VBR modeling method of twelve-phase synchronous motor

Abstract

The invention relates to the technical field of power system simulation, in particular to a twelve-phase synchronous motor VBR modeling method, which comprises the steps of motor grouping modeling; integrating a magnetic saturation effect; and interphase coupling modeling. The method has the advantages that the VBR model is successfully expanded to a twelve-phase (four-group three-phase) synchronous motor, the limitation that a traditional three-phase model and a traditional six-phase model cannot be directly applied to a high-phase-number asymmetric motor is overcome, the coordinate transformation problem caused by a complex winding structure is effectively solved through the multi-stator modeling framework and the projection matrix technology, and the method is suitable for large-scale popularization and application. An effective modeling solution is provided for high-phase-number motor simulation; by introducing a differential inductance matrix and a magnetization coupling matrix into a dq0 coordinate system, the model can accurately describe electromagnetic characteristics of the motor in different saturation states; the model considers the mutual coupling effect among all three-phase units, solves the problem that a traditional model is difficult to accurately simulate complex magnetic saturation and interphase coupling of a twelve-phase motor, and remarkably improves the simulation precision.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of power system simulation, and particularly relates to a VBR modeling method for a twelve-phase synchronous motor. BACKGROUND

[0002] Synchronous motors are key devices in power systems and industrial applications, and their performance directly affects the operational stability and energy efficiency of the entire system. With the rapid development of power electronics technology, multi-phase motor drive systems, and high-reliability application scenarios, higher requirements are placed on the power density, waveform quality, and fault tolerance of motors. Compared with traditional three-phase motors, multi-phase motors (such as six-phase and twelve-phase motors) have lower phase currents when outputting the same power, thereby reducing the current stress of power devices and improving system reliability. In addition, multi-phase motors can continue to operate in the event of partial winding faults due to their inherent phase redundancy, thereby enabling fault-tolerant control and making them widely used in fields with extremely high reliability requirements, such as electric vehicles, aerospace, ship power propulsion, offshore wind power, and rail transportation.

[0003] However, current modeling research on twelve-phase synchronous motors is still relatively lacking, and existing methods have obvious limitations in terms of precision, applicability, and practicality, mainly in the following aspects: 1. Complex coordinate transformation and poor universality: The winding structure of a twelve-phase motor is complex, with a large number of phases and spatial asymmetry. The coordinate transformation methods used in traditional three-phase or six-phase motor modeling cannot be directly applied, and a mathematical model suitable for high-phase-number, non-symmetrical windings must be re-established, increasing the difficulty of modeling.

[0004] 2. Difficulty in accurately representing magnetic saturation and inter-phase coupling effects: The magnetic circuit saturation phenomenon of a twelve-phase motor is significant during operation, and the electromagnetic coupling relationship between the windings of each phase is complex. Traditional multi-phase motor models (such as models based on vector space decomposition or multi-stator simplified models) usually ignore saturation effects or fail to accurately describe complex inter-phase coupling under asymmetric configurations, resulting in deviations between simulation results and actual performance.

[0005] 3. Application limitations of existing VBR models: Voltage-current-magnetic flux linkage (VBR) models have shown good numerical stability and simulation efficiency in six-phase synchronous motor modeling, but existing research has not effectively extended them to twelve-phase or higher-phase-number systems. At the same time, these models often do not fully consider the influence of magnetic saturation characteristics or do not adapt to non-symmetrical winding structures, limiting their application in accurate simulation and fault-tolerant control strategy development.

[0006] The above limitations restrict the accuracy and efficiency of the high-phase-number motor drive system in design, analysis and optimization. Therefore, developing a high-efficiency modeling method which can accurately describe the electromagnetic characteristics of a twelve-phase synchronous motor, especially effectively handle magnetic saturation and complex inter-phase coupling, has important theoretical significance and engineering application value. SUMMARY

[0007] The purpose of the present application is to provide a VBR modeling method for a twelve-phase synchronous motor, which successfully extends the VBR model to a twelve-phase system, constructs a multi-stator modeling framework and uses projection matrix technology to solve the complex problem of coordinate transformation of high-phase-number asymmetric motors; by introducing the differential inductance matrix and magnetization coupling matrix, the magnetic saturation characteristics and inter-phase coupling effects are accurately represented, and the limitations of the simulation accuracy of the traditional model are overcome, providing an effective solution for motor simulation analysis in high-reliability application fields.

