Electrical Systems 2 Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 1 / 36
Active Components The power in electrical circuits is supplied by active components they are able to generate a power P ( t ) = e ( t ) i ( t ) that is supplied to passive components. According to the power sign convention already presented in Figure, positive power is supplied by active components and absorbed by passive components. Active components include the ideal current generator , the ideal voltage generator , and the operational amplifier (also called op-amp). The term “ideal” that associated to the generators means that only the main electrical characteristics of these elements are modelled; such aspects or parameters dealing with the technology of power generation are neglected. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 2 / 36
Ideal current generator The ideal current generator whose symbol is illustrated in Figure, is an active circuit component that supplies a current I ( t ) that is independent of the voltage e ( t ) at its ports; the supplied power is therefore P ( t ) = I ( t ) e ( t ) where I ( t ) does not change according to the circuit dynamics, but according to some externally given value, as, for example, a 50 Hz sinusoidal current with a given RMS value. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 3 / 36
Ideal voltage generator The ideal voltage generator , whose symbol is illustrated in Figure, is an active circuit component that supplies a voltage E ( t ) that is independent of the current i ( t ) flowing from its ports; the supplied power is therefore P ( t ) = i ( t ) E ( t ) where E ( t ) does not change according to the circuit dynamics, but according to some externally given value, as, for example, a 50 Hz sinusoidal voltage with a given RMS value. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 4 / 36
Ideal operational amplifier The ideal operational amplifier whose symbol is illustrated in Figure is an active device that under normal operating conditions behaves like an high-gain linear voltage generator. In practice it is a complex integrated circuit with several components, including transistors, but it has a fairly simple input-output electrical characteristic. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 5 / 36
+ E s positive supply voltage − E s negative supply voltage e + non-inverting input voltage e − inverting input voltage i + non-inverting input current i − inverting input current e 0 output voltage B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 6 / 36
The purpose of the supply voltages ± E s is to provide the power required by the op-amp to function, but they do not enter in the definition of the electrical interaction with the other parts of the circuit, i.e., they are external data necessary only for the op-amp operation, and often they are omitted from the graphical symbol of an op-amp, as in Figure. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 7 / 36
The input-output relationship in the linear range of operation is sketched in Figure where � < E s � e + − e − � A 10 − 3 V � A is the op-amp gain often in the range of 10 4 . B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 8 / 36
In the linear range of operation, the op-amp is characterized by the following approximations: the two currents i + , i − are zero or very small; the two input voltages shall be approximately equal ( e + − e − ) < ε ; the op-amp gain is a constant, independent of all input frequencies. With these approximations, the output voltage is expressed as e 0 ( t ) = A ( e + − e − ) Therefore the op-amp can be considered a dependent ideal voltage source: the output voltage depends only on the values of the difference of the input voltages. B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 9 / 36
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, 2014-15 10 / 36
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. From now on, 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, 2014-15 11 / 36
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, 2014-15 12 / 36
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, 2014-15 13 / 36
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 )) ∂ W ∗ c ( e ( q )) = e ( t ) and = q ( t ) ∂ q ∂ e B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 14 / 36
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, 2014-15 15 / 36
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, 2014-15 16 / 36
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, 2014-15 17 / 36
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 ) ∂ W ∗ i ( i ) = i ( t ) e = λ ( t ) ∂λ ∂ i B. Bona (DAUIN) Electrical Systems 2 Semester 1, 2014-15 18 / 36
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 i ( i ) = λ ( i ) d i = W ∗ 0 0 and � λ � λ λ 2 λ L d λ = 1 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, 2014-15 19 / 36
Recommend
More recommend
