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Engineering, Technology & Applied Science Research Vol. 12, No. 6, 2022, 9536-9545 9536 
 

www.etasr.com Nasir: Determination of the Harmonic Losses in an Induction Motor Fed by an Inverter 

 

Determination of the Harmonic Losses in an 
Induction Motor Fed by an Inverter 

 

Bilal Abdullah Nasir 

Department of Electrical Techniques 

Northern Technical University 
Mosul, Iraq 

bilalalnasir@ntu.edu.iq 
 

Received: 24 April 2022 | Revised: 16 May 2022 | Accepted: 19 May 2022 

 

Abstract-The advancement, development, improvement, and 

increased use of power electronic converters led to the efficient 

speed control of electrical drives. The most famous three-phase 

induction motor-related control to Pulse-Width Modulation 

(PWM) technique is used to operate multilevel inverters such as 

variable-frequency or six-step Voltage Source Inverter (VSI). 

Switching devices of the inverter are used in the drive systems 

and act as the main source of harmonics. When the induction 

motor is fed from the PWM inverter, it will be supplied by low 

order (5th, 7th, 11th) time harmonic voltage. The motor 

performance is affected by the presence of these time harmonic 

components because the additional losses generated in the motor 

defect its performance, generate pulsating torque, and reduce 

efficiency. In this work, the analysis of a dynamic model of an 

induction motor in transient and steady-state operation is 

developed, considering the effect of time-harmonic voltages 

generated by the inverter, skin effect, skew effect, temperature 

rise effect, iron core loss, stray load loss, and magnetic saturation 

on the motor performance. The performance of the motor is 

studied by the time-harmonic equivalent circuit and by the 

fundamental equivalent circuit. The motor performance in terms 

of efficiency and power factor is compared with the experimental 

results for both sinusoidal and VSI motor feeds in order to 

validate the model accuracy. 

Keywords-dynamic modeling; additional losses; 3-phase bridge 

inverter; 3-phase diode rectifier; harmonic equivalent circuit; 

switching functions 

I. INTRODUCTION  

It has been observed that the electrical machine losses 
increase if the supply voltage has harmonics. This gained 
attention at the early '70s, as the machines were supplied with 
static six-step inverters. The additional Ohmic losses were 
produced in the machine windings due to the flowing of 
harmonic currents. The Ohmic losses in the rotor windings of 
the induction machines can increase considerably due to the 
skin effect. There is also an increase in the machine iron losses 
due to the flowing of harmonic currents [1-5]. At low harmonic 
frequency (< 2.5kHz), the harmonic losses are also increased 
with the machine load due to the slight reduction of the 
machine inductances caused by the saturation effect [6-7]. 
Measuring the harmonic losses in electrical machines is a very 

complex task, and its determination by subtracting the output 
power from the input power is very sensitive to errors of 
measurement [8]. At the beginning of this century, the 
calculation of harmonic losses was extended by numerical 
methods with computer analysis based on the Finite Element 
Method (FEM) [9-11]. In numerical methods, the 
computational process is a heavy burden and the complete 
structure of the machine has to be known [12]. The early 
adopted six-step inverters showed a defect in machine 
performance due to the pulsating torques and increasing 
harmonic losses [13-14]. In modern semiconducting 
techniques, the switching frequency and modulation index of 
PWM variable speed drives can be allowed to be further 
increased and eliminate all harmonics below the switching 
frequency. Although this technique can increase the machine 
performance and reduce noise emissions, the machine 
harmonic losses cannot be reduced considerably by increasing 
the switching frequency above 10kHz [15-24]. To model the 
harmonic losses of the induction machines with different power 
ratings, a complete equivalent circuit including all harmonic 
loss components was first presented in [5]. In this circuit, the 
resistances are connected in parallel to the leakage inductances 
accounting for the harmonic stray load losses in the machine 
windings. The harmonic losses due to the skew-effect and skin-
effect are not included in this equivalent circuit model. All the 
equivalent circuit impedances must be frequency-dependent to 
take effect on the skin [25-28]. 

