Engineering, 2010, 2, 328-336
doi:10.4236/eng.2010.25043 Published Online May 2010 (http://www.SciRP.org/journal/eng)
Copyright © 2010 SciRes. ENG
A New Method for Finding the Thevenin and Norton
Equivalent Circuits
George E. Chatzarakis
Department of Electrical Engineering Educators, School of Pedagogical and Technological Education,
Athens, Greece
E-mail: geaxatz@otenet.gr, geaxatz@mail.ntua.gr, gea.xatz@aspete.gr
Received December 3, 2009; revised February 7, 2010; accepted February 12, 2010
Abstract
The paper presents a new pedagogical method for finding the Thevenin and Norton equivalent circuits of a
linear electric circuit (LEC) at the n-different pairs of terminals simultaneously, regardless of the circuit to-
pology and complexity. The proposed method is appropriate for undergraduate electrical and electronic en-
gineering students leading to straightforward solutions, mostly arrived at by inspection, so that it can be re-
garded as a simple and innovative calculation tool for Thevenin equivalents. Furthermore, the method is eas-
ily adapted to computer implementation. Examples illustrating the method’s scientific and pedagogical reli-
ability, as well as real test results and statistically-sound data assessing its functionality are provided.
Keywords: Circuit Analysis, Equivalent Circuits, Inspection, Mesh Analysis, Nodal Analysis, Thevenin and
Norton Equivalent Circuits
1. Introduction
One of the principal and fundamental topics taught to
undergraduate Electrical and Electronic Engineering
students within their Electric Circuits course is finding
the Thevenin and Norton equivalent circuits of a linear
electric circuit (LEC) at a specific pair of terminals.
The ability to find these parameters will proves to be
necessary to the students, when the focus is on a particu-
lar part of the circuit (i.e. finding the voltage, current or
power dissipated to a resistance) or when faced with
problems of load matching.
However, the methods developed and recorded in text-
books so far [1-8], address the finding of Thevenin and
Norton equivalent circuits of a LEC at one pair of termi-
nals only. This means that finding the said circuits at a
different pair of terminals, necessitates repetition of the
whole procedure all over again. This is a time-consuming
approach which may lead to wrong calculations and
which has a negative effect upon students who, in order
to find the new parameters, have to go through the same
procedure again.
Many times the question is posed by the students
themselves: is there a relationship between the Thevenin
and Norton equivalent circuits at different pairs of ter-
minals since the circuit topology remains largely the
same? And, consequently, is there a way to calculate
these parameters at different pairs of terminals simulta-
neously?
Given that the circuit remains the same, the question
arises whether there could be a way for finding the
equivalent circuits at different pairs of terminals simul-
taneously.
The question led the author of this paper to seek and
substantiate a method for finding the equivalent circuits
at more than one pair of terminals simultaneously, based
on a new approach for finding the Thevenin equivalent
circuit in combination with the mesh or nodal analysis
developed by Chatzarakis et al. [9].
In addition and as it is shown in the following sections
the proposed method can be adopted for all kinds of
LECs including or not dependent sources, coupled circuit
components and sinusoidal sources of different frequen-
cies (using superposition principle) so that it can prove a
very powerful and innovative tool for the students and
possibly for field engineers that would like to proceed to
fast and accurate calculations under certain circum-
stances.
2. Method Description
Consider the dc LEC and the n-different pairs of terminals
nn
,,22,11 , shown in Figure 1(a).
G. E. CHATZARAKIS329
)a(
dc
LEC
1
1
2
2
n
n
)b(
dc
LEC
1
1
2
2
n
n
n
R
n
I
2
R
2
I
1
I
1
R
m
I
Figure 1. dc LEC.
According to the method proposed by Chatzarakis et al.
[9], resistances are connected to the ter-
minals , respectively, as shown in
Figure 1(b).
12
,,,
n
RR R
nn
,,2
2,11
Since a dc LEC may be either planar or nonplanar, the
method to be used for calculating the currents through
and the voltages across the resistances in
each of the two cases may be mesh or nodal analysis for
the former [1-8], and nodal analysis for the latter
[1-8,10].
12
,,,
n
RR R
Consider now the following three possible cases:
Case 1) If mesh analysis is used for planar circuits,
defining the currents of the loops made by connecting the
above mentioned resistances at the n-different pairs of
terminals as 12
,,,
n
I
II, and the currents at the nm
remaining loops, possibly existing into the circuit, as
12
,,
nn ,
m
I
I

