Introduction
Fractional calculus, though three centuries old, remains underappreciated in the science and engineering communities Its unique characteristic is that fractional derivatives and integrals reflect non-local properties, capturing historical and distributed effects, which may align more closely with the complexities of nature This discipline enhances our understanding of fundamental natural phenomena, suggesting that it could be the language through which we communicate with nature While traditionally the domain of mathematicians, fractional calculus has recently gained traction in various applied fields, including engineering and economics Current research is exploring the definition of fractional derivatives as local operators within fractal science The next decade is expected to witness a surge in applications stemming from this subject, which can be viewed as an extension of traditional calculus This book focuses on real-number fractional orders and fixed fractional order differintegrals, leaving variable order differintegration for future exploration Aimed at making fractional calculus accessible for practical applications, this introductory chapter includes a table of fractional derivatives for commonly encountered functions, minimizing rigorous mathematical complexities.
Birth of Fractional Calculus
In a letter dated 30th September 1695, L’Hopital wrote to Leibniz asking him a particular notation that he had used in his publication for the nth derivative of a function
S Das, Functional Fractional Calculus for System Identification and Controls 1
Fractional Calculus a Generalization of Integer Order Calculus
1.3 Fractional Calculus a Generalization of Integer Order Calculus
When considering an integer \( n \), we typically visualize \( x^n \) as \( x \) multiplied by itself \( n \) times However, the concept extends beyond integers, as fractional derivatives, such as \( \frac{d^\pi}{dx^\pi} f(x) \), exist even if they are difficult to visualize Just as real numbers exist between integers, fractional differintegrals exist between conventional integer-order derivatives and \( n \)-fold integrations This leads to the generalization where \( x^n = x \cdot x \cdot x \ldots \) for integer \( n \), and for real numbers, we express it as \( x^n = e^{n \ln x} \) Additionally, the factorial \( n! = 1 \cdot 2 \cdot 3 \ldots (n-1) \) applies to integers, while for real numbers, it is represented by the Gamma function \( \Gamma(n+1) \), defined as \( \Gamma(x) = \int_0^\infty e^{-t} t^{x-1} dt \).
The generalization from integers to non-integers broadens the concept of the number line, allowing for the inclusion of fractional differintegrals As illustrated in Figure 1.1, the number line extends in both directions: the negative side represents integration, while the positive side signifies differentiation, encompassing various orders of derivatives and integrals such as \( f, \frac{d f}{dt}, \frac{d^2 f}{dt^2}, \frac{d^3 f}{dt^3}, \) and so on.
Historical Development of Fractional Calculus
The Popular Definitions of Fractional Derivatives/Integrals in
(n−1)≤α 0, involves a complex parameter a represented as a = |a|exp(jφ) His analysis revealed that the stability of this function, which can either decay to zero or increase to infinity as z rises, is contingent upon the selection of parameters a and q Specifically, the function remains bounded for increasing values of z when the condition |φ| ≥ q π/2 is met.
2.2.2.1 One-Parameter Mittag-Leffler Function
E α (z)= ∞ k = 0 z k Γ(αk+1) The expanded form is the infinite series that is as follows:
This function was introduced by Mittag-Leffler in 1903.
2.2.2.2 Two-Parameter Mittag-Leffler Functions
Two-parameter Mittag-Leffler function plays a very important role in fractional calculus This function type was introduced by R P Agarwal and Erdelyi in 1953–1954.
The two-parameter function is defined as follows:
E α, 1(z)= ∞ k =0 z k Γ(α k +1) ≡E α (z) is one-parameter Mittag-Leffler function.
