This document discusses estimating the efficiency of electric machines using magnetic flux circuits. It presents the development of a generalized 4th order dq model equation of an electrical machine expressed in an arbitrary reference frame. The model equations are transformed to use all flux linkage variables instead of current variables. Results using this flux linkage approach are compared to measured machine quantities and good estimations of efficiency are obtained. Magnetic equivalent circuit modeling is an effective method for simulating electric machine performance and estimating efficiency variations in both transient and steady states.

1. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.321-328
Efficiency Estimation Of Electric Machine Using Magnetic
Flux Circuits
I. I. OKONKWO* S. S. N. OKEKE**
P. G. Scholar In Dept. Of Electrical Engineering Anambra State University, Uli. Nigeria*
Professor In Dept. Of Electrical Engineering Anambra State University, Uli. Nigeria**
Abstract:
Improving efficiency of electrical machine is of two decades, magnetic equivalent circuit modeling
paramount importance to optimization of operational method for electrical machines has been further
and resource management. Efficiency method which do developed and applied to modeling many other types of
not account for iron loss could undermine the effective machines, apart from induction machines: synchronous
efficiency estimation. This paper simulates the efficiency machines by Haydock [8], inverter–fed synchronous
variations of electric machine in both transient and motors by Carpenter [10], permanent magnet motors and
steady state, magnetic flux linkages are used instead of claw-pole alternators by Roisse [11], and 3-D Lundell
the traditional current method, though also in d-q axis. alternators, which are main source of electric energy in
Only magnetic flux linkages are used in an arbitrary internal combustion engine automobiles, by Ostovic' [9].
reference frame while machine efficiency in various Sewell developed Dynamic Reluctance Mesh
reference frames namely; rotor, stator and the software for simulating practical induction machines
synchronous reference are investigated. Results are based on the previous work using magnetic circuit
compared with measured quantities and excellent models for electrical machines [7]. This algorithm was
estimations are obtained. developed from rigorous field theory. Combined with
prior known knowledge of electrical machine behavior
Keyword: electrical machine efficiency, machine losses, determined from practical experience, this model is
magnetic circuit, generalized machine equation simplified in a physically realistic manner and results in
a procedure that can efficiently simulate 3D machines
1 INTRODUCTION within an iterative design environment without requiring
Estimation of electric machine performance extensive computational resources. Its computation time
through efficiency estimation enables electric machine was minimized by direct computation of the rate of
engineers to device control that could lead to improvement change of flux, thus the evaluation of time and rotor
in electric machine efficiency, for example in electric position dependent inductances are avoided. Yet there is
machine losses control [1]. Without a robust and good a similar dynamic mesh modeling method but with some
simulation method, neither good estimation of efficiency additional developments in modeling and improvements
nor deep insight about loses variations could be obtained. in the solution process [12].
Many researches have been carried out using magnetic Many different efficiency methods have been
circuit simulation techniques, in 1968 carpenter [2] developed with a field suitably as a goal. These
introduced and explored the possibility of modeling eddy – estimation methods have been enlisted in a thorough
current using magnetic equivalent circuits, which are study on efficiency estimation techniques performed by
analogous to electric equivalent circuits. Magnetic terminal J. S. Hsu et al [13] and [14], which divides efficiency
are introduced which provide a useful means of describing techniques into the following classes: nameplate
the parameters in electromagnetic devices, particularly method, slip method, current method, statistical method,
those operation depends upon induced current behavior. equivalent circuit method, segregated loss method, air
The virtual – work [3 – 4] principle was suggested for gap torque method and shaft torque method. In this
torque calculations and exploited in [7]. Generally, the paper efficiency is estimated through simulation of
magnetic equivalent circuit modeling method can be used to magnetic flux linkages with a generalized arbitrary
analyze many types of electrical machine in terms of reference equation transformed to all flux linkage
physically meaningful magnetic circuits. For example, variables which are used in formulae of loss
Ostovic' evaluated the transient and steady state minimization techniques [1, 13 – 14].
performance of squirrel cage induction machines by using
this magnetic equivalent method [5] 2 All Flux Linkages Generalized Machine
[6], taking into account the machine geometry, the type of Equation In An Arbitrary Reference Frame
windings, rotor skew, the magnetizing curve etc. In the past
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Vol. 2, Issue 4, July-August 2012, pp.321-328
In developing the conventional machine model for
transient equation, the assumptions made were stated in 2.2Model Equation
[15] as thus The generalized 4th order dq model equation of electrical
1. The machine is symmetrical with a linear air-gap and machine [20] expressed in an arbitrary reference frame
magnetic circuit rotating with arbitrary velocity ωc are the following,
2. Saturation effect is neglected with the stator and rotor fluxes as state variables:
3. Skin-effect and temperature effect are neglected
Vqs  rs i qs  pλqs  ωc λds
4. Harmonic content of the mmf wave is neglected
5. The stator voltages are balanced Vds  rs i ds  pλds  ωc λqs
4 
Vqr  rr i qr  pλqr  c  ω r )λdr
(ω
The differential equations governing the transient
performance of the induction machine can be described in Vdr  rr i dr  pλdr  c  ω r )λqr
(ω
several ways; they only differ in minor detail and in their where flux equations
suitability for use in a given application. The conventional λds  Ls i ds  Lm i dr
machine model is developed using the traditional method of
λqs  Ls i qs  Lm i qr
reducing the machine to a two-axis coil (d-q axis) model on
5 
both the stator and the rotor as described by Krause and λdr  Lr i dr  LLm i ds
Thomas [16].
λqr  Lr i qr  Lm i qs
2.1 D-Q Axis Transformation Theory
Stator variables (currents, voltages and flux 3 Model equations in all flux variables
linkages) of a synchronous machine were first related with When dq current variables ids, idr, iqs and iqr are
variables of a fictitious windings rotating with rotor by park solved for in equation (5) and substituted in equation (4)
[17]. This method was further extended by Keyhani [18] accordingly, a generalized model equation in an all flux
and Lipo [19] to the dynamic analysis of induction linkages of arbitrary reference is realized as follows.
machines. By these methods, a polyphase winding can be For current variables as function of linkage fluxes
reduced to a set of two phase-windings with their magnetic
L λ  Lm λdr
axes aligned in quadrature as shown in Figure 1 i ds  r ds
Lr Ls  Lm 2
Ls λdr  Lm λds
i dr 
Lr Ls  Lm 2
Lr λqs  Lm λqr
6 
i qs 
Lr Ls  Lm 2
Ls λqr  Lm λqs
i qr 
Lr Ls  Lm 2
Figure 1: Polyphase winding and d-q equivalent.
The d-q axis transformation eliminates the mutual When equation (6) is substituted in equation (4) and
magnetic coupling of the phase-windings, thereby making simplified accordingly, a generalized electric machine
the magnetic flux linkage of one winding independent of magnetic flux based equation is obtained as
the current in the other winding. For three phases the 
λ  aλ  cλ  ω λ  V
qs qs qr c ds qs
transformations are follows