Variable speed drives are commonly used in many 
industrial applications. To supply the motor with variable 
frequency, the induction motor is fed from the six-step 
uncontrolled rectifier and the rectifier output is connected to 
the six-step Voltage Source Inverter (VSI). Two capacitors are 
connected across the rectifier output to remove the ripple. Due 
to the motor being fed from the switching inverter, the dynamic 
model developed to study the motor performance must be valid 
for any voltage and current waveform. A complete dynamic 
model is required for harmonics analysis. The model must 
include the main parameters of the machine for transient and 
steady-state operation. In this paper, an accurate dynamic 
model of the induction motor fed from a six-step VSI is 
presented using the switching functions for inverter modeling 
and the D-Q axes synchronously rotating reference frame 

Corresponding author: Bilal Abdullah Nasir



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www.etasr.com Nasir: Determination of the Harmonic Losses in an Induction Motor Fed by an Inverter 

 

model for induction motor modeling. In this comprehensive 
modeling, the harmonics, as additional losses, are calculated 
and analyzed easily. The accuracy of the model has been 
verified through Matlab simulations and the harmonic 
equivalent circuit of the machine, which is presented in this 
work as a powerful tool for motor performance calculation. 

II. MODELING OF THE INDUCTION MOTOR 

An accurate estimation parameter of the induction motor-
equivalent circuit is required to implement these parameters for 
high-performance determination. The stator and rotor winding 
temperatures can be determined according to the IEEE standard 
112-B by measuring the stator no-load and load currents, and 

the rotor current is determined at different load conditions. The 
skin effect on the stator parameters can be determined from the 
stator geometry. The iron core resistances are determined from 
the no-load test considering the effect of slip and core 
temperature. The magnetizing reactance is calculated from the 
no-load test at different supply voltages and rated frequencies 
to consider the saturation effect. The stator and rotor winding 

resistances �� and �� � depend on temperature variation and skin 
effect. The stator and rotor leakage reactances �ℓ�  and �� ℓ�  
depend on skin effect and saturation. The magnetizing 
inductance ��  depends on the magnetic saturation. The iron 
core resistance ��	  depends on motor slip and iron core 
temperature. 

 

 
Fig. 1.  The proposed fundamental circuit of induction motor. 

The proposed equivalent circuit in Figure 1 contains the 
additional resistances ����ℓ  and ����ℓ  that model the stray 
losses in the stator and rotor circuits. The stray losses depend 
on the voltage drops due to the leakage reactance, the iron core 
resistances, and stator and rotor resistance. Also, the proposed 
equivalent circuit can deal with the effect of rotor skewing on 
the parameters of the rotor circuit. The iron core resistance is 
varied with the temperature and air-gap voltage and is 
calculated from the no-load test of the motor as [25]: 

��	 = ��
 ∕ ��� ∗ (��ℓ − ��
 ∗ �� )�   (1) 
�� = �� ∗ �� = �� ∗ �� ∗ sin (ϕ�)    (2) 

�� =  !"ℓ#$%&∗'ℓ(') *
+&
    (3) 

where �� is the magnetizing voltage per phase and is calculated 
in terms of magnetizing reactance �� , magnetizing current �� , 
no-load current �� , and no-load sine of power factor angle 
sinφο. The iron core resistance was obtained in terms of air-gap 
voltage from the no-load test and curve-fitting technique [25]. ��= (1-D) is the temperature coefficient of the iron core loss, 
and D is the iron core power loss varying rate per , �  
determined as [25]:  

- = .��	 (�/) − ��	 (�)0 ��	 (�/)1     (4) 
where ��	 (�/)  and ��	 (�) represent the iron core power losses 
at ambient temperature and at any temperature. These iron core 
power losses are measured from the no-load test. ��ℓ  is the 
active no-load power loss per phase, ��� is the stator resistance 
per phase at ambient temperature, 2�ℓ is the reactive no-load 

power loss per phase, and �ℓ�  is the stator leakage reactance 
per phase. The rotor load resistance ��� �(1 − 4)/4 is derived in 
terms of rotor phase resistance �� �  and rotor series stray loss 
resistance ����ℓ  as: 

��� �(1 − 4) 4⁄   = 1  1 78� 9(:#�)� − ����ℓ; − 1 �<=⁄1 *1     (5) 
where �<=  is the mechanical loss resistance. The nonlinearity 
of the machine iron core and mechanical resistances due to the 
saturation of the magnetic core is considered by the dynamic 
model to improve the accuracy of simulation results. The 
resistances ��	 and �<= are determined practically from the no-
load test at zero slip to be modeled by polynomial-curve fitting 
as: 

