# Chapter 9. Glossary of SYMBOLS

Table of Contents

magnetical axis of phase a in the stator

aphase a in a three-phase system

arotational unit vector of 120°

rotational unit vector of 240°

[A]phase transformation matrix

bphase b in a three-phase system

cphase c n a three-phase system

Ccondenser capacity

Dfixed of stator-oriented direct axis

Ddiode

[D]rotational matrix operator of a two-phase axis system

arbitrarily rotating of field-oriented direct axis

air-gap field-oriented direct axis

resultant rotor field-oriented direct axis

resultant stator field-oriented direct axis

rotor-oriented direct axis

space phasor of the electromotive force

stator voltage and stator current frequency

stator current oriented direct axis

space vector or space phasor of a current

armature current in a DC machine

active component of the current space phasor

DC link current

magnetizing current

rotor-flux based magnetizing current

stator-flux based magnetizing current

Rereal axis of the complex plane and real part of the complex expression

Imimaginary axis of the complex plane and imaginary part of the complex expression

rotor current

reactive component of the current space phasor

stator current

jrotational unit vector of 90°

[j]rotational matrix operator of 90° for a two-phase system of magnitudes without zero-sequence component

Jtotal equivalent moment of inertia

proportional coefficient between the magnitude of the stator current space phasor and the d.c. link current

Ktorque constant of a DC machine

[K]rotational matrix operator of 90° for a three-phase system of magnitudes

space phasor definition (constant) coefficient

referring constant of the currents and magneto motoric force

torque constant of an AC machine

torque constant of the induction machine containing the rotor leakage coefficient

torque constant of the induction machine containing the stator leakage coefficient

power coefficient matrix

referring constant of the resistances and inductances

scale coefficient of the space phasor

reffering constant of the voltages and electro motoric force

inductance of the main magnetic circuit

useful inductance of the rotor-phase winding

total per-phase inductance of the rotor

useful inductance of the stator-phase winding

total per-phase inductance of the stator

rotor’s own per-phase leakage inductance

stator’s own per-phase leakage inductance

inductance of the thyristor commutation coil

DC link current filtering inductance

three-phase resultant mutual (useful) inductance

three-phase resultant (total) rotor inductance

zero-sequence inductance of the rotor

three-phase resultant (total) stator inductance

zero-sequence inductance of the stator

resultant leakage inductance of the rotor

resultant leakage inductance of the stator

mmutual inductance between a stator and a rotor phase

electromagnetic torque

dynamic torque

resistent load torque

mutual leakage inductance between two rotor phases

mutual leakage inductance between two stator phases

number of rotor phases

number of stator phases

[q]matrix of the oscillator output quantities

pinstantaneous power or power losses

instantaneous mechanical power

projection of a space vector or space phasor

qfixed or stator-oriented quadrature axis

[Q]rotational matrix of 90° for a two-phase system of magnitudes including the zero-sequence component

stator-current oriented quadrature axis

arbitrarily rotating or field-oriented quadrature axis

air-gap field-oriented quadrature axis

resultant rotor field-oriented quadrature axis

resultant stator field-oriented quadrature axis

rotor-oriented quadrature axis

Rereal axis of the complex plane

Rereal part of a complex expression

resistance of diode

rotor per-phase resistance

stator per-phase resistance

resistance of thyristor or transistor

ttime

[t]three-phase projection matrix

[T]rotational matrix operator of three-phase axis system

space phasor of voltage

voltage drop

DC link output/inverter input voltage

rotor voltage

rectifier output/DC link input voltage

stator voltage

Nturn number of winding

number of the poli-pairs

angular position of the magnetizing-current space phasor

angular position of the rotor-current space phasor

angular position of the stator-current space phasor

angular position of an arbitrarily rotating direct axis or of a field-oriented axis

angular position of the air-gap resultant flux space phasor

angular position of the rotor resultant flux space phasor

angular position of the stator resultant flux space phasor

electrical angular position of the rotor axis

mechanical angular position of the rotor axis

resultant (total) leakage coefficient

rotor leakage coefficient

stator leakage coefficient

space vector or space phasor of flux linkage

space vector of armature reaction flux in DC machine

space vector of compensating flux in DC machine

space vector of exciting flux

air-gap flux

resultant rotor flux

resultant stator flux

rotor leakage flux

rotational matrix of three-phase axis system, when the zero-sequence component is missing

time constant of the rotor

time constant of the stator

electrical angular speed of the rotor

synchronous electrical angular speed

mechanical angular speed of the rotor

rotational speed of the magnetizing-current space phasor

rotational speed of the rotor-current space phasor

rotational speed of the stator-current space phasor

angular speed of an arbitrarity rotating axis system or of a field-oriented axis

rotational speed of the air-gap flux space phasor

rotational speed of the resultant rotor flux space phasor

rotational speed of the resultant stator flux space phasor

 NAME Symbols within the text Symbols within the formulae describing implementations and simulations Stator and rotor currents iS, iR Stator current three-phase co-ordinates iS_a, iS_b, iS_c two-phase co-ordinates iS_alpha, iS_beta field co-ordinates iS_d, iS_q Magnetising current ImR Stator voltage US Rotor flux Psi_R Rotor, rotor flux and slip angular velocity , omega_Rotor, omega_Flux, omega_Slip Reference speed Omega_ref Rotor, rotor flux and slip angle , , eps_Rotor,eps _Flux eps_Slip Load torque torque_L Stator and rotor resistance and Time constants , , Rs, Rr, Ts, Tr, Stator, rotor and main inductance , , Ls, Lr, Lm Leakage factor Sigma Number of poles P Moment of inertia J System noise variance - Output/Measurement noise variance - Kalman gain K K Estimated motor state variable i_alpha_est; i_beta_est Psi_R_alpha_est; Psi_R_beta_est i_d_est; i_q_est -

## 9.1. Mathematical Symbols

• cross vectorial product

• complex conjugate

• element of (contained in)

• not element of (not contained in )

• null set

• implication

• biimplication

• negation (complement)

• identity

• equivalence

• union (disjunction)

• OR

• intersection (conjunction)

• AND

• ordered pair

• Cartesian product

• fuzzy intersection operator (fuzzy AND), T-norm

• fuzzy union operator (fuzzy OR), S-norm, T-conorm

• subset of (A’ÍB’ means that A’ is subset of B’)

• proper subset of (A’ÌB’ means that A’ is proper subset of B’)

• not proper subset of

Ttranspose

Xuniverse of discourse (sample space)

ççx - yúú Euclidean distance between vectors x and y

mmembership function, MF (degree of belongingness)

x1, x2, …symbol 125 \f "Symbol" \s 10crisp set of numbers

## 9.2. General abbreviations

AI artificial intelligence

AIB artificial-intelligence-based

ANN artificial neural network

ASM asynchronous motor

CSI current source inverter

DSP digital signal processor

DTC direct torque control

EKF extended Kalman Filter

ELO extended Luenberger Observer

FLC fuzzy logic controller

FLEOC fuzzy logic efficiency optimizer controller

FLSfuzzy logic system

GA genetic algorithm

IM induction machine

KFKalman Filter

LO Luenberger Observer

MRAC model reference adaptive control

NRAS model reference adaptive system

NARMAX non-linear auto regressive moving average model

p.m. permanent magnet

PWM pulse-width modulator/modulation

SRM switched reluctance machine

SYRM synchronous reluctance machine

TDL tapped delay line

VSI voltage source inverter

## 9.3. Abbreviations for fuzzy sets

NL negative large

NM negative medium

NS negative small

ZE zero

PS positive small

PM positive medium

PL positive large