Electromechanical Systems 1 Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2015-16 B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 1 / 39
Introduction In a system that includes interacting mechanical and electromagnetic components, or that has an intrinsic electromechanical behaviour, as piezoelectric, electro-rheological, and magnetostrictive materials, magnetic shape memory alloys, and others, it is possible to define an electromechanical Lagrangian function L em as the sum of the Lagrangian functions related to the different components. The Lagrangian function of the electromagnetic part of the system may be defined using either the charge or the flux approach. We the index ‘ e ’ to indicate the electromagnetic quantities, and the index ‘ m ’ to indicate the mechanical quantities. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 2 / 39
Generalized charge coordinates In this approach we choose the electrical charges q e as the generalized coordinates for electromagnetical subsystems, and the linear/angular displacements q m as the generalized coordinates for mechanical subsystems. Global generalized coordinates, global generalized velocities, and global generalized forces are therefore � � � � � � q m q m ˙ F m ˙ q = , q = , F = q e q e ˙ F e The total kinetic co-energy is the sum K ∗ em ( q , ˙ q ) = K ∗ m ( q m , ˙ q m ) + W ∗ i (˙ q e , q m ) and the total potential energy is the sum P em ( q ) = P m ( q m ) + W c ( q e , q m ) Observe that the kinetic co-energy and the potential energy of the electrical subsystem are influenced by the mechanical coordinates. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 3 / 39
The total Lagrange function is q ) = K ∗ L em ( q , ˙ em ( q , ˙ q ) − P em ( q ) = K ∗ m ( q m , ˙ q m ) − P m ( q m ) + W ∗ i (˙ q e , q m ) − W c ( q e , q m ) and the Lagrange equations are � ∂ L em � − ∂ L em d = F m i = 1 , . . . , n m d t ∂ ˙ q m ∂ q m � ∂ L em � d − ∂ L em = F e i = 1 , . . . , n e ∂ ˙ q e ∂ q e d t B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 4 / 39
Mechanical equations � ∂ K ∗ � T � ∂ K ∗ � T � ∂ W ∗ � T m ( q m , ˙ q m ) m ( q m , ˙ q m ) i (˙ q e , q m ) d − − d t ∂ ˙ q m ∂ q m ∂ q m � ∂ P m ( q m ) � T � ∂ W c ( q e , q m ) � T + + = F m ∂ q m ∂ q m Electrical equations � ∂ W ∗ � T � ∂ W c ( q e , q m ) � T i (˙ d q e , q m ) + = F e ∂ ˙ d t q e ∂ q e where ∂ W ∗ i (˙ q e , q m ) ∂ W c ( q e , q m ) and ∂ q m ∂ q m are respectively the change of magnetic/inductive kinetic co-energy due to a change in the mechanical coordinates, and the change of capacitive potential energy due to a change in the mechanical coordinates. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 5 / 39
Generalized flux coordinates In this approach we choose the magnetic flux linkages λ as the generalized coordinates for electromagnetical subsystems, and the linear/angular displacements q m as the generalized coordinates for mechanical subsystems. Global generalized coordinates, global generalized velocities, and global generalized forces are � � � ˙ � � � q m q m F m q = q = ˙ F = , , ˙ λ F e λ The total kinetic co-energy is the sum c ( ˙ K ∗ q ) = K ∗ q m ) + W ∗ em ( q , ˙ m ( q m , ˙ λ , q m ) and the total potential energy is the sum P em ( q ) = P m ( q m ) + W i ( λ , q m ) Observe that the kinetic co-energy and the potential energy of the electrical subsystem are influenced by the mechanical coordinates. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 6 / 39
The total Lagrange function is L em ( q , ˙ q ) = K ∗ em ( q , ˙ q ) − P em ( q ) c ( ˙ = K ∗ q m ) − P m ( q m ) + W ∗ m ( q m , ˙ λ , q m ) − W i ( λ , q m ) and the Lagrange equations are � ∂ L em � d − ∂ L em = F m i = 1 , . . . , n m ∂ ˙ d t q m ∂ q m � ∂ L em � − ∂ L em d = F e i = 1 , . . . , n e ∂ ˙ d t ∂ λ λ B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 7 / 39
Mechanical equations � � T � ∂ K ∗ � T � ∂ K ∗ � T c ( ˙ m ( q m , ˙ m ( q m , ˙ ∂ W ∗ d q m ) q m ) λ , q m ) − − ∂ ˙ q m ∂ q m ∂ q m d t � ∂ P m ( q m ) � T � ∂ W i ( λ , q m ) � T + + = F m ∂ q m ∂ q m Electrical equations � � T c ( ˙ � ∂ W i ( λ , q m ) � T ∂ W ∗ λ , q m ) d + = F e ∂ ˙ d t ∂ λ λ where c ( ˙ ∂ W ∗ λ , q m ) ∂ W i ( λ , q m ) and ∂ q m ∂ q m are respectively the change of magnetic/inductive kinetic co-energy due to a change in the mechanical coordinates, and the change of capacitive potential energy due to a change in the mechanical coordinates. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 8 / 39