��	 = −0.0001985 ∗ ��C + 0.111 ∗ ��
 − 23.11 ∗ �� + 840    (6) 
�<H = −6.167 ∗ ��K + 0.00419 ∗ ��C − 1.074 ∗ ��
 +127.4 ∗ �� + 560   (7) 

The resistance of the iron core ��	 can be referred to on the 
stator and rotor circuits as a voltage drop to keep the dynamic 
4

th
 order model fixed while the effect of the iron core loss in 

the dynamic model of Figure 2 is considered. The reflecting 
resistances on the stator circuit ����	  and on the rotor circuit ����	 are calculated as [26]:  

����	 = (��	 ∗ L� ∗ MN ) ∕ ���	
 + (L� + MN )
�    (8) 
����	 = 4 ∗  ����	     (9) 

where 4 is the motor slip and MN  is the magnetizing inductance 
per phase obtained from the no-load test as: 



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MN = �� ∕ L�    (10) 
where L� = (2 ∗ O ∗ P) is the angular frequency in rad/s.  

The modified magnetizing inductance M�  of the dynamic 
model (Figure 2) is calculated as: 

M� = MN ∗ ��	
 ∕ ���	
 + (L� ∗ MN )
�    (11) 
The stray load loss resistances vary with the variation of 

stator and rotor resistances due to the temperature variation. 
These resistances are derived and introduced in the dynamic 
model as series resistances in the stator and rotor circuits [27]: 

����ℓ = (L� ∗ Mℓ� )
 ∗ �� ∕ ���
 + (L� ∗ Mℓ� )
�    (12) 
����ℓ = Q 
 ∗ (L� ∗ M´ℓ� )
 ∕ �´�     (13) 

where ����ℓ  and ����ℓ  are the series stator and rotor winding 
stray-load loss resistances respectively, Q is the motor slip, and Mℓ�  is the rotor leakage inductance per phase. 

All induction motors and generators operate in the 
saturation region and their characteristics are non-linear. The 
variation of magnetizing reactance ��  is the main factor in 
producing the magnetizing voltage ��, which is the main factor 
that produces the iron-core losses, rotor current, mechanical 
losses, and output power. The saturation reduces �� and tends 
to increase. The saturation effect is taken by the relation  �� = P(��). This function is obtained practically from the no-
load test by varying the supply voltage from 125% to 25% of 
the rated value. Then the measurements of ��  and ��  can be 
interpolated by the curve fitting technique. 

 

(a) 

 

 

(b) 

 

Fig. 2.  (a) D-axis and (b) Q-axis equivalent circuit models of the induction motor in the synchronously rotating frame. 

The rotor phase leakage reactance ��� ℓ�  is corrected by the 
skew factor ��S  [25]:  

��� ℓ� = ')T(U& V1 − ��S
 W + '
� ℓ9

T(U&     (14) 

��S = 4XY Z[∗\]!9 ^ ∕ VO ∗ �_ ∕ 2� W    (15) 
where �_  is the machine magnetic pole-pair, and 2�  is the 
number of rotor slots (bars). 

The rotor phase resistance is corrected by the skew factor 
as: 

��� � = �� � ∕ ��S
     (16) 

The rotor phase resistance and leakage reactance are 
corrected by the skin effect as [25]:  

���� � = ��� � ∗ . 0.5 + 0.5 ∗ `Q� Q�⁄   0    (17) 
���� ℓ� = ��� ℓ� ∗ . 0.4 + 0.6 ∗ `Q� Q�⁄   0    (18) 

where Q�  is the rated machine slip and Q� is the machine slip at 
maximum torque which can be given as [27]: 

Q� = (�� � + ����ℓ) ∕ .(�� + ����ℓ)
 + (�ℓ� + �� ℓ� )
0½    (19) 
Also, the machine slip can be corrected due to the 

temperature effect as [25]: 



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Q��,� = Q�,� ∗ (c� + �d ) ∕ (c�e< + �d )    (20) 
where c�  is the rotor bar temperature at any load, c�e<  is the 
temperature of insulation class level, which can be obtained 
from the insulation class table, and �d  is a constant equal to 
225 for aluminum rotor bar type. 

