A mathematical model for rectifier circuits using semiconductor diodes

tial inclusions are considered by analyzing the 
mathematical model received. This is significant in the mathematical point of view, because 
describing and studying the stability of solutions of differential inclusions is much easier and 
more explicit than the discontinuous differential equations. 
Based on the results of this study, we hope to get more profound results in further studies 
and investigate an optimal process for an assembly line of rectifiers in electrical engineering. 
Keywords. rectifier circuit, differential inclusions, semiconductor diode, mathematical model. 
 Received: 19/5/2020 Accepted: 1/6/2020 Published online: 14/06/2020 
I. INTRODUCTION 
The emergence of mathematical models has addressed a large number of applied problems, 
such as mechanics, electricity, theory of automation and control, struggle for survival in 
ecological systems, .... Mathematics is the tool for describing changes in each domain as 
dynamic systems, through which one can indicate their characteristics. Currently, the research in 
this area is still very developed. One of the problems that attracts attention is to study by 
mathematical modelling an operation of rectifier circuits (see [1] - [7]). 
As we already know, most electrical installations use direct current, but the power source is 
alternating current. Therefore, rectifiers are very important, indispensable and widely used in the 
electrical industry. A rectifier is an electric circuit consisting of electrical components used to 
convert alternating current to direct current. This research has led to many interested results (see 
[8], [9]). 
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 92 
In this paper we will research a mathematical model for rectifier circuits using diodes. The 
rectifier circuit has the following general form: 
 Figure 1. The RLDE circuit 
The model describes the operation of the circuit will be presented by differential inclusions 
that is defined as 
( ) ,
,
K
dX
F t X N X
dt
X K
 − − 
 (1) 
where, a set K is a cone in the space ( );n X t is an unknown function whose values belong 
to 
n at moment ;t  is a known constant square matrix of order ( );n F t is a known 
continuous vector function with its values in 
n
 and the set 
KN X is called the normal cone 
which is defined by 
 ( ) : , 0, .= −  nKN X Z Z X K  (2) 
We know that, the theory of differential inclusions and their applications is an intensively 
developed field of mathematics since the mid-19th century to now. There have been many 
studies showing that differential inclusions are equivalent to some differential equations with 
discontinuous right hand sides, such as in [1]. These studies help to find solutions of differential 
inclusions. At that, the solution of the system (1) is understood as a locally absolutely function 
which satisfies (1) almost everywhere. 
The main content of this paper is showed in a theorem that gives a mathematical model for 
rectifier circuits. At that, the model is presented by differential inclusions of the form (1). 
II. THE MATHEMATICAL MODEL FOR RECTIFIER CIRCUITS 
Based on circuit theory (see [6], [7]), as we know, branch of a circuit diagram is two 
terminals of an element; point of connection between two or more branches is called node. 
Moreover, if ,i u are currents and voltages across branches of any selected tree and ,I U 
are currents and voltages across branches complementing the tree to the original circuit 
INTERNATIONAL COOPERATION ISSUE OF TRANSPORTATION - Special Issue - No. 10 
 93 
diagram, then we have 
,T
U Mu
i M I
= 
= − 
 (3) 
where 
TM is the transposition matrix of .M 
Let us consider an electrical circuit having a circuit diagram S and including resistances, 
inductances and a diode converter .D At that, the diode converter D contains m diodes. In 
each diode, positive current readily goes from the anode to the cathode. We denote by 
, ( 1, ),j jx y j m= respectively, the current and the voltage across the j th− diode. Assume that 
diodes are ideal, that is, their currents 
jx and voltages jy are satisfied by 
0
0 ; 1, .
0
j
j
j j
x
y j m
x y
 = 
= 
 (4) 
Note ( )1 2, ,..., mx x x x= and ( )1 2, ,..., my y y y= then from (4), it easily follows 
 ,+ − 
m mx y and ( ), 0x y = . (5) 
Now, we formulate and prove a theorem called the theorem on the mathematical model for 
rectifiers circuits. 
Theorem. The mathematical model for rectifier circuits is presented by differential inclusions of 
the form (1) in which a function ( ) ,F t a matrix  and a set K are defined in the proof 
process of the theorem. 
Proof: In the circuit diagram S all nodes are numbered in some order from 0 to .n We 
denote by ( ), , 0,k ki u k n= respectively, the current passing the k th− node, the voltage 
between the node k and node 0. After that, we are interested in vectors ( )1 2, ,...,D ni i i i= and 
( )1 2, ,...,D nu u u u= (the vectors 0i and 0u are not interested, because they are presented 
through, respectively, other currents, other voltages). In order to show this, let 
1 denote a tree 
consists of all nodes. By the first Kirchhoff’s law, we have
1
, 1,
m
kj j k
j
a x i k n
=
= = ; consequently, 
DAx i= (6) 
where ( )kj n mA a = is a matrix whose elements receive values 1, 1− and 0, respectively, 
if j th− diode’s anode is connected with the k th− node, j th− diode’s cathode is connected 
INTERNATIONAL COOPERATION ISSUE OF TRANSPORTATION - Special Issue - No. 10 
 94 
with the k th− node and in other cases: ( )
1
1 ; 1, ; 1, .
0
kja k n j m
= − = = 
 (7) 
We can see that A is a matrix of one linear operator ( ). : →m n , satisfies (6). We note by 
