Electrical Machines Basic Theory

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Knowledge to make your life easier

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Electrical Machines Basic Theory ©

Jorge Cardenas

First Edition

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2 Editorial: Adneli Consultant, S.L.

www.adneli.com [email protected]

“All rights to this book are retained by Adneli Consultant, S.L. This book is allowed to reproduce individual figures (except the ones belonging to others and mentioned in the specific references) in other publications, and their use for any third party provided proper acknowledgment is given to the author.”

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Electrical Machines – Basic Theory

3

About the Author

I received my Engineering Degree from the Universidad de Ingeniería (Peru) in 1977 and my MBA from the Universidad Politécnica de Madrid (Spain) in 1998.

I began my career with the Utility Electroperu (Peru) in 1978, as a Protection & Control engineer. In 1987 I moved to ABB (Spain) as HV equipment Sales Engineer and was then promoted to a Control Design Engineer. In 1989 I joined General Electric (GE), where I have held several positions until 2020.

Currently, I work as Advisor for Adneli Consultant in Spain. I have authored and co-authored more than 70 articles (English, Spanish and Russian) on protective relaying and topics related to Grid Modernization (papers published in some of the most important congresses around the world: Europe, Asia, USA): See some of them in https://www.researchgate.net/profile/Jorge_Cardenas11. I am a member of the CIGRÉ WG B5.31 and WG B5.43 and a contributor of the magazines GE P&C Journal, Pacworld (USA), IET (UK), and Energía (Spain). I am a regular speaker in congress and conferences in Europe. I have made several contributions in the design of new products related to Generator, Bus, Line, Transformer, Motor, feeders, Network Protection. I have two co-patents: the first one on a new Power System inter-area oscillation detector and the second one on an Early warning system for Cybersecurity alert and automatic protective actions. I have also written a book on Philosophy See:

www.adneli.wordpress.com, a book on “Relay Protection, Control, and Information Management in the Modern Power Systems,” and I am glad to present this book entitled “Electrical Machines – Basic Theory.”

I have worked with some of the major Utilities in the world as PG&E (USA), Scottish Power (UK), Enel Terna (Italy), Iberdrola, Gas Natural, Red Electrica & Endesa (Spain), CFE (Mexico), МЭС Сибири (Rusia), CTEEP (Brazil), IEC (Israel), Fingrid (Finland), TenneT (Netherlands), RTE (France), EDP & REN (Portugal), ESKOM (South Africa), WAPDA (Pakistan), TEIAS (Turkey), Sonelgas (Algeria), ONE (Morocco). I have worked also with some of the major Industries in Oil &Gas as Rasgas (Qatar) and SWCC (Saudi Arabia).

I have worked as well in projects with some of the main laboratories in the world as CESI (Italy), Kema (Holland), CEPRI & NCEPRI (China) and with Universities as “La Sapienza” in Rome, Italy, and UPV in the Basque Country, Spain.

Specialties: Negotiation, Project Management, Consulting in Electrical Engineering focused on Power System Protection and Assets Management. In recent years I have led R&D in the synchrophasor application for Utilities and now I work on projects related to Grid Modernization.

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4

Foreword

This book is a summary of the basic theory of Electrical Machines and aims to support other books such as the one entitled. “Relay Protection, Control, and Information Management in the Modern Power Systems,” where despite the existence of an extensive theoretical description, it does not include the complete basic theory (some has been already developed and explained), because it is not the objective of an application book to be used by engineers with a certain level of training and experience.

The technical aspects that the author considers to be the most important for aid in the better understanding of other application books on electrical protections and other disciplines have been included. Some of the topics developed in application books sometimes present difficulties in

understanding due to not having available some basic theoretical aspects that are sometimes forgotten.

The author hopes to achieve the desired goals and expects to receive any suggestions that allow to improve the content. Be free to send them to [email protected]. Thanks.

Jorge Cardenas

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5

Chapter I-ELECTROMAGNETISM

1.1. Basic definitions

There are four fundamental interactions known to exist (fundamental forces, are the interactions that do not appear to be reducible to more basic interactions): 1) the gravitational, 2) the electromagnetic 3) the strong and 4) the weak interactions. The first two produce significant long-range forces and the last two produce forces at minuscule, subatomic distances and govern nuclear interactions.

1.2. The second fundamental interaction – Electromagnetism

Electromagnetism is a branch of physics involving the study of electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields.

Figure 1.1 Electromagnetic field.

Let´s assume a current circulating through a conductor. This current will create a magnetic field surrounding the conductor. We can express the mathematical relationship of the magnetic field in a point with the following equation:

= ∮ ⃗ ∙ ⃗ 1.1

Where

⃗: magnetic field vector (other definitions are Magnetic flux density and Magnetic induction as well) I: current circulating through a conductor

: vacuum permeability. Air permeability is assumed the same.

As ⃗// ⃗ and B is uniform in all point of the area under analysis:

= ∮ . 1.2

1

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8

= . 2 1.3

= 1.4

Electromagnetic phenomena can be defined in terms of the electromagnetic force, by the Lorentz force law in its vector form¹:

⃗ = ⃗ + (!⃗ X ⃗) 1.5

Where

⃗: force on the particle.

: particle electric charge.

⃗: electric field vector.

!⃗: particle´s velocity.

⃗: magnetic field vector (other definitions are Magnetic flux density and Magnetic induction as well)

⃗: Electric force.

$!⃗ ∙ ⃗%: Magnetic force.

X: operator to multiply phasors.

1.2 Force generated by the interaction of the magnetic field with a particle.

Using the definition of the cross product, the magnetic force can also be written as a scalar equation:

&'()*+,-.= / 0123 1.6

Where 3 is the angle between the velocity of the particle and the magnetic field.

