On-line Copper Loss Minimization Control Methodof Induction and PM Motors with Periodic Fluctuation Load
Masakazu Kato, Jun-ichi Itoh
Nagaoka University of Technology
1603-1 Kamitomioka-cho
Nagaoka city Niigata, Japan
Tel., Fax: +81 / (258) – 47.9533.
E-Mail: katom@stn.nagaokaut.ac.jp
URL: http://itohserver01.nagaokaut.ac.jp/itohlab/index.html

Keywords«Permanent magnet motor», «Induction motor», «Vector control»

Abstract
This paper discusses the efficiency of permanent magnet motors (PM motors) and induction motors
when periodic fluctuation occurs in the load. PM motors generate magnet torque by permanent magnet
in the rotor. The magnet torque of PM motor is independent to d-axis current. However, if d-axis
current of PM motor change rapidly in order to achieve high efficiency control, output torque alsochange rapidly without transient current because the reluctance torque is proportional to d-axis current.
Thus, the torque response is quick according to electric time constant of the PM motor. On the other
hand, induction motors generate output torque by the secondary flux. There is a relationship of firstorder lag between the secondary flux and the excitation current in the induction motors. Consequently,
the secondary flux cannot be rapidly changed by the load torque. When the load torque varies in faster
speed, the high efficiency control method increases the copper loss because the transient current flows
in the excitation current. This paper clarifies the minimum copper loss control method for induction
motors with torque fluctuation. When the load fluctuation is faster than the secondary flux of theinduction motor, the high efficiency control method using the effective value of the load torque is used.
The copper loss in the load fluctuation condition is analyzed in order to derive the boundary condition
that copper loss is minimized. The validity of the boundary condition is confirmed by the experimental
results using 3.7-kW induction motor.

I. Introduction
Electric motors are widely applied to industry applications. Permanent magnet synchronous motors(PM motors) have some advantages such as high efficiency, high power factor compared withinduction motors. In contrast, induction motors have some advantages such as low cost and highreliability in electric motors. In order to improve the efficiency of the electric motor, high efficiency
control methods have been proposed based on the vector control [1-4]. Basically, the high efficiency
control methods for the electric motor are discussed under constant torque or step torque conditions[5][6].
However, the load torque is not just constant value depending on applications. Some loads havetorque fluctuations such as belt conveyors, compression machines, and so on [7][8]. In copper loss
minimization control methods for PM motor, the armature current is controlled depending on the load[5]. In contrast, in the copper loss minimization control methods for the induction motor, the magnetic
flux is controlled by the excitation current depending on the load torque [6]. PM motor generates
output torque by permanent magnet in the rotor. The magnet torque of PM motor is independent to daxis current. However, if d-axis current of PM motor change rapidly in order to achieve highefficiency control, output torque also change rapidly because the reluctance torque is proportional tod-axis current. On the other hand, induction motor generates output torque by secondary flux. There isa relationship of first order lag between the secondary flux and the excitation current. Consequently,
the secondary flux cannot be rapidly changed by the load torque. When the copper loss minimization
control method is applied to the induction motor in which the load torque fluctuation occurs, thecopper loss increases. This is because the transient current of the excitation current increases thecopper loss when the load torque varies faster than the rotor time constant. In order to compensate forthe secondary flux using excitation current, the torque current is drastically increased by the highefficiency control in case of the torque fluctuation. As a result, the copper loss increases because theeffective value of the torque current is increased. In order to solve this problem, an average highefficiency control method has been proposed [9]. Consequently, in order to achieve high efficiency,
different kinds of high efficiency control methods should be used from the relationship between thetime period of the load torque fluctuation and the rotor time constant. However, the boundary
condition depends on the relationship has not been reported in past works.
In this paper, the boundary condition which achieves minimum copper loss is derived in order toachieve high efficiency for induction and PM motors. The copper loss by each condition is analyzed inorder to obtain the boundary condition which achieves minimum copper loss. If it is assumed that theload fluctuation is sinusoidal wave, then the boundary condition is decided from the relationshipamong the fluctuation frequency, the rotor time constant, the torque amplitude and its average. The
remainder of this paper is organized as follows; first, the high efficiency control is introduced based on
the vector control. Second, the copper losses in the constant torque and the fluctuation torque are
analyzed in order to derive the boundary condition. Finally, the experimental results are shown inorder to confirm the validity of the designed boundary condition.