The following identities follow from the definition:
24 2 Functions Used in Fractional Calculus The above equation have the general form as follows:
The trigonometric and hyperbolic functions are also manifestations of the two- parameter Mittag-Leffler function, which is indicated below:
Generalized hyperbolic function of order n is represented below: h r (z,n)= ∞ k = 0 z nk + r −1
(nk+r−1)!=z r − 1 E n , r (z n ), (r =1,2,3, ,n) and the generalized trigonometric function of order n is also represented below: k r (z,n) ∞ m =0
Mathematical handbooks describe er f c(z) as follows:
The error function is defined as er f (z)= 2
0 e − t 2 dt and is represented by series as er f (z)= 2
The complimentary error function is defined as er f c(z)=1−er f (z)=1− 2
2.2 Functions for the Fractional Calculus 25 The series asymptotic expansion of complimentary error function is er f c(z)= e − z 2 z√ π
2.2.2.3 Variants of Mittag-Leffler Function ξ t (ν,a)=t ν
(at) k Γ(ν+k+1) =t ν E 1,ν+1 (at) This function is important for solving fractional differential equations. α (β,t)=t α
∞ k = 0 β k t k(α+1) Γ({k+1}{α+1})=t α E α+ 1 ,α+ 1(βt α+1 ) This function is called Rabotnov function, and a special variant too.
(−1) n z (2−α)n+1 Γ({2−α}n+2)=z E 2−α,2 (−z 2 −α ) is the fractional sine function form-I.
(−1) n z (2 −α )n Γ({2−α}n+1)=E 2 −α, 1(−z 2−α ) is the fractional cosine function form-I. sin λ,μ (z) ∞ k = 0
(−1) k z 2k+1 Γ(2μk+2μ−λ+1) =z E 2μ,2μ−λ+1 (−z 2 ) is the fractional sine function form-II, and cos λ,μ (z) ∞ k =0
(−1) k z 2k Γ(2μk+μ−λ+1)=E 2μ,μ−λ+1 (−z 2 ) is the fractional cosine function form-II.
Generalization of the Mittag-Leffler function to two variables was suggested and were further extended by Srivastava to the following type of symmetric form.
26 2 Functions Used in Fractional Calculus ξ α,β,λ,μ ν,σ ∞ m =0
Several manifestations including several variables representing Mittag-Leffler have been made for multi-dimensional studies on fractional calculus.
2.2.2.4 Laplace Transforms of Mittag-Leffler Function
The following expressions give some identities for Laplace transforms pairs of Mittag-Leffler functions t α k +β−1 E α,β (k) (at α )↔ s α−β k!
For k >0 the operation is differentiation of Mittag-Leffler function, and for k 0 where the powers of t are integers.
Robotnov–Hartley Function
To effect the direct solution of the fundamental linear fractional order differential equations, the following function was introduced by Robotnov and Hartley (1998)
This function is the “impulse response” of the fundamental fractional differential equation and is used by control system analysis to obtain the forced or the initialized system reaction.
Miller–Ross Function
In 1993, Miller and Ross proposed a function that serves as the foundation for solving fractional order initial value problems This function is defined as the v-th integral of the exponential function.
28 2 Functions Used in Fractional Calculus
Generalized R Function and G Function
Developing a generalized function that remains unchanged under fractional differentiation or integration is highly valuable Similar to exponential, trigonometric, and hyperbolic functions in integer order calculus, the definitions of generalized Mittag-Leffler functions play a crucial role in fractional calculus While earlier sections highlighted some variants of the Mittag-Leffler function, this discussion introduces more generalized R and G functions, expanding the understanding of fractional calculus applications.
Here t is independent variable and c is the lower limit of fractional differintegration. Our interest in this function will be normally for the range t>c.