1Wikipedia

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9 1.3 Rotating spire with i current circulating through.

If instead a single conductor, we have a spire rotating counterclockwise with a continuous current flow (because of the particle movement) through a conductor of length l exposed to a uniform magnetic field will give us an equivalent expression for the magnetic force.

⃗ = 4⃗ X ⃗ 1.7

When the spire is rotating because of the torque produced by the magnetic force, a voltage is induced between the terminals of the spire. The value of this voltage at each side of the spire is given by the equation:

5⃗ = 6⃗ X ⃗ 1.8

1.3. MFF Waveforms

In physics, the magnetomotive force (MMF) is a quantity appearing in the equation for the magnetic flux in a magnetic circuit. It is the property of certain substances or phenomena that give rise to magnetic fields:

The magnetic flux can be expressed through the following equation:

7 = X 8 = ∙ 9 ∙ :;03 1.9

Where B is the magnitude of the magnetic flux density, S is the area of the surface crossed by the magnetic flux, and 3 is the angle between the magnetic field lines and the normal (perpendicular) to S.

The magnetic flux is related to the current

7 = < ∙_ℜ^-_>= Magnetic flux 1.10

And the magnetomotive force (MMF) will be:

?&& = < ∙_ℜ^-

>∙ ℜ_@= < ∙ 1 1.11

Where

1: current circulating through the spire

<: number of turns in the coil.

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10 1.4. Main Applications in Electricity

1.4.1. Transformers

Electromagnetic induction, the principle of the operation of the transformer, was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Only Faraday furthered his experiments to the point of working out the equation describing the relationship between

Electromotive Force (emf) and Magnetic Flux now known as Faraday's law of induction²:

/ =^AB_A,^C 1.12

where

/ : Magnitude of the emf in volts.

7D: Magnetic Flux through the circuit in webers.

A transformer is a passive electrical device that transfers electrical energy from one electrical circuit to another, or multiple circuits by magnetic coupling with no moving parts. An alternating current in any winding of the transformer produces a time-varying magnetic flux in the transformer's core, which induces a time-varying electromotive force (voltage) across any other windings wound around the same core. Transformers are used to convert electrical energy between high and low voltages, to change impedance, and to provide electrical isolation between circuits³.

A simple transformer consists of two electrical conductors called the primary winding and the

secondary winding. These two windings can be considered as a pair of mutually coupled coils. Energy is coupled between the windings by the time-varying magnetic flux that passes through (links) both primary and secondary windings.

Figure 1.4 Basic Transformer^1-3.

1.1.4.1. Analysis of Magnetically coupled circuits.

From Figure 1.4

7E = 7EF+ 7E'+ 7 ' 1.13

2Wikipedia.

3Paul C. Krause, Oleg Wasynczuk, Scott D. Sudhoff. “Analysis of Electric Machinery and Drive Systems.” IEEE Press Power Engineering Series.

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11

7 = 7 F+ 7 '+ 7E' 1.14

Important Definitions assuming linear magnetic circuits:

7EF = <E∙_ℜ^-^G

GH: Leakage flux coil 1 1.15

7 _F = < ∙_ℜ^-^I

IH: Leakage flux coil 2 1.16

7E' = <E∙_ℜ^-^G

>: Magnetizing flux coil 1 1.17

7 '= < ∙_ℜ^-^I

>: Magnetizing flux coil 2 1.18

ℜ = ^F_∙J: Reluctance. 1.19

: mean of the equivalent length of the magnetic path.

K: Cross-sectional area of the magnetic path.

: permeability

ℜEF: Reluctance of leakage path 1 ℜ F:Reluctance of leakage path 2

ℜ@: Reluctance of the path of the magnetizing fluxes

< ∙ 1: magnetomotive force (MFF). It represents the potential that a hypothetical magnetic charge would gain by completing the loop. Ampere-turns.

7E: Flux linking coil 1 7 : Flux linking coil 2 We can write:

7E = <E∙_ℜ^-^G

GH+ <E∙_ℜ^-^G

>+ < ∙_ℜ^-^I

> 1.20

7 = < ∙_ℜ^-^I

IH+ < ∙_ℜ^-^I

>+ <E∙_ℜ^-^G

> 1.21

The voltage equations can be expressed as:

/E= E1E+^AL_A,^G 1.22

/ = 1 +^AL_A,^I 1.23

Where

ΨE= 7E<E= <E ∙_ℜ^-^G

GH+ <E ∙_ℜ^-^G

>+ <E∙ < ∙_ℜ^-^I

> : Flux linkage in coil 1 1.24 Ψ = 7 < = < ∙_ℜ^-^I

IH+ < ∙_ℜ^-^I

>+ <E∙ < ∙_ℜ^-^G

> : Flux linkage in coil 2 1.25 N_EE= <_E ∙_ℜ^E

GH+ <_E ∙_ℜ^E

O= N_EF+ N_E': self-inductance of coil 1 1.26 N = < ∙_ℜ^E

IH+ < ∙_ℜ^E

O= N F+ N ': self-inductance of coil 2 1.27 N_E = <_E∙ < ∙_ℜ^E

> : Mutual inductance in coil 1 1.28

N E = < ∙ <E∙_ℜ^E

> : Mutual inductance in coil 1 1.29

NE =^P_P^I

G∙ NE'=^P_P^G

I∙ N ' 1.30

NEF: Leakage inductance in coil 1 N F: Leakage inductance in coil 2

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12 The inductance in matrixial form can be written as:

Q = RN^EE NE

N _E N S = T

NEF+ NE' ^P_P^I

G∙ NE' PG

PI∙ N ' N F+ N 'U 1.31

And the flux linkages:

ΨE= NEF∙ 1E+ NE'(1E+^P_P^I

G∙ 1 ) 1.32

Ψ = N F∙ 1 + N '(1 +^P_P^G

I∙ 1E) 1.33

And in matrixial form:

V = Q ∙ W 1.34

From Equations 1.22, 1.23, 1.31 and 1.34

/E= E1E+ NEE∙ ^-_X^G+NE ∙ ^-_X^I 1.35

/ = 1 + N ∙ ^-_X^I+N E∙ ^-_X^G 1.36

The obtained equivalent circuit is shown in Figure 1.5

Figure 1.5 Equivalent circuit for a single-phase transformer

We can select the coil 1 or coil 2 as a reference, then choosing coil 1 as a reference and moving equations from coil 2 we have:

1^Y =^P_P^I

G1 1.37

/^Y =^P_P^G

I/ , and 1.38

Ψ^Y =^P_P^G

IΨ 1.39

ΨE= NEF∙ 1E+ NE'(1E+ 1^Y) 1.40

Ψ^Y = N^Y_F ∙ 1^Y + N_E'(1_E+ 1^Y) 1.41

Y = Z^P_P^G

I[ 1.42

N_F^Y = Z^P_P^G

I[ NF 1.43

/E= E1E+^AL_A,^G 1.44

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13

/^Y = ^Y1^Y +^A\_A,^I^] 1.45

The new equivalent circuit referred to as coil 1 is shown in Figure 1.6 L’_2l r’₂ L_1l

r₁

i₁

v₁ L_1m

v’₂ i’₂

Figure 1.6 Transformer equivalent circuit referred to coil 1.

1.1.4.2. Effect of saturation

In the development of the above equations, we have assumed parameters being linear, but when the magnetic flux circulates through a ferromagnetic material such as iron, the magnetic flux density exceeds the maximum relative permeability ^{^} , where is the vacuum permeability. In such circumstances an increase in applied external magnetic field H cannot increase the magnetization of the material further, so the total magnetic flux density B more or less levels off. The magnetization flux density remains nearly constant despite of increment of H (proportional to the current circulating through the circuit where the magnetic field is produced) and is said to have saturated.

From Figure 1.6 and assuming no saturation, we can obtain the following equations:

_E= NEF∙ 1E+ _' 1.46

_^Y = N^Y_F ∙ 1^Y + _' 1.47

_'= NE'∙ (1E+ 1^Y) 1.48

In saturation _' is not linear but depends on the saturation characteristic of the material being used as the core of the magnetic circuit. We can define a new term `(_') associated with the saturation curve and _' will become:

_'= NE'∙ (1E+ 1^Y) − `(_') 1.49

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14

Ѱ

i

f(Ѱ^m)

L_1m(i₁+ i’₂) Ѱ^m

Figure 1.7 Typical Saturation curve³. 1.4.2. Overhead Transmission lines

Balanced single-circuit three-phase lines can be studied using the same equations used for

transformers, but in this case, we have flux interactions produced by three currents shifted 120º being the air the magnetic material where we will assume a permeability equal to ₀.

Figure 1.8 Model representing self and mutual impedances in a transposed transmission line.

Nc: self inductance N_': mutual inductance

It is customary to describe the voltage drop along a transmission line in the form of partial differential equations, e.g., for a single-phase line as:

−^Ad_Ae= f. 1 + N^A-_A, 1.50

To facilitate the analysis, the voltage drops can be expressed in the form of phasor equations for AC steady state conditions at a specific frequency where:

g_c= f_ee+ h2 ` ∙ N_c 1.51

g' = fei+ h2 ` ∙ N' 1.52

Where

fee = self-resistance of the respective phase (aa, bb, cc).

fei= mutual resistance between phases (ab, bc, ca).

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15 In the case of the three-phase balanced and transposed line in Figure 1.8, the impedance matrix becomes:

jkl = Tfmm + hnNmm fmo + hnNmo fm: + hnNm:

fmo + hnNmo foo + hnNoo fo: + hnNo:

fm: + hnNm: fo: + hnNo: f:: + hnN::U=pgc g' g'

g' gc g'

g' g' gc

q 1.53

If we assume symmetrical space between conductors or transposed lines, then:

fmm = foo = f:: m2 fmo = fo: = fm: 1.54

The voltages in the three phases can be calculated using the equation 1.55

jrstul = pgc g' g'

g' gc g'

g' g' gc

q jvstul 1.55

1.4.3. Rotating machines

The spire of Figure 1.3 produces an MFF that is a square wave, as we can observe in Figure 1.9

Figure 1.9 MFF produced by the spire of Figure 1.3

By adding more spires uniformly distributed, the MMF wave distribution can evolve from a square wave toward a sinusoid as we can see in Figure 1.10.

Figure 1.10 MMF waveform due to several spires⁴.

4Kundur (1994).

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16 Finally, with multiple spires uniformly distributed we can have each phase producing a sinusoidally distributed MMF wave in the three phases.

The MFF due the three phases shifted 120º can be described with the following equations:

??&(= w ∙ 1(∙ cos (3 ) 1.56

??&(= w ∙ 1(∙ cos (3 − 2 /3) 1.57

??&(= w ∙ 1(∙ cos (3 + 2 /3) 1.58

With balanced sinusoidal phase currents and time origin when 1( is maximum or minimum (reference):

1₍= _'∙ cos (|_cX) 1.59

1} = '∙ cos (|cX − 2 /3) 1.60

1} = '∙ cos (|cX − 2 /3) 1.61

Where

|c= 2 `: angular frequency of stator currents in electrical radian/s.

Figure 1.11 Evolution of the rotational magnetic field with multiple spires in each winding.

Having sinusoidal waveforms, the total MFF due to the three-phase is given by: (Kundur, 1994).

