Modeling of Four Nonlinear Electronic Circuits

Recent Patents on Electrical Engineering 2010, 3, 35-42

35

Modeling of Four Nonlinear Electronic Circuits Genaro Ochoa, Germán Gutierrez, José de Jesús Rubio*, Raúl Rivera and Jaime Pacheco Instituto Politécnico Nacional - ESIME Azcapotzalco. Sección de Estudios de Posgrado e Investigación. Av. de las Granjas No. 682 Col. Sta. Catarina. Del. Azcapotzalco, CP 02250, México Received: July 23, 2009; Accepted: October 9, 2009; Revised: October 23, 2009

Abstract: There are some researchers who work in the design of the electronic circuits. In this paper, the mathematical models of four different and nonlinear electronic circuits are presented. The electronic models are nonlinear, because the model of the diode is nonlinear. In addition the comparison between the real circuits and the simulation of the mathematical models of the circuits are presented. The future research is to work in the design of the presented electronic circuits. The article presented some promising patents on the modeling of four nonlinear electronic circuits.

Keywords: Mathematical model, electronic circuits, state space system. INTRODUCTION There are some researches as are [1-8] who worked in the design of the electronic circuits. In [1], systems and methods are disclosed to fabricate an electronic device on a substrate by genetically engineering first, second, third, and fourth viruses, each having a plurality of selective binding sites. In [2], an improved HE LINAC-based ion implantation system is disclosed utilizing direct digital synthesis (DDS) techniques to obtain precise frequency and phase control and automated electrode voltage phase calibration. In [3], a method for forming a hybrid active electronic and optical circuit by using a lithography mask is presented. In [4], a method and apparatus for an electronic substrate having a plurality of semiconductor devices is described. In [5], a printed circuit board is defined having only one circuit element, with the said circuit element having one or more leads and each lead being in electrical communication to one end of a trace on the printed circuit board, wherein the said trace has a second end terminating at a pad, wherein the said pad is in electrical communication with a receptacle capable of receiving and retaining a wire. In [6], a method is disclosed for defining a dynamic array section to be manufactured on a semiconductor chip, the method includes, defining a peripheral boundary of the dynamic array section. In [7], a semiconductor chip is provided to including one or more distinct but functionally interfaced dynamic array sections. Each dynamic array section follows a dynamic array architecture that requires conductive features to be linearly defined along a virtual grate in each of a plurality of levels of the semiconductor chip. In [8], a method is provided for designing a semiconductor chip having one or more functionally interfaced dynamic array sections. A virtual grate is laid out for conductive features used to define a gate electrode level of a dynamic array section. All these researches are very interesting, but all the work is related *Address correspondence to this author at the Instituto Politécnico Nacional - ESIME Azcapotzalco. Sección de Estudios de Posgrado e Investigación. Av. de las Granjas No. 682 Col. Sta. Catarina. Delegación AzcapotzalcoMéxico; Tel: (+52)55-57296000; Ext: 64497; Fax: (+52)55-57296000; Ext: 64497; E-mail: [email protected] 1874-4761/10 $100.00+.00

to the design of electronic circuits and none of them work with the mathematical design and the validation of their electronic circuits. In books as [9-13] they presented some approximated mathematical models for linear and nonlinear systems but they did not present the graphs for making the comparison of these models with the real systems. The resistors, the capacitors and the coils are linear devices [10], but the diodes are nonlinear devices [9]. In [9] they presented that there are three kind of models of the diodes, the lineal model where the diode lets the pass of the current at 0 volts, the semi-real model is the most used of the approximation, where the diode lets the pass of the current at 0.7 volts, and the least used model is the real model, where the diode presents a nonlinear behavior. In this study, the semi-real model will be used. In this paper, the mathematical models of four different and nonlinear electronic circuits are presented. The electronic models are nonlinear because the model of the diode is nonlinear. In addition the comparison between the behavior of the real circuits and the simulation of the mathematical models of the circuits are presented. MATHEMATICAL MODEL OF THE FIRST CIRCUIT Let us construct the electronic circuit given in Fig. (1):

Fig. (1). the first electronic circuit. © 2010 Bentham Science Publishers Ltd.

36 Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1

Ochoa et al.

Where two constant resistors denoted as R , one constant capacitor C1 , one constant coil L , two diodes D1 and D2 ,

•

one constant source V1 and one sine source Vi are used.

The electric current in the capacitor C is iC and is given

There are two cases, the first one is when Vi > V1 +VD (where

as:

VD is the voltage of the diode) and the second one is when

C

Vi
V1 +VD .

