Maximum Torque per Ampere Control Method
for IPM Synchronous Motor based on V/f Control
Jun-ichi Itoh, Yuki Nakajima, Masakazu Kato
Dept. of Electrical, Electronics and Information Engineering
Nagaoka University of Technology
Nagaoka Niigata, Japan
itoh@vos.nagaokaut.ac.jp, nyuuki@stn.nagoakaut.ac.jp, katom@stn.nagaokaut.ac.jp
Abstract— This paper proposes a maximum torque per ampere(MTPA) control method based on V/f control for the Interior
Permanent Magnetic Synchronous Motor (IPMSM). The V/fcontrol is inherently a position sensorless method. Therefore it issimpler than the conventional methods such as the sensorless
vector control method. In addition, the MTPA control can beachieved by controlling the reactive power without the
information of magnet pole position. The proposed MTPA
control achieves high efficiency because the copper loss can be
reduced. In this paper, the proposed method is demonstrated by
the by the experiments. Then, the output current can be reducedby 76 % in compared to the V/f control without MTPA. The
validity of the proposed method is confirmed by experimental
results.

I.

INTRODUCTION

In recent years, IPMSMs have been intensively studied
because of its attractive features; high efficiency, light weightand high speed rotation.
The vector control is one of the common control methods
to drive the IPMSM. However, the vector control method
requires a position sensor, which is composed from an encoder
and a resolver, in order to detect the magnet pole position. The
position sensor is one of the high cost components in an
IPMSM drive system. Besides, it is difficult to put the position
sensor depending on applications.
In order to not use the position sensor, many kinds ofsensorless vector control methods have been studied andproposed for IPMSM [1-6]. The sensorless vector control
method estimates the magnetic pole position of the IPMSM byusing the motor voltage, current and motor parameters.
However, the calculation for the position estimation iscomplicated and the decision of the control parameter in orderto stabilization is difficult. In addition, the error of the position
estimation occurs easily when the parameters are mismatching
between the controller and motor. Especially the motor
parameters will be changed by the motor temperature, flux

saturation and so on. As a result, it is difficult to certify thestability of sensorless vector control.
On the other hands, open loop control, which has no
current regulator, such as V/f control methods have beenstudied and investigated for the IPMSM [7-9]. The V/f control
method can be considered as an open loop control, because it
does not require the information of the pole position. In theV/f control, the control algorism is implemented on γδ-flame
where the δ-axis is corresponding to the output voltage vectorof an inverter. The control algorithm is easier than that of thesensor less vector control. Especially, the motor parameters
are not used in the control. In Ref [9], the MTPA (id = 0)
control method based on the V/f control for Surface
Permanent Magnetic Synchronous Motor (SPMSM) haveproposed. However, in the IPMSM, the maximum torque per
efficiency control is not achieved at id = 0 because thereluctance torque cannot be used effectively. Therefore, it isseems that the MTPA for IPMSM have not been discussed indetail.
In this paper, a MTPA control method for IPMSM basedon V/f control method is proposed. The proposed method isdoes not need a position sensorless due to open loop control.
Furthermore, the propose method can be achieved bycontrolling the reactive power to regulate with the current
phase.
This paper is organized as follows; first the principle of theV/f control is introduced. Next, MTPA for IPMSM is
proposed. After that, the MTPA control method based on V/fcontrol method is confirmed by simulation. In the simulation,
it is confirmed that the output current in dq-frame can becorresponded to the MTPA operation point. Furthermore, The
proposed MTPA is demonstrated in the experiment using a
1.5-kW IPMSM with a 2-level inverter. As a result, theproposed MTPA reduces the output current by nearly 76 %compared to not use MTPA.

II.

V/F CONTROL BASED ON THE OUTPUT VOLTAGE
VECTOR OF THE INVERTER

A. Basic Principle
Fig. 1 shows the relationship between γδ-frame and dqframe. Generally, in the dq-frame of the IPMSM control, thed-axis is defined as the direction of the flux vector by thepermanent magnet and the direction of the q-axis is defined as
the electromotive force vector. Therefore, it is very important
to identify the flux vector in the vector control. However, theV/f control method is implemented on the γδ-frame, and the δaxis is defined as the direction of the output voltage vector of
the inverter. Therefore, the δ-axis means active powercomponent and the γ-axis means reactive power component.
The voltage equation of IPMSM based on dq-frame isgiven by (1).

