Fractional Transmission Line with Losses José Francisco Gómez-Aguilar

(1)

Fractional Transmission Line with Losses

José Francisco Gómez-Aguilar^aand Dumitru Baleanu^b,c,d

aDepartment of Solar Materials, Renewable Energy Institute, National Autonomous University of Mexico, Priv. Xochicalco s/n. Col. Centro, Temixco Morelos, Mexico

bCancaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, Balgat 0630, Ankara, Turkey

cDepartment of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

dInstitute of Space Sciences, P.O. Box MG-23, R 76900, Magurele-Bucharest, Romania Reprint requests to J. F. G.; E-mail:jgomez@ier.unam.mx

Z. Naturforsch.69a, 539 – 546 (2014) / DOI: 10.5560/ZNA.2014-0049

Received January 23, 2014 / revised June 2, 2014 / published online August 13, 2014

In this manuscript, the fractional transmission line with losses is presented. The order of the Caputo derivative is considered as 0<β≤1 and 0<γ≤1 for the fractional equation in space and time domain, respectively. Two cases are solved, with fractional spatial and fractional temporal derivatives, and also numerical simulations were carried out, where there are taken both derivatives in simultaneous form. Two parametersσxandσtare introduced and a physical relation between these parameters is reported. Solutions in space and time are given in terms of the Mittag–Leffler functions. The classic cases are recovered whenβ andγare equal to 1.

Key words:Transmission Line; Fractional Calculus; Caputo Derivative; Fractional Components.

1. Introduction

Mathematical and physical considerations in favor of the use of models based on derivatives of non- integer order are given in [1–16] and the references therein. The telegrapher’s equations describe the voltage and current on an electrical transmission line with distance and time. The model demonstrates that the electromagnetic waves can be reflected on the wire and that wave patterns can appear along the line. The concept of fractional curl operator and the fractional paradigm in electromagnetic theory have been introduced by Engheta in [17]. In [18], an alternative fractional construction for the electromagnetic waves in terms of the fractional derivative of the Caputo type is presented. The authors considered the propagation of electromagnetic waves in an infinitely extended homogeneous media characterized by the permittiv- ityε and permeability µ. Orsingher and Beghin [19]

discussed the time-fractional telegraph equation with Brownian time. Chen et al. [20] analyzed and derived the solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions, using the method of separating variables. Ors-

ingher and Zhao [21] obtained the Fourier transform of its fundamental solution and presented a symmet- ric process with discontinuous trajectories. This tran- sition function satisfies the space-fractional telegraph equation. Momani [22] suggested analytic and ap- proximate solutions of the space- and time-fractional telegraph differential equations by means of the so- called Adomian decomposition method. In [23], the so-called general space-time fractional telegraph equations were discussed by the methods of differential and integral calculus. Huang [24] derived the analyt- ical solution for three basic problems of the so-called time-fractional telegraph equations. A systematic way to construct fractional differential equations for physical systems has been proposed in [25]. This method- ology consists in analyzing the dimensionality of the ordinary derivative operator and trying to bring it to a fractional derivative operator consistently.

In this work, the idea proposed in [25] for con- structing fractional differential equations is applied in the fractional transmission line with losses represen- tation. The order of the derivative being considered is 0<β≤1 and 0<γ≤1 for the fractional equation in space- and time-domain, respectively.

(2)

The paper is organized as follows. The second section presents the basic definitions of the fractional calculus, the third section analysis the equation of the fractional transmission line with losses, and the fourth section shows our conclusions.

2. Basic Definitions

We used in this work the Caputo fractional derivative (CFD). For these definition, the derivative of a constant is zero and the initial conditions for the fractional- order differential equations can be given in the same manner as for the ordinary differential equations with the known physical interpretation. The CFD for a function f(t)is defined as follows [5]:

C0D^γ_t f(t) = 1 Γ(n−γ)

Z t 0

f⁽ⁿ⁾(η) (t−η)^γ−n+1dη, (n−1<γ≤n),

(1)

whereΓ(·)is the gamma function,n=1,2, . . .∈N. The Laplace transform to CFD is given by [5]

Lh

C

0D^α_t f(t)i

=S^αF(S)−

m−1

∑

k=0

S^α−k−1f^(k)(0). (2) Some common Laplace transforms are

1

s^α+a=t^α−1E_α,α(−at^α), s^α

s(s^α+a)=E_α(−at^α), a

s(s^α+a)=1−E_α(−at^α).