The dynamic electrical equations of the induction motor are 
derived from the proposed equivalent circuit in Figure 2 as:  

f�g = (�� + ����ℓ + ����	) X�g +  ����	 ∗  h́�g − L� ∗ M� ∗ h́�S    (21) 
f�S = L� ∗ (Mℓ� + M�) ∗ X�g + L� ∗ M� ∗  h́�g    (22) 

j = ����	 ∗ X�g + Z���� � + ����ℓ + ����	^ ∗ h́�g + L�ℓ ∗
ZM��� ℓ� + M�^ h́�S    (23) 

j = L�ℓ ∗ M� ∗ X�g + L�ℓ ∗ ZM��� ℓ� + M�^ ∗ h́�g +
Z���� � + ����ℓ + ����	^ ∗ h́�S     (24) 

where L�ℓ  = L� − L�  =  4 ∗  L�  is the slip speed in rad/s, L� = �_ ∗ L�  is the electrical angular speed of the motor in 
rad/s, and k�g  , k�S  , k� �g  and k� �S are the stator and rotor flux 
linkages in the D-Q axes of Figure 2 respectively, defined as : 

k�g = Mℓ� ∗ X�g + M� ∗ X�g     (25) 
k�S = Mℓ� ∗ X�S + M� ∗ X�S     (26) 
k� �g = M��� ℓ� ∗ h́�g + M� ∗ X�g     (27) 
k� �S = M��� ℓ� ∗ h́�S + M� ∗ X�S     (28) 

where X�g , h́�g , X�S, and h́�S are stator and rotor currents in the D 
and Q axes respectively. 

The rotor mechanical speed L� is calculated from the 
dynamic mechanical model as: 

g=)
gl = :m Vce − cℓ − �< ∗ L�W    (29) 

where ce  is the electromagnetic torque of the motor, calculated 
as [27]: 

ce = ZC
^ ∗ �_ ∗ M� ∗ (X�g ∗ h́�S − X�S ∗ h́�g )    (30) 
where cℓ  is the motor load torque (N.m), n is the moment of 
inertia (��. o
),  �_ is the magnetic pole-pair number, and �<  is 
the viscosity resistance factor (N.m/(rad/s)). 

III. MODELING OF THE SIX-STEP VOLTAGE SOURCE 
INVERTER 

The Matlab simulation tool was utilized for the design and 
analysis of the power inverter system. The switching function 
method can be used to represent the inverter itself. Inverter 
harmonics and switching losses can be calculated through 
simulations. The switching function method is used when the 
exact physical forms of the power electronic switches such as 
transistors and thyristors are not of concern. Using the concept 
of the switching function, inverters can be modeled for 
performance calculation. The transfer function model has 
several advantages in modeling VSIs. The power circuit of the 
inverter can be simplified into input and output variables and 
calculated easily, the inverter topologies can be derived easily, 
and the implementation of gating control pulses are simpler. 
Based on the theory of the transfer function of the VSI, the DC 
input voltage �g	  to the inverter and the output currents (Xd, Xp, X	 ) are independent variables, while the input current �g	  and 
the output line voltages (qdp , qp	 , q	d ) are dependent variables. 
These variables are related as [20]: 

[qdp  , qp	  , q	d  ] = [TF] [�g	 ]    (31) 
[�g	 ] = [TF]   [ Xd , Xp , X	 ]    (32) 

where [TF] is the transfer function of the VSI, expressed in the 
form of switching functions of the three-phase bridge VSI. 
Figure 3 shows the schematic diagram of the whole system 
containing the VSI fed 3-phase induction motor system. The 
sinusoidal 3-phase supply feeds the 3-phase uncontrolled diode 
bridge rectifier. A DC link with two capacitances is connected 
across the output diode rectifier to remove the ripple. The VSI 
behaves as a 3-phase voltage source to the induction motor. 

 

 
Fig. 3.  Rectifier-inverter-induction motor system. 

