An Improved Direct Torque Control Method for PMSM

Xin Qiu, Wenxin Huang and Feifei Bu

Jiangsu Key Laboratory of New Energy Generation and Power Conversion
Nanjing University of Aeronautics and Astronautics
Nanjing, China
Email: qiuxin@nuaa.edu.cn

Abstract—This paper presents an improved direct torque

II. PMSM MACHINE EQUATIONS
control (DTC) method for permanent magnet synchronous
motors. The proposed DTC method generates the diver signal The model of a PMSM in the rotor reference frame can be
with SVPWM for the fixed switching frequency and low torque expressed as
ripple relative to conventional hysteresis controllers. Different
with the popular SVM-DTC, the proposed method obtains
amplitude and angle of reference voltage vector directly by the ⎧λ = λ + L i
torque and flux linkage loop respectively without complicated ⎪⎪ d f d d

calculations that reproduces the simplicity and robustness of λ

⎨ q = Lq qi (1)
conventional DTC. Furthermore, the observation and control of ⎪
⎪⎩λs = λd + λq
2 2
torque angle in the proposed method ensures the PMSM
operating stably. The validity of the control method is verified
by experimental tests.

I. INTRODUCTION ⎪⎧ud = Rs id + pλd − ωλq

⎨ (2)
High performance control strategies for PMSMs mainly ⎪⎩uq = Rs iq + pλq + ωλd
include the vector control (VC) and direct torque control
(DTC) [1]-[5]. The VC transforms the PMSM into an
equivalent DC motors with the coordinate transformation, and 3
need the rotor position signal. Te = P (λd iq − λq id ) (3)
2
The DTC possesses several advantages such as lesser
parameter dependence, needlessness of position sensor and where ud, uq, id, iq, λd, λq are the stator voltage, stator current,
fast dynamic response. Traditional DTC employs hysteresis and amplitudes of stator flux linkages on d and q axes; Ld, Lq
controllers and switching tables, which bring variable are the direct and quadrature inductances; λs, λf are the
switching frequency, large torque and flux ripples and amplitudes of stator and rotor flux linkages; Rs is the stator
unsatisfactory low speed performance. The two former could resistance; ω is the rotor electrical speed in rad/s; P is the
be solved as the DTC introduces the space vector pulse width number of pole pairs; p is the differential operator.
modulation (SVPWM) and proportional–integral (PI) In the stationary stator reference frame, the model of
regulators, which is also known as the SVM-DTC [3]-[4]. PMSM in the rotor reference frame can be expressed as
Reference voltage vectors of the SVM-DTC are obtained by a
series of calculations about flux linkage and speed, which
eliminates the direct specialty of DTC more or less [5]. A ⎧λα = (uα − Rs iα ) dt
simple DTC method using vectors with variable amplitude and ⎪ ∫
angle is proposed in article [6], however, the calculations of ⎪
amplitude and angle are still complex. ⎨λβ = ∫ (u β − Rs iβ ) dt (4)
⎪
This paper presents an improved SVM-DTC, which ⎪⎩λs = λα2 + λβ2
simplifies existing SVM-DTCs and regains the directness of
conventional DTC. In the proposed method, the amplitude and
angle of reference voltage vector are directly determined by 3
torque and flux linkage PI controllers respectively without Te = P(λα iβ − λβ iα ) (5)
additional flux and speed calculations. Real-time observation 2
of the torque angle ensures the PMSM operating in linear zone.
where uα, uβ, iα, iβ, λα, λβ are the stator voltage, stator
current, and amplitudes of stator flux linkages on α and β axes.

978-1-4799-2325-0/14/$31.00 ©2014 IEEE 2421

 To unite the proposed SVM-DTC scheme, the PMSM’s where es is the stator back EMF; us and is are the stator voltage
torque is analysis from the flux linkage and torque angle. The and current, respectively. The variation of the stator flux
electromagnetic torque of a PMSM can be expanded as linkage in one control period Ts is shown in Fig.2.

β
3Pλs G
Te = [2λ f Lq sin δ + λs ( Ld − Lq )sin 2δ ] (6)
4 Ld Lq λs (n) G G
Δλs (n) = us ( n)Ts
G
where δ is the torque angle, which is the electrical angle λs (n − 1)
between stator and rotor flux linkages. Fig 1 graphs the
expression, where G G
u s ( n) λ f (n)
Δδ ( n ) G
3Pλsλ f λ f ( n − 1)
Te 0 = (7) Δδ (n − 1)
2 Ld α

Fig. 2. The relationship between torque angle and voltage vector.

