Electrical Systems 2 Basilio Bona DAUIN Politecnico di Torino - - PowerPoint PPT Presentation

Electrical Systems 2

Basilio Bona

DAUIN – Politecnico di Torino

Semester 1, 2014-15

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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

• r parameters dealing with the technology of power generation are

neglected.

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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

• f 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.

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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

• f 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.

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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.

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+Es positive supply voltage −Es negative supply voltage e+ non-inverting input voltage e− inverting input voltage i+ non-inverting input current i− inverting input current e0

• utput voltage
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The purpose of the supply voltages ±Es 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

• mitted from the graphical symbol of an op-amp, as in Figure.
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The input-output relationship in the linear range of operation is sketched in Figure where

• e+ − e−

< Es A 10−3 V A is the op-amp gain often in the range of 104.

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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 e0(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.

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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:

• ne 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.

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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 W (t) = t P(τ)dτ = t e(τ)i(τ)dτ 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.

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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) = dq(t) dt = ˙ q(t). The voltage across the capacitive component is proportional to the charge e(t) = 1 C q(t)

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The electrostatic energy stored in the capacitive element, also called the capacitive energy Wc(q) is defined as Wc(q(t)) = t e(q) i dt = q e(q)dq 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

W ∗

c (e) = qe − Wc(q) =

e q(e)de 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.

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The energy differential is dWc(q) = e(q) dq and the co-energy differential is dW ∗

c (e) = q(e) de

from which the following relations are established ∂Wc(q(e)) ∂q = e(t) and ∂W ∗

c (e(q))

∂e = q(t)

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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 W ∗

c (e) =

e q(e)de = e Ce de = 1 2 Ce2 and Wc(q) = q e(q)dq = q q C dq = 1 2 q2 C In this case the capacitive energy and co-energy are equal, and the characteristic function is a straight line.

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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) = dλ(t) dt = ˙ λ(t). The current flowing into the inductive component is proportional to the flux i(t) = 1 Lλ(t)

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The electromagnetic energy stored in the inductive element, also called the inductive energy Wi(λ), is defined as Wi(λ) = t i(λ) e dt = λ i(λ)dλ 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

W ∗

i (i) = iλ − Wi(λ) =

i λ(i)di 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.

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The energy differential is dWi(λ) = i dλ while the co-energy differential is dW ∗

i (i) = λ di

from which the following relations are established ∂Wi(λ) ∂λ = i(t) e ∂W ∗

i (i)

∂i = λ(t)

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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 W ∗

i (i) =

i λ(i)di = i Li di = 1 2 Li2 and Wi(λ) = λ i(λ)dλ = λ λ Ldλ = 1 2 λ2 L In this case the inductive energy and co-energy are equal, and the characteristic function is a straight line.

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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 Le, is the difference between the “kinetic” co-energy C∗

e(ξ, ˙

ξ) and the “potential” energy Pe(ξ): Le(ξ, ˙ ξ) = Ce(ξ, ˙ ξ) − Pe(ξ) The energy/co-energy functions are different if we use the flux or the charge coordinates.

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Charge coordinates and Lagrange function

Using the charge coordinates ξ = q and velocities ˙ ξ = ˙ q = i, we write Le,charge(ξ, ˙ ξ) = W ∗

i (˙

q) − Wc(q) where the “kinetic” co-energy coincides with the inductive co-energy stored into the inductive element: C∗

e (ξ, ˙

ξ) ≡ W ∗

i (˙

q) = W ∗

i (i)

and the “potential” energy coincides with the capacitive energy stored into the capacitive elements Pe(ξ) ≡ Wc(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.

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Since the energies are additive, assuming a linear circuit with Ni inductors and Nc capacitors, we can write C∗

e (ξ, ˙

ξ) = W ∗

i (i) = 1

2 Ni

• k=1

Lki2

k

• where ik is the current flowing into the k-th inductive component with

inductance Lk. Similarly Pe(ξ) = Wc(q) = 1 2 Nc

• k=1

q2

k

Ck

• where qk is the charge stored into the k-th capacitive component with

capacity Ck.

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Flux coordinates and Lagrange function

Using the flux coordinates ξ = λ and velocities ˙ ξ = ˙ λ = e, we can write Le,flux(ξ, ˙ ξ) = W ∗

c ( ˙

λ) − Wi(λ) where the “kinetic” co-energy coincides with the capacitive co-energy stored into the capacitive element: Ce( ˙ λ) ≡ W ∗

c ( ˙

λ) = W ∗

c (e)

and the “potential” energy coincides with the inductive energy stored into the inductive elements: Pe(λ) ≡ Wi(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.

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Since the energies are additive, assuming again a linear circuit with Ni inductors and Nc capacitors, we can write C∗

e (ξ, ˙

ξ) = W ∗

c (e) = 1

2 Nc

• k=1

Cke2

k

• where ek is the voltage across the k-th capacitive component with

capacity Ck. Similarly Pe(ξ) = Wi(λ) = 1 2 Ni

• k=1

λ2

k

Lk

• where λk is the flux across the k-th inductive component with inductance

Lk.

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In conclusion Le,flux = 1 2 Ni

• k=1

Lki2

k − Nc

• k=1

q2

k

Ck

• and

Le,charge = 1 2 Nc

• k=1

Cke2

k − Ni

• k=1

λ2

k

Lk

• Notice that, for ideal linear components,

Le,flux = −Le,charge

• r

Le,flux + Le,charge = 0

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Dissipative Elements

The dissipative elements are represented by the resistors. The dissipated power is Pr = R i2(t) = e2(t) R In order to include the dissipative effects into the Lagrange equations, it is customary, as it has been in mechanical systems with the definition of the dissipative function, to define an electric dissipative function De as follows De,charge(˙ q) = 1 2

Nr

• k=1

Rk ˙ q2

k = 1

2

Nr

• k=1

Rki2

k

• r

De,flux( ˙ λ) = 1 2

Nr

• k=1

1 Rk ˙ λ2

k = 1

2

Nr

• k=1

1 Rk e2

k

where Nr is the total number of dissipative elements in the circuit, i.e., the resistors, and Rk their resistance.