Let Q:, Q
, QC, QK, Qr, and Qs be the switching functions for 
the three legs (a, b, c). Q: and QK for pole (a), QC and Qs for pole 
(b), and Qr  and Q
  for pole (c). The switching function is 
defined as 1 when the switch is ON and 0 when the switch is 
OFF. By applying Kirchhoff's voltage law to the closed-loop in 

the circuit in Figure 3, taking the conditions corresponding to 
the switching functions for each leg and after series 
manipulation, the motor (load) phase voltages become [29]: 

qd/ =  tuvs (2Q: + Q
 − QC − 2QK − Qr + Qs)    (33) 



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qp/ =  tuvs (2QC + Q
 − Q
 − 2Qs − Qr + QK)    (34) 
q	/ = tuvs (2Qr + QK − QK − 2Q
 − QC + Qs)    (35) 

The input DC (�g	 ) to the 3-phase inverter is expressed in 
terms of switching functions ( Q: ⟶ Qs)  and inverter output 
currents (Xd , Xp , X	) as : 

�g	 = (Q: − QK). Xd + (QC − Qs). Xp + (Qr − Q
). X	     (36) 
IV. MODELING OF THE 3-PHASE BRIDGE RECTIFIER 

By using the 3-phase bridge diode rectifier, the 3-phase AG 
input power supply ( qd , qp , q	 ) is converted into the DC 
voltage (�g	 ) output power supply to the inverter. The input 
supply voltages are given as: 

qd = ��  . sin (Lx)    (37) 
qp = ��  . sin (Lx − 2π/3)    (38) 
q	 = ��  . sin (Lx + 2π/3)    (39) 

where ��  is the maximum supply voltage. 
The switching functions ( Q: − Qs ) of the diode bridge 

rectifier have the same expression as the voltage source 
inverter. The input and output variables of the 3-phase diode 
bridge rectifier are related as: 

zXdXpX	
{ = zQ:QCQr

− QKQsQ
{  . ��g	 �    (40) 

��g	 � = �(Q: − QK). (QC − Qs). (Qr − Q
)�. z
qdqpq	 {    (41) 

The three-phase supply voltages are converted to D-Q axis 
components using the transformation matrix as: 

|�g�S } =  
| cos(Lx)   cos(Lx + 2π/3)   ��4(Lx − 2π/3)− sin(Lx)  −sin(Lx + 2π/3)  − 4XY(Lx − 2π/3)} . z

qdqpq	 {    (42) 
where �g  and �S  are the components of the synchronously 
rotating D-Q axes of the supply voltages. The output DC 
voltage of the bridge rectifier is expressed as: 

�g	 =  C.√C[ . .�g
 + �S
0
+&    (43) 

V. HARMONIC EQUIVALENT CIRCUIT AND PERFORMANCE 
CALCULATION 

The proposed fundamental equivalent circuit of the 
induction motor is shown in Figure 1, where �� , �ℓ�, and ����ℓ 
are the stator winding resistance, leakage reactance, and stray 

load loss resistance respectively, while �� � , ����ℓ, and �� ℓ�  are 
rotor winding resistance, stray loss resistance, and leakage 
reactance respectively, ��  is the magnetizing reactance, ��	 is 
the iron core loss resistance, �<=  is the friction and windage 
(mechanical) loss resistance, and ��� � (1 − 4) 4⁄  is the motor 
load resistance, considering the effect of rotor stray load 

resistance ����ℓ and mechanical resistance �<= into account, as 
calculated by (5). 

There are two types of harmonics generated in the induction 
motor. The first type is the space harmonic generated in the air 
gap as pulsating torques due to the magnetic interaction 
between different phases to produce the revolving magnetic 
field in the air gap. Space harmonics are reduced by improved 
design procedures and optimization [30]. These harmonics 
affect the motor starting and generate noise and vibration. The 
second type is the time-harmonics, which are generated due to 
the applications of variable frequency drives in modern power 
systems using six-step VSIs. Time harmonics increase the 
motor heat by increasing the motor temperature. The time-
harmonic losses are generated due to the inverter switching and 
are impressed in the machine as additional losses. The 
additional time-harmonic losses are not load-dependent. For a 
balanced 3-phase inverter output voltage supply, only the odd 
harmonics should be taken into account. The third and its 
multiple harmonics are in the same phase and should also not 
be considered. Considering only the non-triple odd harmonics 
(5

th
, 7

th
, 11

th
, etc.), the Fourier series of the phase voltages are 

given as [27]: 

qd = �:� . sin(L� x) +  
�r� . sin(5L� x) + ���  . sin(7L� x) + ...    (44) 

qp/ = �:�  . sin(L�x − 120°) + �r�  . sin(5L� x − 120° ) +���  . sin(7L�x − 120°) + …. (45) 
q	/ = �:�  . sin(L�x + 120°) + �r�  . sin(5L�x + 120° ) +���  . sin(7L�x + 120°) + …. (46) 

where �:� , �r� , ���  are the maximum values of the 
fundamental, 5

th
, and 7

th
 harmonic components and L�  is the 

angular frequency of the supply voltage. 

