Maximum Efficiency Per Ampere Control of
Synchronous Reluctance Motor Sensorless Drives
Chengrui Li
School of Mechanical Engineering and
Automation
Harbin Institute of Technology
(Shenzhen)
Shenzhen, China
lichengrui@hit.edu.cn
Jiaxiao Shi
Institute of Future Technology
School of Mechanical Engineering
Harbin Institute of Technology
Lu Wang
Centre for Advanced Low Carbon
Propulsion Systems
Coventry University
Harbin, China
2021110630@stu.hit.edu.cn
United Kingdom
lu.wang@coventry.ac.uk
Dianxun Xiao
Sustainable Energy and Environment
Thrust
The Hong Kong University of Science
and Technology
Guangzhou, China
dianxunxiao@ust.hk
Gaolin Wang
School of Electrical Engineering
Dianguo Xu
School of Electrical Engineering
Harbin Institute of Technology
Harbin Institute of Technology
Harbin, China
wgl818@hit.edu.cn
Harbin, China
xudiang@hit.edu.cn
Abstract—The maximum efficiency per ampere (MEPA)
control strategy is a highly effective technique for enhancing the
efficiency of a synchronous reluctance motor (SynRM) drive.
This paper proposes a novel efficiency optimization control
strategy that considers the motor and inverter as a unified
system, enabling maximum efficiency during operation. Unlike
conventional id = iq or maximum torque per ampere control, the
approach takes into account both iron losses and inverter losses,
together with cross-coupling effects. It is achieved by using
virtual signal injection to extract optimum operation points in
real-time, combined with an accurate iron loss analytical
approach. Experimental results demonstrate the effectiveness of
the proposed method.
Keywords—Maximum efficiency per ampere (MEPA),
Synchronous reluctance motor (SynRM), virtual signal injection
I.
INTRODUCTION
Synchronous Reluctance Motor (SynRM) is characterized
by its simple structure, low cost, and high reliability, making
it widely used in various fields such as fans, water pumps, and
more. In comparison to the widely used Permanent Magnet
Synchronous Motor (PMSM), SynRM does not contain
permanent magnets, which significantly reduces its cost and
eliminates the impact of rare-earth material price fluctuations,
as well as the risk of demagnetization. However, a notable
drawback of the SynRM is its relatively low power density.
Consequently, in applications where high power density is not
a critical requirement, the SynRM can be used as a
replacement for the PMSM, such as in non-road vehicle
traction systems and auxiliary power systems, thus further
expanding the range of applications for SynRM [1, 2].
Currently, there are two main control strategies to improve
the efficiency of the SynRM drive system. The first one aims
to maximize the output torque while keeping the current
amplitude to a minimum, which is also known as Maximum
Torque per Ampere (MTPA) control [3-5]. The objective of
This work was supported by the Research Fund for the National Natural
Science Foundation of China (52107041), Shenzhen Science and
Technology
Program
(Grant
No.
RCBS20221008093122056,
JSGG20210802152540008), Shenzhen talent start-up fund, Shenzhen
postdoctoral Fellowship Program, Guangdong Basic and Applied Basic
Research Foundation (2022A1515110939, 2022A151540050) and
Guangzhou Science and Technology Program (2023A04J1034).
MTPA control is to operate the motor at the point with the
lowest current while delivering the desired torque. This
approach effectively reduces motor losses and improves
motor efficiency, making it widely used in synchronous
motors. Motor parameters play a crucial role in the MTPA
control model. However, the inductance parameters of
SynRM exhibit nonlinear variations due to factors such as
magnetic saturation and temperature changes. Modeling these
variations accurately presents a challenge for achieving
precise MTPA control in SynRM. Two methods, the lookup
table approach based on offline measurement results and the
online search method, can be used to obtain the optimal
current phase angle for MTPA control, and both methods do
not require precise motor parameter information. Compared to
the conventional control strategy, which utilizes equal
orthogonal axis currents, the MTPA control strategy can
indeed improve the overall efficiency of the drive system.
However, it should be noted that MTPA control only focuses
on minimizing the copper losses of the motor system, and
therefore, it does not achieve overall efficiency optimization,
which is known as Maximum Efficiency per Ampere (MEPA)
control. MEPA control aims to achieve the highest possible
efficiency for the motor system, considering not only the
copper losses but also other losses such as iron losses,
mechanical losses, and other relevant energy losses. By
considering these additional factors, MEPA control can
optimize the motor's overall performance and energy
efficiency under various operating conditions [6, 7].