I, leads to the following mesh-current
equations by inspection:
11 121111
21 222222
12
12
nm
nm
nn nnnmn
mmmn mmm
cR RRRIV
RcRRR IV
RR cRRIV
RRRRI V












 

 

n
m









n





(1)
To calculate the currents, flowing
through the resistances respectively, the determi-
nants and should be found. Hence,
k
I1, 2,,k
k
R
k
DD
11 1211
212222
12
12
nm
nm
nn nnn
m mmnmm
cR RRR
RcRRR
DRR cRR
RRR R






m
1111 1
2122 2
1
1
nm
nm
nnnn
mmmn
k colum
cRVR R
RVR
DkRVcR
RVR








nm
mm
R
R
R
which lead to (2) and (3), respectively. Thus,
11
0
11
11
1
nn
n
nn
iji j
jj
ii
j
Da RaRdR
R
 
 
 




 





j
0
(2)
where are real constants, and
0
,,,
j
aa b

11
()()() ()
0 0
11
11
111
nn
n
kk k
ki i
j
jj
kjk
jk jk
nn
ii
DaRaRdR b
RRR
 
 
 






 







k
jj
0
k
(3)
where are real constants. Hence, for any
() ()()
0
,,,
kk
j
aa bnk
1 Equations (2) and (3) give
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
330


11
()()() ()
0 0
11
11
11
00
11
11
111
1
nn
n
nn
kk k
ij ijj
jj
ii
kjk
jk jk
k
knn
n
nn
iji jj
jj
ii
j
aRaRdR
RRR
D
ID
aRa RdRb
R
 
 
 



 
 
 






















k
b
(4)
and


11
()()() ()
0 0
11
11
11
00
11
11
111
1
nn
n
nn
kk k
ij ijj
jj
ii
kjk
jk jk
k k
nn
n
nn
ijijj
jj
ii
j
aRaRdRb
RRR
VR
aRaRdRb
R
 
 
 



 
 
 





















k
(5)
But, since and , (4) and (5) give
kjR
R
k
SC
k
N
k
j
k
imIII


,
0

 

n
ii
jR
k
kjR
R
k
OC
k
TH
kimVimVVV
1
1,



11
()()() ()
0 0
11
11 ()
0
011
,
00
11
11
111
1
k
j
nn
n
nn
kk k
ij ijj
jj
ii k
kjk
jk jk
N
knn
Rk
n
nn
Rjk
ijijj
jj
ii
j
aRaRdRb
RRRa
Iim a
aRa RdRb
R
 
 
 



 
 
 
 






















k
(6)
and


1
11
()()() ()
0 0
11
11 ()
0
0
11
00
11
11
111
1
n
i
i
nn
n
nn
kk kk
ij ijj
jj
ii k
kjk
jk jk
TH
k k
nn
Rn
nn
iji jj
jj
ii
j
aRa RdRb
RRRa
Vim R
a
aRaRdRb
R
 
 
 



 
 

 






















(7)
Thus,
0
TH
THN k
kkN
k
Va
RR a
I
 
k
(8)
Equations (6), (7) and (8) define the Thevenin and Nor-
ton equivalent circuits at the n-different pairs of termi-
nals simultaneously. Note that the cal-
culation, which is of great importance for the transmis-
sion and distribution lines of electric energy, is made
possible through Equation (8) by finding the determinant
D only (Equation (2)), which is explained by the fact that
the determinant D contains all the necessary information
for finding the said equivalent resistances.
nkRTH
k1,
Example 1
For the circuit in Figure 2(a), obtain the Thevenin and
Norton equivalents as seen from terminals A-B, B-C, and
C-E (taken from [2], p. 158).
Solution: In Figure 2(b), mesh analysis gives by in-
spection
1
1
2
2
3
3
4
5004 0
05 0510
24
0055
14
45514
I
R
I
R
I
R
I



















  


Hence, it is a matter of elementary calculations to find
10014514510045
455414
32121
3132321