⎡vd ⎤ ⎡ Ra + pLd⎢v ⎥ = ⎢ ω L⎣ q ⎦ ⎣ re d

− ω re Lq ⎤ ⎡id ⎤ ⎡ 0 ⎤+Ra + pLq ⎥⎦ ⎢⎣iq ⎥⎦ ⎢⎣ω reψ m ⎥⎦

ω re

(L d i d )2 + (L q iq )2

Fig. 1. Relationship between γδ-frame and dq-frame. The δ-axis is definedas the direction of the output voltage vector of the inverter, and the
direction of the q-axis is defined as the electromotive force vector.

(1)

*
rm

Pf

*
re

HPF

T = Pf ψ m i q = Pf ψ m (iγ sin θ + iδ cos θ )pω re =

Pf (T − TL )
J

(3)
(4)

where Pf is the pairs of poles, T is the output torque, TL is theload torque, and J is the inertia of the motor. The gap angularθ between γδ-frame and dq-frame is given bypθ = ω1 − ω re(5).

In the steady state, the rotating speed in either the γδ-frameor the dq-frame is the same value. However, in transient state,
the gap angular between γδ-frame and dq-frame occurs due tothe differential in the rotating speed. Here, the differentiationof θ is equaled to the difference between ω1 and ωre.
From the definition of the γδ-frame, the output voltage
vector vδ is given in the δ-axis. In addition, the electromotive
force ωreψm occurs on the q-axis. Therefore, the gap angular θ
between the γ-axis and dq-frame is subjected to the load
angular.

*1+
K1
Damping
control

-

/3

v

+

vv*
vw*

+

Proposed
MTPA
Control− ω1 Lq ⎤ ⎡iγ ⎤⎡ sin θ ⎤+ ω reψ m ⎢
⎥
⎥
⎢
⎥ (2)
Ra + pLq ⎦ ⎣iδ ⎦⎣cos θ ⎦where vγ(δ) is the γ(δ)-axis voltage, iγ(δ) is the γ(δ)-axis current,and ω1 is the rotating speed of γδ-frame. Here, the equation of
the torque and the relationship (ignored the viscosity
resistance) between the electric angular velocity and thetorque on γδ-frame is given by (3) and (4), respectively.

v

f/v conv.

where vd(q) is the d(q)-axis voltage, id(q) is the d(q)-axis current,ωre is the electric angular frequency, p is the differential
operator, Ra is the armature resistance, Ld(q) is the d(q)-axis
synchronous inductance, and ψm is the flux linkage of thepermanent magnet. Then, the voltage equation of IPMSM
based on γδ-frame is obtained by (2).
⎡vγ ⎤ ⎡ Ra + pLd⎢v ⎥ = ⎢ ω L⎣ δ⎦ ⎣ 1 dvu*v *= 0

v

1
s

*
re

iu

i
i

3 /

iv
iw

Fig. 2. V/f control method based on γδ-frame. The motor is controlled by
V/f control method with damping control.

B. Damping Control
Fig. 2 shows the block diagram of the V/f control method
based on γδ-frame. When a motor is controlled by simple V/fcontrol method which means an open loop control, the torquevibration is generated due to the resonance between thesynchronous the reactance and the moment of inertia of the
IPMSM as indicated in Ref. [9].
In Ref. [9], the V/f control method based on γδ-frame with
damping control was proposed. The torque vibration causes
the vibration of the δ-axis current because the δ-axis current is
active current. In another words, the vibration component ofthe δ-axis current can assume the vibration component of thetorque. The damping control suppresses the torque vibration
with feedback the δ-axis current into the electric angularfrequency reference value ωre* in order to cancel the load
angular vibration. When the damping control is applied, thedamping coefficient is a function to the feedback gain K1,
therefore, the stable operation can be achieved.
III.

MAXIMUM TORQUE AMPERE CONTROL METHODBASED ON V/F CONTROLMTPA (id = 0) control can achieve high efficiency for the
SPMSM. On the other hands, the MTPA is not achieved at id =

0 in of IPMSM due to the reluctance torque. Therefore, in thischapter, MTPA control based on V/f control for IPMSM is
discussed. In order to achieve the MTPA in IPMSM, thereactive power is used.
The reactive power Qdq on dq-frame is given by (6)Qdq = vqid − vd iq(6)

By substituting (1) into (6), the reactive power Qdq can be
expressed as (7).