Efficiency optimization control, considering all the losses in
the drive system, helps maximize the overall efficiency of the
SynRM drive system. This approach ensures the motor
operates at its peak efficiency across various conditions,
leading to reduced energy consumption, lower operating
costs, and improved environmental friendliness. The
comprehensive efficiency optimization further enhances the
advantages of the SynRM drive system, making it an attractive
choice for applications where energy efficiency and reliability
are crucial factors.
Efficiency optimization control strategies based on online
MEPA operating point tracking can be further classified into
methods that utilize loss models [8], online search methods [9,
10], and the hybrid optimization method [11, 12]. The
efficiency optimization control method based on loss models
allows for online retrieval of optimal reference current
commands. It involves establishing an online lookup table or
high-order polynomial loss model based on results obtained
from pre-offline testing. For model-based approaches, the
accuracy of the mathematical models for copper and iron
losses is a critical factor in determining the effectiveness of
efficiency optimization control. Moreover, the loss modelbased methods require detailed system loss information to
cover all operating states of the SynRM. This necessitates a
large number of repetitive experiments to obtain a complete
online lookup table, resulting in drawbacks such as timeconsuming and computationally demanding processes [8, 13].
Furthermore, most efficiency optimization control strategies
utilize fixed-parameter models or nonlinear parameter models
obtained from offline testing to mathematically describe the
losses in the SynRM system. However, these models cannot
be updated in real-time based on motor operating conditions
and operating conditions, which reduces the accuracy and
real-time effectiveness of the loss models [14]. Reference [15]
proposed an online update method for inductance parameters
based on the Extended Kalman Filter (EKF). However, it did
not consider the variations of stator resistance and equivalent
iron loss resistance under different operating conditions.
To address the aforementioned issues, researchers have
proposed the online search method, which allows for online
searching of the minimum power consumption operating point
for the SynRM drive system. Compared to the efficiency
optimization control method based on loss models, the online
search method can effectively track the current phase angle
corresponding to the MEPA operating point regardless of
variations in motor parameters. However, the online search
method generally suffers from slow convergence speed and is
typically suitable for steady-state operations. During transient
states, continuous adjustments of the current phase angle are
required, which can degrade the system dynamic
performance, leading to oscillations or steady-state errors.
Additionally, the accuracy of the search results is influenced
by the current sampling precision, and current or voltage
harmonics can affect the actual control performance [16].
Recently, researchers have proposed a Virtual Signal
Injection-based MTPA control method for PMSM. This
method allows obtaining the optimal current phase angle for
MTPA from the virtual motor torque model [17]. This method
only requires injecting virtual high-frequency square or
sinusoidal signals, which do not affect the actual output torque
and losses of the motor. Additionally, it imposes a reasonable
computational burden on the processor and exhibits a certain
level of robustness to parameter variations. Due to its
simplicity, it can be used for real-time updating of the online
MTPA current reference commands. When using the square
wave signal injection mode, it demonstrates good dynamic
performance [18, 19]. However, there is still limited research
on efficiency optimization control for SynRM based on virtual
signal injection [5, 20]。
To improve the control robustness to the parameter
nonlinearity, a MEPA control strategy based on virtual signal
injection is proposed in this paper. In addition, the full speed
range sensorless control is achieved based on a combination
of signal injection and extended back-EMF based methods.
This article is composed of four sections, respectively. The
adopted SynRM model is discribed and discussed in Section
II. The proposed MEPA control scheme is introduced and
illustrated in Section III. The proposed method is validated by
the experimental results presented in Section IV. The
conclusion is given in the last section of the article.
II.
SYNRM DRIVE SYSTEM MATHMATICAL MODEL
A. SynRM Model
The reference frame of SynRM is shown in Fig.1, where
superscript s, r and e represent stationary, rotary and estimated
reference frame, respectively. The SynRM dynamic voltage
equations in the synchronous rotating d-q reference frame can
be expressed as:
ud   Rs + pLdh
 u  = ω L + pL
dq
 q  e d
−ωe Lq +pLdq  id 
Rs + pLqh   iq 
(1)
where u d and u q are the stator voltages, id and iq are the
stator currents, Rs is the stator resistance, ωe is the electrical
angular speed, Ld and Lq are the d-q axis apparent selfinductances, Ldh, Lqh and M h are the d-q axis incremental selfinductances and mutual inductance, respectively. Since the
cross-coupling effect is relatively severe for SynRMs, it
cannot be omitted as that of interior PMSMs. The dynamics
equation between the motor mechanical speed ωm and the
electromagnetic torque Te can be expressed as:
dωm Te − Tl − Bωm
=
dt
J
(2)
where Tl is the load torque, J is combined moment of inertia
of the drive system, and B is the friction coefficient.