RRRRR
RRRRRRRD
32321 48020056 RRRRD
200890200210 31312 
RRRRD
48094648026621213
RRRRD
Thus, it is clear that , ,
014a154a245a
,
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS331
V24
2
A
B
C
E
3
1
4
V10
5
)a(
V24
2
A
B
C
E
3
1
4
V10
5
)b(
1
I
2
I
1
R
2
R
3
R
4
I
3
I
Figure 2. Circuit for Example 1.
345a,, , and, con-
sequently,
(1)
056a(2)
0210a(3)
0266a
54 45
3.857 3.214
14 14
56 210
1.037A4.667A
54 45
56 210
4V 15V
14 14
TH TH
AB BC
NN
AB BC
TH TH
AB BC
RR
II
VV



 
 

45 3.214
14
266 5.911 A
45
266 19 V
14
TH
CE
N
CE
TH
CE
R
I
V



Case 2) If nodal analysis is used for nonplanar circuits
specifically, following a similar procedure for finding the
conductance matrix determinant D, the voltages , and
the currents ,
k
V
k
I nk
1(as in Case 1), the following
functions are always derived:

11
00
11
11
1
nn
n
nn
iji jj
jj
ii
j
Dq GqGwGb
G
 
 
 




 





(9)


11
()()() ()
0 0
11
11
11
00
11
11
111
1
nn
n
nn
kk k
ij ijj
jj
ii
kjk
jk jk
knn
n
nn
iji jj
jj
ii
j
qGq GwGb
GGG
V
qGq GwGb
G
 
 
 



 
 
 





















k
(10)


11
()()() ()
0 0
11
11
11
00
11
11
111
1
nn
n
nn
kk k
ij ijj
jj
ii
kjk
jk jk
k k
nn
n
nn
iji jj
jj
ii
j
qGqGwGb
GGG
k
I
G
qGq GwGb
G
 
 
 



 
 
 






















0
(11)
where , are real constants.
00
,,,
j
qq b0
() ()()
,,,
j
kk k
qq b
But, since and , (10) and (11) give
0
,0
0
1


n
ii
j
kG
k
kjG
G
k
OC
k
TH
kimVimVVV 
kjG
G
k
SC
k
N
k
j
k
imIII
 

,0


1
11
()()() ()
0 0
11
11 ()
0
00
11
00
11
11
111
1
n
i
i
nn
n
nn
kk k
ij ijj
jj
ii k
kjk
jk jk
TH
knn
Gn
nn
iji jj
jj
ii
j
qGq GwGb
GGGb
Vim b
qGq GwGb
G
 
 
 



 
 
 






















k
(12)
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
332
and


11
()()() ()
0 0
11
11 ()
0
0, 11
00
11
11
111
1
k
j
nn
n
nn
kk kk
ij ijj
jj
ii k
kjk
jk jk
N
k k
nn
Gk
Gjk n
nn
iji jj
jj
ii
j
qGqGwGb
GGG b
Iim G
w
qGq GwGb
G
 
 
 



 
  

 






















(13)
Hence,
0
b
w
I
V
RR k
N
k
TH
k
N
k
TH
k (14)
Equations (12), (13) and (14) define the Thevenin and
Norton equivalent circuits at the n-different pairs of ter-
minals. It is worth mentioning again that calculating the
is made possible through Equation (14)
by finding determinant D only (Equation 9).
,1
TH
k
Rkn
Example 2
For the circuit in Figure 3, obtain the Thevenin and
Norton equivalents as seen from terminals A-B, C-B, and
E-C.
Solution: Since this circuit is nonplanar, the method to
be used is the nodal analysis. Hence, in view of Figure 4,
nodal analysis gives by inspection




8
5
5
25.15.05.0
5.04.18.0
5.08.07.1
3
2
1
33
332
1
V
V
V
GG
GGG
G
Hence, it is a matter of elementary calculations to find
12312 23
13 123
1.25 1.7
1.651.5 1.8751.115 1
D GGGGGGG
GG GGG