Electromechanical Two-port Networks We consider a particular class of electromechanical systems, the two-port networks , whose inner structure may be represented by characteristics that can be either mainly inductive or mainly capacitive. An electrical two-port network is a network that presents two ports; one is the input port the other is the output port . When the ports can be reversed, i.e., the input becomes the output and vice-versa, the network is called a reversible two-port network . If inside the network no electrical power sources are present, the two-port is called passive two-port network , otherwise it is an active two-port network . The physical quantities at the two ports are always a quantity called effort s ( t ) and a quantity called flux φ ( t ). The effort-flux couples ( s ( t ) , φ ( t )) may be electrical or mechanical at both ports, or different at the two ports; we consider this last class of two-port systems. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 9 / 39
The electrical port is characterized by a voltage s ( t ) = e ( t ) and a current φ ( t ) = i ( t ), while the mechanical port is characterized by a force s ( t ) = f ( t ) and a velocity φ ( t ) = v ( t ). The port power is the product P ( t ) = s ( t ) φ ( t ). Figure shows a two-port systems, where the input port is located on the left side and the output port on the right side. P i ( t ) is the inflowing input power, P o ( t ) is the outflowing output power. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 10 / 39
Inductive two-port networks In this type of two-port network the power conversion is obtained by an inductive element characterized by a flux linkage λ ( t ). B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 11 / 39
The two-port system is characterized by the following constitutive relations i ( t ) = i ( λ ( t ) , x ( t )) f ( t ) = f ( λ ( t ) , x ( t )) and e ( t ) = d λ ( t ) = ˙ λ ( t ) d t v ( t ) = d x ( t ) = ˙ x ( t ) d t where x ( t ) represents a displacement. These relations are generic; they must be specified according to the type of electromagnetic interactions of the considered system. B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 12 / 39
The power absorbed by the system is given by the difference between the input and the output power P λ ( t ) = P i ( t ) − P o ( t ) = e ( t ) i ( t ) − f ( t ) v ( t ) = ˙ λ ( t ) i ( t ) − f ( t ) ˙ x ( t ) Considering the energy, we can write P λ d t = d W i ( λ, x ) = i d λ − f d x and, since d W i ( λ, x ) = ∂ W i ( λ, x ) d λ + ∂ W i ( λ, x ) d x , ∂λ ∂ x comparing the above relations one obtains i ( λ, x ) = ∂ W i ( λ, x ) f ( λ, x ) = − ∂ W i ( λ, x ) and ∂λ ∂ x B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 13 / 39
Considering the co-energy, we can write W ∗ i ( i , x ) = λ i − W i ( λ, x ) and taking the differential work, we have d W ∗ i ( i , x ) = i d λ + λ d i − d W i ( λ, x ) Since we can write i d λ − d W i ( λ, x ) = f d x we obtain d W ∗ i ( i , x ) = λ d i + f d x Recalling that i ( i , x ) = ∂ W ∗ i ( i , x ) d i + ∂ W ∗ i ( i , x ) d W ∗ d x ∂ i ∂ x it follows that λ ( i , x ) = ∂ W ∗ i ( i , x ) f ( i , x ) = ∂ W ∗ i ( i , x ) ; ∂ i ∂ x B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 14 / 39
We can summarize the above relations as follows d W ∗ d W i ( λ, x ) = i d λ − f d x i ( i , x ) = λ d i + f d x i ( λ, x ) = ∂ W i ( λ, x ) λ ( i , x ) = ∂ W ∗ i ( i , x ) ∂λ ∂ i f ( λ, x ) = − ∂ W i ( λ, x ) f ( i , x ) = ∂ W ∗ i ( i , x ) ∂ x ∂ x The constitutive relations are now defined as the partial derivatives of the energy; the flux linkage is the partial derivative of the co-energy, and the force f can be expressed as a function of λ ( t ) or i ( t ). B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 15 / 39
Linear flux Let us assume that the flux is linear with respect to the current i ( t ) λ ( i , x ) = L ( x ) i ( t ) where L ( x ) represents the magnetic circuit inductance, in general function of a mechanical displacement x ( t ). We can write i ( i , x ) = 1 2 L ( x ) i 2 ( t ) W ∗ and λ 2 f ( λ, x ) = 1 d d x L ( x ) L 2 ( x ) 2 or f ( i , x ) = 1 2 i 2 d d x L ( x ) B. Bona (DAUIN) Electromechanical Systems 1 Semester 1, 2015-16 16 / 39
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