II. Copper loss minimization control method
A. Periodic fluctuation load
Figure 1 shows a periodic fluctuation of the load torque. As first step, the torque is fluctuated at asingle frequency without harmonic components in this paper. Note that a is the ratio between thetorque amplitude Tm and the average torque T0. There are two methods to achieve high efficiency inthe periodic fluctuation load as follows; the high efficiency control by controlling the excitation
current based on instantaneous value or constant value.
Tc : Constant value of
Tm : Torque amplitude
load torque

TorqueTc
T0

T0 : Average torqueT  T0  Tm sin t T0 1  a sin t 

where a 

Tm
T0

Time

Figure 1. Fluctuating load torque. The fluctuation of load torque can be analyzed
as sinusoidal functions. Note that, a is the ratio between the torque amplitude Tm
and the average torque T0.

B. Copper loss minimization control for PM motors
Figure 2 shows the equivalent circuit of PM motor. In the copper loss minimization control methodof PM motor, the armature current is adjusted by the load torque. When the excitation current isdecreased, the copper loss can be decreased at same time. In this paper, the iron loss is not taken into
consideration for simplicity. Note that, the iron loss minimization is an important challenge for thehigh efficiency control. However, in heavy load, the copper loss is dominant compared to the iron loss.
Therefore, this paper considers the copper loss minimization in order to improve the efficiency in theheavy load. The instantaneous copper loss pc_PM in Figure 2 is given by (1).

2

pc _ PM  3Ra I a 



3
2
2
Ra id  iq
2



(1)

where, Ra is the armature winding resistance, id and iq are the d- and q-axis components of thearmature current. The q-axis current is given by (2).
iq TPf y m  Ld  Lq id (2)where, Pf is the number of pole pairs, Ld and Lq are the d- and q-axis components of the armature selfinductance, and ym is the flux linkage of the permanent magnet. It is noted that pc_PM is calculated bysubstituting (2) into (1). The condition of minimizing copper loss can be derived by differentiating
pc_PM with respect to id and equating the derivatives to zero. As a result, the copper loss minimization
condition is given by (3).
id 

T2
Pf

2

Ld  Lq(3)

y  L  L i 

3

m

d

q

d

When the torque T is given, the current which achieves minimum copper loss can be derived from(3). If PM motor has not saliency, the current id is zero.
The current iq is decided by substituting the derived optimal current id. If the current id is derived byinstantaneous torque, the copper loss is not changed transiently. This is because there is no
relationship of first order lag. Therefore, in case that the load torque fluctuation frequency is slower
than natural angular frequency of ACR, the high efficiency control method for PM motors is achievedby d-axis current control according to load torque fluctuation.

id

iq
Ld

Ra

(a) d-axis equivalent circuit

Ra

Lq

reym

(b) q-axis equivalent circuit

Figure 2. Equivalent circuit of PM motor. In this paper,
the iron loss is not taken into consideration for simplicity.

C. Copper loss minimization control method for induction motors
Figure 3 shows the equivalent circuit of induction motor. In the high efficiency control method, theexcitation current is adjusted by the load torque. When the excitation current is decreased, the copper
loss can be decreased at same time. The instantaneous copper loss pc_IM in Figure 3 is given by (4).
pc _ IM 

3R1  R2 iq 2  R1id 2
2



(4)

where R1 is the primary winding resistance, R2 is the secondary winding resistance and id is theexcitation current. where L2 is the self-inductance, Pf is the number of pole pairs, M is the mutual
inductance, and 2d is the magnetic flux at secondary side.
The torque current is given by (5).
iq 