(3)

The inverse Laplace transform requires the introduction of the Mittag–Leffler function defined by the se- ries expansion as [5]

Ep,q(t) =

∞ m=0

∑

t^m

Γ(pm+q), (p>0), (q>0), (4) whenp=1 andq=1, then from (4), we obtain e^t, the exponential function.

3. An Alternative to Fractional Transmission Line with Losses

Some authors replace the integer derivative by a fractional one on a pure mathematical basis without

considering that the physical parameters involved in the equations have the correct dimensionality.

To be consistent with the dimensionality measured in practice and following [25], we introduce parame- tersσ_xandσ_tin the following way:

d dx→ 1

σx^1−β

· d^β

dx^β , n−1<β≤n, (5) d

dt → 1 σ_t^1−γ

· d^γ

dt^γ, n−1<γ≤n, (6) wheren=1,2, . . .∈N. When β =1 andγ =1, the expressions (5) and (6) become a classical derivative, which is true if the parameterσ_xhas the dimension of length (meters) andσthas the dimension of time (seconds). This expression represents a space and temporal derivative since their dimensions are inverse meters and inverse seconds. The parametersσxandσt char- acterize the fractional space and fractional temporal structures (components that show an intermediate behavior between a system conservative and dissipative) of the fractional space or fractional temporal operator [25]. In the following, we apply this idea for the case of the transmission line with losses.

The equivalent circuit for the transmission line with losses is represented in Figure1.

Applying the Kirchhoff circuit laws in the node A and using (5) and (6), we get the equations of the fractional transmission lines with losses for the voltage and current, respectively:

1 σx^2(1−β⁾

· ∂^2β

∂x^2βV(x,t)− LC σ_t^2(1−γ)

· ∂^2γ

∂t^2γV(x,t)

−RC+GL σ_t^1−γ

· ∂^γ

∂t^γV(x,t)−GRV(x,t) =0, 0<β≤1, 0<γ≤1,

(7)

1 σx^2(1−β⁾

· ∂^2β

∂x^2βI(x,t)− LC σ_t^2(1−γ)

· ∂^2γ

∂t^2γI(x,t)

−RC+GL σ_t^1−γ

· ∂^γ

∂t^γI(x,t)−GRI(x,t) =0, 0<β≤1, 0<γ≤1,

(8)

whereRdenotes the resistance of the conductors,Lthe inductance due to the magnetic field around the wires, Cthe capacitance between the two conductors, andG the conductance of the dielectric material separating the conductors.

(3)

Ldx

A

Cdx

Rdx

Gdx

x

V(x,t) V(x+dx,t)

i(x+dx,t) i(x,t)

Fig. 1. Equivalent circuit for the transmission line consider losses.

3.1. Fractional Space Transmission Line with Losses In this work, considering (7) and assuming that the space derivative is fractional (5) and the time derivative is ordinary, the spatial fractional equation is

∂^2(β)

∂x^2(1−β)V(x,t)−LCσx^2(1−β)· ∂²

∂t²V(x,t)

−(RC+GL)σ_x^2(1−β)· ∂

∂tV(x,t)−GRσ_x^2(1−β)

·V(x,t) =0, 0<β ≤1.

(9)

Supposing the solution is

V(x,t) =V₀·e^iωtu(x) (10) and substituting (10) in (9), we obtain

d^2βu(x) dx^2β +

(LCω²−GR)

−iω(RC+GL)

σx^2(1−β)u(x) =0.