??&,~,= ??&(+ ??&}+ ??&. 1.62

= w ∙ _'∙ jcos(|_cX) ∙ cos(3 ) + cos •3 −2

3 € ∙ cos •|^cX −2

3 € + cos •3 +2

3 € ∙ cos •|^cX +2

??&,~,=^•w ∙ '∙ cos(3 − |cX) 1.63 3 €l

The MFF has a sinusoidal special distribution with a constant amplitude and a space-angle |cX whia ch is function of time. The MFF is continuously rotating at constant speed |c electrical radians/s called synchronous speed.

1.4.3.1. Complementary definitions

The synchronous speed is a multiple of the effective mechanical speed of the machine, and it depends on the number of poles p in the machine. Both are related as follows:

|c'=_‚|c: mechanical speed in radians/s.

Vendors usually give this value in rpm where:

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17 2c= ^ƒ |c'=^E_‚ `: rpm

We need to point out that for balanced operation the MMF wave to stator currents is stationary with respect to the rotor.

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18

Chapter II-TRANSFORMATIONS

2.1. Definition

Transformations are mathematical tools that facilitate the solution of complex problems in Engineering. In the case of the analysis of electromagnetic phenomena, one of the variables that introduce a great difficulty in the solution of the equations is the time variable. Some transformations allow creating an alternative space where time or its influence can be eliminated, facilitating the solution of complex equations. Other transformations allow us to analyze asymmetric phenomena as unbalanced currents, decomposing circuits into a set of new ones associated with balance currents that together represent the original one.

The simplest transformation is the one that converts a periodic sinusoid into a phasor

Figure 2.1 Phasor equivalent of a periodic sinusoid.

Table 2.1 Phasor representation.

Polar Form Rectangular form

Complex form Exponential form

Phasor form

|…|∠3 ‡ + hˆ |…|(:;0 3 + h012 3) |…|‰^Š‹ … or …

Being |…| = Œ‡ + ˆ where h = √−1

Other transformations can be classified as follows:

• Sequence components in a three-phase power system.

• Clarke and Park transformations.

• Modal Analysis transformation.

• Laplace transformation.

2

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19 These transformations have applications in the different components of the Power System (lines, transformers, generators and motors mainly).

2.2. Sequence components in a Three-Phase Power System

To facilitate the analysis in unbalanced faults, the three-phase power system can be broken down into several symmetrical sequence components (Dr. Charles Fortescue. 1918), and so, applying the superposition principle, several important relations can be found:

2.2.1. Symmetrical Components⁵

Since the angles between phasors revolving at the same rate are fixed, a set of three voltage or current phasors, •(, •}, •., and the components which are to replace them, can be represented in the same vector diagram with any current or voltage phasor revolving at the same rate, as reference phasor. A balanced and symmetrical set of three phasors have equal magnitude with phase shift of 120º.

VA

VC

VB N

Sequence a-b-c

Figure 2.2 Balanced and symmetrical three-phase phasors.

In any unbalanced or non-symmetrical part of the system, the symmetrical components methodology applies to any polyphase system (this methodology is based on “create three balanced networks”

representing together the unbalance network). Because of the widespread use of three-phase systems and the greater familiarity which electrical engineers have with them, they are used in practically all the applications related to short circuit calculations (the methodology used is now called the “method of symmetrical components”). Symmetrical component equations were developed by Dr. Charles Fortescue and reported in a paper in 1918.

2.2.1.1. Sequence components-fundamentals

Let us assume three phasors of an unsymmetrical but balanced system, “rotating counterclockwise”.

The sum of the three phasors is zero. When three or more phasors are in a balanced system, they can be represented in a polygon. See Figure 2.3.

Figure 2.3 A balanced system can be represented as a polygon.

Let´s now assume three phasors of an unsymmetrical and unbalanced system. The sum of the three phasofrom is different to zero. We can make a “trick” by creating an artificial vector •_•to build a polygon.

5Edith Clarke: “Circuit Analysis of AC Power Systems”, General Electric Series, 1961

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20 Vb Va

Vc Va

Vb Vc

Figure 2.4 An unbalanced system cannot be represented as a polygon.

0 V R

Figure 2.5 Artificial 3• phasor to balance the system “creating” a polygon.

Add them up, as in Figure 2.5, i.e.,

•(+ •}+ •.= •• 2.1

So, we see that,

•(+ •}+ •.− •• = 0 2.2

defining

• =^E_••_• 2.3

Then

•(+ •}+ •.− 3• = 0 2.4

(•(− • ) + (•}− • ) + (•.− • ) = 0 2.5

Defining

•′(= (•(− • )

•′} = (•}− • ) 2.6

•′. = (•.− • ) Then

•′(+ •′}+ •′.= 0 2.7

Conclusion: We obtain an unsymmetrical set of balanced voltages that sum zero by subtracting • from each original phasor, where • is 1/3 of the resultant phasor, illustrated in Figure 2.5.

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21 Figure 2.6 Subtracting V0 from unsymmetrical phasors.

Now, the system has become an unsymmetrical balanced system.

Figure 2.7 A balanced system where the sum of the three phasors is zero.

This balanced system can break down into two a-b-c symmetrical sets or in an a-b-c and in an a-c-b system, both symmetrical.

Figure 2.8 Adding 2 symmetrical sets.

The three a-b-c phasors will be called positive sequence and the a-c-b ones, negative sequence.

The sum of these unsymmetrical phasors is zero, since we just added two phasor sets that sum zero, i.e.,

•′(E+ •′}E+ •′.E= 0

•′( + •′} + •′. = 0 2.8

V₍^Y + V_}^Y + V_.^Y = 0

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22 Conclusion: In this example, we can represent any unsymmetrical set of 3 phasors that sum zero as the sum of 2 constituent symmetrical sets:

• A positive (a-b-c) sequence set.