L2 T
L2 T

Pf M 2 d Pf M 2 id

(5)

It is noted that pc_IM is calculated by substituting (5) into (4). In addition, the excitation current id isobtained by differentiating Pc. Further, the zero point of the differential value is calculated. As a result,
id_min at the minimum loss is expressed as (6).
id _ min  4

R1  R2
R1

L2T
Pf M 2(6)

According to (6), the excitation current at the minimum loss, which depends on the load torque, iscontrolled by the copper loss minimization control method.
The excitation current using the instantaneous torque T(t) is given by (7).
id 

1 R1  R2
4
M
R1

L2T0 1  a sin t 
Pf

(7)

Further, there is a relationship of first order lag between the excitation current id and the secondary
flux 2d. According to (7), the relationship of id and 2d is expressed as (8).
2R  R2d
 2 d   2 d  Mid  4 1
dt
R1

L2T0 1  a sin t 
Pf

(8)

where, 2 is the rotor time constant. When it is defined as non-dimensional time function k(t), thesecondary flux 2d is obtained by (10).
dk (t )  k (t )  1  a sin tdt
R  R2 T0 L2
 2 d  k t 4 1
R1
Pf

(9)

2

(10)

In order to solve for k(t), it is necessary to calculate the right formulas in (9) by Fourier transform.
Additionally, k(t) is solved by calculating the frequency response, that the amplitude of each frequency
component is used. Thus, the torque current iq is expressed (11).
iq 

k iq
L2 T
R1
4

Pf M  2 d M R1  R2

k iq 

T0 L2
Pf

(11)

1  a sin t
k t 

(12)

where kiq is the non-dimensional time function. In addition kiq is depending on 2. As a result, thecopper loss by copper loss minimization control method which considers the instantaneous value ofthe load torque, Pc_ins can be expressed as (13).
Pc _ ins  R1 R1  R2 



L2T0
2
1  k iq _ rms
pM 2



(13)

It is noted that kiq_rms is the effective value of kiq. Therefore, the copper loss by the copper loss
minimization control method, which uses the instantaneous torque, is calculated by the motor
parameters and kiq_rms.
Figure 4 shows the relationship between 2 and non-dimensional parameter kiq_rms. The nondimensional parameter kiq_rms is a function of 2 and a. Therefore, the copper loss Pc_ins depends on
2 and a because Pc_ins depends on kiq_rms.

a = 1.0
iq
R2

l1

R1

M

l2
1 s
R2
s

id

a = 0.9
a = 0.8

kiq_rmsi1

a = 0.5
a = 0.2

2
Figure 3. Equivalent circuit of induction motor.
In this paper,the iron loss is not taken into
consideration for simplicity.

Figure 4. Relationship between 2 andnon-dimensional parameter kiq_rms. The copper
loss Pc_ins is depending on 2 and a because
Pc_ins is depending on kiq_rm.

D. Copper loss minimization control method based on constant value of instantaneous
torque for induction motors
The excitation current by using a constant torque Tc and the torque current are expressed as (14) and(15), respectively.
id 

1 R1  R2
4
M
R1

L2Tc
Pf

(14)

iq 

R1
1
4
M R1  R2

T0 L2
1  a sin t 
Pf

(15)

In addition, the copper loss of the copper loss minimization control based on the constant value ofthe instantaneous torque Pc_const can be expressed as (16).
Pc _ const  R1

1 2 2
1 2 2i d dt  R1  R2 
i q dt2 02 0 R1 R1  R2 T
L2 
 T  rms
2  c
Tc
Pf M 

2

(16)






where Trms is the effective value of the load torque T(t). The effective value is defined by (17).
Trms 

T
1 2
22
T t  dt  To  m
2 0
2

2

(17)

Figure 5 shows the relationship between the copper loss of the copper loss minimization control Pc
and the ratio between the constant torque Tc and the average torque T0. The average torque T0 is set to0.2, whereas the load characteristic a is set to 0.6. The function of the copper loss is concave up. In
addition, the point where Tc equals to Tm does not achieve the minimum copper loss. The constanttorque is obtained by differentiating Pc. Furthermore, the zero point of the differential value iscalculated. Consequently, the constant torque value that can be achieved the minimum copper loss isexpressed as (18) and (19).
dp c
0
dTc

(18)

Tc  Trms(19)

According to (19), the constant value achieve the minimum copper loss is the effective value of theload torque. The minimum copper loss using the effective value of the load torque is can be expressedas (20).