1A− a matrix of an inverse operator ( )
1
. ,
− TA is the transpose matrix of A. 
On the other hand, using (3) we obtain 
 .T DA u y= (8) 
Now, we denote by 
2 a tree containing resistances ,R inductances L and a supply source. 
Then, branches complementing the tree 
2 to the original circuit diagram are included 
resistances ,r inductances l and the diode converter .D By the second Kirchhoff’s law, we 
have 
1 ,R rU M u= (9) 
 ( )2 3 4 ,L r l DU M u M u M u E t= + + + (10) 
where, ( )E t depends on the voltage ( )e t of the supply source; 1 2 3 4, , ,M M M M are 
matrices that depend on the research circuit; ,L RU U are potential difference ,L R , 
respectively; ,l ru u are potential difference ,l r , respectively. 
From (9) and (10), it implies that 
( )
1
2 3 4
00 0
. .
r
R
l
L
D
u
U M
u
M M M E tU
u
= + 
Using the last equation and (3), we get 
1 2
3
4
. .0
0
T T
r
RT
l
LT
D
M Mi
I
i M
I
i M
= − 
Where , ; ,L R l rI I i i respectively are current intensity through ,L R ; ,l r . 
From here, we get 
4 ,
T
D Li M I= − (11) 
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 95 
3 ,
T
l Li M I= − (12) 
2 1 .
T T
r L Ri M I M I= − − (13) 
In order to find the mathematical model for rectifier circuits, we use obvious following 
equations 
,R RU RI= (14) 
 ,r ru ri= (15) 
 ,L L
dI
L U
dt
= (16) 
 .l l
di
l u
dt
= (17) 
Where , ,L I R and r are diagonal matrices whose diagonal elements are positive values. To 
solve the system (6) - (17), we consider 
LI and Du as the main unknowns. Further, by (12) and 
(17) we obtain 
3 3 3 .
T L
l
dI
M u M l M
dt
= − (18) 
On the other hand, from equations (9), (13), (14) and (15), it implies 
 ( ) 12 1 2 1 1 .T T T Tr r L R L ru r i r M I M I r M I r M R M u−= = − + = − − 
Thus, with I is the identity matrix, we have 
 ( )
1
1
1 1 2 .
T T
r Lu I r M R M r M I
−
−= − + (19) 
Using (10), (16), (18) and (19), characteristics of the research circuit are represented by 
 ( )4
L
L D
dI
I M u E t
dt
 + − = (20) 
here 
3 3:
TL M l M = + and ( )
1
1
2 1 1 2: .
T T
LM I r M R M r M I
−
− = + 
Note that 
 ( )
1 1
2 2
4; .L DX I Y M u
−
=  = − (21) 
then, the equation (20) is written by 
 ( ) ,
dX
X Y F t
dt
+ + = (22) 
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 96 
( ) ( )
1 1
2 2
1
2
,
.F t E t
− −
−
 =   
 =  
 (23) 
In order to finish the proof of the theorem we will study properties of X and .Y 
First, by (21) we obtain 
 ( ) ( ) ( )
1 1 1 1
2 2 2 2
4 4, , , .
T
L D L DX Y I M u I M u
− − 
 =   − =   − 
Since the matrix  is diagonal, we can see: ( ) ( )4, , .L DX Y I M u= − 
Furthermore, using (11), (6) and (8), we have 
 ( ) ( ) ( ) ( ) ( ) ( )4, , , , , , .T TL D D D D D DX Y M I u i u Ax u i A u x y= − = = = = 
And, by (5) we also obtain 
 ( ), 0.X Y = (24) 
On the other hand, from (6), (8) and (5) it directly implies 
 .+ 
m
Di A (25) 
Additionally, note that 
 ( ) ( )
1
1
2
4
−
+=  −
T mK M A (26) 
then, by using (25), (21) and (11) we have 
 .X K (27) 
Finally, we will prove that .KY N X For this, we estimate a value ( ), , .Y X K −  From 
(24), (26), (21) and (8) we get 
( ) ( ) ( ) ( )
1 1
2 2
4 4, , ,
T
DY X Y M u M Aa 
− 
− = =  −  − 
( ) ( ) ( ) ( )( )
( )( )( ) ( ) ( ) ( )
1 1
1 1
2 2
4 4 4 4
1
4 4
, ,
, , , , ,
T
T T
D D
T T T
D D D
M u M Aa M u M Aa
u M M Aa u Aa A u a y a
− − −
−
 =   − − = − − 
= − − = = =
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 97 
where + 
ma such that ( )
1
1
2
4 .
TM Aa 
−
 − = 
Consequently, ( ) ( ), , 0,Y X y a K − =  (because + 
ma and − 
my ). From here 
and the definition 
KN X by (2), it implies that 
 .KY N X (28) 
Thus, using (22), (27) and (28), we obtain 
 ( ) . − − K
dX
F t X N X
dt
where ( ) , ,F t K are defined by (23) and (26). That completes the proof of the theorem. 
To illustrate the results of the study, we consider a electric circuit of a following figure known 
as a full wave rectifier; it contains 4 diodes, a source, resistance R and inductance L . In a 
supply circuit there is a source including a voltage ( )e t , resistance r and inductance l . This 
case is used to show the mathematical model of the form (1) for the considered rectifier circuit. 
For this, the choice of positive voltage is marked by indicators and on the other all nodes are 
numbered by 0, 1, 2, 3 as the figure 2. 
Figure 2. The full wave rectifier 
We can see from (25) and (7) that 
( ) ( ) 41 2 3 1 2 3, , ; , , += = D Du u u u i i i i A 
with the matrix A is determined by 
1 1 0 0
0 0 1 1 .
0 1 1 0
A
− 
= − 
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 98 
One can easily see that a voltage of the supply circuit (between nodes 1 and 2) equals 
1 2u u− 
and a voltage of the load circuit (between nodes 3 and 0) equals 
3.u Then, by the Kirchhoff’s 
laws we have 
2 3 1i i i= − and 
( )1 1 1 2
3
3 3 0.
di
l ri u u e t
dt
di
L Ri u
dt
+ + − = 
 + + =
 (29) 
From (21), with 4
0 1 1 0
; ,
0 0 0 1
l
M
L
− 
 = − = 
 the system (29) is rewritten according to 
the equation of form (22), where 
0
0
r
l
R
L
 = 
 and ( )
( )
1
2 .
0
e t
F t
l
= 
Now, we have to find a set K such that X K and .KY N X 
First, we establish the set K is defined by (26): ( )
1
1
2
4 .
−
+
−  = 
T mM K A 
So, for every 
1
2
,
X
X K
X
= 
 there exists ( ) 4
4 1
, 1,4+ = =ix x i such that 
1
21
32
4
1
1 1 0 01 0 0
. . 0 0 1 1 . .1 0
1
0 0 1 1 00 1
x
xXl
xX
l x
− = −− 
So, in conjunction with (6) we deduce 
1
1 2 1
1 3 4 2
32 3
2
1
1
.
1
X
l
x x i
X x x i
l
ix x
X
L
 − 
− = − + = 
 + 
 (30) 
From here, we have 
1 1 2 2 3 4 3 2 3 1 2, , , ,i x x i x x i x x i i= − = − + = + = − 
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 99 
3 3 1 3 1
3 1
0, 0, 0,
.
 + − 
i i i i i
i i
 (31) 
Then, with 1 1 2 3, ,X i l X i L= = we obtain 
1 2
2 2 1 2 1
2
: 0, , .
  