[0008] To achieve the above purpose, the present application realizes the following technical scheme: A VBR modeling method for a twelve-phase synchronous motor, comprising: S1, motor grouping modeling: According to the winding structure and phase distribution of the motor winding, the synchronous motor is composed of three-phase units, each three-phase unit contains three phases, and there is a spatial phase difference between the windings of each group, wherein is an integer not less than 4; S2, magnetic saturation effect integration: The voltage equation and flux linkage equation of each three-phase unit are established in the dq0 synchronous rotating coordinate system; S3, inter-phase coupling modeling: Through inverse rotation transformation and Clarke inverse transformation, the electromagnetic equation in the dq0 coordinate system is converted to the abc coordinate system corresponding to each three-phase unit; The inductance matrix and back electromotive force matrix are constructed, and the corresponding expressions contain the differential inductance term and the saturation state variable; The current coupling relationship between different three-phase units is processed by projection matrix.

[0009] In S1, the twelve-phase synchronous motor is an asymmetric twelve-phase interior permanent magnet motor, which is composed of four three-phase units, and the spatial difference between the windings of each group is 15°, .

[0010] In S2, the voltage equation and flux linkage equation of each three-phase unit are established in the dq0 synchronous rotating coordinate system, including: For the first three-phase unit, the voltage equation is established, and the expression is: (1); In formula (1) : represents a stator voltage vector, unit: V; represents the first group stator resistance, unit: Ω; represents a stator current vector, unit: A; represents a stator flux linkage vector, unit: Wb; represents a complex vector matrix, ; represents the electrical angular velocity of the shaft, unit: rad / s; The flux linkage equation is established, and the expression is: (2); In formula (2) : represents a permanent magnet flux, unit: Wb; represents the first group stator current vector, unit: A; represents a stator flux linkage vector, unit: Wb; represents a stator leakage inductance, unit: H; represents a permanent magnet flux linkage, unit: Wb; represents the first group stator current component in the shaft, unit: A; The final voltage equation and flux linkage equation of the first group in the dq0 coordinate system are derived, which contains all current coupling terms of the third group three-phase unit ; represents a matrix describing the magnetization coupling relationship in the dq0 coordinate system, and the expression is: (3); In formula (3) : represents the self-induction matrix of the shaft, unit: H; represents the mutual inductance matrix of the shaft, unit: H; denotes axis and mutual inductance matrix between axes, unit: H; denotes self inductance matrix of axis, unit: H.

[0011] In S3, including: Convert the dq0 coordinate equation to the αβ0 stationary reference frame by inverse rotation transformation; Convert to the αβ0 stationary reference frame by Clarke inverse transformation; group of phase coordinate system; The current coupling relationship is processed by using a projection matrix, and the expression is: (4); In formula (4): denotes the group of abc phase current vectors, unit: A; denotes the projection matrix; is denotes the phase difference between the α axis and the group of phases, unit: rad; denotes the projection vector of the group of currents in the group of abc coordinate systems, unit: A.

[0012] The expression of the projection matrix is: (6); In formula (6): ; denotes the rotor position angle, unit: rad.

[0013] denotes the phase difference between the group of phases and the α axis, unit: rad.

[0014] The inductance matrix and the back electromotive force matrix construction method includes: Use the differential inductance term to represent the saturation characteristic; Introduce the saturation state variable in the voltage equation; Correct the magnetization curve by fitting the experimental data matrix.

[0015] The projection matrix satisfies an orthogonality constraint, and the expression is: is (7); In formula (7): denotes a unit matrix; denotes a transpose matrix of the projection matrix.