λ  aλds  cλdr  ωc λqs  Vds
i qd0  Tabc i abc 
1 ds
7 
also 
λqr  bλqr  dλqs  fλdr  Vqr
Vqd0  TabcV abc 2  
λdr  bλdr  dλds  fλqr  Vdr
Equations (1-2) can be applied in any reference frame by where
making a suitable choice for theta (θ) in equation (3).
rs Lr rr Ls
a 2
, b ,
If theta equals θr, then equations (1-2) lead to an expression Lr Ls  Lm Lr Ls  Lm 2
for voltage in the rotor reference frame. Also, if θ is equal rs Lm r r Lm
c  2
, d  , 8 
Lr Ls  Lm 2
to zero, then equations (1-2) apply to a frame of reference
Lr Ls  Lm
rigidly fixed in the stator (i.e. stationary reference frame).
Otherwise, for θ equal to ωt in equations (1-2), a synchr- f  c  ωr )
(ω
onously rotating reference frame results
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3. I. I. OKONKWO, S. S. N. OKEKE / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.321-328
4 The Efficiency Of Electric Machine In Dq – Flux
Linkage Variable
Pcr  3rr i r 2 
3
2

r r i dr 2  i qr 2 
2
Three major losses that are accounted for in this 3  1 
analysis are stator winding losses, rotor winding losses and  r  
2 L L L 2 
iron losses. The various expressions of all these losses are  r s m 
given [1] as functions of dq – currents, also the stator input
power is the total power. To find the efficiency in dq – flux
 2 2
(L s λdr  Lm λds )  s λqr  Lm λqs )
(L  16 
2
linkage variables all the dq – current variables must be
eliminated as follows Pi  3ri i i 2 
3  i m ωs
ri 
2  ri


 i dm 2  i qm 2
 
 
Stator power losses 2
 2
  
 3 r  i m ωs   1
Pcs  3rs i s 2
3

 rs i ds 2  i qs 2
2
 9  i   
 2  r i   Lr Ls  Lm 2 
   


 
 