12323
510.2512.15 10.05DGG G G 
21313
32.2511.66.875DGGG G  
31213 123
838.7 16.1 11.613.17DGGGGGGG  
32 1212
810.95 16.16.295DD GGGG
Thus, it is clear that , ,
1
0b5.1
1w875.1
2
w
295.6
)3(
0b
,
, , ,
and, consequently,
115.1
3w05.10
)1(
0b875.6
)2(
0
b
1.5 1.875
1.5 1.875
11
10.05 6.875
10.05V6.875 V
11
10.05 6.875
6.7 A3.667A
1.5 1.875
TH TH
AB BC
TH TH
AB BC
NN
AB BC
RR
VV
II



 
 
 
1.115 1.115
1
6.295 6.295V
1
6.295 5.646 A
1.115
TH
EC
TH
BC
N
BC
R
V
I



Case 3) By a similar procedure, the proposed method
holds also for circuits containing dependent voltage
and/or current sources (active circuits), as well. The
A5
A
B
2
2
5.2
25.1
10
A8
4
C
E
Figure 3. Circuit for Example 2.
A5
A
B
2
2
4
25.1
5.2
10
A8
1
V
2
V
3
V
2
R
C
1
R
3
R
E
Figure 4. Method application circuit for Example 2.
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS333
difference lies in the fact that in this case the determinant
D is taken to be the final determinant of the linear system
which results after taking into consideration the relations
describing the dependent sources. Hence, the determi-
nant is no longer resistance or conductance determinant
(Case 1 and Case 2, respectively). This fact, however,
does not affect the proposed method as regards the cal-
culation of the Thevenin and Norton equivalent circuits
at more than one pairs of terminals simultaneously.
It must be noted that students often find it difficult to
treat dependent sources in active circuits. The traditional
serial procedure requires the solution of two circuits, one
for finding the and one for finding the , for
each pair of terminals separately. This approach demands
special attention as regards the dependent sources, since
one/some of them may be zero-valued (in the second
circuit), also influencing other circuit components that
must eventually be removed. The proposed method ad-
dresses the above issues by treating the circuit only once
and without having to remove any of its components,
which constitutes a great advantage over the traditional
approach.
TH
VN
I
Example 3
For the circuit in Figure 5(a), obtain the Thevenin and
Norton equivalents as seen from terminals A-B and K-L
(taken from [3], p. 310)
Solution: In Figure 5(b), mesh analysis gives by in-
spection
1
1
2
22
3
22
4
8044 20
0044
44120
40880
x
I
R
I
RR
v
I
RR
I







 






 


 


Taking into account the fact that the
above matrix equation becomes
23
4( )
x
vII
1
1
2
22
23
3
22
4
804420
0044
4( )
44120
408 80
I
R
I
RR
I
I
I
RR
I







 







 


 



or, equivalently,
1
1
2
22
3
22
4
8044 20
04 40
0
48160
80
408
I
R
I
RR
I
RR
I






 





 

 


Hence, it is a matter of elementary calculations to find
12 1 2
160256 6401024DRRRR
12
10880 15360DR
421
2560 10240DD R
KL
4
4
4
x
v
4
4
V100
V20
A
B
x
v
)a(
KL
4
4
4
x
v
4
4
V100
V20
A
B
x
v
)b(
3
I
1
I
2
I
1
R
2
R
4
I
Figure 5. Circuit for Example 3.
Thus, it is clear that, ,
0160a1640a2256a
,
, and consequently
(1)
010880a(2)
025a60
640 256
41
160 160
10880 2560
17 A10 A
640 256
10880 2560
68 V16 V
160 160
TH TH
ABK L
NN
ABK L
TH TH
ABK L
RR
II
VV



.6
 
 
 