[0016] Compared with the prior art, the present application has the following beneficial effects: 1. The VBR model is successfully extended to a twelve-phase (four groups of three-phase) synchronous motor, overcoming the limitation that the traditional three-phase and six-phase models cannot be directly applied to high-phase-number asymmetric motors, and effectively solving the coordinate transformation problem caused by complex winding structure through a multi-stator modeling framework and a projection matrix technology, thereby providing an effective modeling solution for high-phase-number motor simulation; 2. By introducing a differential inductance matrix and a magnetization coupling matrix in the dq0 coordinate system, the model can accurately describe the electromagnetic characteristics of the motor under different saturation states; the model considers the mutual coupling effect between all three-phase units, thereby solving the problem that the traditional model cannot accurately simulate the complex magnetic saturation and inter-phase coupling of the twelve-phase motor, and significantly improving the simulation accuracy; 3. Modular design is adopted, supporting configuration of any number of three-phase units (N≥4), and by adjusting the phase difference parameters in the projection matrix, the model can be flexibly applied to asymmetric motor structures with different phase numbers and different spatial phase distributions, and has good universality and expansion capability; 4. The model is established in the abc phase coordinate system, and can directly simulate various internal fault conditions such as single-phase or multi-phase open-circuit fault, thereby providing a reliable simulation platform for studying the fault characteristics of the motor and developing effective fault-tolerant control strategies, and being particularly suitable for high-reliability application fields such as aerospace and electric vehicles; 5. While maintaining the good numerical stability of the VBR model, the simulation accuracy of the model in the dynamic process is ensured by introducing saturation state variables and accurate current coupling processing, and the model structure is suitable for implementation in commonly used simulation software, thereby providing a practical simulation tool for engineering applications. BRIEF DESCRIPTION OF DRAWINGS

[0017] Figure 1 is a schematic diagram of a twelve-phase motor coordinate system.

[0018] Figure 2 is an equivalent circuit diagram of a four-path three-phase synchronous motor in a dq-axis rotating coordinate system. DETAILED DESCRIPTION

[0019] The present invention will now be described in detail with reference to the accompanying drawings, but it should be noted that the implementation of the present invention is not limited to the following embodiments.

[0020] A VBR modeling method for a twelve-phase synchronous motor, the specific steps of which include: Step S1: Group modeling of motors; Based on the winding structure and phase distribution of the twelve-phase synchronous motor, the multiphase synchronous motor is decomposed into multiple independent three-phase winding units. For example, the twelve-phase motor is decomposed into four three-phase units. Each group of three-phase units contains three phases, and there is a spatial phase difference between each group of windings. Each group of three-phase elements is modeled independently in the rotating dq coordinate system, and the electromagnetic mutual coupling effect between elements is described by the magnetic flux coupling matrix.

[0021] Step S2, magnetic saturation effect integration; In the synchronous rotating coordinate system dq0, establish the voltage equations and flux linkage equations for each group of three-phase units, including: For the Assemble a three-phase unit and establish the voltage equation, the expression is: (1); In formula (1): This represents the stator voltage vector, with units of V; This represents the stator resistance of the kth group, in Ω; This represents the stator current vector, with units of A. This represents the stator flux linkage vector, in units of Wb; Represents a complex vector matrix. ; Represents the electric angular velocity along the d-axis, in rad / s; The flux linkage equation is established, and its expression is: (2); In formula (2): This represents the magnetic flux of a permanent magnet, measured in Wb. Indicates the first Stator current vector, in A; This represents the stator flux linkage vector, in units of Wb; This indicates stator leakage inductance, expressed in ohms (H). This indicates the flux linkage of a permanent magnet, measured in Wb. Indicates the first Stator current in The components of the axis, in amperes (A); The derivation yields the first The final voltage and flux linkage equations of the set in the dq0 coordinate system contain all Current coupling terms of a three-phase unit ; The matrix representing the magnetization coupling relationship in the dq0 coordinate system is expressed as: (3); In formula (3): express Axial self-inductance matrix, in units of H; express Mutual inductance matrix of axes, in units of H; express shaft and The mutual inductance matrix between axes, in units of H; express The axis self-inductance matrix, in units of H.

[0022] Step S3: Phase coupling modeling; Using projection matrix The current from different three-phase units is transferred from abc. g Coordinate system transformation to abc k A coordinate system is used to accurately characterize the effect of phase shift between three-phase units on magnetic coupling; S3-1. Transform the equations of the dq0 coordinate system to the αβ0 stationary reference system through inverse rotation transformation; S3-2, Transformed to the [number]th [section] using Clarke inverse transform. group Phase coordinate system; S3-3. Construct the inductance matrix and the back electromotive force matrix. The corresponding expressions include differential inductance terms and saturation state variables. a. Construction of the inductance matrix and back electromotive force matrix, including: Using the differential inductance term Characterizing saturation characteristics, which is the phenomenon that the magnetic induction intensity of the motor magnetic circuit no longer increases linearly with the magnetic field strength when the magnetic field strength is high; Introducing saturation state variables into the voltage equation ; Correcting by fitting magnetization curve with experimental data Matrix.