For each harmonic component, the motor equivalent circuit 
components are approximately represented by multiplying 
these components by a constant factor k, which represents the 
order of the harmonic. The fundamental circuit of Figure 1 is 
converted to a time-harmonic equivalent circuit as shown in 
Figure 4, taking the effect of core loss resistance ��	  as a 
voltage drop in the stator and rotor circuits. Resistances ����	� 
and ����	�  are reflected series resistances of the iron core 
harmonic resistance ��	� at the stator and rotor windings [25]. 

 

 
Fig. 4.  Proposed equivalent circuit for time-harmonic calculation. 

Equations (44)-(46) of the output inverter phase voltage 
indicate that the 5

th
 harmonic voltage component has a negative 

phase sequence, whereas the 7
th
 harmonic has a positive phase 



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sequence like the fundamental component. The magnetic flux 
generated in the air gap due to the 5

th
 harmonics opposes that 

due to the 7
th
 harmonics. The rotor speed is related to the 

fundamental frequency, and it appears practically constant 
concerning the harmonic field is given as [31]: 

Q� = (�#:)���  for forwarding rotating fields    (47) 
Q� = (��:)#��  for backward rotating fields    (48) 

From Figure 4, if �. �� is assumed to approach infinity and �. (�ℓ� + �� ℓ� ) is greater than ��� , ����ℓ , ��� , and ����ℓ , then 
the current of the k-th harmonic component is: 

�� = �� .� .  (�ℓ� + �� ℓ� )0 = t]��&.('ℓ(�'� ℓ9) 1     (49) 
The corresponding expression of the RMS harmonic ripple 

current ��  is obtained by the superposition principle as [18]: 
�� = ��r
 + ��
 + �::
 + �:C
 + �:�
 + ⋯ �+&    (50) 

The total stator and rotor windings copper losses and stray 
load losses due to the fundamental and harmonic ripple 
currents are obtained as: 

Ρ	��� = 3. V�_��
 + ��
W. (�� + ����ℓ )    (51) 
Ρ	��� = 3. V��_��
 + ��
W. V�� � + ����ℓW    (52) 

The stray load losses are generated in the machine 
essentially due to hysteresis and eddy currents induced by the 
stator and rotor windings leakage fluxes. 

If �_��  is the fundamental stator phase current of the motor 
and ��  is the total harmonic ripple of the phase current, and by 
applying the superposition theorem, the stator input phase 
current ��  when the motor is fed from the six-step VSI is 
obtained as : 

�� = .�_��
 + ��
0+&    (53) 
The total input power to the motor is: 

Ρ�� = 3. �_�  . ��  . ��4ϕ    (54) 
The friction and windage (mechanical) power loss due to 

the harmonic mechanical resistance �<=�  depends on the air-
gap magnetizing voltage V� and is obtained as: 

P<=� = 3. ��
 (�
 . R<=� ) 1     (55) 
The iron core loss of the k-th harmonic is obtained as: 

�	� = 3. ��
 ��	�⁄     (56) 
Then, the superposition principle is applied to find the total 

harmonic core losses as: 

Ρ�	� = ∑ Ρ�	���r ,�,…     (57) 
The fundamental iron core losses are obtained as: 

�	 = 3. ��
 ��	⁄     (58) 
Then, the total iron core loss is obtained as: 

Ρ�	l = Ρ�	 + Ρ�	�    (59) 
The simplified formula for the k-th time-harmonic 

electromagnetic torque ce�  is derived by [32]: 
T�� =  C.$�9�&  .8� 9�� .=(� = C.$��

&  .8� 9
�� .=(�     (60) 

where L�� = �. L� is the angular speed of the k-th harmonic. 
The k-th rotor harmonic current is equal to the k-th stator time-
harmonic current because �. ��  is very large. The harmonic 
torque is generated by the interaction between the harmonic 
flux of the stator and the rotor flux. The harmonic torque is 
smaller than the fundamental motor torque because the 
harmonic current and fluxes are comparable to the normal and 
the torque generated by the harmonic components is partially 
canceled by the harmonic torque that moves in the negative 
direction. The 5

th
 harmonic torque opposes the 7

th
 harmonic 

torque, and a similar opposition happens between the 11
th
 and 

13
th
 harmonics [18]. The k-th time-harmonic mechanical output 

power from the motor is derived as: 

P�� =  C.$�9�&  .8�� 9 .(:#��)��     (61) 
The total harmonic mechanical output power is obtained by 

the superposition theorem as: 

Ρ�� = ∑ Ρ����r ,�,::,…     (62) 
Then, the net mechanical output power is determined as: 

�� =  C.$�9& .8�� 9 .(:#�)� − P��     (63) 
The efficiency of the motor fed from the six-step VSI is: 

�% = �)��" . 100    (64) 
The efficiency of the motor will drop when the machine is 

operated with VSI due to increasing losses and heat. The losses 
are dissipated as heat. The harmonic currents increase the 
copper and iron-core losses. This leads to a higher temperature 
in the stator winding, iron core, and rotor bars. The power 
factor cos(�) is defined as the ratio of the output mechanical 
power to the apparent input power �d  as: 

�. � = cos(�) = \)\�" = �)√C.tℓ.$ℓ    (65) 
where �ℓ and �ℓ are the line supply voltage and the current of 
the motor. 

The lagging power factor can be corrected by introducing 
capacitors. Since VSI has capacitors on the DC bus, this 
provides isolation of the induction motor as the load from the 
network, to improve the overall power factor of the system. 
When the VSI starts the induction motor, low frequency 
applies to the motor and high starting currents will be avoided. 
After that, the applied voltage and frequency are increased to 
accelerate the motor to the steady-state speed without drawing 
excessive current [33]. 

VI. RESULTS AND DISCUSSION 

To examine the validity of the dynamic model of the whole 
system and the harmonic equivalent circuit of the motor when 



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fed from the six-step VSI, the motor performance is simulated 
in transient and steady-state operation in Matlab and the results 
are compared with the ones measured when the motor is fed 
from the sinusoidal power supply. The parameters of the 
fundamental equivalent circuit at 50Hz supply are given in 
Table I. 

TABLE I.  FUNDAMENTAL EQUIVALENT CIRCUIT PARAMETERS 

Machine 

power 

(KW) 

�  (¡) ¢� £  (¡) ¤¥¦ (¡) ¤� ¥£ (¡) ¤§ (¡) 
Moment 

of inertia 

(kg.m
2
) 

2.0 5.1 3.5 5.0 7.5 88.0 0.03 

 
The iron-core and the mechanical loss resistances ��	 and �<=  are varied with air-gap voltage ��  and their relations are 

generated by the curve-fitting technique as given in (6)-(7). The 
iron core resistance ��	  is reflected on the stator and rotor 
circuits to stay in the dynamic model of the 4

th 
order, although 

the iron core is considered in the model. The reflecting 
resistances ����	  and ����	  are derived by (8)-(9). The stator 
and rotor stray load resistances are varied with stator and rotor 
phase resistances and are derived by (12)-(13). 

A. Results of the Dynamic Model 

Figures (5)-(8) show the starting-up motor speed, 
electromagnetic torque, stator phase current, and rotor phase 
current when the motor is fed from a 50Hz six-step VSI supply. 
The results show that there are harmful harmonic currents in 
the stator and rotor. These harmonic currents affect the motor 
performance by introducing pulsating torques, overheating, and 
noise.  

 

 
Fig. 5.  Start-up motor speed with inverter supply. 

 
Fig. 6.  Start-up motor torque with inverter supply. 

 
Fig. 7.  Start-up motor-stator phase current with inverter supply. 

 
Fig. 8.  Start-up motor-rotor phase current with inverter supply. 

 
Fig. 9.  Start-up motor speed with sinusoidal supply. 

 
Fig. 10.  Start-up motor torque with sinusoidal supply. 



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www.etasr.com Nasir: Determination of the Harmonic Losses in an Induction Motor Fed by an Inverter 

 

 
Fig. 11.  Start-up motor-stator phase current  with sinusoidal supply. 

 
Fig. 12.  Start-up motor-rotor phase current with sinusoidal supply. 