(11)

From (11), we can expect the wave vectork to have a realδ and an imaginaryϕpart. This vector indicates a propagation with attenuation,

δ=LCω²−GR, (12)

ϕ= (RC+GL)ω, (13)

where

k=δ−iϕ, (14)

k²=

(LCω²−GR)−iω(RC+GL)

σx^2(1−β⁾. (15)

k² is the fractional dispersion relation in presence of fractional space components, andkis the wave vector without that presence.

Substituting (15) in (11), we have

∂^2βu(x)

∂x^2β +k²u(x) =0. (16) Thus, the solution is written as

u(x) =E_2β −k²x^2β

. (17)

The particular solution of (17) is

V(x,t) =V₀·e^iωt·E_2β(−k²x^2β), (18) whereE_2β(−k²x^2β)is the Mittag–Leffler function.

Consideringk= (δ−iϕ)and substituting this expression in (15) we have

(δ−iϕ)²=δ²−2iδ ϕ−ϕ² (19) where

δ²−2iδ ϕ−ϕ²=

(LCω²−GR)−iω(RC+GL)

σx^2(1−β⁾. (20) Solving this equation forϕ, we obtain

ϕ=RCω+GLω

2δ σx^2(1−β⁾, (21)

and forδ δ =p

LCω²−GR (22)

·

"

1 2±1

2 s

1+(RCω+GLω)² (LCω²−GR)²

#¹₂ σ_x^1−β. Now substituting (22) in (21), we get

ϕ=RCω+GLω

2 (23)

· 1

p(LCω²−GR)

1 2±¹₂

r

1+^(RCω+GLω)²

(LCω²−GR)²

¹₂ σ_x^1−β.

Equations (22) and (23) describe the real and imaginary part of the wave vector in terms of the frequency ω, the parametersR,L,C,G, and the fractional space structureσx. Equations (22) and (23) are the fractional Helmholtz equations and indicate propagation with

(4)

attenuation, caused by the losses. Now we analyze the cases whenβ takes different values.

Case I.Forβ=1, we havek= (δ−iϕ), where substituting this expression in (22), we obtain forδ

δ= q

(LCω²−GR)

·

"

1 2±1

2 s

1+(RCω+GLω)² (LCω²−GR)²

#¹₂ ,

(24)

and forϕ, we have ϕ=RCω+GLω

2

· 1

p(LCω²−GR)

1 2±¹₂

r

1+^(RCω+GLω)²

(LCω²−GR)²

¹₂

. (25)

Equations (24) and (25) describe the real and imaginary part of the wave vector in terms of the frequency ω and the parametersR,L,C, andG.

From (18), we have

V(x,t) =V₀·e^iωt·E2(−k²x²), (26) where

V(x,t) =V₀·e^iωt·cosh p

−k²x²

=cos(−ikx) =Re(e^−ikx).

(27) In (27), Re indicates the real part, andk=δ−iϕ is the wave vector, δ andϕ is given by (24) and (25), respectively. Substituting these expressions in (27), we have

V(x,t) =Re

V₀·e^i(ωt−δ^x)·e^−ϕx

. (28)

Equation (28) represents the classic solution for the wave equation in a transmission line with losses.

Case II.Forβ =¹₂, we havek= (δ−iϕ)σ_x, where substituting this expression in (22), we obtain forδ

δ=p

LCω²−GR

·

"

1 2±1

2 s

1+(RCω+GLω)² (LCω²−GR)²

#¹₂ σx,

(29)

and forϕ, we have

ϕ=RCω+GLω

2 (30)

· 1

p(LCω²−GR)

1 2±¹₂

r

1+^(RCω+GLω)²

(LCω²−GR)²

¹₂ σx.

Equations (29) and (30) describe the real and imaginary part of the wave vector in terms of the frequency ω and the parametersR,L,C, andG in presence of fractional space components represented byσ_x.