The Laplace transforms of R function are
2.2 Functions for the Fractional Calculus 29
2.2.7.2 Relationship of R Function to Other Generalized Function
30 2 Functions Used in Fractional Calculus
{(−r )(−1−r ) .(1−j−r )}(−a) j t (r +j )q−v−1 Γ(1+j )Γ({r+j}q−v) function time expression f (t ) Laplace transform F(s)
Mittag-Leffler E q (at q ) = ∞ n = 0 a n t nq Γ(nq + 1) s q s(s q − a)
List of Laplace and Inverse Laplace Transforms Related to Fractional
2.3 List of Laplace and Inverse Laplace Transforms Related to Fractional Calculus 31 log s 2 + a 2 s 2
32 2 Functions Used in Fractional Calculus
J n Bessel function of first kind
√ 1 s 2 − a 2 I 0 (at), modified Bessel function of the first kind zero order
Concluding Comments
This chapter introduces key basis functions essential for studying fractional order systems, primarily focusing on the Mittag-Leffler function, which serves as the generalized exponential function Just as the exponential function is fundamental in integer order calculus, the Mittag-Leffler function plays a similar role in fractional calculus Additionally, various compact forms of the Mittag-Leffler function are discussed, highlighting their applications in solving fractional differential equations All these functions are characterized by power-series expansions and are applicable to a range of power law processes.
In conclusion, the circuit theory experiment involving a low pass filter created with lumped resistance and capacitance leads us to question whether the observed step response truly reflects a pure exponential reaction, as expected from the unique time constant derived from these components While we often attribute deviations from this expected behavior to factors such as non-linearity, instrument errors, and component drifts, a re-evaluation using a Mittag-Leffler type power series function suggests that the underlying circuit behavior may be better described by fractional order differential equations rather than traditional integer order equations This perspective challenges our understanding of circuit dynamics and invites further exploration into the reality of circuit responses.
Fractional calculus reveals that pure capacitors exhibit no resistance, while pure resistors lack lumped characteristics, highlighting the reality of distributed parameters This concept applies to various relaxation processes, including diffusion, reactor kinetics, and electrochemistry Over the past four decades, several variants of Mittag-Leffler functions have been developed, with the potential for future advancements to further elucidate the physical processes found in nature.
Observation of Fractional Calculus in Physical System Description
Introduction
Fractional calculus provides a compact representation and solution for spatially distributed systems, enhancing the understanding of these systems While the concepts of fractional integrals and derivatives have been known since the inception of regular calculus, fractional calculus remains largely unfamiliar to many engineers Despite this, it has been explored by prominent mathematicians and scientists in operational calculus Unfortunately, much of the existing research is presented in advanced analytical language, making it less accessible to the broader engineering and scientific community Notably, several systems exhibit fractional order dynamics, with the semi-infinite lossy (RC) transmission line being one of the first recognized examples, where the current is proportional to the half derivative of the applied voltage, illustrating the concept of impedance.
The study of fractional order systems has been significantly advanced by researchers like Heaviside, who in 1871 utilized operational calculus to explore the mathematical complexities between complete differentiations and integrations He emphasized the validity of fractional operators, highlighting their real-world applications An illustrative example is the diffusion of heat in semi-infinite solids, where the boundary temperature corresponds to the half integral of the heat rate Other notable systems exhibiting fractional order dynamics include viscoelasticity, colored noise, electrode–electrolyte polarization, dielectric polarization, boundary layer effects in ducts, and electromagnetic waves Given that many of these systems are influenced by specific material and chemical properties, a diverse range of fractional order behaviors can be expected across different materials.
Temperature–Heat Flux Relationship for Heat Flowing
The thermocouple consists of two pairs of dissimilar metals with a common junction point Because the wires are long and insulated, they will be treated as
“semi-infinite” heat conductors Figure 3.1 represents one such wire of thermocouple
S Das, Functional Fractional Calculus for System Identification and Controls 35
36 3 Observation of Fractional Calculus in Physical System Description
In a semi-infinite wire thermocouple pair, heat flow is illustrated, with the thick line representing the semi-infinite heat conductor The thermocouple wire measures the temperature at the furnace wall located at x = 0.
T sur f (t), which dynamically varies with the time The initial temperature is denoted by T 0
The heat conduction issue in thermocouple wire is primarily one-dimensional This analysis demonstrates the application of fractional calculus in connecting the conduction heat flux within semi-infinite thermocouple wire to the temperature at the origin The equation involves the product of specific heat capacity (c), density (ρ), and the rate of temperature change (∂T).