[0023] b. The current coupling relationship is processed by using a projection matrix, and the expression is: (4); In formula (4): Iabc represents the vector of the first group abc-phase current, and the unit is A; Iabc represents the vector of the first group abc-phase current, and the unit is A; Iabc represents the vector of the first group abc-phase current, and the unit is A; Iabc represents the vector of the first group abc-phase current, and the unit is A; Iabc represents the vector of the first group abc-phase current, and the unit is A. c. The expression of the projection matrix is:

[0024] (6); In formula (6): Iabc represents the vector of the first group abc-phase current, and the unit is A.

[0025] Iabc represents the vector of the first group abc-phase current, and the unit is A. Iabc represents the vector of the first group abc-phase current, and the unit is A.

[0026] d. The projection matrix satisfies the orthogonality constraint, and the expression is: Iabc represents the vector of the first (7); In formula (7): Iabc represents the vector of the first group abc-phase current, and the unit is A. Iabc represents the vector of the first group abc-phase current, and the unit is A.

[0027] The following examples are implemented on the premise of the technical solutions of the present application, and detailed implementation modes and specific operation processes are given, but the protection scope of the present application is not limited to the following examples. The methods used in the following examples are all conventional methods unless otherwise specified.

[0028] Example 1 A VBR modeling method of a twelve-phase synchronous motor, the steps are as follows: Step S1: Configure the winding structure and phase distribution of the twelve-phase synchronous motor according to the motor winding configuration.

[0029] The motor is an asymmetrical twelve-phase built-in permanent magnet (IPM) motor, consisting of four sets of three-phase units. The windings are spatially separated by 15°. The three sets of windings are labeled A, B, and C, and each set contains four phases: A1, A2, A3, A4; B1, B2, B3, B4; C1, C2, C3, C4. (See...) Figure 1 .

[0030] The VBR model is constructed using the multi-stator (MS) method for modeling multiphase motors; the multi-stator (MS) method is a method of independently modeling each phase unit and coupling the flux linkage.

[0031] Each three-phase unit is modeled independently, while simultaneously considering electromagnetic coupling effects within the three-phase unit and between different three-phase units. See Figure 2 .

[0032] To make the method universal, consider any number of three-phase units N.

[0033] Step S2: Establish the voltage equation and flux linkage equation in the twelve-phase coordinate system under the dq0 coordinate system.

[0034] According to the unified theory of motors, the stator winding in the dq0 coordinate system is obtained as follows: Voltage equations and flux linkage equations for a group (k=1,...,N): where the unified theory of motors is a general mathematical framework for describing the electromagnetic relationships of multiphase motors; For the Assemble a three-phase unit and establish the voltage equation, the expression is: (1); In formula (1): This represents the stator voltage vector, with units of V; Indicates the first Stator resistance, unit is Ω; This represents the stator current vector, with units of A. This represents the stator flux linkage vector, in units of Wb; Represents a complex vector matrix. ; express The electric angular velocity of the shaft, measured in rad / s.

[0035] The flux linkage equation is established, and its expression is: (2) ; wherein: λ m is the permanent magnet flux linkage; λ MG is the permanent magnet flux, whose expression is: (3) ; (4) ; wherein, M dq is a matrix describing the magnetization coupling relationship in the dq0 coordinate system, whose expression is: ; wherein; represents the self-induction matrix of the d-axis, with the unit of H; represents the mutual-induction matrix of the d-axis, with the unit of H; represents the mutual-induction matrix between the d-axis and the q-axis, with the unit of H; represents the self-induction matrix of the q-axis, with the unit of H.

[0036] All windings jointly affect the generation of the magnetic flux, while the zero sequence component is decoupled in nature.

[0037] and are the stator resistance and leakage inductance of the kth group, respectively; is a complex vector matrix, ; is the electrical angular velocity of the q-axis.

[0038] The formula (2) is derived by taking the derivative of the flux linkage equation with respect to time: (5) ; wherein, represents a constant, whose derivative is zero.

