] Next, an integral path is selected. To this end, an integral path consisting of three directed line segments (DLS) is defined over which to evaluate the integral giving coenergy. For the purposes of this disclosure, the dummy variable of integration is ξ while phase current variables (i_a,i_b,i_c,i_α,i_β) are in lower case. Their associated final values, with respect to path integrals, are (I_a,I_b,I_c,I_α,I_β).

[0123] Directed line segment one:

[0124] i_α=0 di_α=0

[0125] i_β=0 di_β=0

[0126] i_fis the variable of integration, ranging from 0 to I_f.

[0127] Directed line segment two:

[0128] i_α=0 di_α=0

[0129] i_β is the variable of integration, ranging from 0 to I_β

[0130] i_f=I_fdi_f=0

[0131] Directed line segment three:

[0132] i_α is the variable of integration, ranging from 0 to I_α

[0133] i_β=I_β di_β=0

[0134] i_f=I_fdi_f=0

[0135] FIG. 3 illustrates an integration path defined as a sequence of directed line segments 51, 52, 53.

[0136] Each of the integrals D₁-D₆are then evaluated over the selected path. Evaluating D₁:,

[0137] D₁=∫λ_adi_α

[0138] This is identically zero over all but the third DLS, hence: $\begin{array}{cc}\begin{array}{c}\begin{array}{c}{D}_1={\int }_0^{I_{\alpha }}\sum _{p=0}^P{\xi }^p·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{apqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{apqrn}}·\cos \left(n·\theta \right)\right)d\xi \\ \mathrm{yielding}:\end{array}\\ D_1=\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{apqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{apqrn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(31\right)\end{array}$

[0139] Evaluating D₂:

[0140] D₂=∫λ_bdi_α

[0141] This is also identically zero over all but the third directed line segment. $\begin{array}{cc}\begin{array}{c}\begin{array}{c}{D}_2={\int }_0^{I_{\alpha }}\sum _{p=0}^P{\xi }^p·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{bpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{bpqrn}}·\cos \left(n·\theta \right)\right)d\xi \\ \mathrm{yielding}:\end{array}\\ D_2=\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{bpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{bpqrn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(32\right)\end{array}$

[0142] Evaluating D₃:

[0143] D₃=∫λ_bdi_β

[0144] This is identically zero over all but the second directed line segment, where all terms other than those where p=0 are zero: $\begin{array}{cc}\begin{array}{c}\begin{array}{c}{D}_3={\int }_0^{I_{\beta }}\sum _{q=0}^Q{\xi }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{b0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{b0qrn}}·\cos \left(n·\theta \right)\right)d\xi \\ \mathrm{yielding}:\end{array}\\ D_3=\sum _{q=0}^Q\frac{I_{\beta }^{q+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{b0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{b0qrn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(33\right)\end{array}$

[0145] Evaluating D₄:

[0146] D₄=∫λ_cdi_α

[0147] This is identically zero over all but the third directed fine segment, where all terms are zero except for those where p=q=0: $\begin{array}{cc}\begin{array}{c}\begin{array}{c}{D}_4={\int }_0^{I_{\alpha }}\left[\sum _{p=0}^P{\xi }^p·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{cpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{cpqrn}}·\cos \left(n·\theta \right)\right)\right]d\xi \\ \mathrm{yielding}:\end{array}\\ D_4=\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{cpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{cpqrn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(34\right)\end{array}$

[0148] Evaluating D₅:

[0149] D₅=∫λ_cdi_β

[0150] This is identically zero over all but the second directed line segment. $\begin{array}{cc}\begin{array}{c}\begin{array}{c}{D}_5={\int }_0^{I_{\beta }}\sum _{q=0}^Q{\xi }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{c0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{c0qrn}}·\cos \left(n·\theta \right)\right)d\xi \\ \mathrm{yielding}:\end{array}\\ D_5=\sum _{q=0}^Q\frac{I_{\beta }^{q+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{c0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{c0qrn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(35\right)\end{array}$

[0151] Evaluating D₆:

[0152] D₆=∫λ_fdi_f

[0153] This term is identically zero over all but the first directed line segment. $\begin{array}{cc}{D}_6={\int }_0^{I_f}\left[\sum _{r=0}^R{i}_f^r·\sum _{n=0}^N\left(g_{\mathrm{f00rn}}·\sin \left(n·\theta \right)+h_{\mathrm{f00rn}}·\cos \left(n·\theta \right)\right)\right]d\xi \mathrm{yielding}\text{:}{D}_6=\sum _{r=0}^R\frac{I_f^r}{r+1}·\sum _{n=0}^N\left(g_{\mathrm{f00rn}}·\sin \left(n·\theta \right)+h_{\mathrm{f00rn}}·\cos \left(n·\theta \right)\right)& \left(36\right)\end{array}$

[0154] Substituting for D₁, i=1, . . . ,6 into Equation (29): $\begin{array}{cc}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{\omega }_c=\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{apqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{apqrn}}·\cos \left(n·\theta \right)\right)\right]\dots +\\ \frac{-1}{2}·\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{bpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{bpqrn}}·\cos \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{-\sqrt{3}}{2}·\left[\sum _{q=0}^Q\frac{I_{\beta }^{q+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{b0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{b0qrn}}·\cos \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{-1}{2}·\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{cpqrn}}·\sin \left(n·\theta \right)+h_{\mathrm{cpqrn}}·\cos \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{\sqrt{3}}{2}·\left[\sum _{q=0}^Q\frac{I_{\beta }^{q+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=0}^N\left(g_{\mathrm{c0qrn}}·\sin \left(n·\theta \right)+h_{\mathrm{c0qrn}}·\cos \left(n·\theta \right)\right)\right]\dots +\\ \sum _{r=0}^R\frac{I_f^r}{r+1}·\sum _{n=0}^N\left(g_{\mathrm{f00rn}}·\sin \left(n·\theta \right)+h_{\mathrm{f00rn}}·\cos \left(n·\theta \right)\right)\end{array}& \left(37\right)\end{array}$

[0155] Substituting Equation (37) into Equation (18): $\begin{array}{cc}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}T=\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=1}^{N}n·\left(g_{\mathrm{apqrn}}·\cos \left(n·\theta \right)-h_{\mathrm{apqrn}}·\sin \left(n·\theta \right)\right)\right]\dots +\\ \frac{-1}{2}·\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=1}^{N}n·\left(g_{\mathrm{bpqrn}}·\cos \left(n·\theta \right)-h_{\mathrm{bpqrn}}·\sin \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{-\sqrt{3}}{2}·\left[\sum _{q=0}^Q\frac{I_{\beta }^{p+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=1}^{N}n·\left(g_{\mathrm{b0qrn}}·\cos \left(n·\theta \right)-h_{\mathrm{b0qrn}}·\sin \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{-1}{2}·\left[\sum _{p=0}^P\frac{I_{\alpha }^{p+1}}{p+1}·\sum _{q=0}^Q{I}_{\beta }^q·\sum _{r=0}^R{I}_f^r·\sum _{n=1}^{N}n·\left(g_{\mathrm{cpqrn}}·\cos \left(n·\theta \right)-h_{\mathrm{cpqrn}}·\sin \left(n·\theta \right)\right)\right]\dots +\end{array}\\ \frac{\sqrt{3}}{2}·\left[\sum _{q=0}^Q\frac{I_{\beta }^{q+1}}{q+1}·\sum _{r=0}^R{I}_f^r·\sum _{n=1}^{N}n·\left(g_{\mathrm{c0qrn}}·\cos \left(n·\theta \right)-h_{\mathrm{c0qrn}}·\sin \left(n·\theta \right)\right)\right]\dots +\\ \sum _{r=0}^R\frac{I_f^r}{r+1}·\sum _{n=1}^{N}n·\left(g_{\mathrm{f00rn}}·\cos \left(n·\theta \right)-h_{\mathrm{f00rn}}·\sin \left(n·\theta \right)\right)\end{array}& \left(38\right)\end{array}$