βs
Is
γ
bs
θr
as/αs
cs
Fig.1 SynRM reference frame.
Since there is no permanent magnet is the SynRM, the
output electromagnetic torque is only composed of reluctance
torque without permanent magnetic torque under fieldoriented control. With cross-coupling effect taken into
consideration, the full order electromagnetic torque
expression can be derived as[21]:
Te = Te 0 + Tec =
3P
( Ld − Lq )id iq + Ldq (id 2 − iq 2 ) 
2 
(3)
where P is the number of pole pairs, Te 0 and Tec are the
conventional electromagnetic torque without considering
cross effect and the torque caused by cross effect, respectively.
In addition, the inner power factor angle γ is defined as the
angle between the q-axis current iq and the given reference
current command is :
id = Is sin(γ )

iq = Is cos(γ )
(4)
For the MTPA control strategy, the optimal angle is
supposed to meet (∂Te / ∂γ ) = 0 , which can be further
deduced as:
2M h
γ = cot (
)
Ld − Lq
−1
(5)
However, it needs to be pointed out that the parameters
including self and mutual inductances are nonlinear along
with the load variation. Especially for the SynRM, the
parameter nonlinearity is relatively severe under various d-q
axis currents, which make the conventional control strategy
adopting constant parameters unavailable for SynRM drives.
In addition, even the precise parameters are adopted, the
maximum efficiency cannot be achieved due to the ignoration
of iron losses and inverter losses.
B. SynRM Drive System Loss Model
Accurate loss model is the basic for efficiency
optimization control strategy. In this section, the loss model
characteristics of SynRM drive system including copper
losses, iron losses and inverter losses are analyzed and
discussed, respectively. The fundamental frequency losses of
SynRM are composed of both copper losses and iron losses.
Normally, the copper losses can be represented as:
PCu =
3
Rs (id 2 + iq 2 )
2
(6)
The iron losses can be expressed with the Bertotti iron loss
formula, the iron losses per volume can be expressed as:
dPFe = kh Bm 2 f +
π 2σ kd 2
6
Bm 2 f 2 + Bm1.5 f 1.5
(7)
where k h and ke are the coefficients of hysteresis losses and
excess losses, respectively. σ is the material conductivity,
and k d is the lamination thickness. The aforementioned four
parameters are relevant to the steel material and can be
referred to the data sheet provided by manufacturers. f is
the current frequency, and
is the magnetic flux density
peak value.
To avoid complex modeling process using FEA, analytical
calculation method is adopted in this paper and will be
illustrated and discussed in detail in future work [8]. The iron
losses can be calculated as:
PFe = dPFetd , qVt + dPFejd , qV j
= k hd (ψ d 2 + ψ q 2 ) + kep (ψ d 1.5 + ψ q1.5 )
= k hd  ( Ld id + Ldq iq ) 2 + ( Lq iq + Lqd id ) 2 
+ kep ( Ld id + Ldq iq )1.5 + ( Lq iq + Lqd id )1.5 
(8)
where k hd and kep are two coefficients associated with the
motor structure.
For the inverter, main losses are composed of switching
losses Psw and conduction losses Pcon , which can be
expressed as:
(9)
Pinv = Psw + Pcon
Psw =
6
π
( Eon + Eoff + Eerr ) f swVdc I s
(10)
1
1
Pcon = 6( Von I s + Ron I s 2 )
π
4
(11)
where Eon , Eoff and Eerr are the energy consumed during
the IGBT turn on, turn off and the diode turn off period,
respectively. f sw is the switching frequency, Von is the
forward threshold voltage, and Ron is the resistance.
Take all the aforementioned losses into consideration, the
total losses for SynRM drive system can be expressed as:
(12)
Pl = PCu + PFe + Pinv
The input power for the SynRM drive system is:
3
Pin = (ud id + uq iq )
(13)
2
where usd and usq are the reference voltages, respectively.
The efficiency of the SynRM drive system is derived as:
P −P
(14)
η = in l
Pin
III. EFFICIENCY OPTIMIZATION CONTROL METHOD
The proposed MEPA control method is introduced in this
section. The virtual high frequency signals are injected to
extract the efficiency optimized current angle based on the
SynRM mathematical model. The proposed method is not
sensitive to the motor parameter variation during the operation.