In the study of heat transfer, key variables include time (t), spatial direction (x), specific heat capacity (c), density (ρ), temperature (T(t,x)), and the coefficient of heat conduction (k) These parameters are essential for understanding how heat flows through materials, with specific heat measured in joules per kilogram per Kelvin (J kg −1 K −1), density in kilograms per cubic meter (kg m −3), and heat conduction coefficient in watts per meter per Kelvin (W m −1 K −1).
Let u(t,x )=T (t,x )−T 0 Substituting this in the above set of equations, we get cρ∂u dt =k∂ 2 u
3.2 Temperature–Heat Flux Relationship for Heat Flowing in Semi-infinite Conductor 37 Taking Laplace transforms for the above equation gives cρ.sU (s,x )=k∂ 2 U (s,x )
∂x 2 −cρs k U (s,x )=0 The bounded solution for x tends to−∞is
; differentiating this, we find dU (s,x ) d x =U (s,0) scρ k exp x scρ k
From these two expressions, we get the following by putting x = 0 and taking the inverse Laplace of s −0.5 F(s) → d −1/2 f (t), i.e., semi-integration, we obtain semi-differential equation in time variable:
∂x cρ k d 1/2 dt 1/2 u(t,0) Returning from u(t,x ) to T (t,x ), we get k∂T (t,0)
Heat flux, denoted as ∂xT(t,0)=Q(t), represents the thermal energy transfer occurring at the interface between the furnace wall and the thermocouple wire at the point of contact This expression quantifies the rate at which heat is conducted through the material, providing essential insights into thermal dynamics.
38 3 Observation of Fractional Calculus in Physical System Description
Single Thermocouple Junction Temperature in Measurement of Heat
The general heat flow equation describes the relationship between heat flux in a semi-infinite conductor and the temperature at the origin, taking into account time-varying constitutive relations.
The semi-derivative is shown for initial time point a When initial forcing conditions (states) are zero, then the operator is
0 D t 1 / 2 ≡ d 1/2 dt 1 / 2 ; hereαis thermal diffusivity; T b is the body temperature, at the point of contact of thermocouple to the furnace wall.
The following equations define the time domain behavior.
The input heat flux to the thermocouple, represented by Q i = h A(T g (t) − T b (t)), originates from the steam temperature and reaches the tip of the thermocouple junction At this junction, the heat flux is distributed into two thermocouple wires, as illustrated in Fig 3.2.
Converting this expression to integral form, we obtain the thermocouple node temperature related to two heat fluxes as
The two semi-infinite heat conductors have constitutive equations in semi- differential form as derived for Fig 3.1, as
Fig 3.2 Thermocouple junction for temperature (heat flux) measurement k 1 , α 1 k 2 , α 2
3.3 Single Thermocouple Junction Temperature in Measurement of Heat Flux 39
The equation √α 2 a D t 1 / 2 T b (t) represents the relationship between the convective heat transfer coefficient (hA) and the surface area, along with the product of mass and specific heat (mc) This constitutive equation is derived by substituting the values of heat transfer (Qs) as hA.
√α 2 a D t 1 / 2 T b (t)=mcd T b (t) dt , after taking Laplace transforms of the constitutive equations, we have the following expression: mcs+ k 1
The transfer function is as follows:
The analysis highlights the significance of fractional calculus, contrasting it with traditional methods that necessitate solving two simultaneous partial differential equations alongside ordinary integer order differential equations The Bode plots reveal two distinct asymptotes: one with a slope of -10 dB/decade, indicative of semi-pole s^(1/2) behavior, and another at -20 dB/decade observed at higher frequencies This information is visually represented in Figure 3.3, where (mc/h A) is set to 0.005.
