# Chapter 9. Glossary of SYMBOLS

${a}_{s}$magnetical axis of phase a in the stator

aphase a in a three-phase system

arotational unit vector of 120°

${\underset{_}{a}}^{2}$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

$d\lambda$arbitrarily rotating of field-oriented direct axis

$d{\lambda }_{m}$air-gap field-oriented direct axis

$d{\lambda }_{r}$resultant rotor field-oriented direct axis

$d{\lambda }_{s}$resultant stator field-oriented direct axis

$d\theta$rotor-oriented direct axis

$\underset{_}{e}$space phasor of the electromotive force

${f}_{s}$stator voltage and stator current frequency

$d{\epsilon }_{s}$stator current oriented direct axis

$\underset{_}{i}$space vector or space phasor of a current

${i}_{a}$armature current in a DC machine

${i}_{A}$active component of the current space phasor

${i}_{d}$DC link current

${i}_{m}$magnetizing current

${i}_{mr}$rotor-flux based magnetizing current

${i}_{ms}$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

$i{}_{r}$rotor current

${i}_{R}$reactive component of the current space phasor

${i}_{s}$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

${k}_{i}$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

${K}_{F}$space phasor definition (constant) coefficient

${K}_{I}$referring constant of the currents and magneto motoric force

${K}_{M}$torque constant of an AC machine

${K}_{Mr}$torque constant of the induction machine containing the rotor leakage coefficient

${K}_{Ms}$torque constant of the induction machine containing the stator leakage coefficient