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Generalized forces - charge approach

Using charge coordinates, the k-th generalized electric force is a generalized voltage, indicates as Fe,k = Ek and can be computed considering the set of k = 1, . . . , NE ideal voltage generators Ek(t) present in the circuit, starting from the general expression of the virtual work δW nc =

NE

• k=1

Fnc

k δqk

and the virtual work related to the electrical system dW =

NE

• k=1

Pkdt =

NE

• k=1

ekikdt =

NE

• k=1

ekdqk we can write δW nc =

NE

• k=1

Ek(t)δqk =

NE

• k=1

Ekδqk

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In order to compute each Ek it is necessary to consider the currents flowing into the ideal voltage generators, compute the product (voltage × current) and assign to each virtual charge variation δqk the contribution of the related voltages. If ideal current generators are present, they do not contribute to the generalized voltages, but act as constraints on the node current summation.

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Generalized forces - flux approach

Using flux coordinates, the k-th generalized electric force is a generalized current, indicates as Fe,k = Ik and can be computed considering the set

• f k = 1, . . . , NI ideal current generators Ik(t) present in the circuit,

starting from the general expression of the virtual work δW nc =

NI

• k=1

Fnc

k δqk

and the virtual work related to the electrical system dW =

NI

• k=1

Pkdt =

NI

• k=1

ikekdt =

NI

• k=1

ikdλk we can write δW nc =

NI

• k=1

Ik(t)δλk =

NI

• k=1

Ikδλk

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In order to compute each Ik it is necessary to consider the voltage across the ideal current generators, compute the product (voltage × current) and assign to each virtual flux variation δλk the contribution of the related currents. If ideal voltage generators are present, they do not contribute to the generalized currents, but act as constraints on the mesh voltage summation.

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Lagrange Equations in Electromagnetic Systems

After having chosen the n generalized coordinates ξ and velocities ˙ ξ, either charge or flux coordinates, we write the n Lagrange equations d dt ∂Le ∂ ˙ ξk

• − ∂Le

∂ξk + ∂De ∂ ˙ ξk = Fk k = 1, . . . , n If the k-th coordinate involves both capacitive and inductive elements, the corresponding equation will in general be a second-order differential equation expressed in terms of ¨ qk or ¨ λk. In all the other cases, e.g., when only resistive elements are present in the circuit, a first-order differential equation or an algebraic equation will result.

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If we collect all equations and organize them in matrix form, we can write the following second-order linear matrix equations Z¨ q(t) + R˙ q(t) + Qq(t) = E(t)

• r

Y¨ λ(t) + G ˙ λ(t) + Λλ(t) = I(t) where Z, R, Q, Y, G, Λ are suitable n × n matrices Z inductance matrix Y = Z−1 reluctance matrix R resistance matrix G = R−1 conductance matrix Q capacitance matrix Λ = Q−1 elastance matrix

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It is often convenient to rewrite the equations as functions of the generalized velocities ˙ ξ = {˙ qk = ik, ˙ λk = ek} In this case we obtain mixed integro-differential equations. If we define the voltage across a capacitor and the current in an inductor respectively as vCk =

• 1

Ck qk dt iLk =

• 1

Lk λk dt the Lagrange equations become Z d dt i(t) + Ri(t) + vC(t) = E(t) (1)

• r

Y d dt e(t) + Ce(t) + iL(t) = I(t) (2)

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Comments

The Lagrange approach, that makes use of the energy formulation to

• btain the differential equations, does not represent the tool commonly

adopted for linear circuits analysis: the Kirchhoff equations applied to meshes and nodes is by far the preferred tool, in conjunction with the Laplace transforms. Nevertheless there are some advantages related to the use of the Lagrangian approach has a general validity, independent of the specific field of application, mechanical or electrical; can be easily applied when nonlinear components are present; is very beneficial when modelling systems where electrical and mechanical parts interact. is a good starting point for state-variable equation. Furthermore, the Lagrange approach allows to introduce some analogies between electrical and mechanical quantities.

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Electrical and mechanical analogies

Recalling the definition of the k-th generalized momentum µk = ∂L ∂ ˙ qk and the generalized velocities ˙ qk(t) = ik(t) ˙ λk(t) = ek(t) we see that the power P(t) can be expressed as the product of two abstract quantities, the first one called effort s(t), the other called flow φ(t), not to be confused with the flux of the magnetic field Φ(t), P(t) = s(t)φ(t) Using these two quantities it is possible to define a number of analogies between the mechanical and the electrical variables, making possible to interpret a quantity in one class with the analogous quantity in the other

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Type ξ ˙ ξ s φ µ C P D F Transl x ˙ x f = m¨ x ˙ x m ˙ x 1 2m ˙ x2 1 2kl∆x2 1 2βl ˙ x2 f Rot θ ˙ θ τ = Γ ¨ θ ˙ θ Γ ˙ θ 1 2Γ ˙ θ2 1 2ka∆θ2 1 2βa ˙ θ2 τ Charge q i e = L¨ q ˙ q = i Li = λ 1 2Li2 1 2C q2 1 2R ˙ q2 E Flux λ e i = C ¨ λ ˙ λ = e Ce = q 1 2Ce2 1 2Lλ2 1 2R ˙ λ2 I Table: Electro-mechanical analogies for one-dimensional linear ideal elements.

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