Maxwell

Electromotive Force (emf) or voltage around C

= Negative of the time rate of increase of the

magnetic flux crossing S bounded by C.

Faraday’s Law, the first EMantra

Gauss’ Law for the Magnetic Field

[1] Dynamic simulation of electric machinery

CHEE-MUN ONG

PRENTICE hALL, New Jersey 1998

[2] Nannapaneni Narayana Rao

http://faculty.ece.illinois.edu/rao/FE/

Referencias Bibliograficas

Ampere circuit law

Ec 3.4

Maxwell’s Equations are elegant and beautiful. As profound as they are, they are actually quite simple to explain and understand.

The Guiding Equations of Electromagnetics

So, why are these poor little guys so perplexed at the sight of Maxwell’s Equations?

ò

S

=

S

ò

S

S

d

×

ò

ò

S

dt

+

S

d

J

=

l

d

ò

S

ò

V

r

S

S

d

B

d

–

=

= Time rate of decrease of magnetic flux crossing S,

= Magnetic flux crossing S,

Voltage around C, also known as electromotive force (emf) around C (but not really a force),

Magnetomotive force (mmf) around C

= Current due to flow of charges crossing S bounded by C

+ Time rate of increase of electric (or displacement) flux crossing S

Ampere’s Circuital Law, the second EMantra

= Displacement flux, or electric flux, crossing S,

= Current due to flow of charges crossing S,

= Magnetomotive force (only by analogy with electromotive force),

= Time rate of increase of displacement flux crossing S, or, displacement current crossing S,

r

Magnetic flux emanating from a closed surface S = 0.

Gauss’ Law for the Magnetic Field, the fourth EMantra

Out of the four EMantras, only the first two, Faraday’s and Ampere’s circuital laws are independent. The fourth Mantra, Gauss’ law for the magnetic field, follows from Faraday’s law, and the third Mantra, Gauss’ law for the electric field, follows from Ampere’s circuital law, with the aid of an auxiliary equation, the law of conservation of charge.

r(t)

Current due to flow of charges emanating from a closed surface S

= Time rate of decrease of charge enclosed by S.

Law of Conservation of Charge, an auxiliary EMantra

The “Mahatmyam (Greatness)” of Maxwell’s Equations

Law of Conservation

of Charge

Gauss’ Law

for E

Faraday’s

Law

Ampere’s

Circuital Law

The “Mahatmyam (Greatness)” of Maxwell’s Equations

Thus, Faraday's law says that a time-varying magnetic field gives rise to an electric field, the space-variation of which is related to the time-variation of the magnetic field. Ampere's circuital law tells us that a time-varying electric field produces a magnetic field, the space variation of which is related to the time-variation of the electric field. Thus, if one time-varying field is generated, it produces the second one, which in turn gives rise to the first one, and so on, which is the phenomenon of electromagnetic wave propagation, characterized by time delay of propagation of signals. In addition, Ampere’s circuital law tells us that an electric current produces a magnetic field, so that a time-varying current source results in a time-varying magnetic field, beginning the process of one field generating the second.

The “Mahatmyam (Greatness)” of Maxwell’s Equations

Hertzian dipole and radiation pattern on the covers of the U.S. and Indian Editions of “Fundamentals of Electromagnetics”

You will have noted that none of the four equations are named after Maxwell. So, the question arises as to why they are known as Maxwell’s equations. It is because of a purely mathematical contribution of Maxwell. This mathematical contribution is the second term on the right side of Ampere’s circuital law. Prior to that, Ampere’s circuital law consisted of only the first term on the right side.

The Contribution of Maxwell

Without the second term on the right side of Ampere’s circuital law, the loop is not complete and hence there is no interdependence of time-varying electric and magnetic fields and no EM waves!

The Contribution of Maxwell

Thus, the purely mathematical contribution of Maxwell in 1864 unified electricity and magnetism and predicted the generation of EM waves owing to the interdependence of time-varying electric and magnetic fields. Only 23 years later in 1887, eight years after his death in 1789, the theory was proved correct by the experimental discovery of EM waves by Heinrich Hertz.

Unifying Electricity and Magnetism

**Faraday**

**Ampere**

**Gauss**

Maxwell’s Equations in Differential Form and the Continuity Equation

Electric Field

Magnetic Field

Electric Charge

Electric charge:

basic property of matter carried by some elementary particles. Electric charge, which can be positive or negative, occurs in discrete natural units and is neither created nor destroyed.

The unit of electric charge in the metre–kilogram–second and SI systems is the coulomb.

One coulomb consists of 6.24 × 10e18 natural units of electric charge, such as individual electrons or protons.