Fig 3.3 Frequency response amplitude frequency Bode plot
40 3 Observation of Fractional Calculus in Physical System Description
This analysis suggests that the traditional method of measuring heat flux in thermal systems using two thermocouples can be simplified Instead, a single thermocouple can be employed, and by analyzing the temperature values over time, the instantaneous semi-differential equation can be solved to effectively estimate the flowing heat flux.
The transfer function T b (s)/T g (s) indicates that the system is a first-order system according to integer order calculus theory Typically, a first-order system exhibits a damped response to a step input, characterized by no oscillation or overshoot However, the inclusion of fractional order terms introduces complexities, as discussed in Chapter 9.9 Despite appearing first-order, the half-order term in the denominator can lead to an oscillatory response with overshoot when subjected to a step input Consequently, the definition of system order in fractional order systems differs from that in integer order calculus.
Heat Transfer
System identification is a crucial aspect of control practice that involves determining the parameters for the mathematical model of a system This is particularly significant in thermal systems with convection, where the heat transfer coefficient is often uncertain and can fluctuate over time due to physical or chemical changes at the heat transfer surface Additionally, parameters such as the thermal capacity of the conductive body, its surface area, and thermal diffusivity must also be accurately identified to ensure effective system modeling and control.
To efficiently analyze heat transfer, it is more practical to estimate a non-dimensional expression that aligns with observed data rather than calculating each parameter individually This approach is illustrated through the cooling or heating of a one-dimensional plane wall of thickness L, which starts with a uniform initial temperature T_i The convective heat transfer coefficient h is applied from one wall to the surrounding fluid at temperature T_∞ The precise solution is derived from a partial differential equation, considering the relevant boundary and initial conditions.
The temperature field is T (x,t), where x is the coordinate measured from the wall, and t is the time The transient heat conduction equation in the wall is given by
∂x 2 , whereαis the thermal diffusivity.
The initial and boundary conditions are
T (x,t)=T i , at t=0 where k is the thermal conductivity of the wall material With change of variable to make dimensionless equation, we get the following transformed unitless variables as ξ = x
T i −T ∞ , and we obtain the dimensionless equation as
∂ξ 2 , with ∂θ ∂ξ =0, atξ =0 and ∂θ ∂ξ +B i θ=0 atξ =1, and forτ =0.
Where the “size-factor” B i =h L/k is called the Biot’s number.
Solution to this dimensionless equation, which is the exact representation of heat transfer, is θ(x, τ)= ∞ n = 1
2λ n +sin (2λ n ) andλ n are positive roots of transcendental expressionλ n tanλ n =B i
The dimensionless mean temperature is θ(τ) 1 0 θ(ξ, τ)dξ
42 3 Observation of Fractional Calculus in Physical System Description
The exact solution is obtained above, by considering various Biot’s number (0.1–10) However, the first way to approximate this heat transfer phenomena is by having spatial average of the temperature, i.e.
T (x,t)d x be taken as dependent variable In terms of this average temperature, the heat bal- ance equation is d T(t) dt + h ρc
In the analysis of heat transfer in wall materials, the density (ρ) and specific heat (c) are crucial parameters The initial and boundary conditions for convective heat transfer are defined as ∂T/∂x = 0 at x = 0 and k(∂T/∂x) + h(T - T∞) = 0 at x = 1, leading to an average expression for approximation At time t = 0, the temperature is set as T = Ti, resulting in T(0) = 1 By employing dimensionless variables similar to those used in the exact solution, the equation dθ/dτ + Biθ = 0 is derived with θ(0) = 1 Numerical experiments reveal that for Bi = 0.1, there is a close match between the approximation and the exact solution, while larger values of Bi result in significant deviations.
An improvement to the above approximation is to write a fractional order differ- ential equation, as (d q θ /dτ q )+p B i θ=0, withθ(0)=1 Here q and p be varied to minimize
0 e(τ) 2 dτ , where e(τ) is the difference between the exact solution and approximate solution of θ(τ).τ maxis the maximum value ofτ to which integration is carried out.