Pc _ rms  2 R1 R1  R2 

L2Trms
Pf M 2(20)

The minimum copper loss using the effective value of the load torque is independent from 2.
Hence, the copper loss is suppressed when the load torque is varied faster than the rotor time constant.
The minimum copper loss control of the induction motor is more effective than that of the PM
motor. Therefore, this paper proposes on-line copper loss minimization method for the induction
motor. In addition, this paper conducts experiments in order to confirm the utility of proposed method.
The function of copper loss Pc_const is concave up.

Pc_const [W]

Pc _ const  R1 R1  R2 

2
T 
L2 
Tc  rms 
Tc 
Pf M 2 Constant torque = average torqueminimum point ofcopper loss

0.8

1.0

Tc

1.2

1.4

T0

Figure 5. Relationship between the copper loss of the copper loss minimization control
and the ratio between the constant torque and average torque. Note that the point that Tc
equivalent to Tm is not minimum copper loss.

III. Proposed copper loss minimization method for periodic fluctuation load
According to (20), Pc_rms is constant value. In contrast, according to (13), Pc_ins depends on therelationship between the rotor time constant 2 and the angular frequency of the load torque fluctuation
. Therefore, the boundary condition of the minimum copper loss based on the relationship between 2and  is calculated from the intersection pint between the function of Pc_rms and the function of Pc_ins.
From (20) and (13), the boundary condition is given by (21).
2k iq _ rms  2 1 

a2
1
2

(21)

From (21), in the periodic fluctuation load, the torque characteristic a and 2 determine theexcitation current which achieves the minimum copper loss.
However, it is necessary to clarify the load characteristic of the induction motor in order to control
the flux by the load torque characteristic. In other words, the real load characteristic of the induction
motor should be measured. In order to solve this problem, an on-line control method is proposed inorder to control the excitation current. This method can decide the excitation current by calculation thefrequency of load torque.
Figure 6 shows the calculation method of load torque frequency. First, the average torque iscalculated by the integration value of the instantaneous torque. Second, subtract the average torquefrom the instantaneous torque. Then, the load frequency is calculated by counting the number of zerocross points N. The load frequency is calculated by (22).
f 

Nt 2  t1

(22)

Figure 7 shows the block diagram of the proposed copper loss minimization control method. The
online excitation current control block calculates the load torque frequency according to figure 6. Inaddition, the online excitation current control block determines the excitation current which achieves
the minimum copper loss according to (21).
Torque(1) Calculation of average value
average value

0

t2

t1

Time

Torque(2) Subtract average value form instantaneous value
instantaneous value
average value

0

Time

Torque(3) Count zero cross point from t1 to t2
0

Time
t1

t2

N

(4) Calculation of load torque frequency f N[Hz]t 2  t1

Figure 6. Calculation method of load torque frequency. (1) Average torque iscalculated by the integration value of the instantaneous torque. (2) Subtract
the average torque from the instantaneous torque. (3)Count zero-crossing
points. (4) Load frequency is calculated by N.
T*

Online exiting
current control

id
ωr

i*
id calc. d +Eq.(5)
Eq. (3)

ACR

i*
+ q
- ACR

+

+
Calculate
slip

id
iq

1
s

vd*

vq*

vu*
dq⇒
vv*
uvw
vw*

q
uvw
⇒dq

iu
iv
iw

Figure 7. Block diagram of the proposed copper loss minimization control method.
The online exiting current control block determines the exciting current which
achieves minimum copper loss according to (19).