[0039] If the magnetization term of is derived with respect to the current, then: (6) ; wherein, represents the differential inductance matrix, which is composed of the self-differential inductance ( , ) and the mutual-differential inductance (​ , It consists of two parts, a differential inductance that changes in saturation state.

[0040] The final voltage equation and flux linkage equation of the kth group in the dq0 coordinate system are simplified as shown in formulas (7) and (8).

[0041] (7); (8); in: This represents the stator voltage vector, with units of V; This represents the stator current vector, with units of A. This represents the rate of change of stator current, expressed in A / s. This represents the stator flux linkage vector, with units of Wb.

[0042] Step S3, in Establish electromagnetic equations in a twelve-phase coordinate system.

[0043] To calculate the k-th group The electromagnetic model in the coordinate system requires inverse rotation transformation and Clarke inverse transformation; Define a common αβ0 stationary reference frame, see Figure 1 This makes the rotor position θ d It is uniquely determined.

[0044] Applying the inverse rotation matrix Transform the voltage and flux linkage equations of the k-th group (see Equations 7 and 8) from the dq0 coordinate system to the αβ0 stationary reference system by inverse rotation of the matrix. The expression is: ; Applying Clarke inverse transform Explicitly represent the k-th group in The electromagnetic model in phase coordinates is expressed as follows: ; in: Indicates the α-axis and a k Phase (k=1, ...,N, representing the first phase) Group The phase difference between phases is shown in the figure. Figure 1 ; No. Group in abc kThe final electromagnetic equation in the coordinate system is expressed as: (9); Wherein: ; ; and The matrixes are shown in equations (10) and (11) and are expressed as: (10); (11); Wherein: represents the inductance difference, and the unit is H; represents the sine function matrix; represents the average inductance, and the unit is H; represents the cosine function matrix; represents the differential inductance difference, and the unit is H; represents the back electromotive force coupling matrix, and the unit is H; and are respectively expressed as: ; ; ; ; ; The inductance intermediate variables in equation (10) , and are respectively expressed as: ; ; ; ; Equation (9) can be expressed as: (12); Wherein: represents the first Phase abc voltage vectors, in units of V; Indicates the first Stator resistance, unit is Ω; Indicates the first The current vector of phases a, b, and c, in A; This represents the stator leakage inductance matrix, in units of H; This represents the coupled inductance matrix, with units of H; Representing the back electromotive force term, the two back electromotive force terms in formula (9): Rotational electromotive force: the back electromotive force generated by the rotor motion. ; Differential inductance electromotive force: derived from the differential inductance term The generated electromotive force .

[0045] In formula (12), the current of the k-th stator needs to be represented in the abc phase coordinate system. This directly represents the resistance voltage drop term and the leakage flux term, and is also crucial for correctly handling the excitation coupling between groups, thus affecting the implementation of the VBR model. Because The phase current is Measured in a coordinate system, therefore the first... The current in the group needs to be properly handled. The solution proposed in this invention is to... The current of the group is from its phase coordinate system ( Projected onto the new Phase coordinate system ( The following conversion formula is applied: ; in: ; This indicates the rotor position angle, in rad. Indicates the α axis and Phase difference between phases, see Figure 1 For the asymmetric twelve-phase motor being analyzed, this value is taken as 0°, 15°, 30°, and 45°.

[0046] The present application successfully extends the VBR model to twelve-phase (four groups of three-phase) synchronous machines, overcomes the limitation that traditional three-phase and six-phase models cannot be directly applied to high-phase-number asymmetric machines, effectively handles the coordinate transformation problem caused by complex winding structure through the multi-stator modeling framework and projection matrix technology, and provides an effective modeling solution for high-phase-number machine simulation; by introducing the differential inductance matrix and magnetization coupling matrix in the dq0 coordinate system, the model can accurately describe the electromagnetic characteristics of the machine under different saturation states; the model considers the mutual coupling effect between all three-phase units, solves the problem that traditional models are difficult to accurately simulate the complex magnetic saturation and inter-phase coupling of twelve-phase machines, and significantly improves the simulation accuracy; modular design is adopted to support the configuration of any number of three-phase units (N ≥ 4), and by adjusting the phase difference parameters in the projection matrix, the model can be flexibly applied to asymmetric machine structures with different phase numbers and different spatial phase distributions, and has good universality and expansion ability; the model is established in the abc phase coordinate system and can directly simulate various internal fault conditions such as single-phase or multi-phase open circuit fault, providing a reliable simulation platform for studying the fault characteristics of the machine and developing effective fault-tolerant control strategies, and is particularly suitable for high-reliability application fields such as aerospace and electric vehicles; while maintaining the good numerical stability of the VBR model, the simulation accuracy of the model in the dynamic process is ensured by introducing saturation state variables and accurate current coupling processing, and the model structure is suitable for implementation in commonly used simulation software, providing a practical simulation tool for engineering applications.