• A negative (a-c-b) sequence set.

Extension: We can represent any unsymmetrical set of 3 phasors as the sum of 3 constituent sets, each having 3 phasors:

• A positive (a-b-c) sequence set.

• A negative (a-c-b) sequence set.

• An equal set.

Three sets we call respectively:

• Positive (•(E, •}E, •.E).

• Negative (•( , •} , •. ).

• Zero (•( , •_} , •_. ) sequence components.

The implication of this is that any unsymmetrical set of 3 phasors •(, •}, •., can be written in terms of the above sequence components in the following way:

•(= •(E+ •( + •(

•}= •}E+ •} + •( 2.8

•_. = •_.E+ •_. + •₍

Equations 2.16 can be standardized, but first, we must describe a mathematical operator that is essential.

2.2.1.2. The “a” operator

The operator “j” is familiar to us, since is used in complex numbers.

“j” is a vector with a magnitude and an angle equal to:

h = 1∠90° 2.9

In the same way, we are going to define the “a” operator as:

m = 1∠120° 2.10

And develop the following relationships:

m = 1∠ − 120° 2.11

m^•= 1∠0° 2.12

m^–= 1∠120° = m 2.13

We also have that:

1 + m = −m = 1∠60° 2.14

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23

1 1+a a

Figure 2.9 Illustration of 1+a.

Note that:

−m = −1∠240° = 1∠60° 2.15

Similarly, we may show that:

1 + m = −m = 1∠ − 60° 2.16

1 − m = √3∠ − 30° 2.17

1 − m = √3∠30° 2.18

m − 1 = √3∠150° 2.19

m + m = 1∠ − 180° 2.20

V0 V0 V0

V0 V0

V0

2.10 Summary of the sequence components.

2.2.2. Clarke and Park Components⁶

On 1943, Edith Clarke in his book “Circuit Analysis of AC Power Systems,” Volume I, Chapter X developed a system called α, β and 0 components. Initial ideas come from a paper written on 1929 by R.H. Park, “Two-Reaction Theory of Synchronous Machines,” where three-phase system AC

6Edith Clarke; “Circuit Analysis of A-C Power Systems, Volume I, Symmetrical and Related Components.” June 1943.

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24 quantities were reduced into two DC quantities shifted both 90º. These components were called 0.

The 0 transform is essentially an extension of the Clarke transform, applying an angle

transformation to convert from a stationary reference frame to a synchronously rotating frame. The synchronous reference frame can be aligned to rotate with the voltage (e.g., used in voltage source converters) or with the current (e.g., used in current source converters).

With phase a reference in a three-phase system, the α, β and 0 components of current and voltages are defined as follows:

• α components in phases b and c are equal; they are opposite in sign and of half the magnitude of α component of phase a.

• β component in phases b and c are equal in magnitude and opposite in sign; in phase a, they are zero.

• 0 components are identical in the three phases (similar to zero sequence components).

Components of current flow into a three-phase circuit in phase a, returning one-half in phase b and one-half in phase c, while β components are circulating currents in phases b and c. 0 components are zero sequence components taken over from symmetrical components without change, except in notation. E. Clarke decided to call 0 components for brevity and to indicate that they are to be used with α and β components.

2.2.2.1. Relations between Phase Currents and Voltages and their α, β and 0 Components The constant coefficients required to express a set of three vectors •(, •}, •. of a three-phase system in terms of their α, β and 0 components are as follows:

T•›

•œ

• U =_•

⎣⎢

⎢⎢

⎡1 −^E −^E

0 ^√• −^√•

E E E

⎦⎥

⎥⎥

⎤ p•(

•}

•.

q 2.21

Defining:

£ =_•

⎣⎢

⎢⎢

⎡1 −^E −^E

0 ^√• −^√•

E E E

⎦⎥

⎥⎥

⎤

2.22

We see that Equation 2.21 can be written as follows:

T•›

•œ

• U = £ p•(

•}

•_.q 2.23

We may also obtain the α, β and 0 quantities from the abc (phase) quantities:

p•(

•}

•.

q = £^¤¥T•›

•_œ

• U 2.24

Where

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25

£^¤¥=

⎣⎢

⎢⎡ 1 0 1

−^E ^√• 1

−^E −^√• 1⎦⎥⎥⎤

2.25

The corresponding equations for currents are as follows:

Tœ^›U = £ p⁽} .

q p ⁽} .

q = £^¤¥T^›œU 2.26

The above Clarke transformation preserves the amplitude of the electrical variables to which it is applied.

Having three-phase symmetrical currents

1(= √2•¦§∙ cos (3) 2.26

1} = √2•¦§∙ cos (3 − _•) 2.27

1. = √2•¦§∙ cos (3 + _•) 2.28

And applying T, we obtain

1_› = √2•¦§∙ cos (3) 2.29

1œ = √2•¦§∙ sin (3) 2.30

1 = 0 2.31

The Park transformation (named after Robert H. Park), where the rotor is the new reference converts vectors from the abc reference frame (stator) to the dq0 reference frame (rotor), to take advantage of DC parameters when vectors from stator reference are converted to rotor reference, that simplifies the calculation during dynamic studies. The primary value of the Park transform is to rotate the reference frame of a vector at an arbitrary frequency. The Park transform shifts the frequency spectrum of the signal such that the arbitrary frequency now appeit’s as "DC" and the old DC appears as the negative of the arbitrary frequency. The Park transformation matrix is as follows:

ª« = pcos (3) sin (3) 0

−012(3) cos(3) 0

0 0 1q 2.32

The “normalized” Park transformation ¬ is the Clarke and Park transforms combined:

Original Park transformation: ¬ = ª« ∙ £ =_•

⎣⎢

⎢⎢

⎡cos (3) cos (3 − •) cos (3 + _•) 012(3) sin (3 − _•) sin (3 + _•)

E E E

⎦⎥

⎥⎥

⎤

And its inverse: ¬^¤¥ =_•·

cos (3) 012(3) 1

cos (3 − _•) sin (3 − _•) 1 cos (3 + _•) sin (3 + _•) 1

¸

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26

£ and ¬ are not unitary because £^¹ ≠ £^¤¥ and ¬^¹ ≠ ¬^¤¥; because that, the transformation has not the power invariant characteristic as we can see below:

»(}. = /(1(+ /}1}+ /.1.= p/(

/}

/.

q

,

p1(

1}

1.

q = T¬^¤¥p/A

/¼

/½

qU ·¬^¤¥p1A

1¼

1½

q¸ 2.33

»(}. = j/^A /¼ /½l¾¬^¤¥¿^¹¬^¤¥p1A

1¼

1_½q 2.34

¾¬^¤¥¿^¹¬^¤¥ =

⎣⎢

⎢⎢

⎡^• 0 0

0 ^• 0

0 0 ^E_•⎦⎥⎥⎥⎤ , then:

»_(}. = j/^A /¼ /½l

⎣⎢

⎢⎢

⎡^• 0 0

0 ^• 0

0 0 ^E_•⎦⎥⎥⎥⎤ p1A

1¼

1½

q =^• (/_A1_A+ /_¼1_¼) +^E_•/_½1_½ 2.35

»(}.= /(1(+ /}1}+ /.1. =^• (/A1A+ /¼1¼) +^E_•/½1½⁷, using the original Park transformation.

The 3/2 factor comes from the constant used in the transformation. Note that the q and d voltages, currents, flux linkages, and electric charges are dependent upon the angular velocity of the frame of referethe nce, being the total power independent of the frame of reference.

For Power invariant transformation, T unitary becomes⁸:

£ = Á_•

⎣⎢

⎢⎢

⎡ 1 −^E −^E

0 ^√• −^√•

E

√ E

√ ⎦⎥⎥⎥⎤

2.36

Using per unit values, the “normalized” unitary power invariant Park transformation ¬ becomes:

¬ = pcos (3) sin (3) 0

−012(3) cos(3) 0

0 0 1q ∙ Á_•

⎣⎢

⎢⎢

⎡ 1 −^E −^E

0 ^√• −^√•

E

√ E

√ ⎦⎥⎥⎥⎤

2.37

¬ = Á_•

⎣⎢

⎢⎢

⎡ cos (3) cos (3 − _•) cos (3 + _•)

−012(3) −sin (3 − _•) −sin (3 + _•)

E

√

E

√

E

√ ⎦⎥⎥⎥⎤

9 2.38

7/½= 3/ and 1½= 31 , where / and 1 are zero sequence components.

8 W.A. Lewis, “A Basic Analysis of Synchronous Machine – Part I,” AIEE Trans, Vol. 77, pp. 436-456, 1958.

9The original Park transformation is not a power invariant transformation and does not result in a d-q-0 reciprocal (symmetric) inductance matrix. Equation A.28 facilitates solution of the normalized equations allowing symmetry of the inductance d-q-0 matrix. The modified zero sequence component would appear to be obtained applying a scale factor of Œ1/3) to its power invariant value. Despite of this advantages, there are several discussions on the fact that original Park transformation reflects more closely the physical features of the machine (Kundur 1994).

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27 Or when d-axis leads the q-axis:

¬ = Á_•

⎣⎢

⎢⎢

⎡cos (3) cos (3 − •) cos (3 + _•) 012(3) sin (3 − _•) sin (3 + _•)

E

√

E

√

E

√ ⎦⎥⎥⎥⎤

2.39

Figure 2.11 Axis d-q-0 referenced to phase a.

Where θ is the instantaneous angle of an arbitrary ω frequency. We will assume a rotating reference frame (rotor) rotating with the synchronous speed nc which will be along the axis of phase a at t=0. If θ is the angle by which the rotor direct axis is ahead of the magnetic axis of phase a, then

3 = ncX + Ã + rad 2.40

Being Ã the synchronous torque angle (rotor angle); value dictated by the power generated.

The main field-winding flux is along the direction of the d axis of the rotor. It produces an efm that lags this flux by 90º. Therefore, the machine efm E is primarily along the rotor q axis.

Being the inverse:

¬^¤¥= ¬^¹ = Á_•

⎣⎢

⎢⎢

⎡ cos (3) sin (3) _√Ê cos (3 − _•) sin (3 − _•) _√Ê cos (3 + _•) sin (3 + _•) _√Ê⎦⎥⎥⎥⎤

2.41

finally, we may write:

W ÄÅ= ¬ ∙ Wstu and Wstu= ¬^¤¥∙ W ÄÅ 2.42

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28 We note that ¬^¤¥ = ¬^¹, which means that the transformation in per-unit values is orthogonal, and also, power invariant, and we should expect to use the same power expression in either the a-b-c or the d-q- 0 frame of reference

Æ = /₍1₍+ /_}1_}+ /_.1_. = !_stu^¹ W_stu= (¬^¤¥! _ÄÅ)^¹(¬^¤¥W _ÄÅ) = !^¹_ÄÅ(¬^¤¥)^¹¬^¤¥W _ÄÅ 2.43 Æ = !^¹_ÄÅ¬¬^¤¥W ÄÅ= !^¹_ÄÅW ÄÅ= /A1A+ /¼1¼+ / 1 2.44 We can write similar expressions for voltages and flux linkages (Ψ = <7 :;0 n X)

! ÄÅ= ¬!stu , V ÄÅ = ¬Vstu 2.45

In the case of generators, we need to consider three particularities:

1. The transformation from the three-phase windings (N) to a system with two-phase windings (N’). Various authors select the ratio N/N’ equal to 2/3, but majority (as Fitzgerald et al., 1992;

Kundur, 1994; Krause et al., 2002) use Œ2 3⁄ . As this is the most used widely, we will consider it as the valid ratio.