Here the effect of B i is to be discussed The fractional order q → 1 and the multiplier of the Biot’s number p→1, as B i →0.
Dynamic system modeling effectively utilizes measurements for simultaneous time-dependent system identification and feedback control error signals In heat transfer scenarios involving walls with simple geometries, a lumped parameter energy balance is often employed, assuming a spatially uniform temperature to model transient conductive systems subjected to convective heat fluxes This approach simplifies the fitting of experimental data and involves solving a first-order differential equation However, for larger Biot numbers, discrepancies arise between the actual temperature field and the spatial average used in the lumped model, necessitating the resolution of partial differential equations for accurate transient analysis.
Driving Point Impedance of Semi-infinite Lossy Transmission Line
Practical Application of the Semi-infinite Line in Circuits
The circuit illustrated in Fig 3.5 serves to semi-integrate the input voltage v_i(t) This process utilizes a half-order element, specifically a semi-infinite lossy line, which is grounded in the principles of the one-dimensional diffusion equation.
The terminal characteristic, or driving point impedance, is represented by the equation v(t) = r√αd - 1/2 i(t) dt - 1/2 + ϕ1(t), and can also be expressed as i(t) = 1/r√αd 1/2 v(t) dt 1/2 + ϕ2(t) This relationship is illustrated through a ladder of discrete resistance and capacitance, as shown in Fig 3.4, with its connection in an operational amplifier circuit depicted in Fig 3.5.
50 3 Observation of Fractional Calculus in Physical System Description
In an electrical circuit, the voltage (Herev(t)) and current (i(t)) at the terminal element are influenced by the resistance per unit length (r) and the product of resistance and capacitance per unit length (α) The initial conditions are dictated by the existing charge, voltage, or current across the infinite array of elements In the case of an operational amplifier with a negative feedback configuration, the relationship is expressed as v i(t) - 0 = i i(t)R.
The transition from the lowercase differential operator to the uppercase version signifies the inclusion of the initialization function A detailed discussion on the initialization of fractional differintegrals will be provided in Chapter 6.
This is the basis of semi-integrator computing element The equivalent (un- initialized) impedance form may also be calculated as Z f =r√ α/s 1/2 ,Z i = R. The transfer function (un-initialized) form is thus is v 0(s) v i (s) = −r√α
For the circuit in Fig 3.6, the negative feedback configuration gives i i (t)= 1 r√ α c D 1/2 t (v i (t)−0)
This formation with the leading coefficients specialized to one is the basis of semi- differential computing element Fig 3.7 gives a practical circuit for semi-integration with operational amplifiers.
3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line 51 Fig 3.6 Semi-differentiator v i i i
The circuit in Fig 3.7 is to realize the fractional order PID analog control system.
In this circuit, the offset adjustment parts are not explicitly shown The semi-integral control will have transfer function as
By replacing s with jω, one gets the relation as
This circuit functions as a constant phase element with a phase angle of -45°, resulting in a consistent 45° phase lag in the output in response to a sinusoidal input By utilizing specific impedance values, the transfer function remains constant.
22×10 3 =9.8, and the transfer function is
Fig 3.7 Practical circuit for semi-integrator
52 3 Observation of Fractional Calculus in Physical System Description
Table 3.1 Practical results from semi-integrator circuit measurement
Table 3.1 presents practical results indicating a nearly constant phase of approximately -55° The circuit, excited by sinusoidal voltage, recorded the phase lag alongside the peak-to-peak amplitude.
Application of Fractional Integral and Fractional
Differentiator Circuit in Control System
A fractional order control system can be designed using either analog or digital methods As illustrated in Figure 3.8, a classical integer order system, such as a DC motor, can be effectively managed by a fractional order feedback controller.