IV. Experimental results
Figure 8 shows the schematic of the experimental system. It is noted that the induction motor isdriven by a 2-level inverter and the switching frequency is 10 kHz. In addition, the load motor is used
as load machine to supply periodic fluctuation loads. Table 1 lists the motor parameters.
Figure 9 shows the copper loss based on experimental results. Note that the ratio between theconstant torque and the average torque a is 0.6. The copper loss of the copper loss minimization
control method using the instantaneous torque depends on the load frequency. On the other hand, thecopper loss of the copper loss minimization control method using the effective value of load torque isconstant. The error between the calculation boundary condition is 6.7%. The excitation current ischanged by the on-line excitation current control considering the load torque condition.
Figure 10 shows the boundary condition of minimum copper loss based on the calculation result andexperimental result. Note that 2 = 1 means the angular frequency of the load torque fluctuation  issame as the rotor time constant 2. The error between calculation result and experimental result is
7.3%. The area achieves minimum copper loss using the instantaneous torque narrow by 2.
Figure 11 shows the waveforms of each copper loss minimization control at operation point A
shown in figure 10 (a = 0.2, 2 = 1.1). Figure 11 (a) shows the copper loss minimization control
method using the instantaneous torque, and (b) shows the copper loss minimization control method

using the effective value of the load torque. In Figure 11 (a), the excitation current is changed inaccording to the torque current. In addition, the phase of the excitation current is as same as the phaseof the torque current. In this condition, the copper loss is 86.8 W. On the other hand, in Figure 11 (b),the excitation current is constant. In this condition, the copper loss is 87.9 W.
Figure 12 shows the waveforms of the each copper loss minimization control at operation point B
shown in figure 10. (a = 0.2, 2 = 2.6). Figure 12 (a) shows the copper loss minimization control
method using the instantaneous torque, and (b) shows the copper loss minimization control method
using the effective value of the load torque. In addition, in Figure 12 (a), the copper loss is 88.7, and in(b), the copper loss is 88.1 W. In Figure 12 (a), the excitation current is ahead of the torque current.
The fluctuation frequency is higher two times than the rotor time constant. As a result, the copper loss
increases because the root mean square value of the torque current is increased by the flux fluctuation.
Figure 13 shows the waveforms of the each copper loss minimization control at operation point C
shown in figure 10. (a = 0.8, 2 = 2.6). Figure 13 (a) shows the copper loss minimization control
method using the instantaneous torque, and (b) shows the copper loss minimization control method
using the effective value of load torque. In Figure 13 (a), the transitional torque current increases
compared to point B. The transitional torque current is due to an increase of a. As a result, theboundary becomes narrow shown in Figure 10.

Grid

InverterIM
Coupling

Inverte
r

Load
motor

Grid

PWM
Converter
Load
Figure 8. Experimental system. The induction motor is driven by
a 2-level inverter and the switching frequency is 10 kHz
DiodeRectifier

Table 1. Motor parameters.
Parameter
Poles
Rated power
Rated voltage
Rated current
Rated speed
Primary resistance R1Secondary resistance R2
Primary leakage inductance l1Secondary leakage inductance l2
Mutual inductance M
Time constant 2

Copper loss [W]Boundary line ofminimum copper loss
(calculation result)
Pc_rms94

92
1.5

1.24 mH
34.3 mH
0.528 sec

Pc_rms (Experimental result)

95

93

Value
4
3.7 kW
188 V
18.0 A
1500 r/min
0.414 W
0.423 W
1.24 mH

Pc_ins (Experimental result)
Online exciting current control
Pc_insOnline exciting
current control
Error : 6.7%

1.6

Boundary line ofminimum copper loss
(experimental result)

1.7

1.8

2
Figure 9. Copper loss of the on-line minimum copper loss control method based on
experimental result. The error between calculation boundary condition is 6.7%.
The exiting current is changed by on-line exiting current control considering
the load torque condition.