2. The active and reactive powers computed in Clarke´s domain with the transformation shown above are not the same as those computed in the standard reference frame. This is because T is not unitary.

3. When a synchronously reference frame is used (to facilitate the analysis in synchronous machines), both stator and rotor fluxes as seen as stationary. An additional transformation (Park) is needed where the rotor is the new reference to transfer three-phase stator and rotor quantities into a single rotating reference frame to eliminate the effect of time-varying inductances and transform the system into a linear time-invariant system¹⁰.

Figure 2.12 Transformation from a three-phase system to a bi-phase system.

The most common representation that facilitates the visualization of parameters analysis is using the d-axis ahead of the q-axis. This changes the sign of the second row in Equation 2.38:

10Wikipedia

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29 Figure 2.13 d-axis ahead of q-axis

With the currents in the stator

1(= √2•¦§∙ cos (|cX) 2.46

1} = √2•¦§∙ cos (|cX − _•) 2.47

1. = √2•¦§∙ cos (|cX + _•) 2.48

Applying the d-q transformation, we have:

T^A¼U =2 3

⎣⎢

⎢⎢

⎢⎡cos (3) cos (3 −2

3 ) cos (3 +2 3 ) 012(3) sin (3 −2

3 ) sin (3 +2 1 3 )

√2

1

√2

1

√2 ⎦⎥⎥⎥⎥⎤ p⁽}

.

q

1_A= √2•¦§∙ cos (|_cX) 2.49

1¼ = −√2•¦§∙ sin (|cX) 2.50

1 = 0 2.51

The inverse transformation

p⁽} .

q =2 3

⎣⎢

⎢⎢

⎡ cos (3) 012(3) 1

cos (3 −2

3 ) sin (3 −2 3 ) 1 cos (3 +2

3 ) sin (3 +2 3 ) 1⎦⎥⎥⎥⎤

T^A¼U

1₍= _A∙ cos(|_cX) + _¼012(|_cX) + 2.52

1} = A∙ cos Z|cX − _•[ + ¼012 Z|cX − _•[ + 2.53

1_. = _A∙ cos Z|_cX + _•[ + _¼012 Z|_cX + _•[ + 2.54

In balanced systems = 0

The current relationships taking into consideration the current equivalent from the three-phase system to bi-phase system: 3 È/2 (peak value):

N′1A=^•<√2 •¦§∙ cos (|cX) 2.55

<^Y1_¼ = −^•<√2 •¦§∙ sin (|_cX) 2.56

1 = 0 2.57

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30 Using _P^P_]= Œ2 3⁄ :

1A= √3•¦§∙ cos (|cX) 2.58

1¼ = −√3•¦§∙ sin (|cX) 2.59

1 = 0 2.60

With currents

1(= √2•¦§∙ sin (|cX) 2.61

1_} = √2•¦§∙ sin (|_cX − _•) 2.62

1. = √2•¦§∙ sin (|cX + _•) 2.63

We will obtain:

1A= √3•¦§∙ sin (|cX) 2.64

1¼ = √3•¦§∙ cos (|cX) 2.65

1 = 0 2.66

2.2.3. Transformation for modal analysis

Propagation of High-frequency signals as traveling waves (TW) occurs in the other conductors, even though only one conductor is energized. Modal analysis is a mathematical tool, similar to the

symmetrical components technique used for analysing unbalanced faults on three-phase (50 or 60 Hz) power systems. Like Clarke components (simile is closer than with symmetrical components), modal analysis is a mathematical tool whose components or “modes” can be generated and measured separately. The modal theory is based on the premise that there are as many independent modes of propagation on multiconductor circuits, as there are conductors involved in the propagation of energy¹¹.

There are five characteristics of natural modes in a three-phase line:

a) Any set of phase-conductor currents or voltages existing at any point on a lossy, reflection-free three-phase line can be resolved into three sets of natural-mode components.

b) At any point on a line, the mode components must add up to the actual phase quantities. Also, the total power derived from phase currents and voltages must be equal to the sum of mode powers c) The ratio of mode voltage to mode current (the mode “characteristic impedance”) is constant on

each phase conductor.

d) Each mode propagates with a specific attenuation per unit length and a specific velocity of propagation.

e) A set of phase components corresponding to one mode only cannot be resolved into other modes.

The modes are independent, and there is no inter-mode coupling on a uniform line.

There are three modes of propagation:

1. Mode 1 is a high attenuation mode that is propagated on all three phases with the ground return.

2. Mode 2 is a medium-attenuation mode that is propagated on one outside phase and returns on the other outside phase. There is no mode current in the center phase.

3. Mode 3 is the least attenuated of the three modes. The energy is propagated on the two outer phases and returns to the center phase.

11Applied Protective Relaying. ®Westinghouse Electric Corporation, 1976.

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31 To use matrix algebra in determining modal quantities, the following assumptions are made:

a) Line is transposed, so the impedance matrix is symmetrical.

b) No effect of frequency.

c) Instantaneous currents (phase or modal quantities) in the three-phase are either in phase or 180º out of phase.

Figure 2.14 Simplified presentation of the propagation modes.