System transfer function of DC motor is
Fig 3.8 Block diagram of fractional order control system
3.5 Driving Point Impedance of Semi-infinite Lossy Transmission Line 53
J is the payload (inertia) Phase margin of the controlled system is Φ m =arg|G( jω)H ( jω)| +π, the controller characteristics is
Here, choose K 2 = T Note that H (s) is composed of a differentiator of frac- tional order (1−α) and an integral controller of orderα This gives constant phase margin as Φ m =arg|G( jω)H ( jω)| +π
The close-loop transfer function is
The step input response will be y(t)=L −1
Iso-damping refers to a control system where the overshoot remains consistent across different payloads, demonstrating both robustness and efficiency This concept is illustrated in Figure 3.9.
Time Fig 3.9 Iso-damping in fractional order controlled system
54 3 Observation of Fractional Calculus in Physical System Description
In control science, it is noteworthy that a system can utilize a fractional order controller even if it is an integer order system By applying fractional calculus, these systems can achieve robust and efficient feedback control This topic is explored in greater detail in Chapter 9.
Semi-infinite Lossless Transmission Line
The article discusses the concepts of semi-differentiation and semi-integration in relation to the driving point impedance of a semi-infinite lossy transmission line A lossless transmission line is characterized by the distributed inductance (L) and capacitance (C) along its length, as illustrated in Figure 3.4.
L will replace R The line considered here is the semi-infinite lossless line whose impedance is constant or an operator of zero order The problem is written as
The voltage (v) and current (i) in a semi-infinite lossy line can be analyzed using time-dependent input variables (v I (t)), inductance per unit length (L), and capacitance per unit length (C) A classical solution to this problem is derived through iterated Laplace transforms, as discussed in Section 3.5 The key findings from this analysis are summarized below.
This contains transfer function of driving point (not impedance) as
√LC s and in time domain v(0,t)= − 1
The equation LC dv(0,t) d x dt + ϕ1(t) represents a time-dependent initial condition The transfer function is composed of two components: the forced response attributed to V*(0,s) and the initial condition response resulting from the voltage distribution in a lossless line By utilizing the provided current expression, we can analyze the system's behavior effectively.
3.6 Semi-infinite Lossless Transmission Line 55 the driving point impedance is obtained as follows:
The voltage in the lossless line consists of two components: the forced response attributed to I(0,s) and the response resulting from the initial voltage distribution Focusing solely on the forced response, we can determine the impedance observed when analyzing this line.
I (0,s) L C which is simply a constant Mathematically, the impedance expressed in time domain as v(0,t) L
The time-dependent initial condition response of Ci (0,t)+ϕ 2(t), influenced by the initial voltage and current distribution, can be derived by applying the inverse Laplace transform to the last three terms of the equation that describes the relationship between V (0,s) and I (0,s).
A simple constant gain operator, or zero-order operator, can incorporate time-varying initial conditions As illustrated in Figure 3.10, a lossless semi-infinite transmission line serves as a zero-order element that outputs the input function unchanged, aside from gain or attenuation However, in generalized calculus theory, the initial distributed charges and voltages will affect the output, resulting in time-varying initial functions The distributed inductance (L) and capacitance (C) along the infinite line create these initialization functions over time Although this zero-order element does not require differintegrations, the initial conditions associated with its distributed characteristics are crucial to the generalized theory of initialized fractional calculus Operational amplifier circuits utilizing zero-order distributed elements provide practical insights into this concept, which is explored further in Chapters 6 and 7.
56 3 Observation of Fractional Calculus in Physical System Description
Fig 3.10 Semi-infinite lossless transmission line
In the configuration shown in Fig 3.6, a lumped resistor (R) serves as the input element while a lumped capacitor (C) functions as the feedback element, creating a lumped integrator circuit The input voltage, denoted as v_i(t), is activated at time a, during which the voltage remains at zero The integration process commences at time t = c (where c > a), indicating that the capacitor has been pre-charged with q(c) Coulombs from time a to c, establishing an initial voltage v_o(c) as the initial condition for the integrator circuit.
