Electrical Systems 2 Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2015-16 B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 1 / 28
Generalized Coordinates in Electrical Systems The Lagrange approach to electrical circuits on the other hand requires the definition of the kinetic co-energies and potential energies, and consequently it is necessary to define a different set of generalized coordinates and velocities. Two formulations are possible: one is called charge formulation and is based on generalized charge coordinates , the other is called flux formulation and is based on generalized flux coordinates . In the first case the generalized coordinates and velocities are charges and currents, in the second case fluxes and voltages. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 2 / 28
According to the choice made, the kinetic co-energies and the potential energies will be different Given the power P ( t ) = i ( t ) e ( t ), the energy or work is given by � t � t W ( t ) = P ( τ ) d τ = e ( τ ) i ( τ ) d τ 0 0 This relation is used to define the Lagrange state function L ( q , ˙ q ), as specified in the following. Unless otherwise stated, we will consider only ideal linear circuits , i.e., circuits where all the involved the components are linear and ideal. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 3 / 28
Generalized charge coordinates The generalized coordinates are the charges q ( t ) stored inside the capacitive components of the circuit. The generalized velocities are the time derivatives of the charges, i.e., the currents flowing into the capacitors i ( t ) = d q ( t ) = ˙ q ( t ) . d t The voltage across the capacitive component is proportional to the charge e ( t ) = 1 C q ( t ) B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 4 / 28
The electrostatic energy stored in the capacitive element, also called the capacitive energy W c ( q ) is defined as � t � q W c ( q ( t )) = e ( q ) i d t = e ( q ) d q 0 0 The capacitive co-energy W ∗ c ( q ) represents the electrostatic energy expressed as a function of the voltage e ( t ). This energy has no clear physical significance, as in the mechanical case, but is useful for the definition of the Lagrange function. The co-energy W ∗ c ( e ) is � e W ∗ c ( e ) = qe − W c ( q ) = q ( e ) d e 0 Both the energy and the co-energy do not depend on time. A time inversion, i.e., time flowing anti-causally in the negative direction, does not affect the results; from a physical point of view this means that capacitive energy/co-energy can be stored or released at will to and from an ideal capacitive element in the circuit. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 5 / 28
The energy differential is d W c ( q ) = e ( q ) d q and the co-energy differential is d W ∗ c ( e ) = q ( e ) d e from which the following relations are established ∂ W c ( q ( e )) c ( e ( q )) ∂ W ∗ = e ( t ) and = q ( t ) ∂ q ∂ e B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 6 / 28
If the element is linear with respect to the capacity i.e., q ( e ) = Ce and C is a constant, the well known relations follow � e � e Ce d e = 1 2 Ce 2 W ∗ c ( e ) = q ( e ) d e = 0 0 and � q � q q 2 C d q = 1 q W c ( q ) = e ( q ) d q = 2 C 0 0 In this case the capacitive energy and co-energy are equal, and the characteristic function is a straight line. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 7 / 28
Generalized flux coordinates The generalized coordinates are the fluxes λ ( t ) generated inside the inductive components of the circuit. The generalized velocities are the time derivatives of the fluxes, i.e., the voltages e ( t ) = d λ ( t ) = ˙ λ ( t ) . d t The current flowing into the inductive component is proportional to the flux i ( t ) = 1 L λ ( t ) B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 8 / 28
The electromagnetic energy stored in the inductive element, also called the inductive energy W i ( λ ), is defined as � t � λ W i ( λ ) = i ( λ ) e d t = i ( λ ) d λ 0 0 It is also possible to express the inductive co-energy W ∗ i ( i ), i.e., the electromagnetic energy expressed as a function of the current i ( t ). This energy has no clear physical significance, as in the mechanical case, but nevertheless is useful for the definition of the Lagrange function. The co-energy W ∗ i ( i ) is defined as � i W ∗ i ( i ) = i λ − W i ( λ ) = λ ( i ) d i 0 Both the energy and the co-energy do not depend on time. A time inversion does not affect the results; from a physical point of view this means that capacitive energy/co-energy can be stored or released at will to and from an ideal inductive element in the circuit. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 9 / 28