Electric current: any movement of electric charge carriers.

http://www.britannica.com/EBchecked/topic/182416/electric-charge

http://www.britannica.com/EBchecked/topic/182467/electric-current

[2012-11-04]

Electric Current

A common unit of electric current is the ampere, a flow of one coulomb of charge per second, or 6.2 × 1018 electrons per second.

http://www.britannica.com/EBchecked/topic/182467/electric-current

[2012-11-04]

Line Integral and Voltage

Surface Integral

Electric relation on materials

r

Electric

Magnetic

**Electric Aspect**

**Thermal Aspect**

**Mechanical Aspect**

**Laws of Thermodynamics**

0. Heat and temperature heat capacity

1. Enthalpy

2. Entropy

3. Absolute rest at 0ºk.

http://www.genchem.net/thermo/laws.html

http://en.wikipedia.org/wiki/Laws_of_thermodynamics

**Heat transfer**

Conduction

Convection

Radiation

http://www.wisc-online.com/Objects/ViewObject.aspx?ID=sce304

http://www.physics.ohio-state.edu/~p670/textbook/Chap_4.pdf

Heat equation

Fluid dynamics

**Newton's laws of motion**

**-Inertia**

-F = m a

-Action-Reaction

-F = m a

-Action-Reaction

Stress analysis

http://en.wikipedia.org/wiki/Stress_analysis

[2012-11-06]

**Integral <-> differential**

**Faraday**

**Ampere**

**Gauss**

**Q**

**Div - Rot**

Hooke's law

**Circuit models**

History of EMF

http://learningtools.arts.ubc.ca/timeline2.0/bin/view.php?id=134792915

**Chemical Aspect**

**Materials**

Diagnostic

Aging

Diagnostic

Aging

2.1.4 Parameter estimation

2.1.5 Extension to higher frequency

2.2 Inrush Current

2.3 Iron Core Modeling

compelemtarios

Model based on an hybrid approach

1. An equivalent magnetic circuit can be used to represent a transformer. An interface between the magnetic model and the electrical network is then required.

2. An equivalent electric circuit derived by duality transformation of the magnetic circuit can be used to model a transformer directly in the electrical domain.

3. Model based on an hybrid approach

approaches for the modeling of transformers in the low frequency

range

The two main components to be considered:

The windings: transfer characteristic of the transformer (short-circuit response).

The iron core: flux balance (no-load response) and the phase-to-

phase coupling (transformer topology).

2.1 Low-frequency Transformer Modeling

Typically occur in gas insulated substations.

Model transformer as a capacitor.

At very high frequency the leakage and magnetizing impedances can be neglected.

Very fast front transients

Lightning strokes on overhead transmission lines and substations. Studies are aimed to the design of transmission lines and substations, and to the protection of equipments.

Transformer are represented by their stray capacitances to ground.

The influence of the magnetic core is usually neglected in the study of high frequency

transients.

fast front transients

They are caused by energization and de-energization of system components. The study of switching phenomena is useful for insulation co-ordination, determination of arrester characteristics, calculation of transient recovery voltages, and establishment of transient mitigation solutions.

Transformer are usually represented by lumped parameter coupled-winding models. (R-L-C)

Switching transients

Slow transients: from 5 Hz to 1 kHz.

Switching transients: from the fundamental power frequency up to 10 kHz.

Fast front transients: from 10 kHz up to 1 MHz.

Very fast front transients: from 100 kHz to 50 MHz

A classification of frequency

ranges of transients

(CIGRE WG 33.02)

The principal part of the document is divided in three main sections:

State of the art of low-frequency transformer.

Modeling, background information on inrush current.

Overview on modeling of a nonlinear hysteretic inductor.

Chapter 2

Nicola Chiesa

Thesis for the degree of Philosophiae Doctor

Trondheim, June 2010

Norwegian University of Science and Technology

Faculty of Information Technology,

Mathematics and Electrical Engineering

Department of Electric Power Engineering

http://ntnu.diva-portal.org/smash/record.jsf?pid=diva2:322811

Power Transformer

Modeling for Inrush Current

Calculation

CURSO DE TRANFORMADORES ELECTRICOS

SEMESTRE 2013-1

INGENIERIA ELECTRICA (Version 3)

Jaime A. Valencia V.

7 de mayo de 2013

CIRCUIT MODEL ON ELECTRIC TRANSFORMER

Examples of slow transients are torsional oscillations, transient torsional torques, turbine blade vibrations, fast bus transfers, controller interactions, harmonic interactions, and resonances.

Classical low frequency transformer model

Models based on an equivalent electric circuit

Models based on a magnetic circuit

Ejemplo: RuleBook -04e

Ejemplo: RuleBook -04e

Ejemplo: RuleBook -04e

Ejemplo: RuleBook -04e

Ejemplo: RuleBook -04e

Ejemplo: RuleBook -04e

Calculo de parametros

Calculo de parametros

Curso Transformadores electricos

Semestre 2013-1

Jaime A. Valencia V.

16 de mayo de 2013

Pruebas de transformadores

y

Calculo de parametros

Se energiza en el lado de baja.

Se alimenta a tension nominal.

Para determinar las perdidas en el nucleo.

Para determinar la caracteristica del nucleo.

Para determinar la inductancia de vacio.

Prueba de circuito abierto

(open circuit)

Modelo implementado en la rama satura del EMTP-ATP.

Puede permitir multiples bibinas secundarias.

Modelo monofasico

Modelo circuital

Se hace en lado de alta tension.

La corriente en baja debe ser la nominal.

Determina las perdidas en cobre.

Prueba Corto Circuito

(short-circuit)