Electrical Systems 1 - polito.it · The basic quantities involved in electrical circuit modelling...

Electrical Systems 1

Basilio Bona

DAUIN – Politecnico di Torino

Semester 1, 2016-17

B. Bona (DAUIN) Electrical Systems 1 Semester 1, 2016-17 1 / 30

Introduction

An electrical network is represented by a closed graph of passive or activeone-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, andthe N voltages ek(t),k = 1, . . . ,N between the component ports.


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 tothe network.

Since energy cannot be created from nothing these elements use otherforms of external power, like hydraulic, mechanical, etc., that aretransformed into electrical power and supplied to the system.


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≡ q̇(t) = i(t)

dλ (t)

dt≡ λ̇ (t) = e(t)

PowerP(t) = e(t)i(t

is a signed quantity, whose sign convention is different for active or passiveelements, as specified in Figure.


a) Power convention for active components: P(t) > 0 when the currentflows from the positive pole

b) Power convention for passive components: P(t) > 0 when the currentflows into the positive pole.


Inductors

An inductor is a one-port passive component that generates a flux linkageλ (i(t)) in response to the port current i(t)

The port voltage is the time derivative of the flux,

e(t) =ddt

λ (t) = λ̇ (t)

The representation of a generic inductor is given in Figure


Examples of the constitutive relations between the current and the flux

λ (t) = λ (i(t))

or between the flux and the current

i(t) = i(λ (t))

in an inductive one-port element, are illustrated in Figure.


Electromagnetic circuit


The current i(t) flowing in the coils produces a magnetic field H(t), and aflux 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 magneticpermeability

µ = µrµ0 = (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.


∮H(t) ·dσ = Ni(t) = F (t) magnetomotive force

In a magnetic circuit the magnetomotive force Ni produces a magneticflux ΦM ,

ΦM(t) =∫

SB(t) ·ds = B(t) ·

∫S

ds = B(t) ·Sn

where S = Sn is the signed surface vector, and n is the unit normal vectorto 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 themagnetic core and in the air gap.


∮H(t) ·dσ = Ni(t) = Hm`+Hah =

(`

µ+

h

µ0

)B =

(`

µ+

h

µ0

)Φ

S

sinceµ0Ha = B; µHm = B

The magnetic reluctance R is the ratio between the magnetomotiveforce and the flux, so

Ni(t) = RΦ(t)

where

R =1

S

(`

µ+

h

µ0

)≈ h

µ0S

since µ � µ0.


The total flux linkage λ is obtained considering the N windings, so wewrite 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 =µ0N

2S

h

is called the (auto-)inductance of the one-port component. In linearcases the port voltage is simply

e(t) = Lddt

i(t)


Capacitors

A capacitor is a passive one-port component that stores a charge q(t) inresponse to an applied port voltage e(t); the port current is the timederivative of the charge,

i(t) =ddt

q(t) = q̇(t)


An example of the constitutive relations between voltage and charge

q(t) = q(e(t))

or between charge and voltage

e(t) = e(q(t))

in an capacitive one-port element are illustrated in Figure


Electrostatic circuit


The electrostatic field E between the two plates, is normal to them withequipotential surfaces parallel to the planar plates (except in the vicinity ofthe 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 establishedbetween the two plates

ΦE (t) =∫

SE(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 .


For a closed surface we have

ΦE (t) =∮

SE(t) ·ds =

q(t)

ε(1)

where q(t) is the total charge, ε = εrε0 = (1 + χ)ε0 is the dielectricpermittivity ε0 is the vacuum permittivity, εr the relative permittivity ofthe 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)


If C is constant, the constitutive relation between voltage and charge inplanar surface capacitors is established as

q(t) =εrε0S

de(t) = Ce(t)

The current i(t) flowing into the capacitor is

i(t) =ddt

q(e) = Cddt

e(t)

so, when e(t) is a constants, as in the case of a battery supply, the currentinto the capacitor is zero.


Resistors

A resistor is a passive one-port component that dissipates the electricalinput power, usually transforming it in heat in response to an applied portvoltage e(t) or an applied current i(t)

The relation between the port current and the port voltage isinstantaneous, i.e., no time derivatives of electrical quantities are involved

e(t) = R(i(t), t) or i(t) = G (e(t), t)

When the resistor is constant and linear we have the well-known Ohm’slaw or its inverse

e(t) = Ri(t) or i(t) = Gi(t) =1

Ri(t)


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


Constitutive relations

e(t) = R(i(t)) = ρ(t)`(t)

S(t)

and

i(t) = G (e(t)) = σ(t)`(t)

S(t)(2)

σ is the conductivity of the material.

When an alternating current flows in the conductor the skin effect makescurrent flow near the boundary of the conductor, reducing the totalcross-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 itwill be correct to write

e(t) = R(i(t)) = ρ(t,T )`(t)

S ′(t)


Active Components

The power in electrical circuits is supplied by active components they areable to generate a power P(t) = e(t) i(t) that is supplied to passivecomponents. According to the power sign convention already presented inFigure, positive power is supplied by active components and absorbed bypassive components.

Active components include the ideal current generator, the idealvoltage generator, and the operational amplifier (also called op-amp).

The term “ideal” that associated to the generators means that only themain electrical characteristics of these elements are modelled; such aspectsor parameters dealing with the technology of power generation areneglected.


Ideal current generator

The ideal current generator whose symbol is illustrated in Figure, is anactive circuit component that supplies a current I (t) that is independentof 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, butaccording to some externally given value, as, for example, a 50 Hzsinusoidal current with a given RMS value.


Ideal voltage generator

The ideal voltage generator, whose symbol is illustrated in Figure, is anactive circuit component that supplies a voltage E (t) that is independentof 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, butaccording to some externally given value, as, for example, a 50 Hzsinusoidal voltage with a given RMS value.


Ideal operational amplifier

The ideal operational amplifier whose symbol is illustrated in Figure isan active device that under normal operating conditions behaves like anhigh-gain linear voltage generator. In practice it is a complex integratedcircuit with several components, including transistors, but it has a fairlysimple input-output electrical characteristic.


+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 output voltage


The purpose of the supply voltages ±Es is to provide the power requiredby the op-amp to function, but they do not enter in the definition of theelectrical interaction with the other parts of the circuit, i.e., they areexternal data necessary only for the op-amp operation, and often they areomitted from the graphical symbol of an op-amp, as in Figure.


The input-output relationship in the linear range of operation is sketchedin Figure where ∣∣e+− e−

∣∣< Es

A10−3 V

A is the op-amp gain often in the range of 104.


In the linear range of operation, the op-amp is characterized by thefollowing 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 theinput voltages.


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