The energy differential is d W i ( λ ) = i d λ while the co-energy differential is d W ∗ i ( i ) = λ d i from which the following relations are established ∂ W i ( λ ) i ( i ) ∂ W ∗ = i ( t ) e = λ ( t ) ∂λ ∂ i B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 10 / 28
If the element is linear with respect to the inductance i.e., λ ( i ) = Li and L is a constant, the well known relations follow � i � i Li d i = 1 2 Li 2 W ∗ i ( i ) = λ ( i ) d i = 0 0 and � λ � λ λ 2 L d λ = 1 λ W i ( λ ) = i ( λ ) d λ = 2 L 0 0 In this case the inductive energy and co-energy are equal, and the characteristic function is a straight line. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 11 / 28
Lagrange Function in Electromagnetic Systems To avoid confusion between the symbol for generalized coordinates and the symbol for charges, we will use the symbol ξ and ˙ ξ for generalized coordinates and generalized velocities, respectively. In the electromagnetic systems the Lagrange function L e , is the difference e ( ξ , ˙ between the “kinetic” co-energy K ∗ ξ ) and the “potential” energy P e ( ξ ): L e ( ξ , ˙ ξ ) = K e ( ξ , ˙ ξ ) − P e ( ξ ) The energy/co-energy functions are different if we use the flux or the charge coordinates. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 12 / 28
Charge coordinates and Lagrange function Using the charge coordinates ξ = q and velocities ˙ ξ = ˙ q = i , we write L e , charge ( ξ , ˙ ξ ) = W ∗ i (˙ q ) − W c ( q ) where the “kinetic” co-energy coincides with the inductive co-energy stored into the inductive element: e ( ξ , ˙ K ∗ ξ ) ≡ W ∗ i (˙ q ) = W ∗ i ( i ) and the “potential” energy coincides with the capacitive energy stored into the capacitive elements P e ( ξ ) ≡ W c ( q ) We notice that the kinetic co-energy does not depend on the generalized coordinates, but only on the generalized velocities ˙ q = i . The vectors q and i are the collection of all the charges on the capacitive elements and all the currents flowing into the inductive elements. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 13 / 28
Since the energies are additive, assuming a linear circuit with N i inductors and N c capacitors, we can write � N i � i ( i ) = 1 e ( ξ , ˙ � L k i 2 K ∗ ξ ) = W ∗ k 2 k =1 where i k is the current flowing into the k -th inductive component with inductance L k . Similarly � N c � q 2 P e ( ξ ) = W c ( q ) = 1 � k 2 C k k =1 where q k is the charge stored into the k -th capacitive component with capacity C k . B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 14 / 28
Flux coordinates and Lagrange function Using the flux coordinates ξ = λ and velocities ˙ ξ = ˙ λ = e , we can write L e , flux ( ξ , ˙ c ( ˙ ξ ) = W ∗ λ ) − W i ( λ ) where the “kinetic” co-energy coincides with the capacitive co-energy stored into the capacitive element: K e ( ˙ c ( ˙ λ ) ≡ W ∗ λ ) = W ∗ c ( e ) and the “potential” energy coincides with the inductive energy stored into the inductive elements: P e ( λ ) ≡ W i ( λ ) We notice that the kinetic co-energy does not depend on the generalized coordinates, but only on the generalized velocities ˙ λ = e . The vectors λ and e are, respectively, the collection of all the fluxes on the inductive elements and all the voltages across the capacitive elements. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 15 / 28
Since the energies are additive, assuming again a linear circuit with N i inductors and N c capacitors, we can write � N c � c ( e ) = 1 e ( ξ , ˙ � C k e 2 K ∗ ξ ) = W ∗ k 2 k =1 where e k is the voltage across the k -th capacitive component with capacity C k . Similarly � N i � λ 2 P e ( ξ ) = W i ( λ ) = 1 � k 2 L k k =1 where λ k is the flux across the k -th inductive component with inductance L k . B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 16 / 28
In conclusion � N i N c � q 2 L e , charge = 1 � L k i 2 � k k − 2 C k k =1 k =1 and � N c N i � λ 2 L e , flux = 1 � � C k e 2 k k − 2 L k k =1 k =1 Notice that, for ideal linear components, L e , flux = −L e , charge or L e , flux + L e , charge = 0 B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2015-16 17 / 28
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