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Magnetization Characteristics and Torque
In terms of the construction the capability of a motor to transform energy and pro duce a torque is determined by the magnetization characteristics, whose wave forms result from the engineering details and properties of the ferromagnetic materials applied. What is meant here is the family of magnetization characteristics in the function of the position of rotor tooth in relation to stator tooth in the range from the unaligned position to the completely aligned one. An example of magnetization characteristic is presented together with a single cycle of converting the energy of the magnetic field into mechanical energy in rotational motion of the rotor.
From the schematic diagram one can conclude about the relation: the more non-linear magnetization characteristic are for the aligned position the greater the co-energy of the magnetic field T´ that is converted into mechanical work of the drive within a cycle of the power supply. At the same time, less energy of the magnetic field Tf is returned to the source during the power diode conduction duty period. On the basis of the cycle of energy conversion presented, it’s possible to assess the motor's torque:
Te = e angle/cycl rotation e power/cycl mechanical
The relation presented here and the interpretation of the cycle of energy conversion are relevant for a simplified case when the particular machine duty cycles are separate and there isn't a period of a common conduction.
In the further part of this section the examples and illustrations will be based on two typical layouts of SRM motors:
- motor A with rated values:
- motor B, for which:
Details of the two machines are summarized. Below is a summary of the characteristics of magnetization and several other characteristics associated with torque generation for motor A.
+-+-+- A family of magnetizing characteristics for motor A, for 50 consecutive rotor position angles from unaligned position (30º) to aligned position (0º)
+-+-+- Saturated inductance curves of motor A for increasing stator phase current values.
+-+-+- Electromagnetic torque curves, for an individual supply of consecutive phase windings of motor A with i = 120 [A] as a function of a rotor position theta_r.
Math Model -- SRM Motor
Mathematical Model-- Foundations / Assumptions
The mathematical model forms the basic tool for conducting simulations of dynamic courses and finding characteristics of the motor as well as a whole drive. It’s also indispensable in the research of drive control systems. The degree of complication and the precision of a given mathematical model is relative to the practical application of a model, i.e. the number, type and relevance range of the characteristics that are determined by its use. The form of the mathematical model is clearly relative to the simplifying assumptions adopted during the statement of the model. It’s also important to note the source of information serving for the purposes of developing a model, i.e. whether the source originates in engineering data or data gained on the basis of measuring and testing existing objects, or data has been gained in some other way. The models based on measurements can either be deterministic in nature or originate on the basis of artificial intelligence methods. The latter, however, have a limited scope of application since they serve in order to examine objects the information of which is often approximated.
This guide is based on the development and application of deterministic models in the form of ordinary differential equations derived as Lagrange's equations, in accordance with the procedure described. The basic application of such a model is in the research of dynamic and static characteristics, selection of control methods and procedures as well as in the simulation of drive operation in desired circumstances.
In the bibliography in this subject there is a considerable number of mathematical models of this kind, which, however, differ in details. For instance, mathematical models of SRM, which apply engineering details and linkage between the windings are presented in, and the ones without linkage. In turn, another model designed for dynamic calculations and simulations of wave forms is presented under the assumption of familiarity of functions of windings inductance and their derivatives. Basically, the majority of models ac count for non-linearity of magnetization characteristics in SRM motor. This is so because it plays significant role with respect to generation of the electromagnetic torque. While some of the mathematical models found in bibliography account for the magnetic linkage between the adjacent phase windings of the motor, a considerable number of studies completely disregard them. This fact of ignoring the linkage between phase windings in a SRM motor is justified by two details.
Firstly, the value of the linkage is small, since in the sense of an upper limit it does not exceed 10% of the self-inductance of the winding and the standard is that their value is from 5% to 6% of this inductance. Secondly, the period of the simultaneous conduction of adjacent phases of a motor is limited and during the time when in one of the phases the current gains its maximum value, in the other one it’s al ready decaying. The presented mathematical model is designed to conduct swift and multiple dynamic calculations of the waveforms of SRM; hence, it’s considerably simplified. It accounts for non-linearity of the characteristics of magnetization of the motor, which is indispensable, but disregards magnetic linkage between the windings. It leads to the simplification of the model concurrently not causing relevant errors as a result of the application of the model. This is confirmed by the characteristics and waveforms gained on the basis of measurements and, in particular, this pertains to characteristics gained for the case of reversing the current in the adjacent phase windings during comparative analysis and measurement of inductance in standard SRM motors. The inconsiderable differences between the characteristics gained on the basis of such measurements indicate that the linkage between phases can be disregarded without the deterioration of the precision of the results collected by the application of this model. The mathematical model of the motor is derived on the basis of Lagrange's equations for an electromechanical system while preserving the notations used. The number of the degrees of freedom in the system is equal to:
... while m denotes the number of the electric degrees of freedom of the system and the electric charges associated with phase windings form the respective variables ... and their time derivatives have the meaning of phase currents of the motor: Concurrently, the remaining degree of freedom is reserved for the mechanical variable of the system ... and denotes the angle of rotation of the rotor. Under the assumption of the lack of magnetic linkage between phase windings, Lagrange's function for this system takes the form:
The virtual work of the system corresponding to the exchange of energy between the system and the surrounding environment is equal to:
… in which the particular terms refer to:
- magnetic flux associated with k-th phase winding, Rk - resistance associated with current flow through k-th phase winding, accounting for resistance of electronic elements and resistance of the supply source u_k - voltage applied to k-th phase winding, J - moment of inertia associated with motor's shaft, D - viscous damping coefficient associated with damping of the motion, Tl - shaft load torque.