Rotor power losses    [λdr m  Ls ) λds m  Lr )] 2 
(L  (L 17 
 2 
Pcr  3rr i r 2 
3
2

rr i dr 2  i qr 2   
10  [λqr m  Ls ) λqs m  Lr )] 

(L  (L


 

Iron losses
3
2 Pin  [Vqs i qs V ds i ds ]
Pi  3ri i i 2 3 i ω
 ri  m s
2  ri


 i dm 2  i qm 2
  11  2
  3   L λ  Lm λqr  
 V qs  r qs  
Power input 
2   Lr Ls  Lm
2
 
 
3  real    
18
Pin  [Vqs i qs Vds i ds ]  
12   L λ  Lm λdr  


2  ds  r ds
V 
2 

  Lr Ls  Lm  
  

where but efficiency η
i s  i ds  jiqs Pin   losses
η 19 
i r  i dr  jiqr Pin
i m  i dm  jiqm  
13 Pin  Power input
i dm  i ds  i dr  losses  Pcs  Pcr  Pi Total Losses
i qm  i qs  i qr the values of Pcs, Pcr, Pi and Pin are found in equation
[15 – 18]
hence
i s 2  i ds 2  i qs 2 5 Transient Simulation Of Electric Machine Dq
– Flux Linkages
i r 2  i dr 2  i qr 2  
14 In this paper transient dq – flux linkages of
2 2 2
im  i dm  i qm electric machine are simulated by transforming the
arbitrary reference frame electric machine equation in in
substituting equation (6) in (9 – 12) and simplifying its magnetic flux equivalent (7) from time domain to
accordingly to get laplace frequency domain, and hence rational function
of the flux linkages in s – domain are the result of such
 
transformation. The inverse transforms of these flux
3
Pcs  3rs i s 2  rs i ds 2  i qs 2 linkages (rational functions in s – domain) are close –
2 form transient time domain solution of electric machine
2
3  1  flux linkages. Discretization can now be achieved at
 rs    desired time point from the close – form continuous
2  Lr Ls  Lm 2 
  time function of flux linkages obtained above.
 2
(L 2
(L r λds  Lm λdr )  s λqs  Lm λqr )  15  The transformation of the magnetic flux
equivalent of the generalized electric machine (7) to
laplace frequency domain may be achieved as follows
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Vol. 2, Issue 4, July-August 2012, pp.321-328
From equation (7), taking the laplace transform of (7(a)) to effects are not as sensitive as that of stator, rotor and the
get magnetizing inductances. The iron loss is still the
minimum of all the losses, though estimation of the iron
λqs  λqs  aλqs  cλqr  ωc λds Vqs
(s) (0) (s) (s) (s) (s) losses does not include the iron resistance in the
(s) (0) 20 
generalized machine equation that calculated the flux
(s  a)λqs  ωc λds  cλqr  0 Vqs  λqs
(s) (s) (s) linkages.
Also The transient variation of the flux time is
 ωc λqs
(s)(s  a)λds
 (s) 0  cλdr
 (s) Vds  (0)21 
(s) λds shown in fig (2) which shows that the stator flux is
 dλqs
(s) 0   b)λqr
 (s (s) fλdr
 (s) Vqr
(s) λqr
 (0) 22  always higher than that of the rotor flux as expected.
The steady state flux linkages in the stator and the rotor
0  dλds
(s) fλqr
 (s)(s  b)λdr
 (s) Vdr
(s) λdr
 (0) 23  windings increase as the slip increase, that is reduced
rotor speed. The results also shows that when the stator
Equation (20 – 23) may be rearranged in compact form to and the rotor flux linkages increase more than the
get optimal values , the operation of the motor results in
s  a ωc c 0  λqs  Vqs  λqs 
(s) (s) (0) reduced efficiency.
 ω s  a 0  λ (s) V (s) λ (0)

