Electrical Systems 1 Basilio Bona DAUIN Politecnico di Torino - - PowerPoint PPT Presentation

Electrical Systems 1

Basilio Bona

DAUIN – Politecnico di Torino

Semester 1, 2016-17

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Electrical Systems 1 Semester 1, 2016-17 1 / 30

Introduction

An electrical network is represented by a closed graph of passive or active

• ne-port electrical components

The circuit behaviour is completely determined by the set of 2N quantities, namely the N currents ik(t),k = 1,...,N flowing into the components, and the N voltages ek(t),k = 1,...,N between the component ports.

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Components

One-port components includes Passive components: that can store or dissipate energy, but not create it. Active components: that can “create” electrical energy and supply it to the network. Since energy cannot be created from nothing these elements use other forms of external power, like hydraulic, mechanical, etc., that are transformed into electrical power and supplied to the system.

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The basic quantities involved in electrical circuit modelling are: Currents i(t) Voltages e(t) Other important electrical quantities related to the previous ones are Electrical charge q(t) Magnetic flux linkage λ(t) dq(t) dt ≡ ˙ q(t) = i(t) dλ(t) dt ≡ ˙ λ(t) = e(t) Power P(t) = e(t)i(t is a signed quantity, whose sign convention is different for active or passive elements, as specified in Figure.

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a) Power convention for active components: P(t) > 0 when the current flows from the positive pole b) Power convention for passive components: P(t) > 0 when the current flows into the positive pole.

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Inductors

An inductor is a one-port passive component that generates a flux linkage λ(i(t)) in response to the port current i(t) The port voltage is the time derivative of the flux, e(t) = d dt λ(t) = ˙ λ(t) The representation of a generic inductor is given in Figure

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Examples of the constitutive relations between the current and the flux λ(t) = λ(i(t))

• r between the flux and the current

i(t) = i(λ(t)) in an inductive one-port element, are illustrated in Figure.

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Electromagnetic circuit

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The current i(t) flowing in the coils produces a magnetic field H(t), and a flux density B(t) B(t) = µH(t) These two quantities are described by the following Maxwell equations rot H = ∇×H = j(t) div B = ∇·B = 0 where j(t) is the current density in the coils and µ is the magnetic permeability µ = µr µ0 = (1+ χm)µ0 µ0 is the air permeability µr the relative permeability χm is the magnetic susceptibility of the material usually χm ≫ 1 in magnetic materials, so that µ ≫ µ0.

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• H(t)·dσ = Ni(t) = F(t)

magnetomotive force In a magnetic circuit the magnetomotive force Ni produces a magnetic flux ΦM, ΦM(t) =

• S B(t)·ds = B(t)·
• S ds = B(t)·Sn

where S = Sn is the signed surface vector, and n is the unit normal vector to the section. If B = B, and B is always orthogonal to the surface S, then write ΦM(t) = BS The flux Φ is the same in every section of the magnetic circuit, both in the magnetic core and in the air gap.

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• H(t)·dσ = Ni(t) = Hmℓ+Hah =

ℓ µ + h µ0

• B =

ℓ µ + h µ0 Φ S since µ0Ha = B; µHm = B The magnetic reluctance R is the ratio between the magnetomotive force and the flux, so Ni(t) = RΦ(t) where R = 1 S ℓ µ + h µ0

• ≈

h µ0S since µ ≫ µ0.

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The total flux linkage λ is obtained considering the N windings, so we write the constitutive relation λ(t) = NΦ(t) and in general λ(i(t)) = N2(t)i(t) R(t) If both N and R are constant parameters we have a linear relation λ(t) = Li(t) where L = µ0N2S h is called the (auto-)inductance of the one-port component. In linear cases the port voltage is simply e(t) = L d dt i(t)

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Capacitors

A capacitor is a passive one-port component that stores a charge q(t) in response to an applied port voltage e(t); the port current is the time derivative of the charge, i(t) = d dt q(t) = ˙ q(t)

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An example of the constitutive relations between voltage and charge q(t) = q(e(t))

• r between charge and voltage

e(t) = e(q(t)) in an capacitive one-port element are illustrated in Figure

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Electrostatic circuit

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The electrostatic field E between the two plates, is normal to them with equipotential surfaces parallel to the planar plates (except in the vicinity of the plate borders). The field norm is E = E = e(t) d where d is the distance between the planar plates. The electric flux ΦE due to the total accumulated charge is established between the two plates ΦE(t) =

• S E(t)·ds

If the flux is constant across the surface, we have ΦE = E·S = SE·ns where ns is the unit norm vector orthogonal to the surface of area S.

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For a closed surface we have ΦE(t) =

• S E(t)·ds = q(t)

ε (1) where q(t) is the total charge, ε = εrε0 = (1+ χ)ε0 is the dielectric permittivity ε0 is the vacuum permittivity, εr the relative permittivity of the dielectric material and χ is the electric susceptibility. The relation between the electric field E and the displacement field D is εΦE(t) = D = εE = q(t) S Since D = q(t)/S, introducing the capacitance C(t) = q(t) e(t), we have ε = εrε0 = q(t) S d e(t) = C(t)e(t) S d e(t) = C(t)d S and C(t) = εrε0S(t) d(t)

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If C is constant, the constitutive relation between voltage and charge in planar surface capacitors is established as q(t) = εrε0S d e(t) = Ce(t) The current i(t) flowing into the capacitor is i(t) = d dt q(e) = C d dt e(t) so, when e(t) is a constants, as in the case of a battery supply, the current into the capacitor is zero.

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Resistors

A resistor is a passive one-port component that dissipates the electrical input power, usually transforming it in heat in response to an applied port voltage e(t) or an applied current i(t) The relation between the port current and the port voltage is instantaneous, i.e., no time derivatives of electrical quantities are involved e(t) = R(i(t),t)

• r

i(t) = G(e(t),t) When the resistor is constant and linear we have the well-known Ohm’s law or its inverse e(t) = Ri(t)

• r

i(t) = Gi(t) = 1 R i(t)

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Two planar conductive plates of equal surface S are separated by a length ℓ of conductive material having an electrical resistivity ρ or specific electrical resistance; it is a measure of the characteristics of the internal molecular structure of the conductive material .

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Constitutive relations e(t) = R(i(t)) = ρ(t) ℓ(t) S(t) and i(t) = G(e(t)) = σ(t) ℓ(t) S(t) (2) σ is the conductivity of the material. When an alternating current flows in the conductor the skin effect makes current flow near the boundary of the conductor, reducing the total cross-section that becomes S′ < S. Similarly, if two conductors are near and both carry an alternating current, their resistances will increase due to the proximity effect. The resistivity usually changes with the temperature T, so at the end it will be correct to write e(t) = R(i(t)) = ρ(t,T) ℓ(t) S′(t)

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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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