Using the direct and the inverse polarization of the diode when Vi > V1 +VD gives the Fig. (2):

(5)

x1 =
RL x1 + 1L x2

dVC dt

(6)

= iC = ii
iL

A detailed analysis of the current in the nodes is given in [9,10,12]. Substituting equation (2) in equation (6) produces: C

dVC

dVC dt

dt

=

Vi
V1
0.7
VC R

=
i
1 C L

1 RC


iL

VC


(7) 1 RC

V1
0.7 + RC

1 RC

Vi

Considering (4) in (7) gives: dx2 dt

=
C1 x1


1 RC

x2


1 RC

u2


0.7 RC

+

1 RC

(8)

u1

Equations (5) and (8) describe the mathematical model of the circuit of Fig. (2) when Vi > V1 +VD . Using the direct and the inverse polarization of the diode when Vi
V1 +VD gives the Fig. (3):

Fig. (2). Circuit produced when Vi > V1 +VD

Considering the loop of the left side and that VL = L didt , VR1 = RiL , VR2 = Rii gives: L

Vi
V1
0.7
RiL
Rii
L diL dt

diL dt

=0

(1)

=
RL iL
RL ii
0.7L
1L V1 + 1L Vi

Where iL is the electric current in the coil, ii is the electric current in the input, VC is the electric voltage in the capacitor and 0.7 volts is the direct polarization of the diode D1 . A detailed analysis of the voltage in the loops is given in

Fig. (3). Circuit produced when Vi
V1 +VD

[9,10,12]. Let us consider again the loop of the left side with Vc inside of VL +VR1 and that VR2 = Rii gives: Vi
V1
0.7
VC
VR2 = 0 ii =

Vi
V1
0.7
VC

(2)

R

dt

=
RL iL + 1L VC

(9)

x 2 =
C1 x1

(3)

Let us define the states as follows: x1 = iL x2 = VC

•

x1 =
RL x1 + 1L x2 •

Substituting (2) in (1) gives: diL

Making similar process than before gives:

(4)

u1 = Vi u2 = V1

A detailed consideration of the definition of the states is given in [13]. Then the equation (3) becomes to:

Equation (9) describes the mathematical model of the circuit of Fig. (3) when Vi
V1 +VD . MATHEMATICAL CIRCUIT

MODEL

OF

THE

SECOND

Let us consider the electronic circuit of Fig. (4). Where two constant resistors R and RF , one constant capacitors C, one coil L, one diode D, one operational

Non Linear Electronic Circuit

Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1 diL dt

37

(11)

=
RL iL + 1L Vi

Considering again Vx = 0 , the electric current in the electric circuit R denoted as iR and in the capacitor C denoted as iC are: iR =

Vx
V0 Rf

iC = C

(

=
R1 V0

(12)

f

dVx dt




)

dV0 dt

=
C

dV0 dt

Getting the current in the node near to Vx is: (13)

iC + iR = iL

Substituting (12) in (13) produces:
C Fig. (4). The second electronic circuit.

dV0 dt

amplifier 741 and one sine source Vi are used. Vi is the same than the given in the circuit 1. Let us consider that the electrical current in the operational amplifier is cero and the voltage in the terminal - denoted as Vx and + denoted as V y is the same [9, 11]. There are two cases, the first one is when Vi >
VD (where VD is the voltage of the diode) and the second one is when Vi
VD . Using the direct and the inverse polarization of the diode when Vi >
VD gives the Fig. (5):

dV0 dt




1 Rf

V0 = iL

(14)

=
C1 iL
CR1 V0 f

Let us define the states as: x1 = iL

(15)

x2 = V0 u = Vi

Then the equation (11) becomes to: •

x1 =
RL x1 + 1L u

(16)

Considering (15), (14) becomes to: •

x 2 =
C1 x1
CR1 x2

(17)

f

Equations (16) and (17) describe the mathematical model of the circuit of Fig. (5). Using the direct and the inverse polarization of the diode when Vi
VD gives the Fig. (6):

Fig. (5). Circuit produced when Vi >
VD

Considering the loop between Vi , R, L and Vx and considering that VL = L didt , VR = RiL , gives: L

Vi
VR
VL
Vx = 0 Vi =
RiL
L

diL dt

(10)

Fig. (6). Circuit produced when Vi
VD .


Vx = 0

Where iL is the electric current in the coil L and VC is the electric voltage in the capacitor C. Considering that V y = Vx = 0 in (10) gives:

Making similar process than before gives: •

x1 =
RL x1 + 0.7L

(18)

•

x 2 =
C1 x1
CR1 x2 f

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Ochoa et al.