{

2

2

Qdq = ωre Ld id + Lq iq + ψ mid}

Fig.3 High Efficiency control method for IPMSM. Qdq command value iscalculated from (10) at the Qdq* calc. block
Table.1 Motor Parameters used in simulations. The IPMSM has a saliency
ratio of 2.0 and the rated power is 1.5 kW(7)Equation (7) can be re-written as (8) by using Ia and β.

{

Qdq = ω re Ld I a sin 2 β + Lq I a cos 2 β − ψ m I a sin β } (8)22where the equation of the current phase β when the MTPA
control [10] is given by
2
⎞
⎞ 1⎟
⎛
ψm
⎟ + ⎟ (9).
⎜⎜ 4 Lq − Ld I a ⎟
2⎟
⎠
⎝
⎠

⎛
⎜
ψm
β = sin ⎜ −+⎜ 4 Lq − Ld I a
⎝
−1

(

)

(

)

Note that the current phase β becomes zero when Ld equal
to Lq i.e. it means SPMSM because 1st term equals to 2nd term.
If it is defined as Ia sin (β) = X, then the equation of thereactive power is obtained by

{

(

)

2

Qdq = ω re Ld X 2 + Lq I a − X 2 − ψ m X

}

(10).

Equation (10) is the reactive power when the MTPA
control is achieved. On the other hands, the reactive power
based on γδ-flame Qγδ is given by

Qγδ = vδ iγ(11).

If the Qγδ as shown in (11) equals to Qdq as shown in (10),the MTPA control can be achieved. In another words, thesatisfaction of (12) achieve the MTPA control during the V/fcontrol.

{

(

)

}

ωre Ld X 2 + Lq iγ 2 + iδ 2 − X 2 − ψ m X = vδ iγ(12)

In order to satisfy the condition in (12), the PI controller isimplemented to regulate the δ-axis voltage vδ.
Fig. 3 show the control block diagram of the proposed
MTPA control method. The control block shown in Fig. 3 issame as the proposed MTPA control block shown in Fig. 2. In
addition, Qdq command value is calculated from (10) at the
Qdq* calc. block in Fig. 3. The proposed MTPA block
conform Qγδ to Qdq* by PI controller. The PI controller outputs
the δ-axis voltage command compensation value of Δvδ*. The
Δvδ* compensate the δ-axis voltage command vδ* in order toaccomplish (12).

Fig.4 Simulation model. The load is a constant torque load.

IV. SIMULATION RESULS
In order to confirm the operation, the proposed method issimulated in PLECS (Plexim GmBH). Table 1 shows the
motor parameters, and the IPMSM has a saliency ratio of 2.0
and the rated power is 1.5 kW. In addition, the switchingfrequency is 10 kHz.
The simulation model uses a 2-level inverter as shown in
Fig. 4. .Note that the dq-frame current is calculated from theinverter output current by using the information of the magnet
pole position in order to the confirming of the MTPA
operation. However, the information of the magnet poleposition is not required in the actual IPMSM system due toopen loop. In addition, the load is a constant torque load.
The voltage command vδ* in the V/f control is calculatedby (13).
vδ * = vf (1 − v bst )ω re * + v bst(13)

where vf is the V/f ratio and vbst is the torque boost voltage
which compensates the voltage drop at the armature resistance.
The V/f ratio is defined as the ratio, where the output voltage
is proportional to the motor frequency. The V/f ratio is defined
as 1, where 90 Hz is equivalent to 180 V in this paper. . The
torque boost voltage is 0.05 p.u. of the rated voltage (180 V)
that is equaled to the percentage of Ra.
Fig. 5 shows the waveforms at the d and q-axis current,
output current and d-axis voltage command with the proposed
MTPA method. The operation point in the simulation are as
follows; the rotating speed command is 0.2 p.u. and the load
torque is 0.2 p.u.. The V/f control with the damping control isCurrent phase β [rad]Fig.6 Variation of reactive power(Load Torque is 0.2 p.u.and Rotation speed is 0.2 p.u.). The reactive
power Qγδ becomes converged as the command Qdq*.

1.2
1

Without MTPA control
With MTPA control

0.8
0.6
0.4
0.2

Theoritical value with MTPA control0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rotating speed ωrm [p.u.]