The area achieve minimum copper loss
using torque of effective value

3.0

・B

2

2.5

Boundary point based on experimental result

・C

Boundary line based on calculation result

Maximum error is
7.3%.

2.0
1.5

The area narrow by 2.

1.0

・A

0.5

The area achieve minimum copper loss
using instantaneous torque

0
0

0.2

0.4

0.6
Tm 

a 
T0 


0.8

1.0

Figure 10. Boundary condition of minimum copper loss based on the calculationand experimental results. The area achieves minimum copper loss using instantaneous
torque narrow by 2.
motor speed [1.0 p.u./div]0

Exiting current vary based on load torquemotor speed [1.0 p.u./div]0

Exiting current is constant value
exiting current [0.2 p.u./div]

exiting current [0.2 p.u./div]00torque current [0.2 p.u./div]

torque current [0.2 p.u./div]00Phase of exiting current is corresponded to phase of torque current
u-phase output current [10.0 A/div]

100 [ms/div]0u-phase output current [10.0 A/div]0(a) Using the instantaneous torque.

100 [ms/div](b) Using effective value of torque.

Figure 11. Waveforms of each copper loss minimization control at operation point A
shown in figure 10.(a = 0.2, 2 = 1.1). In (a), the excitation current is changed inaccording to the torque current. In addition, the phase of exiting current is as same as
the phase of torque current. On the other hand, in (b), the exiting current is constant.

motor speed [1.0 p.u./div]

motor speed [1.0 p.u./div]0

Exiting current vary based on load torquephase difference between exiting current and torque current
exiting current [0.2 p.u./div]000
torque current [0.2 p.u./div]00u-phase output current [10.0 A/div]

040 [ms/div]0(a) Using the instantaneous torque.

Exiting current is constant value
exiting current [0.2 p.u./div]

torque current [0.2 p.u./div]

u-phase output current [10.0 A/div]

40 [ms/div](b) Using effective value of torque.

Figure 12. Waveforms of each copper loss minimization control at operation point B
shown in figure 10.(a = 0.2, 2 = 2.6). In (a), the exiting current is ahead of the torquecurrent. As a result, the copper loss increases because the root mean square value ofthe torque current is increased by flux fluctuation.

motor speed [1.0 p.u./div]0

Exiting current vary based on load torqueexiting current [0.2 p.u./div]0

torque current [0.2 p.u./div]0u-phase output current [10.0 A/div]

040 [ms/div](a) Using the instantaneous torque.

motor speed [1.0 p.u./div]000

Exiting current is constant value
exiting current [0.2 p.u./div]

torque current [0.2 p.u./div]

u-phase output current [10.0 A/div]

040 [ms/div](b) Using effective value of torque.

Figure 13. Waveforms of each copper loss minimization control at operation point C.
(a = 0.8,  2 = 2.6). The transitional torque current increases compared to figure 12.
The transitional torque current is due to an increase of a. Therefore, the boundary becomes narrow.

V. Conclusion
This paper discusses the on-line copper loss minimization control by the excitation current control
when the periodic fluctuation occurs in the load. The high efficiency control of PM motors, if the load
torque fluctuation frequency is slower than natural angular frequency of ACR, the high efficiency
control of PM motors is achieved by d-axis current control independently of load torque fluctuation
frequency. Besides, the high efficiency control of induction motors has two control methods according
to the relationship between the load torque fluctuation frequency and the rotor time constant. In thispaper, the copper loss by each control method is theoretically calculated in order to design theboundary, where the copper loss is minimized. The error between calculation result and experimental
result is 7.3%. From the experimental result, the trend of the boundary is confirmed. In addition, theon-line control method is proposed in order to control the excitation current. This method can decide
the excitation current by calculation the frequency of the load torque. The error between thecalculation boundary condition and the experimental boundary condition is 6.7%.The proposed
method achieves the copper loss minimization by calculation the excitation current depends on thetorque fluctuation.

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