INTRODUCTION
With lots of advantages, such as high torque density, small size and low maintenance
cost, the Permanent Magnet Synchronous Motors (PMSMs) are widely used in industrial
and domestic fields (Liu et al., 2009; Zaher,
2008; Yan et al., 2008). Performance improvement
and cost reduction of the motor control system have been always hot topics because
PMSM is a high order, nonlinear, strong coupling and timedependent system.
The main modern motor control theory and technology can be divided into two
categories: vector control and direct torque control. Current, voltage and magnetic
chain are regarded as vectors which provides a very convenient approach to motor
analysis and control. Many fundamental and improved algorithms of vector control
and direct torque control for the AC machine have been proposed and discussed
by many scientists, such as Wang et al. (2006)
and Tang (2004). But transform methods and formulae are
different in form of expression in literature, although the essence is same.
Based on the literature (Liu et al., 2009; Zaher,
2008; Yan et al., 2008; Wang
et al., 2006, 2009; Xie,
2003; Tang, 2004), the study discusses comparisons
between two common transforms in PMSM vector control. The computer simulation
is a powerful tool to assess the control system and always be used. With the
performance of computer and simulation software greatly improved, the computer
simulation has become more and more powerful and easier and easier. The simulation
makes engineers and scientists discover the essence behind the phenomena and
evaluate their design quickly, so the research and development cost and cycle
can be significantly reduced (Zaher, 2008). The literature
(Liu et al., 2009; Zaher,
2008; Yan et al., 2008; Wang
et al., 2006, 2009; Xie,
2003; Tang, 2004) discussed how to build and analyze
PMSM model and the control system. The powerful simulation software package
MATLAB/Simulink has been widely utilized in electrical engineering and power
electronics simulation. In MATLAB/ Simulink/ SimPowerSystems toolbox, many basic
blocks can be directly used, such as PMSM model block, Park transform block
and Clarke transform block. Because of difference between different reference
coordinate systems, some blocks cannot always used directly. Xie
(2003) discussed the problem of PMSM model in the power system block set
in MATLAB 5.3 and given a modification method. Karabacak
and Eskikurt (2012), Choi and Lee (2012), Elsayed
et al. (2012), ShouQuan et al. (2011)
and Chen et al. (2011) discussed computer simulation
methods to PMSM control system.
VECTOR DEFINITION AND TRANSFORM
Coordinate system: Three coordinate systems are always used in vector
control, as shown in Fig. 1 (Wang et
al., 2006; Holmes and Lipo, 2003). The static
frame ABC derives from axes of the three stator windings. The other two are
the static frame αβ and the rotor frame dq. The axis α is consistent
with the axis A. The axis β has a 90° leading phase to α. The
axis d is oriented by the rotator flux linkage and q has a 90° leading phase
to d.
Vector definition: In the threephase PMSM, the stator steady state current can be expressed as:
where, I_{1} is the phase current peak value, ω is the electrical angular velocity and n_{0} is the initial phase angle.

Fig. 1: 
Diagram of reference frames and vectors 
The current can be expressed using a vector as:
The current vector is a rotary one with constant amplitude which is 3/2 times of the phase current peak value. This is same to the firstharmonic magnetic motive force vector and voltage vector.
Always two types of definitions of current, voltage or flux linkage are adopted
(Wang et al., 2006; Tang, 2004):
So:
Coordinate transformation: In frame αβ, currents can be expressed as:

Fig. 2: 
Relation between static frame ABC, αβ and rotor
frame dq 
In frame dq, currents can be expressed as:
where, θ_{δ} is the electrical angle between the current vector and d axis, as shown in Fig. 1.
It can be found Eq. 3 is the equal modulus transform but Eq. 4 is not. That is to say, T1 makes the peak amplitudes of A, B, C, α and β equal. The magnetic linkage vector and voltage vector are similar to the current vector.
The transform relations between three coordinates can be shown in Fig.
2. Always the combination transform of Clarke and Park is called ABC2dq
transform (the block is named abc_to_dq0 in MATLAB, 3ph>RRF in Plecs) and
the inverse transform called dq2ABC (dq0_to_abc in MATLAB, RRF>3ph in Plecs).
From Eq. 68, the transform formulae can
be gotten. Equation 9 and 10 are from definition
T1 and Eq. 11 and 12 from T2:
PMSM SIMULATION MODELLING METHOD
PMSM mathematical model: Based on some assumptions (Wang
et al., 2006), the stator voltage vector equation can be expressed
as:
where, u_{s} is the voltage vector, R_{s} is the stator resistance, L_{s} is the stator inductance and ψ_{f} is the flux induced by rotator magnet.
ψ_{f0} is the amplitude of the flux in the stator sing phase winding induced by rotator magnets. In definition T1:
In definition T2:
In reference frame dq, Eq. 13 can be expressed as:
where, u_{d}, u_{q} are the voltage in d, q, respectively; L_{d}, L_{q} are the inductance in d, q; ω_{e} is the electrical angle speed.
In T2, the electromagnetic torque can be expressed as:
where, p_{n} is the pairs of poles.
In T1, the electromagnetic torque can be expressed as:
From Eq. 5 and 15, it can be found that
Eq. 17 and 18 are same in essence.
The mechanical equations are:
where, ω_{m} is mechanical angle speed, J is rotary inertia, F is frication factor, θ is mechanical angle and T_{m} is load torque:
PMSM simulation model: Almost all simulation models have the load torque
input port T_{m} and the measurement port m (Fig. 3)
that outputs PMSM states such as the stator current, rotator position, speed
and electromagnetic torque.