The influence of Rs is neglected for facilitating the

analysis. From Fig. 2, the most efficient and quick approach to
change torque angle (or torque) is regulating the amplitude of
voltage vector in the vertical direction of stator flux linkage.
The increment of torque angle is

G G
us (n) Ts ω (n − 1) λs (n − 1) Ts
Δδ (n) ≈ G − G (10)
λs (n) λs (n)

In addition, for regulating the amplitude of stator flux

linkage, the voltage vector changes its direction, as shown in
Fig. 1. Torque characteristic of PMSM.
Fig. 3.
In Fig. 1, a peak is inherent with δ increasing, and a nadir
appears with λs increasing. The nadir is forbidden because it G G
would destroy the control monotonicity. For insuring PMSMs G Δλs 3 ≈ u3 Δt
λs 3 G G
working in monotonic zone, two prerequisites as follows need G Δλs 2 ≈ u2 Δt
to be satisfied [7]. λs 2 G G
β G Δλs1 ≈ u1Δt
λs1
⎧ Lq Lq G
⎪ + ( )2 + 8 λs
⎪ dTe L q − Ld Lq − Ld
⎪ = 0 ⇒ δ < arccos[ ]
⎨ dδ 4
⎪ dT Δδ
Lq δ G
⎪ e δ =0 > 0 ⇒ λs < λf
⎪⎩ d δ Lq − L d θ λf
(8)
α
III. PROPOSED DTC SCHEME Fig. 3. Voltage vectors with different directions.
Based on the aforementioned analysis, Te is proportional to
δ, which means that the torque can be controlled through the By adding voltage vectors with different directions, there
control of torque angle. are three possible states of the stator flux linkage,
With the stator flux linkage equation expressed in
stationary frame G G G
⎧ λs1 = λs + Δλs1
⎪⎪ G G G
⎨ λ s 2 = λ s + Δλ s 2 (11)
λs = ∫ es dt = ∫ (us − Rs is )dt (9) ⎪G G G
⎪⎩ λs 3 = λs + Δλs 3

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 G G IV. EXPERIMENTAL RESULTS
⎧ λs1 < λs
⎪ G G The method is proved with the testing platform shown in
⎪
⎨ λs 2 ≈ λs (12) Fig. 5.
⎪ G G
⎪⎩ λs 3 > λs

Thus, the improved SVM-DTC is obtained, in which the

amplitude and angle of reference voltage vector are calculated
directly by the torque and flux PI regulators respectively.

∠u

Fig. 5. Experimental platform.

The platform contains a MAGTROL 2PB115-IS powder

dynamometer and its controller DSP6001. The pole-pairs of
PMSM foe testing is 3, the rated speed is 500rpm. The direct
and quadrature axis inductances are both 3mH, the rotor flux
is 0.317Wb.
The bus voltage is 90V, and the base speed is about
500rpm. The stator flux linkage waveforms of 500rpm,
700rpm and 950rpm are shown in Fig. 6. The flux linkage is
Fig. 4. Block diagram of the proposed strategy. under control and the proposed method has field-weakening
ability.
The block diagram of the proposed method is shown in
Fig.4. The Te* is given torque, which is obtained from the
speed regulator. The voltage vector’s amplitude can be
obtained by

G
u ( n) = L p (Te* ( n ) − Te ( n)) + Li ∑ Ts (Te* ( n) − Te ( n)) (13)

where Lp and Li are proportional and integral coefficients

respectively, and Ts is the sampling time of discrete system.
The voltage vector’s angle can be obtained by PI flux linkage
regulator

∠u ( k ) = M p ( λ * ( n ) − λ ( n ) ) + Fig. 6. Stator flux linkage waveforms.

(14)
M i ∑ Ts ( λ * ( n) − λ ( n) ) + θ (n) + π / 2

where Mp and Mi are proportional and integral coefficients

respectively, and θ is the angle of flux linkage in real time.
As the amplitude and angle of voltage vector has been
acquired, voltage components need by SVM are obtained with
the trigonometric transformation simply. The usage of polar
coordinate is beneficial to insure the voltage vector less than
max voltage.
Compare with existing SVM-DTCs, the improved SVM-
DTC dispenses with calculations of transforming flux errors to
voltage vectors, and brings a simpler structure. Consequently,
the improved SVM-DTC is ‘direct’ and retains the advantages
of low ripples and fixed switching frequency with the help of
PI regulators and the SVM. Fig. 7. Steady state experiment waveforms.

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