= = −  
 
X L L
K X X X X X X
X l l
Finally, we have to show that .KY N X From (8), we have 
( )( )
( )( )
( )( )
( )( )
1
2
3
4
, 1,0,0 0
, 1,0,1 0
, 0, 1,1 0
, 0,1,0 0,
D
D
D
D
y u
y u
y u
y u
 = 
= − 
= − 
= 
and we also have: 
1 3 1 3 2 2 30, , , 0, 0.u u u u u u u 
From here it follows 
( )
( )
3 1 2
3 2 1
1 1
,
1 1
.
u u u
L L
u u u
L L
 − 
 −
 (32) 
Moreover, using the note ( )
1
2
4 ,DY M u
−
=  − we obtain 
1 2
2
3
.
− 
 = 
u u
l
Y
u
L
 (33) 
By (32) and (33), we have 
1 2
2 2 1 2 1
2
: 0, , , .
  
 = −   
 
K
Y l l
Y N X Y Y Y Y Y X K
Y L L
Thus, the characteristics of the full wave rectifier are presented by differential inclusions of the 
form (1). 
III. CONCLUSION 
Mathematical simulation of engineering systems from which to study in an overview, the 
INTERNATIONAL COOPERATION ISSUE OF TRANSPORTATION - Special Issue - No. 10 
 100 
nature of their operating principle is one of the most important applications of mathematics. The 
characteristics of the rectifier using semiconductor diodes have been investigated in this paper 
by establishing a mathematical model that describes these characteristics and analyzes the 
mathematical models received. We have also considered a concrete case to illustrate the result of 
the study. 
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