3. Evaluation of the Suggested Method
The theoretical development of the proposed method
(Section 2), and, particularly, its results enable the author
to support the pedagogical value of its findings.
Evidence of the effectiveness of the proposed method
on the student learning outcomes came from data gath-
ered during an evaluation process encompassing fourth
cycles of testing the students’ reactions to the suggested
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
Copyright © 2010 SciRes. ENG
334
Figures 6-8 show the number of students who re-
sponded to each of the two methods within the allotted
time, managing, at the same time, to arrive at an accurate
solution in each of the 4 problem-solving situations. The
outcomes show that there is a definite advantage to the
proposed method as to the degree of its retention and its
easy retrieval from memory, even if too much time has
passed since its last exemplification and/or application
(tests conducted up to 1 year later).
method in comparison with the traditional serial proce-
dure. The sample was taken from ASPETE, a School of
Pedagogical & Technological Education, located in
Athens, GR, and consisted of 4 groups of Electrical and
4 groups of Electronic Engineering students. Each group
was limited to a maximum of 25 (n = 200), and there
were no significant changes between cycles as to the
group’s internal arrangements.
The first cycle comprised six 60-minute tests, corre-
sponding to the three examples examined within Cases1),
2) and 3) above, conducted 2 weeks after each method’s
presentation in class (two 60-minute tests (×) three ex-
amples = six 60-minute tests).
It is worth noticing that the decrease rate in the num-
ber of students who responded successfully in the 4 cy-
cles of tests remains constant between successive tests in
the case of proposed method, i.e.,
175/143143/116116 / 921.2
 (Figure 6)
The cycle was repeated three more times (5 weeks, 1
semester and 2 semesters later), with problem-solving
situations of similar complexity, and with no prior notice
given to the cohorts.
132/101101/7777/ 601.3
 (Figure 7)
156 /125125/100100 / 801.25
 (Figure 8)
175
143
116
92
80
55
28
12
0
20
40
60
80
100
120
140
160
180
200
2 weeks
la ter
5 weeks
la ter
1 semester
la ter
2 semesters
la ter
No. of Students
Proposed method
Classical method with
serial procedure
Case 1)
Figure 6. Accuracy of answers with the two methods (Case 1).
132
101
77
60
68
40
20
7
0
20
40
60
80
100
120
140
160
180
200
2 weeks later5 weeks later1 semester
later
2 semesters
later
No. of Students
Proposed method
Classical method with
serial procedure
Case 2)
Figure 7. Accuracy of answers with the two methods (Case 2).
G. E. CHATZARAKIS335
156
125
100
80
44
15 41
0
20
40
60
80
100
120
140
160
180
200
2 weeks later5 weeks later1 semester
later
2 semesters
later
No. of Students
Proposed method
Classical method with
serial procedure
Case iii)
3
Figure 8. Accuracy of answers with the two methods (Case 3).
On the contrary, this rate increases significantly in the
case of classical method with serial procedure, i.e.,
28 /1255 /2880/551.451.2 (Figure 6)
20 / 740 / 2068 / 401.71.3 (Figure 7)
4/1 15/444/152.9 1.25  (Figure 8)
Semi-structured interviews with students and teachers,
administered by the author at the end of the observation
cycles, cross-checked the above outcomes. The respon-
dents not only validated the above findings, they also
demonstrated an extremely positive attitude towards the
proposed method in comparison with the traditional se-
rial procedure. More particularly, students across the 8
participating groups and faculty members that were
asked to apply the proposed method in their own classes
appeared to have identical views regarding the fact that
the proposed method “requires much less time to apply”,
and that the students feel “less anxious about” and, hence,
“more confident” and/or “competent” in working with it
than with the traditional serial procedure. When asked to
comment on the method’s effectiveness regarding stu-
dent performance in terms of real test results, student and
teacher comments validated qualitative impressions re-
corded by the author as field notes and/or empirical data.
4. Conclusions
The study presented in this paper, pertaining to the si-
multaneous finding of the Thevenin and Norton equiva-
lent circuits of a dc LEC at the n-different pairs of ter-
minals, has led to the following conclusions:
Its application by the undergraduate students of the
Departments of Electrical and Electronics Engineering
does not require a high level of mathematical knowledge.
Basics of linear algebra pertaining to the solution of lin-
ear systems could be enough. In any case, the specific
subject area is included in the taught courses of first year
study programs in almost all universities.
It is literally applied by simple inspection, regard-
less of circuit complexity and topology. This reduces the
amount of anxiety experienced by the students regarding
the way to the solution of the problem.
Correctness of results is ensured using either calcu-