The solution for (18) is

V(x,t) =V₀·e^iωt·e^−k²^x, (31) wherek²=

(LCω²−GR)−iω(RC+GL) σx. We have a direct relation between the parameterσ_x and the wave lengthλgiven by the orderβof the fractional differential equation:

β=kσ_x=σx

λ , 0<σx≤λ. (32) We can use this relation in order to write (18) as

V(x,˜t) =V₀·e^iωt·E_2β −β^2(1−β⁾x˜^2β

, (33)

where ˜x= ^x

λ is a dimensionless parameter.

3.2. Fractional Time Transmission Line with Losses In (7), assuming that the space derivative is integer and the time derivative is fractional (6), the temporal fractional equation is

∂^2γ

∂t^2γV(x,t) +RC+GL

LC σ_t^1−γ· ∂^γ

∂t^γV(x,t)

−σ_t^2(1−γ) LC · ∂²

∂x²V(x,t) +GR LCσ_t^2(1−γ)

·V(x,t) =0, 0<γ≤1.

(34)

Supposing the solution

V(x,t) =V₀eⁱ^˜kxu(t), (35) where ˜k is the wave vector, and substituting (35) in (34), we obtain

d^2γu(t)

dt^2γ +RC+GL

LC σ_t^1−γd^γu(t) dt^γ +˜k²+GR

LC σ_t^2(1−γ)u(t) =0.

(36)

(5)

The solution of (36) may be obtained by applying (2) and the inverse Laplace transform. Taking the solution (35), we have

V(x,t) =V₀eⁱ^˜kx·Eγ

−(RC+GL)σ_t^1−γ

2LC t^γ

·E2γ −

"

k˜²+GR

LC −(RC+GL)² 4L²C²

#

σ_t^2(1−γ)t^2γ

! .

(37)

For the underdamped case, we have (^k^˜²_LC^+GR −

(RC+GL)²

4L²C² )=0,R=^{2 ˜}^k

√ LC+GL

C . ConsidererR=^{2 ˜}^k

√ LC+GL

C

andV(0) =V₀in (37),ω₀²=^k^˜²_LC^+GR is the undamped natural frequency, andα²=^RC+_2LC^GL is the damping factor.

Forγ=1, from (37), we have V(x,t) =V₀e^{i ˜}^kx·e⁻^(RC+GL)t^2LC

·cos

sk˜²+GR

LC −(RC+GL)² 4L²C² t

! .

(38)

Equation (38) represents the classic case and the well- known result. This equation describes an ordinary underdamped system characterized by a constant of time τ = _RC+GL^2LC and undamped natural frequency ω0= qk+˜ GR

LC . From (38), we see that there is a physical relation betweenγandσ_tgiven by

γ=

k˜²+GR

LC −(RC+GL)² 4L²C²

¹₂ σt, 0<σ_t≤ 1

_˜

k²+GR

LC −^(RC+GL)²

4L²C²

¹₂ .

(39)

Then the solution (37) for the underdamped case R<^{2 ˜}^k

√ LC+GL

C orα<ω₀takes the form V(x,t) =˜ V₀e^{i ˜}^kx

·E_γ

− RC+GL 2LC

qk˜²+GR

LC −^(RC+GL)²

4L²C²

γ^1−γt˜^γ

·E_2γ

−γ^2(1−γ)t˜^2γ

, (40)

where ˜t= (^k^˜²^+GR_LC −^(RC+GL)²

4L²C² )¹²tis a dimensionless parameter.

Due to the conditionR<^{2 ˜}^k

√ LC+GL

C , we can choose an example:

RC+GL 2LC

qk˜²+GR

LC −^(RC+GL)²

4L²C²

=1 3,

0≤ RC+GL

2LC

qk˜²+GR

LC −^(RC+GL)²

4L²C²

<∞.

(41)

So, the solution (37) takes its final form V(x,t) =˜ V₀e^{i ˜}^kx·Eγ

−1 3γ^1−γt˜^γ

·E2γ

−γ^2(1−γ)t˜^2γ

.