Claims

1. A method of VBR modeling of a twelve-phase synchronous machine, characterized in that, The application relates to a method for modeling a multi-phase motor, comprising the following steps: S1, motor grouping modeling: According to the winding structure and phase distribution of the twelve-phase synchronous motor, the winding structure of the twelve-phase synchronous motor is configured as Each group of three-phase units comprises three phases, and there is a spatial phase difference between the groups of windings, wherein is an integer not less than 4; S2, magnetic saturation effect integration: In a dq0 synchronous rotating coordinate system, voltage equations and flux linkage equations of each three-phase unit are established; S3, phase-to-phase coupling modeling: Through inverse rotation transformation and Clarke inverse transformation, electromagnetic equations in the dq0 coordinate system are converted into the abc coordinate system corresponding to each three-phase unit; An inductance matrix and an electromotive force matrix are constructed, and corresponding expressions contain differential inductance items and saturation state variables; The current coupling relationship between different three-phase units is processed through a projection matrix.

2. The VBR modeling method of a twelve-phase synchronous machine according to claim 1, characterized in that, In S1, the twelve-phase synchronous motor is an asymmetric twelve-phase built-in permanent magnet motor, which is composed of four groups of three-phase units, and the space between each group of windings is 15°, .

3. The VBR modeling method of a twelve-phase synchronous machine according to claim 1, wherein, In S2, the voltage equations and the flux linkage equations of each three-phase unit in the dq0 synchronous rotating coordinate system are established, comprising: For the first The voltage equation for the third group of three-phase units is established and expressed as （1）； In formula (1): V represents the stator voltage vector, in V; represents the first set of stator resistances in ohms; is the stator current vector in A; represents the stator flux linkage vector in Wb; represents a complex vector matrix, ; denotes electrical angular velocity of the shaft in rad / s; The flux linkage equation is established, and the expression is: （2）； In formula (2): represents the permanent magnet flux, in Wb; represents the first set of stator current vectors in A; represents the stator flux linkage vector in Wb; This indicates stator leakage inductance, expressed in ohms (H). φpmrepresents the permanent magnet flux linkage, in Wb; represents the first group stator current in axis component, in A; The final voltage equation and flux linkage equation of the group in the dq0 coordinate system are derived The final voltage equation and flux linkage equation of the group in the dq0 coordinate system are derived The current coupling term of the three-phase unit of the group ; Magnetic coupling relationship described in the dq0 coordinate system is represented by a matrix, and the expression is as follows: （3）； In formula (3): denotes shaft self-induction matrix in H; denotes the mutual inductance matrix in H; representing axes and mutual impedance matrix between the axes, in H. represents shaft self-induction matrix in H.

4. The method of claim 1, wherein In S3, comprising: The dq0 coordinate system equation is converted into an alpha-beta-zero static reference system through inverse rotation transformation; converted to the first group of phase coordinate system; The current coupling relationship is processed through a projection matrix, and the expression is: （4）； In formula (4): represents the first abc-phase current vector, in A; denotes a projection matrix; For represents the phase difference between the α-axis and the first group axis, in rad; represents the first group current in the projection vector of the first group abc coordinate system, in A.

5. The method of claim 4, wherein, The expression of the projection matrix is: （6）； In formula (6); ； represents the rotor position angle in rad; represents the first group phase difference between the phase and the a-axis, in rad.

6. The method of claim 1, wherein The inductance matrix and the electromotive force matrix construction method comprises: Using a differential inductance term Characterizing saturation characteristics; Introducing saturation state variables in voltage equations ; Correcting the magnetization curve by fitting the experimental data matrix.

7. The method of claim 1, wherein The projection matrix satisfies an orthogonality constraint, and the expression is: for (7); In formula (7): denotes the identity matrix; denotes the transpose of the projection matrix.

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