lators that can treat large matrices with symbolic vari-
ables or inexpensive math software for personal com-
puters. Furthermore, the method is easily adapted to
computer implementation.
Correctness of results is also ensured by the fact
that the method does not allow for an increase in the er-
rors that are likely to be made from one pair of terminals
to the other etc due to the repeated application of almost
the same procedure. In other words the student can arrive
at correct results for a number of terminals within the
time period required for the calculation of these parame-
ters for only one pair of terminals.
Since the calculations are mostly made by inspec-
tion, the students can easily apply the proposed method
at a much later time than the time the specific subject
area of electric circuits was taught to them.
From a purely educational point of view, the pro-
posed method enables the students and their instructors
to calculate the said equivalent circuits at more than one
pairs of terminals without having to repeat calculations
right from the beginning. Hence, the approach may be
said to hold a particular interest for both instructors and
students, since they do not have to think and/or act de-
pending on the dc LEC topology; the method leads them
to an easy, systematic, time-saving and absolutely accu-
rate calculation of the above parameters at more than one
pairs of terminals simultaneously.
We are able to know the behavior of a given circuit
at any pair of terminals, with regards to the open-circuit
voltage, the short-circuit current, and the equivalent re-
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
336
sistance. This means that in theory we are able to know
the pair of terminals with respect to which we may drive
the maximum power the specific circuit can supply, since
the matching loads are equal to the Thevenin equivalent
resistances, and the voltages equal to the corresponding
Thevenin voltages at every pair of terminals.
In practice, however, connecting loads to a circuit is
limited to specific pairs of terminals which are not many.
This, in combination with the fact that the load is known,
enables us to select from the few pairs of terminals the
one with respect to which we may achieve the best pos-
sible matching. The criterion for determining the pair of
terminals where the specific load L
R will be connected,
is its minimum divergence from the Thevenin equivalent
resistances, i.e. the appropriate pair kk
, where the
load connection will cause the minimum power reflec-
tions, is determined by the
L
TH
kRR min. In view of
this, a major contribution of this study is the fact that in
real world applications, it is very useful for the engineer
to know the Thevenin equivalent circuit at more than one
pairs of terminals simultaneously, in order to confront
possible faults.
The approach holds equally well for ac LECs, the
sources of which operate at the same frequency, noting
that the coefficients giving the values of the equivalent
circuits parameters are all complex numbers.
Finally, the accuracy and the effectiveness of the
proposed method can also be verified in a laboratory
environment of electric or electronic circuits where the
time students spend on a weekly basis can not be very
long. Following this method, students can obtain easily
and fast Thevenin equivalents of multi-terminal LECs,
either for practice or on purpose, that could validate the
theoretic results and justify the time savings achieved.
The author hopes that the reader will be inspired to
re-examine his or her own text from a more critical
viewpoint and perhaps add additional critical contribu-
tions to the literature. The author also hopes he has ef-
fectively affirmed that research (both technical and
pedagogical) in circuit analysis can be both interesting
and profitable.
5
. References
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ing, Addison Wesley Publishing Company, 5th Edition,
MA, 1996.
[2] C. K. Alexander and M. N. O. Sadiku, “Fundamentals of
Electric Circuits,” McGraw-Hill, New York, 2000.
[3] G. E. Chatzarakis, “Electric Circuits,” Tziolas Publica-
tions, Thessaloniki, Vol. 1, 1998.
[4] G. E. Chatzarakis, “Electric Circuits,” Tziolas Publica-
tions, Thessaloniki, Vol. 2, 2000.
[5] W. H. Hayt and J. E. Kemmerly, “Engineering Circuit
Analysis,” McGraw-Hill, New York, 1993.
[6] C. A. Desoer and E. S. Kuh, “Basic Circuit Theory,”
McGraw-Hill, New York, 1969.
[7] R. A. Decarlo and P.-M. Lin, “Linear Circuit Analysis,”
Oxford University Press, New York, 2001.
[8] D.E. Johnson et al., “Electric Circuit Analysis,” Pren-
tice-Hall, Upper Saddle River, 1997.
[9] G. E. Chatzarakis et al., “Powerful Pedagogical Ap-
proaches for Finding Thevenin and Norton Equivalent
Circuits for Linear Electric Circuits,” International Jour-
nal of Electrical Engineering Education, Vol. 41, No. 4,
October 2005, pp. 350-368.
[10] G. E. Chatzarakis, “Nodal Analysis Optimization Based
on Using Virtual Current Sources: A New Powerful
Pedagogical Method,” IEEE Transactions on Education,
Vol. 52, No. 1, February 2009, pp. 144-150.
Copyright © 2010 SciRes. ENG