Figures (9)-(12) show the starting-up motor speed, torque, 
stator phase current, and rotor phase current when the motor is 
fed from a 50Hz sinusoidal supply using the parameters of the 
fundamental equivalent circuit of the motor. The results show 
there are no harmonic currents in the stator and the rotor of the 
motor. 

B. Results of the Harmonic Equivalent Circuit 

Figure 13 compares the measured efficiency of the motor 
fed from the sinusoidal and VSI supplies with that produced 
from the simulations in Matlab. The supply current of the VSI 
contains high harmonics, which contribute to additional motor 
losses and cause the motor efficiency to decrease by 3% 
compared to that of the sinusoidal supply voltage at full load 
operation. Similarly, Figure 14 compares the measured power 
factor from the VSI supply with that measured from the 
sinusoidal supply. Due to the harmonic currents in the VSI 
supply, the power factor is reduced by 0.08 compared to that of 
the sinusoidal supply, while the power factor in Matlab 
simulation is higher than that of the sinusoidal supply operation 
by 0.05 at full load, because the harmonic currents are not 
considered in the fundamental equivalent circuit of the motor. 

Figure 15 compares the measured total motor losses fed 
from the sinusoidal and VSI supplies with that of Matlab 
simulations predicted from the harmonic equivalent circuit and 
the fundamental equivalent circuit. Due to the VSI supply 
which contains high harmonic currents, the measured losses 
from the VSI fed motor are higher than those measured from 
the sinusoidal supply. The generated additional losses decrease 

the motor's efficiency. The predicted total losses from the 
harmonic equivalent circuit are 13% higher than that of the 
fundamental equivalent circuit. Figure 16 shows the detailed 
losses of the induction motor fed from the VSI supply 
predicted by the Matlab simulation of the harmonic equivalent 
circuit. 

 

 
Fig. 13.  Measured motor efficiency. 

 
Fig. 14.  Measured motor power factor. 

Since the iron core loss is a function of air-gap voltage and 
frequency, the harmonic currents add extra losses to the iron 
core loss. The iron core loss reached 50% of the total losses at 
full load operation. Also, the presence of harmonic currents 
causes a rise in the losses of stator and rotor windings, 
according to (51)-(52). The stray load losses in the stator and 
rotor windings are also increased due to the harmonic currents. 
The friction and windage losses are assumed to be fixed and 
not affected by the harmonic currents. 

VII. CONCLUSION 

The recent developments in power electronics and semi-
conductor material technologies improved power electronic AC 



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drive systems. Different circuit topologies of inverters, such as 
the six-step VSI have been used. In the six-step VSI fed 
induction motor, low order time harmonics (5

th
, 7

th
, 11

th
) that 

lead to additional motor losses are generated. These additional 
losses are not load-depended and are estimated as the 
difference between the no-load losses with VSI and the 
sinusoidal supply. The increase of total losses in the VSI 
supply by nearly 20% is caused by the time-harmonic currents 
generated by the VSI and the conduction and switching losses 
of the inverter. As expected, the motor efficiency with PWM-
VSI supply is lower than that with the sinusoidal supply by 
about 3%. 

 

 
Fig. 15.  Measured total motor losses. 

 
Fig. 16.  Details of motor losses with VSI supply. 

The stray load loss is increased by 30% with the VSI supply 
in comparison with the sinusoidal supply. The core losses at 
rated power are increased by 50% with the VSI supply, because 
these losses depend on the air-gap voltage and supply 
frequency. When the motor is operated with a VSI supply 
instead of a sinusoidal supply, the motor power factor is 
reduced by 0.08 and the stator and rotor copper losses are 

increased by 30% due to the harmonic current and the skin 
effect. The additional losses generated with the VSI supply lead 
to a higher temperature in the stator and rotor windings. The 
harmonic currents lead to extra reactive currents resulting to a 
decrease of the motor power factor. When the motor operates 
by a VSI supply, it applies low frequency and voltage to the 
motor in order to avoid the high inrush current. The motor 
supply voltage must be kept in the same proportion with the 
supply frequency for all motor speeds to maintain the rated 
torque constant and to avoid magnetic saturation. By good 
design of the motor structure and the output passive filter, and 
increasing the inverter switching frequency, the harmonics can 
be mitigated. 

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