$${a}_{s}$$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}$$projection of a space vector or space phasor
qfixed or stator-oriented quadrature axis
[Q]rotational matrix of 90° for a two-phase system of magnitudes including the zero-sequence component
$$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}$$rotational speed of the resultant stator flux space phasor
NAME |
Symbols within the text |
Symbols within the formulae describing implementations and simulations |
Stator and rotor currents |
$$\overline{i},\text{}{\overline{i}}_{R}$$ |
iS, iR |
Stator current three-phase co-ordinates |
$${i}_{a},\text{}{i}_{b},\text{}{i}_{c}$$ |
iS_a, iS_b, iS_c |
two-phase co-ordinates |
$${i}_{\alpha},\text{}{i}_{\beta}$$ |
iS_alpha, iS_beta |
field co-ordinates |
$${i}_{d},\text{}{i}_{q}$$ |
iS_d, iS_q |
Magnetising current |
$${i}_{mR}$$ |
ImR |
Stator voltage |
$${u}_{S}$$ |
US |
Rotor flux |
$${\Psi}_{R}$$ |
Psi_R |
Rotor, rotor flux and slip angular velocity |
$${\omega}_{r},\text{}{\omega}_{Flux},\text{}{\omega}_{Slip}$$, |
omega_Rotor, omega_Flux, omega_Slip |
Reference speed |
$${\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}$$ |
- |
Kalman gain |
K |
K |
Estimated motor state variable |
$${\widehat{i}}_{\alpha},{\widehat{i}}_{\beta}$$ $${\widehat{\Psi}}_{R\alpha},{\widehat{\Psi}}_{R\beta}$$ $${\widehat{i}}_{d},{\widehat{i}}_{q}$$ $${\widehat{R}}_{r}$$ |
i_alpha_est; i_beta_est Psi_R_alpha_est; Psi_R_beta_est i_d_est; i_q_est - |
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
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
NL negative large
NM negative medium
NS negative small
ZE zero
PS positive small
PM positive medium
PL positive large