1

Fig.7 Variations of Current phase with/without MTPA control method(Load torques are 0.2 p.u.). In the case the MTPA control is applied, thecurrent phase is corresponded to the theoretical current phase that canachieve the MTPA control operation.

reduced from 0.57 p.u. to 0.20 p.u. The effectiveness of the
proposed method is demonstrated in the simulation result.
In addition, the d-axis current is -0.016 p.u. and the q-axis
current is 0.199 p.u. at the time 6.0 s. Therefore, the current
phase β is 0.08 rad. and this value is equivalent to the β ,which is calculated from (9).. Therefore, it is confirmed that
the MTPA control can be achieved.

Fig.5 Operation of MTPA control method based on V/f control (Load
Torque is 0.2 p.u. and Rotation speed is 0.2 p.u.). The amplitude of theoutput current is reduced from 0.57 p.u. to 0.20 p.u. with proposed
MTPA control.

applied from 0 s to 2.0 s, and then the V/f control and the
MTPA control (including the damping control) is applied from
2.0 s and onwards.
The d and q-axis currents decrease when the MTPA
control is applied because the δ-axis voltage command value
is compensated by the high efficiency control system asshown in Fig. 2. The q-axis current decreasing cause themagnet torque decreasing. Note that the output torque remains
unchanged even d and q-axis current have changed. Therefore,
the reluctance toque compensates for the magnet torquedecreasing. In addition, the amplitude of the output current is

Fig. 6 shows the relationship between the reactive powercommand value Qdq* and the detected value Qγδ when the
MTPA control is applied. The operation point is similar to Fig.
5. The reactive power becomes converged as the commandhas decrease because the PI control is implemented. Therefore,
it is confirmed that the regulation of the reactive power can beachieved.
Fig. 7 shows the differences in the current phase between
the case with and without applying the MTPA control. The
operation points are as follows: the rotating speed is from 0.2
p.u. to 1.0 p.u. and the output torque is 0.2 p.u. constantly.
Without the MTPA control, the current phase varies insubjected to the load condition, due to the difference in theload angular. In addition, the current phase is different fromthe theoretical current phase that can achieve the MTPA
control operation. Therefore, the V/f control method without
the MTPA control cannot control the reluctance torque. On theother hands, in the case the MTPA control is applied, thecurrent phase is corresponded to the theoretical current phasethat can achieve the MTPA control operation. Thus, thereluctance torque can achieve its maximum by applying the
proposed method.

V.

EXPERIMENTAL RESULTS

Fig. 10 shows the schematic of the experimental system.
The IPMSM is driven by a 2-level inverter and the switchingfrequency is 10 kHz. In addition, the load motor is used asload machine to supply a constant torque. In this experiment,in order to confirm the effectiveness of the proposed MTPA
control, the dq-frame current is calculated from the inverter
output current using the information of magnet pole position
witch is obtained from Hall Effect sensor. However, theinformation of magnet pole position is not used in the actual
motor drive system. In addition, the motor parameters inshown Table 1.
Fig. 11 shows the experimental result of the IPMSM drive
operation using a 2-level inverter with the MTPA control
method based on the V/f control method. The experimental
conditions are follows; the rotating speed command is 0.4 p.u.
and the load torque is 0.2 p.u.. From Fig. 13, the output
current can be reduced by 68 % (from 0.72 p.u. to 0.23 p.u.)
after applying the MTPA control method. If the proposed
MTPA control is disabled, the reluctance torque cannot reachits maximum and the output current becomes large. On theother hands, if the proposed MTPA control is enabled, thereluctance torque can reach its maximum and the output
current becomes its minimum.
Fig. 12 shows the step response waveform with the MTPA
control method based on the V/f control method. The rotating
speed is 0.2 p.u.. At the 0.8 s, the load torque is varied from
0.2 p.u. to 0.4 p.u.. When the output torque increases, theoutput current increases due to the increasing the load power.
In addition, the d-axis current decreases and the output current
increases. It means that the current phase β is varied due to thevariation of the output current amplitude Ia in (9). Furthermore,
it is confirmed that the proposed MTPA control is stableagainst the step load variation.
Fig. 13 compares the amplitude of the output current when
the reactive power is varied by adding the error Qdqref_vary as
(14).
*

{

(

2

)

}

Q dq = Q dqref_vary + ω re L d X 2 + L q I a − X 2 −ψ m X (14)Current phase β [rad]Fig. 9 shows the decreasing rate of the output current with theproposed MTPA. Here, the percentage is represented in a form
that, 100% - ( Ia (with MTPA) / Ia (without MTPA) ), andtherefore the larger the percentage means that the output
current can be reduced largely. At the operation point of therotating speed and output torque are 0.2 p.u., the output
current is reduced by more than 60%. The effectiveness of the
proposed method and its characteristics are demonstrated
under various speed conditions.