From Figure 2.14, we can define a transformation matrix, as in the case of Clarke components, but for propagation modes:

8 = p1 1 1

1 0 −2

1 −1 1 q and 8^¤¥=^E_•p 1 1 1 3/2 0 −3/2

1/2 −1 1/2q 2.67

We can apply this transformation to a special case with balanced impedances, i.e., the impedance matrices kstu are of the form:

k = k_stu= pgc g' g'

g_' g_c g_' g' g' gc

q 2.68

where

gc= Self-impedance of each conductor (assumed equal in the three conductors) g_' = Mutual impedance (three-phase line transposed) is equal between conductors.

The matrix for modal impedances is the same as the matrix impedances obtained with other transformations as the Clarke or symmetrical components ones.

kÉ= pgc+ 2g' 0 0

0 gc− g' 0

0 0 gc− g'

q 2.69

2.2.4. Laplace Transformation

The Laplace transformation, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of the algebra. After the algebraic equation has been combined (added, multiplied) with other algebraic equations, the result to a new differential equation can then be solved by applying the inverse Laplace transformation.

ℒË`(X)Ì = Í `(X)‰^Î ^¤c, X = &(0) 2.70

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32 Figure 2.15 Time domain and frequency domain transformation⁶

This transformation has several properties, some of them are as follows¹²: a) Linearity

ℒËm`(X) + oÏ(X)Ì = mℒË`(X)Ì + +oËÏ(X)Ì = m&(0) + oÐ(0) 2.71 ℒ^¤EËm&(0) + oÐ(0)Ì = mℒ^¤EË&(0)Ì + oℒ^¤EËÐ(0)Ì = m`(X) + oÏ(X) 2.72

b) Derivatives of `(X)

ℒË`′(X)Ì = 0ℒË`(X)Ì − `(0) = 0&(0) − `(0) 2.73

⋮ ℒÒ`^(*)(X)Ó = 0^(*)&(0) − 0^(*¤E)`(0) − 0^{(*¤ )}`^Y(0) − ⋯ − `^(*¤E)(0) 2.74

c) Derivatives of F(s)

ℒ^¤EË&′(0)Ì = −X`(X) 2.75

⋮ ℒ^¤EÒ&^(*)(0)Ó = (−1)^*X^*`(X) 2.76

d) Integrals of `(X)

ℒ ÕÍ `(Ö) Ö^, × =^E_c&(0) 2.77

⋮

ℒ ÕÍ … Í `(Ö)( Ö)^, ^, ^*× =_c^EÙ&(0) 2.78

e) Integrals of F(s)

ℒ^¤EÒÍ &(Ö) Ö^Î Ó =^E_,`(X) 2.79

⋮ ℒ^¤EÒÍ … Í &(Ö)( Ö)^Î ^Î ^*Ó =_,^E_Ù`(X) 2.80

12https://www.efunda.com/math/laplace_transform/rules.cfm

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33 f) Convolution

ℒ^¤EË&(0)Ð(0)Ì = Í `(Ö)Ï(X − Ö) Ö = `(X) ∙ Ï(X)^, 2.81 Some examples:

i. Constant function `(X) = 1; the Laplace transformation is ^E_c. ii. Linear function `(X) = X; the Laplace transformation is _c^EI

iii. The power function `(X) = X^*; the Laplace transformation is _c_ÙÛG^*!

iv. We can express phasor of Figure 2.1 in exponential form |…|‰^Š‹ with C=1, then:

‰^Š‹= :;03 + h0123 2.82

Equation 2.82 is also known the Euler´s formula¹³

If we have a function `(X) = sin(mX), using Equation 6.36 we obtain:

sin(mX) =⁺^ÜÝÞ^¤+_Š^ßÜÝÞ 2.83

The Laplace transformation of sin(mX) then becomes:

ℒËsin (mX)Ì = ^E_ŠÍ $‰^Î ^Š(,− ‰^¤Š(,%‰^¤c, X =₍_I⁽_àc_I 2.84

ℒËcos (mX)Ì =₍_I^c_àc_I (ℝ‰(0) > 0) 2.85

In a similar way and using Equation 2.70 and the properties, we can obtain the Laplace transformation of other functions.

13 Kim Thibault https://mathvault.ca/laplace-transform/

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34

Chapter III-SYNCHRONOUS MACHINE PARAMETERS’

DERIVATION. PART 1

3.1. Synchronous Machine Derivation

Synchronous Generators or Alternators are used to convert mechanical power derived from steam, gas, hydraulic-turbine, or wind to AC electric power. Largest single-unit electrical machines in production (up to 2000 MVA). Synchronous generators are the primary source of electrical energy we consume today, and large AC power networks rely almost exclusively on synchronous generators.

A conductor winding, through which a DC current circulates continuously, mechanically propelled by an external element, moves across producing a rotating electro-magnetic field. This electromagnetic field interacts with the stator winding (named sometimes alternator), and in this way, sinusoidal voltage (AC) is produced across the terminals of the induced circuit (Stator). See Figure 3.1.

The rotor magnetic field may be produced by induction (in a "brushless" generator), by permanent magnets (usually in very small machines), or by a rotor winding energized with direct current through slip rings and brushes. Automotive alternators invariably use brushes and slip rings, which allow control of the alternator-generated voltage by varying the current in the rotor field winding.

Permanent magnet machines avoid the loss due to magnetizing current in the rotor but are restricted in size owing to the cost of the magnet material. Since the permanent magnet field is constant, the terminal voltage varies directly with the speed of the generator. Brushless AC generators are usually larger machines than those used in automotive applications.

Generators, due to their construction and to the needs of the industry itself, generally produce three- phase voltage. We will therefore limit ourselves to an analysis of three-phase circuits, fundamentally in the operating frequency of 50 and 60 Hz.

A rotating generator has two basic components:

• Armature or Stator.

• Field circuit or Rotor.

a) Stator winding b) Rotor Winding

Figure 3.1 Basic Parts of a Synchronous Motor (Source: Wikipedia).

Electrical Machines Basic Theory