The describing equations for this configuration is as follows: v i (t)−0=i f R
C c D t − 1 i f (t). c D −1 t is integer order one integration process starting from time t = c which includes the initialization, process that is charging of the capacitor C from time t =a to t=c which is represented as ψ / (t) c a i f dt = 1
Therefore, the total process is un-initialized integration starting from time t = c, that is: c d t −1 i f t c i f dt plus initialization integration process from a to c, that is: a d c − 1 i f c a i f dt =ψ / (t)
3.6 Semi-infinite Lossless Transmission Line 57 These equations yield the final result by putting i i (t)=i f (t) v o (t)= − 1
RC c D t − 1 v i (t), withψ(t) = −RCv o (t) This is classical integer order calculus, with initialization as constant.
In the circuit depicted in Fig 3.6, the input component is substituted with a semi-infinite lossless (LC) transmission line, classified as a zero-order element, while the feedback component is replaced by a lumped capacitor, denoted as C_f The terminal equation for the transmission line is reformulated as i(t) C.
Lv(t)+ϕ(t), withϕ(t) as initial charge distribution on the distributed element.
The defining equations of this circuit are i i (t) C
C f t t = c i f (t)dt+[−v o (c)], as done for the lumped integrator case above i i (t)=i f (t) Therefore solving forv o (t), we obtain v o (t)=−
Here the initialization function is not a constant, but a function of time.
This expression resembles the classical integer order integrator with lumped parameters; however, it is implemented using distributed elements A key distinction lies in the initialization function values, as the distributed element integrator incorporates the influence of past history, which is not limited to a constant value \( v_o(c) \).
The observation of fractional calculus in physical systems reveals that the charge on the capacitor \( C_f \) is influenced by the distributed charge represented in the function \( \psi(t) \) along a semi-infinite line Additionally, it is noted that the zero-order input element behaves in accordance with a wave equation.
The propagation of perturbations along a semi-infinite line, as described by ∂x², results in variations in the input voltage v_i(t) that never return to the circuit, particularly during terminal charging In contrast, side charging, which involves arbitrary voltage distribution along the line, can introduce an additional time function that may affect the circuit output based on the initial voltage distribution When the circuit, as shown in Fig 3.6, is configured with a lumped capacitor (C) as the input element and a lumped resistance (R) as the feedback element, it functions as an integer order differentiator, governed by the equation v_i(t) - 0 = 1.
The equation governing the differentiation process is given by \( v_i(t) = -RC \cdot \frac{d}{dt} v_i(t) + \psi(t) \), where the initialization term \( \psi(t) = \frac{d}{dt} v_i(c) \) is typically assumed to be zero However, if there is an initial charge present in the input capacitor, it results in an impulse output at the beginning of the differentiation process when \( t = c \).
By altering the circuit in Fig 3.6 to include a capacitor \( C_i \) as the input element and a distributed LC zero-order element as the feedback component, we achieve an integer-order differentiator transfer function This modification is grounded in the principles of initialization functions and generalized calculus, leading to the defining equation \( v_i(t) - 0 = 1 \).
3.6 Semi-infinite Lossless Transmission Line 59 For the distributed feedback zero-order elements, the expression in the circuit is
Putting i i (t)=i f (t), yields the final result as v o (t)= −
Generalized differentiation necessitates an initialization function In the case of terminal charging for integer order differentiation, this initialization is set to zero Conversely, for side charged transmission lines, an additional time function is outputted by the circuit.
A simple gain (memory) less zero-order operator can be implemented by setting up the circuit shown in Fig 3.6, where R i acts as a lumped resistor at the input and R f serves as a lumped resistor in the feedback loop Consequently, the transfer characteristics are expressed as v o (t) = −R f.
,withψ(t)=0, clearly this circuit has no memory.
Zero-order circuit may be realized by employing semi-infinite distributed lossless transmission lines at input leg and one lumped resistor R at feedback, of circuit of Fig 3.6.
The input leg equation with LC line is i i (t) C
The feedback leg equation is
60 3 Observation of Fractional Calculus in Physical System Description