Line:Theoretical value
Point:Simulation value
0.3

T=1.0 [p.u.]
T=0.8 [p.u.]

0.2

T=0.6 [p.u.]
T=0.4 [p.u.]0.1

T=0.2 [p.u.]0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rotating Speed ωrm [p.u.]

Fig.8 Comparison of current phase between simulation results and ideal
values. The theoretical results are almost identical to the simulation
results.

Current decreasing rate [%]Fig. 8 shows the current phase of the theoretical value andsimulation result when the MTPA control is applying. The
operation point is as follows: the rotating speed and output
torque are varied from 0.2 p.u. to 1.0 p.u. respectively. As theresult, at all of the simulation conditions, the theoretical results
are almost identical to the simulation results. It is confirmedthat the proposed method can achieve the MTPA control atvarious kind of load conditions.

70
T=0.2 [p.u.]
60
T=0.4 [p.u.]
50
T=0.6 [p.u.]
T=0.8 [p.u.]
40
T=1.0 [p.u.]
30
20
10
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rotating Speed ωrm [p.u.]

Fig.9 Current decreasing rates with MTPA control. The percentage isrepresented in a form that, 100% - ( Ia (with MTPA) / Ia (without MTPA)).
Grid

InverterInverter
IPM
motor

Diode
RectifierCoupling

Load
motor

Grid

PWM
Converter
Load
Fig.10 Experimental system. The IPMSM is driven by a 2-level inverter
and the switching frequency is 10 kHz.

Fig.11 Experimental results (load torque is 0.2 p.u. and rotating speed is
0.4 p.u.). The output current can be reduced by 68 % (from 0.72 p.u. to
0.23 p.u.) after applying the MTPA control method.

In the Fig. 13, the horizontal axis shows the difference
between the reactive power command and the theoreticalAs a result, the effectiveness of the proposed MTPA method
can be observed.
Fig. 14 shows the amplitude of output current when theload condition is varied. From the Fig. 15, even if the loadcondition is varying, the proposed method can reduce theoutput current. In addition, the output current can be reducedby 76 % (from 0.96 p.u. to 0.23 p.u.).

Fig. 12 Experimental results (load torque is from 0.2 p.u. to 0.4 p.u. at 0.8s, and rotating speed is 0.4p.u.). It is confirmed that the proposed MTPA
control is stable against the step load variation.

Output current Ia [p.u.]

0.6
0.5

VI. CONCLUSIONSThis paper proposed a maximum torque per ampere(MTPA) control method based on the V/f control and thevalidity of the proposed method was confirmed byexperimental results. The magnet pole position information is
not necessary because the proposed method uses the reactive
power control which is calculated on γδ-frame. From theexperimental result, it is confirmed that the output current canbe reduced by 76 % compared to not use the proposed MTPA.

0.4

REFERENCES
[1]

0.3
0.2

-0.08 -0.04
00.04 0.08 0.12
Variation of reactive power command value
Q dqref_vary [p.u.]

Fig.13 Relationship between Qdqref and Ia (load torques are 0.2 p.u. androtating speeds are 0.2 p.u.). It is confirmed that the amplitude of theoutput current becomes minimum at 0.23 p.u. when the reactive powercommand value is equivalent to the theoretical value.

Output current Ia [p.u.]

1
Withot MTPA control
0.9
0.8
0.7
0.6
0.5
With MTPA control
0.4
0.3
0.2
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rotating speed ωrm [p.u.]1

Fig.14 Variation of output current with/without MTPA control method(Load torques are 0.2 p.u.). Even if the load condition is varying, theproposed method can reduce the output current.

value Qdqref. From Fig. 5, it is confirmed that the amplitude of
the output current becomes minimum at 0.23 p.u. when thereactive power command value is equivalent to the theoreticalvalue. It means that the reactive power command from (10)can minimize the output current at the constant load torque.

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