$\left[{K}_{p}\right]$power coefficient matrix

${K}_{R}$referring constant of the resistances and inductances

${K}_{SC}$scale coefficient of the space phasor

${K}_{u}$reffering constant of the voltages and electro motoric force

${l}_{m}$inductance of the main magnetic circuit

${l}_{r}$useful inductance of the rotor-phase winding

${l}_{rr}$total per-phase inductance of the rotor

${l}_{s}$useful inductance of the stator-phase winding

${l}_{ss}$total per-phase inductance of the stator

${l}_{\sigma r}$rotor’s own per-phase leakage inductance

${l}_{\sigma s}$stator’s own per-phase leakage inductance

${L}_{B}$inductance of the thyristor commutation coil

${L}_{F}$DC link current filtering inductance

${L}_{m}$three-phase resultant mutual (useful) inductance

${L}_{r}$three-phase resultant (total) rotor inductance

${L}_{ro}$zero-sequence inductance of the rotor

${L}_{s}$three-phase resultant (total) stator inductance

${L}_{so}$zero-sequence inductance of the stator

${L}_{\sigma r}$resultant leakage inductance of the rotor

${L}_{\sigma s}$resultant leakage inductance of the stator

mmutual inductance between a stator and a rotor phase

${m}_{e}$electromagnetic torque

${m}_{j}$dynamic torque

${m}_{r}$resistent load torque

${m}_{\sigma r}$mutual leakage inductance between two rotor phases

${m}_{\sigma s}$mutual leakage inductance between two stator phases

${N}_{r}$number of rotor phases

${N}_{s}$number of stator phases

[q]matrix of the oscillator output quantities

pinstantaneous power or power losses

${p}_{m}$instantaneous mechanical power

${p}_{roj}$projection of a space vector or space phasor

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

$q{\in }_{s}$stator-current oriented quadrature axis

$q\lambda$arbitrarily rotating or field-oriented quadrature axis

$q{\lambda }_{m}$air-gap field-oriented quadrature axis

$q{\lambda }_{r}$resultant rotor field-oriented quadrature axis

$q{\lambda }_{s}$resultant stator field-oriented quadrature axis

$q\theta$rotor-oriented quadrature axis

Rereal axis of the complex plane

Rereal part of a complex expression

${R}_{D}$resistance of diode

${R}_{r}$rotor per-phase resistance

${R}_{S}$stator per-phase resistance

${R}_{T}$resistance of thyristor or transistor

ttime

[t]three-phase projection matrix

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

$\underset{_}{u}$space phasor of voltage

$\Delta u$voltage drop

${u}_{d}$DC link output/inverter input voltage

${u}_{r}$rotor voltage

${u}_{red}$rectifier output/DC link input voltage

${u}_{s}$stator voltage

Nturn number of winding

${z}_{p}$number of the poli-pairs

${\epsilon }_{m}$angular position of the magnetizing-current space phasor

${\epsilon }_{r}$angular position of the rotor-current space phasor

${\epsilon }_{s}$angular position of the stator-current space phasor

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

${\lambda }_{m}$angular position of the air-gap resultant flux space phasor

${\lambda }_{r}$angular position of the rotor resultant flux space phasor

${\lambda }_{s}$angular position of the stator resultant flux space phasor

$\theta$electrical angular position of the rotor ${a}_{r}$ axis

${\theta }_{r}$mechanical angular position of the rotor ${a}_{r}$ axis

$\sigma$resultant (total) leakage coefficient

${\sigma }_{r}$rotor leakage coefficient

${\sigma }_{s}$stator leakage coefficient

$\underset{_}{\psi }$space vector or space phasor of flux linkage

${\underset{_}{\psi }}_{a}$space vector of armature reaction flux in DC machine

${\underset{_}{\psi }}_{c}$space vector of compensating flux in DC machine

${\underset{_}{\psi }}_{e}$space vector of exciting flux

${\psi }_{m}$air-gap flux

${\psi }_{m}$resultant rotor flux

${\psi }_{s}$resultant stator flux

${\psi }_{\sigma r}$rotor leakage flux

$\left[\tau \right]$rotational matrix of three-phase axis system, when the zero-sequence component is missing

${\tau }_{r}$time constant of the rotor

${\tau }_{s}$time constant of the stator

$\omega$electrical angular speed of the rotor

${\omega }_{o}$synchronous electrical angular speed

${\omega }_{r}$mechanical angular speed of the rotor

${\omega }_{{\epsilon }_{m}}$rotational speed of the magnetizing-current space phasor

${\omega }_{{\epsilon }_{r}}$rotational speed of the rotor-current space phasor

${\omega }_{{\epsilon }_{s}}$rotational speed of the stator-current space phasor

${\omega }_{\lambda }$angular speed of an arbitrarity rotating axis system or of a field-oriented axis

${\omega }_{{\lambda }_{m}}$rotational speed of the air-gap flux space phasor

${\omega }_{{\lambda }_{r}}$rotational speed of the resultant rotor flux space phasor

${\omega }_{\lambda }$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 ${i}_{mR}$ ImR Stator voltage ${u}_{S}$ US Rotor flux ${\Psi }_{R}$ Psi_R Rotor, rotor flux and slip angular velocity , omega_Rotor, omega_Flux, omega_Slip Reference speed ${\omega }_{R}^{ref}$ Omega_ref Rotor, rotor flux and slip angle ${\epsilon }_{R}$, ${\epsilon }_{Flux}$, ${\epsilon }_{Slip}$ eps_Rotor,eps _Flux eps_Slip Load torque ${m}_{L}$ torque_L Stator and rotor resistance and Time constants $R$, ${R}_{R}$ ${T}_{S}$, ${T}_{R}$ Rs, Rr, Ts, Tr, Stator, rotor and main inductance $L$, ${L}_{R}$, ${L}_{m}$ Ls, Lr, Lm Leakage factor $\sigma$ Sigma Number of poles $p$ P Moment of inertia $J$ J System noise variance ${R}_{\mathrm{w}}$ - Output/Measurement noise variance ${R}_{v}$ - Kalman gain K K Estimated motor state variable ${\stackrel{^}{i}}_{\alpha },{\stackrel{^}{i}}_{\beta }$ ${\stackrel{^}{\Psi }}_{R\alpha },{\stackrel{^}{\Psi }}_{R\beta }$ ${\stackrel{^}{i}}_{d},{\stackrel{^}{i}}_{q}$ ${\stackrel{^}{R}}_{r}$ 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

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