Fig. 3(ad): 
Different inputs and outputs of permanent magnet synchronous
motor (PMSM) simulation model (a) Threephase voltage input in frame ABC
with constant PMSM parameters, (b) Twophase voltage input in frame αβ
with constant PMSM parameters, (c) Twophase voltage input in frame dq with
constant PMSM parameters and (d) Twophase voltage input in frame dq with
timevarying PMSM parameters 
The difference between the four models in Fig. 3 is the
input voltage. MATLAB/Simulink and Plecs adopt the model as Fig.
3a. In Fig. 3c, the PMSM parameters such as R_{s},
L_{d}, L_{q}, ψ_{f}, are constant but this is too
ideal to simulate the parameter timevarying case. So the study discusses how
to build a PMSM model as shown in Fig. 3d. The model has some
advantages as follows:
• 
PMSM parameter P (a vector) can be changed in simulation.
It is very useful to analyze the effect of parameter variety on PMSM performance.
The model also can be used to verify PMSM parameter identification algorithms 
• 
PWM technology is widely in motor control. In the classical three closedloop
control system, the inner current loop gives u_{d} and u_{q}.
Based on u_{d} and u_{q}, control signals to switch tubes
such as IGBT are generated. But this is timeconsuming or even simulation
fails in MATLAB/Simulink. Figure 3bd
consider the inverter ideal 
The simulation model is based on Eq. 1621.
The model is divided into three components: electrical model, mechanical model
and measurement model, as shown in Fig. 4a. The mechanical
model is shown in Fig. 4c.

Fig. 4(ac): 
Permanent magnet synchronous motor (PMSM) simulation model
with parameters that can be changed in simulation (a) Structure of the model
(b) Electrical model and (c) Mechanical model 
In Fig. 4b, iq and id models are derived from Eq.
16, electromagnetic torque function from Eq. 18 and mechanical
model from Eq. 1921.
SIMULATION RESULTS AND DISCUSSION
The simulation model using hysteresis current control is shown in Fig.
5. This model structure is derived from power_pmmotor.mdl in MATLAB Demos.
The parameters are stator resistance Rs = 2.875 Ω, inductance Ld = Lq =
0.0085 H, flux induced by magnets ψ_{f} = 0.175 Wb, inertia J =
0.0008 kg m^{2}, frication factor F = 0 and pairs of poles p_{n}
= 4. The Step block with Step time 0.04 sec, Initial value 1 and Final value
3, is used to apply the load. The speed, stator current and torque are shown
in Fig. 6. From the figures, the built PMSM simulation model
can be easily used and produce excellent application effectiveness. The SVPWM
(space vector pulse width modulation) model and inverter model are negligible
in simulation which can significantly improve the simulation speed with high
precision. From this respect, the PMSM simulation in the study is better than
that in the literature (Xie, 2003; Chen
et al., 2011). But it should be noticed that the SVPWM model and
inverter model must be used if the simulation focuses effect assessment of harmonic
and dead time to PMSM. If the hysteresis current control is substituted by vector
control or direct torque control, this simulation model still holds good.
In the simulation, Eq. 3 is adopted. If Eq.
4 is adopted, flux induced by magnets
and the coordinate transforms should adopt Eq. 11 and 12
in PMSM block and simulation model. The simulation results on speed, stator
current and torque are consistent with Fig. 6, because speed,
stator current and torque are actual physical quantities in motor control system.
The big difference between different transforms lies in the artificial definition.
The vectors, such as current, voltage and magnetic chain, are artificial physical
quantities, although they come from the actual ones.

Fig. 5: 
Simulation experiment model using motor model in Fig.
4 and hysteresis current control method 

Fig. 6(ac): 
Simulation results of Fig. 5 with parameters
of stator resistance 2.875 Ω, d and q inductance 0.0085 H, flux induced
by magnets 0.175 Wb, inertia 0.0008 kg m^{2}, frication factor
0 and 4 pairs of poles (a) Speed response, (b) Stator current and (c) Torque
response 
Different definitions must lead to different transform methods and formulas.
So in the literature (Chen et al., 2011; Choi
and Lee, 2012; ShouQuan et al., 2011;
Elsayed et al., 2012; Holmes
and Lipo, 2003; Liu et al., 2009; Tang,
2004; Wang et al., 2006; Yan
et al., 2008) the transform and equation have to be coordinated with
the definition.
CONCLUSION
The study introduces coordinate systems and vectors in PMSM control and compares the difference of two transforms and definitions. Although, there is phenomenal difference, the essence is same. A new PMSM model is built in which parameters can change with time. This model is more practical and has other advantages which is verified using simulation.