Equation (18) describes the mathematical model of the circuit of Fig. (6) when Vi
VD .

x1 = iL x2 = VC1

(20)

u1 = V1

MATHEMATICAL CIRCUIT

MODEL

OF

THE

THIRD

Let us construct the electronic circuit given in Fig. (7):

u2 = V2

A detailed consideration of the definition of the states is given in [13]. Then the equation (19) becomes to: •

x1 =


R1 L

(21)

x1
1L x2 + 1L u1
0.7L

Considering the loop of the right side gives: V2
i23 (R2 + R3 )
VC1
0.7 = 0 i23 =

V2
VC 1
0.7 R2 +R3

=

V2 R2 +R3

VC 1







R2 +R3

(22)

0.7 R2 +R3

Where i is the electric current in R and R. The electric current in the capacitor C is iC1 and is given as: C1

dVC 1 dt

(23)

= iC1 = iL + i23

A detailed analysis of the current in the nodes is given in [1, 2, 4]. Substituting equation (22) in equation (23) produces:

Fig. (7). The third electronic circuit.

Where 3 constant resistors R, R and R, one constant capacitor C, one constant coil L, two diodes D and D, one constant source V and one sine source V are used. There are two cases, the first one is when V
(-5 volts,10 volts) and the second one is when V
(-5 volts,10 volts). Using the direct and the inverse polarization of the diode when V
(-5 volts,10 volts) gives the Fig. (8):

C1

dVC 1

dVC 1 dt

dt

= iL +

V2 R2 +R3




VC 1

V

0.7 R2 +R3




R2 +R3

(24)

V

0.7 = iL + ( R +R2 )C
( R +RC 1 )C
( R +R )C 2

3

1

2

3

1

2

3

1

Considering (20) in (24) gives: • 0.7 x 2 = x1
( R +R1 )C x2 + ( R +R1 )C u2
( R +R )C 2

3

1

2

3

1

2

3

(25)

1

Equations (21) and (25) describe the mathematical model of the circuit of Fig. (8) when V
(-5 volts,10 volts). Using the direct and the inverse polarization of the diode when V
(-10 volts,-5 volts) gives the Fig. (9): Making similar process than before gives: •

x1 =


R1 L

x1
1L x2 + 1L u1
0.7L

(26)

•

x 2 = x1 +

1 ( R2 +R3 )C1

x2 +

1 ( R2 +R3 )C1

u2 +

0.7 ( R2 +R3 )C1

Equation (26) describes the mathematical model of the circuit of Fig. (9) when V
(-10 volts,-5 volts).

Fig. (8). Circuit produced when V
(-5 volts,10 volts).

Considering the loop of the left side gives: V1
R1iL
L diL dt

=


R1 L

diL dt


VC1
0.7 = 0

(19)

iL
1L VC1 + 1L V1
0.7L

Where iL is the electric current in the coil, VC1 is the electric voltage in the capacitor and 0.7 volts is the direct polarization of the diode D. A detailed analysis of the voltage in the loops is given in [9, 10, 12]. Let us define the states as follows:

Fig. (9). Circuit produced when V
(-10 volts,-5 volts).

Non Linear Electronic Circuit

MATHEMATICAL CIRCUIT

Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1

MODEL

OF

THE

FOURTH

39

Then the equation (31) becomes to: •

x1 =
C 1R x1 + C 1R x2
C 1R u

Let us consider the electronic circuit of Fig. (10)

3 5

3 4

(33)

3 4

On the other hand, the current iC 2 is: iC 2 = C2

dVC 2

(34)

dt

From (28) and (34) produces: C2 dVC 2 dt

dVC 2 dt

=
R1 VC 2 +

=


1 R4

Vi

1 R4C2

Vi

4

1 R4C2

VC 2 +

(35)

Considering (33), (35) becomes to: •

x 2 =
R 1C x2 + R 1C u 4 2

Equations (33) and (36) describe the mathematical model of the circuit of Fig. (10).

Fig. (10). The fourth electronic circuit.

Where two constant resistors R and R, two constant capacitors C and C, one operational amplifier 741 and one sine source Vi are used. Let us consider that the electrical current in the operational amplifier is cero and the voltage in the terminal - denoted as Vx and + denoted as V y is the same [9, 11]. Considering the loop between Vi , R, C and Vx gives: (27)

Vi
iC 2 R4
VC 2
Vx = 0

(36)

4 2

EXPERIMENTAL RESULTS In this section a comparison is made between the models of the real circuits with the mathematical model of this circuits. In all the following Fig. they are shown the variable voltage in the input and the states of the mathematical model. Example 1: Figure 11 shows the simulation result for

x1 = iL of the circuit of Fig. (1) and Fig. (12) it is shown its experimental result.