Motor Motion Equations
The generalized form of equations of motion in accordance with Lagrange's model is in the form where generalized force... is the force acting along its j-th generalized coordinate calculated as the respective partial derivative of virtual work dA. For the electric circuits of the motor in accordance with the assumptions of disregarding mutual phase linkages, we obtain the total of m equations in the form: which is proportional to the current in the winding and the angular speed of the rotor. This term determines the similarity between SRM machine and series wound DC motor since the phase current ik plays here the same role as the excitation current in the DC machine. The equations, in the consideration of the lack of linkages, can be arranged in the standard form:
Function of Winding Inductance:
A more detailed insight into the mathematical model, given by equations is associated with the need of noting the functions of windings inductances, which play a key role in this model. In the examined model the following form of the inductance function has been provided.
Such inductance function notation in the form of a product may seem complicated; however, it has a number of advantages. The term M(_k) presents the unsaturated inductance of the winding in the form of the function _k, which is the rotor's angular position reduced to the pitch of the teeth tr = 2p/Zr: ... where the term frac(x) denotes the fractional part of argument x. Concurrently, the second term in ( __), presented in the graphical form, is responsible for the magnetic saturation and introduces the adequate functional relation from rotor's angular position and current in the winding. Such a presentation of inductance coefficient makes it possible to study the effect of saturation as well as engineering changes on inductance as well as estimate the parameters of the motor on the basis of measured characteristics. The respective partial derivatives of the inductance take the form:
+-+-+-Unsaturated inductance coefficient of a phase winding for SRM motor A in a function rotor position angle _
+-+-+- Saturation factor ?( theta_r,i) of a phase winding inductance, for position angle values theta_r = 0°,1.5°,3°,…30°, as a function of winding's current.
This way of representing the inductance of the winding and its derivatives affects the detailed form of the expression of electromagnetic torque, which takes the following form ...
The first integral cla(?k,ik) is presented in the graphical form in the function of the current for various values of the ?k angle. Evaluation of this integral indicates that ...
The first term of the expression, which defines electromagnetic torque of the motor, reminds one of the classical expression denoting torque of reluctance origin diminished by the influence of saturation, which contrasts with unsaturated inductance function, which is relative only to the angle of rotation. This term de notes the basic component of the torque. Concurrently, the other term of the torque in the form of an integral clafi(?k,ik) is presented for positive values of the machine's angle of rotation. Since it’s an odd function and the motor's mode of operation takes place for negative values of angle ?k, the second term in the expression determines decrease of the basic torque computed by use of the first term. However, one can note that the change of the torque associated with non-linearity of magnetization characteristics plays a more important role for large values of the current. For the case of motor A, whose characteristics are presented here, it demonstrates for i > 60 [A], i.e. the value of the current that is higher than the rated current.
This is illustrated, which presents both components of the torque with respect to the angle of rotation, i.e. the basic term relative to the derivative of the inductance - as component I and the other term that is relative to the change of the saturation- as component II. This decomposition of the electromechanical torque of a motor is presented for high value of the phase current in the motor i = 120 [A] and for the current i = 50 [A], i.e. for small magnetic saturation. As one can see from the illustrations, the less important component II of the torque is the smaller with the smaller saturation of the magnetic circuit. For the phase current i = 120 [A] it’s equal to around 30% of the value of the basic torque, while for i = 50 [A] it corresponds only to 5% of this torque.
+-+-+- Particular components of SRM's electromagnetic torque ( __). Component I - basic torque originated from inductance derivative on a position angle; component II - reflecting a change in saturation: a) for a high saturation i = 120 [A] b) for a low saturation i = 50 [A]