(42)

In the overdamped case,α>ω0orR>^{2 ˜}^k

√LC+GL

C ,

the solution of (37) has the form V(x,˜ t) =V˜₀e^{i ˜}^kx·E_γ

−(RC+GL)σ_t^1−γ

2LC t^γ

·E_γ

−

(RC+GL)² 4L²C² −

k˜²+GR LC

¹₂ σ_t^1−γt^γ

.

(43)

Forγ=1, from (43), we have

V(x,˜ t) =V˜₀e^{i ˜}^kx·e

−^RC+GL_2LC

1+

r

1−^2LC(^k^˜²^+GR)

(RC+GL)2

t

, (44)

where ˜V(0) =V˜₀is the initial voltage in (x=0,t=0).

The solution (44) represents the change of the voltage V(x,˜ t)on the wave. This represent the classic case and the well-known solution.

Taking into account that the relation betweenγ and σtis

Fig. 2. Simulation for (33) and (42) forβ=1 andγ=1.

(6)

Fig. 3. Simulation for (33) and (48) forβ=1 andγ=1.

γ=

(RC+GL)²

4L²C² −k˜²+GR LC

¹₂ σt, 0<σt≤ 1

_(RC+GL)2

4L²C² −^k^˜²_LC^+GR¹₂ .

(45)

Then the solution (43) takes the form V(x,˜ t) =˜ V˜₀e^{i ˜}^kx

·E_γ

− RC+GL 2LC

q(RC+GL)²

4L²C² −^k^˜²^+GR_LC γ^1−γt˜^γ

·E_γ

−γ^1−γt˜^γ

, (46)

where ˜t =_(RC+GL)2

4L²C² −^k^˜²^+GR_LC ¹₂

t is a dimensionless parameter.

Due to the conditionR>^{2 ˜}^k

√ LC+GL

C , we can choose an example

RC+GL 2LC

q(RC+GL)²

4L²C² −^k^˜²_LC^+GR

=3,

1< RC+GL 2LC

q(RC+GL)²

4L²C² −^k^˜²^+GR_LC

<∞.

(47)

Then, the solution (43) can be written in its final form V(x,˜ t˜) =V˜₀e^{i ˜}^kx·Eγ

−3γ^1−γtγ˜

·E_γ

−γ^1−γt˜^γ .

(48)

Fig. 4. Simulation for (33) and (42) forβ=0.9 andγ=0.9.

3.3. Space-Time Fractional Transmission Line with Losses

Now considering (7) and assuming that the space and time derivative are fractional, then the order of the time-space fractional differential equation is represented by 0<β ≤1 and 0<γ ≤1. σ_xhas the dimension of length andσt of time. Figures2–5show numerical simulations of (33) – (42) and (33) – (48) for different values ofβ andγ arbitrarily chosen.

4. Conclusions

The fractional transmission line with losses from the point of view of fractional calculus is presented. Two types of fractional differential equations have been ex- amined separately; the fractional spatial derivative and the temporal fractional derivative. The parametersσx

andσ_t are introduced characterizing the existence of the fractional space and time components, respectively.

These parameters represent components that show an intermediate behavior between a system conservative and dissipative. Finally, numerical simulations where taken for both derivatives in simultaneous form. The general solutions of the fractional differential equations depending only on the parameter β andγ and given in the form of the multivariate Mittag-Leffler functions preserve the physical units of the system studied.

For the spatial case, solution (18) corresponds to the spatial generalized solution of the voltage in the

(7)

Fig. 5. Simulation for (33) and (48) forβ=0.9 andγ=0.9.

transmission line. In the case β =1, (28) represents the classic case. For β = ¹₂, (31) represents the voltage equation with fractional spatial componentsσx. In this case exists a physical relation between the param- eterσxand the wave lengthλ given by the orderβ of the fractional differential equation. For the case of relation (32), the periodicity with respect to space xis broken and the wave behaves like a voltage wave with spatial-decaying amplitude (33).