Where iC 2 is the electric current in the capacitor C and

VC 2 is the electric voltage in the capacitor C. Considering that V y = Vx = 0 in (27) gives: iC 2 =

Vi
VC 2

=
R1 VC 2 +

R4

4

1 R4

(28)

Vi

Considering again Vx = 0 the electric current in the electric circuit R denoted as iR5 and in the capacitor C denoted as iC 3 are: iR5 =

Vx
V0 R5

iC 3 = C3

=
R1 V0

(29)

5

(

dVx dt




dV0 dt

)

=
C3

dV0 dt

Getting the current in the node near to Vx is: (30)

iC 3 + iR5 = iC 2

Fig. (11). Simulation result for x1 = iL

Substituting (28) and (29) in (30) produces:
C3 dV0 dt

dV0 dt

=





1 R5

1 C3 R5

V0 =
R1 VC 2 +

V0 +

4

1 C3 R4

VC 2


1 R4

Vi

1 C3 R4

Vi

(31)

Let us define the states as: x1 = V0 x2 = VC 2 u = Vi

(32)

From Fig. (11) and (12), it can be seen that both signals are similar. Figure 13 shows the simulation result for x2 = VC of the circuit of Fig. (1) and in Fig. (14), it is shown its experimental result.

40 Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1

Fig. (12). Experimental result for x1 = iL

Fig. (13). Simulation result for x2 = VC

Ochoa et al.

Fig. (15). Simulation result for x1 = iL

Fig. (16). Experimental result for x1 = iL

From Fig. (15) and (16), it can be seen that both signals are similar. Figure 17 shows the simulation result for x2 = V0 of the circuit of Fig. (4) and Fig. (18) shows its experimental result.

Fig. (14). Experimental result for x2 = VC

From Fig. (13) and (14) it can be seen that both signals are similar. Example 2: Figure 15 shows the simulation result for x1 = iL of the circuit of Fig. (4) and in Fig. (16) it is shown

the experimental current of this coil.

Fig. (17). Simulation result for x2 = V0

Non Linear Electronic Circuit

Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1

41

From Fig. (19) and Fig. (20) it can be seen that both signals are similar. Figure 21 shows the simulation result for x2 = VC1 of the circuit of Fig. (7) and in Fig. (22) shows its experimental result.

Fig. (18). Experimental result for x2 = V0

From Fig. (17) and (18) it can be seen that both signals are similar. Example 3: Figure 19 shows the simulation result for x1 = iL of the circuit is of Fig. (7) and Fig. (20) shows the experimental voltage of this coil (the oscilloscope does not give the measure of currents).

Fig. (21). Simulation result for x2 = VC1

Fig. (19). Simulation result for x1 = iL Fig. (22). Experimental result for x2 = VC1

From Fig. (2) it can be seen that both signals are similar. Example 4: Figure 23 shows the simulation result for

x1 = V0 of the circuit of Fig. (10) and Fig. (24) shows its experimental result. From Figure 23 and Fig. (24), it can be seen that both signals are similar. Figure 25 shows the simulation result for x2 = VC 2 of the circuit of Fig. (10) and in Fig. (26) it is shown its experimental result. From Fig. (25) and Fig. (26), it can be seen that both signals are similar. Fig. (20). Experimental result for x1 = iL

42 Recent Patents on Electrical Engineering, 2010, Vol. 3, No. 1

Ochoa et al.

Fig. (23). Simulation result for x1 = V0 Fig. (26). Experimental result for x2 = VC 2

CURRENT & FUTURE DEVELOPMENTS It will be studied in the future that other electronic circuits, it will be presented the comparison between the real circuits and its simulation in real time and it will be considered the voltage of the diodes as a nonlinear function (in this paper the voltage of the diode is considered as a constant). The future research is to work in the design of the presented electronic circuits as in [1-8]. ACKNOWLEDGMENT

Fig. (24). Experimental result for x1 = V0

The authors thank the Secretaria de Investigación y Posgrado and the Comisión de Operación y Fomento de Actividades Académicas del IPN for their help in this research. CONFLICT OF INTEREST The authors declare no conflicts of interest. REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10]

Fig. (25). Simulation result for x2 = VC 2

CONCLUSION In this paper the mathematical models of four different and strange electronic circuits were presented. In addition the comparison between the real circuits and the simulation of the mathematical models of the circuits were presented.

[11] [12] [13]

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