 c   ds   ds  ds 
 c   24  8 Conclusion
 d 0 s  b f   λqr  Vqr
(s) (s) λqr 
 (0)
     Simulation software has been formulated which
 0 d  f s  b   λdr  Vdr
(s) (s) λdr 
 (0)
estimates the efficiency the electric motor. The optimal
The expression for a, b, c, d, e and f are given in equation parameters of the electric motor which are responsible
(8) for maximum efficiency was able to be estimated
through simulation and the results agreed with the
6 Steady State Simulations Of Dq Flux Linkages designed parametric configurations of the machine. The
For steady state simulation equation (24) is speed versus flux linkages of the machine are in the
modified for use in the steady state solution of the dq flux agreement with the expected results. The formulated
linkages. Modification is by replacing all the (s) in the simulation program has shown to be useful in accessing
matrix operator of equation (24) by (jω) and also the optimum configuration of electrical machine
transforming all the source voltages to time domain parameters especially as they affect the efficiency of the
equivalent and all the initial values set to zeroes. Hence machine at no load. For example, other parameters of
 jω  a ωc c 0  λqs  Vqs 
(t) (t) the simulated induction motor are found to be fully
 ω jω  a 0  λ (t) V (t)

 c   ds   ds 
 optimized with only the stator winding resistance which
 c
 25  when change from 4.85 Ω to 5.4 Ω improves the no load
 d 0 jω  b f   λqr  Vqr 
(t) (t) efficiency only but little figure (6)
    
 0 d f jω  b   λdr  Vdr 
(t) (t) Improvement in the method of the estimation of
the machine efficiency may include the inclusion of the
The expression for a, b, c, d, e and f are given in equation
iron resistance Ri in the generalized machine equation
(8)
that simulated the d – q flux linkages of the magnetic
circuit equation.
7 Simulation Results
Efficiency variations of the induction motor are
simulated in steady state while flux linkages in the windings TABLE 1 MACHINEPARAMETER
are also simulated during transient. All the simulations are Machine Parameter Actual Estimated
done with MATLAB 7.40 mathematical tools, the results of Stator resistance 4.85 Ω 5.4 Ω
the simulation of the electrical induction motor are shown Rotor resistance 3.805 Ω 3.8 Ω
in fig. (2) thru fig. (13). Efficiency of the motor is simulated Iron loss resistance 500 Ω 1000+ Ω
at various motor parameter variations, optimal machine Mutual inductance 0.258 H 0.259 H
parameters are obtained and compared with the designed Stator inductance 0.274 H 0.272 H
parameters, and complete results are summarized in table Rotor inductance 0.274 H 0.272 H
(1). These results are of that of the rotor reference frame, Rotor inertia 0.031 Kg/m2 – –
there were no much differences observed in the results from Friction coefficient 0.008 Nm.s/rd – –
other reference frames Output power 1.5 Kw – –
The optimal slip of the motor (0.26) with Poles 2x2 – –
maximum efficiency of (97 %) is the result of the Voltage 220/380V 220 V
simulation fig (4). The efficiency varies as other machine’s Current 3.64/6.31 A – –
parameter varies with exemption of the terminal voltage Rated speed 1420 rev/min – –
which shows no significant efficiency changes at variations Frequency 50 Hz 50 Hz
at no – load. Though, stator and the rotor resistance Maximum efficiency – – 97.04 %
variations affects the efficiency of the induction motor, their
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Vol. 2, Issue 4, July-August 2012, pp.321-328
Nomenclature
s, r : stator and rotor indices
xdq : components in the synchronous reference frame
dq axes
Rs, R r : stator and rotor resistances
Ri: Iron loss resistances
Ls, Lr: stator and rotor inductances Figure 3: Steady State Stator And
ls, lr: stator and rotor leakage inductances Rotor Slip Versus Flux
Response
Lm: magnetizing flux
s, r: stator and rotor flux Steady State Efficiency Response Simulation
Graphs
m: magnetizing flux
Ω s: stator pulsation (rd/s)
Ωr: rotor speed (rd/s)
p: number of pole pairs
Pcs, Pcr: stator and rotor copper losses
Pi: Iron losses
Steady And Transient State Flux Response
Simulation Graphs
Figure 4: Slip Versus Efficiency
Response
Figure 2: Transient State Stator And
Rotor Time Versus Flux
Response
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Figure 5: Stator Voltage Versus
Efficiency Response
Figure 9: Magnetizing Inductance
Versus Efficiency Response
Quick survey
Figure 6: Stator Resistance Versus
Efficiency Response
Figure 10: Stator Inductance Versus
Figure 7:Rotor Resistance Versus Efficiency Response
Efficiency Response
Optimal point survey
Figure 8: Iron Resistance Versus Figure 11: Stator Inductance Versus
Efficiency Response Efficiency Response
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