The detailed method will be illustrated. The proposed MEPA
control scheme is shown in Fig.2.
ωref
+ ωerr
−
ωˆ e
Speed
regulator
γ out
isref
γ ref +
+
α
d-q axes
reference
current
calculation
idref
iqref
Current
regulator
Coordinate
transformation and
high frequency
signal injection
θˆe
uαβ ref
Virtual high
frequency signal
injection
MEPA tracking
regulator
udqref
SVPWM
ia , c
γ fdb
θˆe
Sensorless
control
SynRM
Fig.2 Block diagram of SynRM efficiency optimization sensorless control
adopting the proposed virtual signal injection based MEPA control strategy.
The injected signals are defined as:
1

kTs ≤ t < (k + )Ts
 ∆,
2
α inj (t ) = 
1
−∆, (k + )T ≤ t < (k + 1)T
s

2
(15)
where ∆ is the amplitude of the injected signals, and the
frequency of the injected signals is selected as the same as the
PWM frequency. After the virtual signal injection, the Taylor
series expansion of the efficiency can be expressed as:
η h (α ) = η h (0) +
∂η h
∂α
α+
α =0
1 ∂ 2η h
2 ∂ 2α
2
α 2 +L
α =0
1∂ η 2
∂η
= η (γ ) + α +
α +L
2 ∂ 2γ
∂γ
. (16)
Since the amplitudes of high order terms in (16) are
relatively lower compared with the DC component and 1st
order, which can be omitted. The (16) could be simplified as:
η h (α ) = η (γ ) +
∂η
α.
∂γ
(17)
Hence, the objective of searching for the efficiency
optimized current angle is to calculate:
∂η
α =0
∂γ
(18)
Since the ∆ is relatively small, the d-q axis current after
the injection can be simplified as:
id h = I s sin(γ + α ) ≈ id + iqα
(19)
iq h = I s cos(γ + α ) ≈ iq − id α
(20)
TABLE I
SYNRM PARAMETERS
Rated power (kW)
Rated torque (N·m)
Rated speed (Hz)
Pairs of poles
Rated voltage (V)
Rated current (A)
Phase resistance (Ω)
4
12.7
3000
2
380
10
0.43
The efficiency of SynRM drive system under various load
at rated speed is shown in . Compared with the conventional
MTPA control strategy, the SynRM efficiency under the
proposed method can be further improved.
The input power for the SynRM drive system after the
signal injection can be expressed as:
3
(usd id h + usq iq h )
2
3
= usd id + usq iq + (usd iq − usq id )α 
2
Pin h =
(21)
The copper loss, iron loss and inverter loss of the SynRM
drive system after the signal injection can be respectively
calculated as:
PCu =
3
3
Rs (id h 2 + iq h 2 ) = Rs (1 + α 2 )(id 2 + iq 2 )
2
2
PFe h = khd (ψ dh 2 + ψ qh 2 ) + kep (ψ dh1.5 + ψ qh1.5 )
Pinv =
6
π
(22)
(23)
( Eon + Eoff + Eerr ) f swVdc I s 1 + α 2
1
+6( Von I s
π
1
1 + α + Ron (1 + α 2 ) I s 2 )
4
(24)
2
where the d-q axis flux can be estimated by:
 h − Rs (iq − id α ) + uq
ψ d =
ωe


ψ h = Rs (id + iqα ) − ud
 q
ωe

(25)
EXPERIMENTAL RESULTS
The proposed scheme is realized on a DSP-based 4kW
SynRM drive experimental platform as shown Fig.3. The
parameters of the SynRM is listed as in Table I.
Fig.3 Experimental SynRM drive test platform.
V.
CONCLUSION
A MEPA control method for SynRM drive system based
on virtual signal injection is proposed and verified in this
paper. Compared with conventional MTPA method, the iron
losses and inverter losses are taken into account along with the
coper losses to further improve the entire drive system
efficiency. In the future work, the iron losses monitoring
method and the dynamic performance of the proposed method
will be further investigated.
VI.
Using a bandpass filter and a lowpass filter, the efficiency
optimized current angle of the drive system can be extracted,
and a simple PI regulator is adopted to tracking the optimum
efficiency operation point to achieve the MEPA control
strategy.
IV.
Fig.4 SynRM drive system efficiency under various control strategy at rated
speed and varying load.
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