For the temporal case, (37) shows the solution for the fractional time transmission line with losses. For the underdamped case,ω0=

qk˜²+GR

LC is the undamped natural frequency, and α =q

RC+GL

2LC is the damping factor. Forγ=1, (38) shows an ordinary underdamped

system characterized by a constant of timeτ=_RC+GL^2LC and undamped natural frequencyω0=

qk˜²+GR LC . This solution represents the classic case, a physical relation betweenγ andσ_t is given by (39). In the overdamped case, the solution of (37) is represented for (43). For γ =1, a physical relation betweenγ and σt is given by (45), and substituting this relation, we obtain the solution (46).

In the case where both derivatives are considered (time-space) simultaneous, Figures2–5exhibit a non- local behavior of the wave voltage interpreted as an existence of memory effects which correspond to the intrinsic dissipation characterized by the exponents of the fractional derivativeβ andγ in the system and is related to the behavior of the wave voltage in a fractal space-time geometry.

Solutions (18) and (37) correspond to space and time generalized solutions of the fractional transmission line with losses. Both solutions are given in terms of the Mittag–Leffler function depending only on the parametersβandγ, respectively. The physical units of the system are preserved.

We believe that with this approach it will be possible to have a better study of the transient effects, analysis, and development of new mechanisms failure in electrical systems.

Acknowledgements

We would like to thank to Mayra Martínez for the interesting discussions. This research was supported by CONACYT.

[1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York 1974.

[2] K. S. Miller and B. Ross, An Introduction to the Frac- tional Calculus and Fractional Differential Equations, John Wiley, New York 1993.

[3] M. Duarte Ortiguera, Fractional Calculus for Scientists and Engineers, Springer, New York 2011.

[4] D. Cafagna, Fractional Calculus: A Mathematical Tool From Past for Present Engineers, IEEE Industrial Elec- tronics Magazine, New York 2007.

[5] I. Podlubny, Fractional Differential Equations, Aca- demic Press, New York 1999.

[6] D. Baleanu, Fractional Calculus Models and Numerical Methods, World Scientific Publishing Company, Singa- pore 2012.

[7] D. Baleanu, New Trends in Nanotechnology and Fractional Calculus Applications, (Eds. Z. B. Gün- venc and J. A. Tenreiro Machado), Springer, New York 2010.

[8] R. Metzler and J. Klafter, Phys. Rep.339, 1 (2000).

[9] F. G´omez, J. Rosales, and M. Gu´ıa, Cent. Eur. J. Phys, 11, 1361 (2013).

[10] M. Gu´ıa, F. G´omez, and J. Rosales, Cent. Eur. J. Phys, 11, 1366 (2013)..

[11] J. F. G´omez-Aguilar and D. Baleanu, Proc. Roman.

Acad. A1, 27 (2014).

[12] F. G´omez, J. Bernal, J. Rosales, and T. C´ordova, J.

Electr. Bioimp.3, 2 (2012).

[13] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore 2000.

(8)

[14] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractional Operators, Springer, Berlin 2003.

[15] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Roddin 2006.

[16] V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, New York 2012

[17] N. Engheta, IEEE Trans. Anten. Propagat. 44, 554 (1996).

[18] J. J. Rosales, M. Gu´ıa, J. F. G´omez, and V. I. Tkach, Discontin. Nonlin. Complex.1, 325 (2012).

[19] E. Orsingher and L. Beghin, Probab. Theory. Rel.

Fields.128, 141 (2004).

[20] J. Chen, F. Liu, and V. Anh, J. Math. Anal. Appl.338, 1364 (2008).

[21] E. Orsingher and X. Zhao, Chin. Ann. Math. B24, 45 (2003).

[22] S. Momani, Appl. Math. Comput.170, 1126 (2005).

[23] R. Figueiredo Camargo, A. O. Chiacchio, and E.

Capelas de Oliveira, J. Math. Phys.49, 33505 (2008).

[24] F. Huang, J. Appl. Math.2009, 890158 (2009).

[25] J. F. Gómez-Aguilar, J. J. Rosales-Garc´ıa, J. J. Bernal- Alvarado, T. Córdova-Fraga, and R. Guzmán-Cabrera, Rev. Mex. Fís.58, 348 (2012).