COMPEL 30,2. Ramazan Rezaeyan Department of Mathematics, Sciences and Research Branch, Islamic Azad University, Tehran, Islamic Republic of Iran, and

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1 The current issue and full text archive of this journal is available at COMPEL 3,2 812 A stochastic perspective of RL electrical circuit using different noise terms Rahman Farnoosh and Parisa Nabati Applied Mathematics Department, Iran University of Science and Technology, Tehran, Islamic Republic of Iran Ramazan Rezaeyan Department of Mathematics, Sciences and Research Branch, Islamic Azad University, Tehran, Islamic Republic of Iran, and Morteza Ebrahimi Applied Mathematics Department, Iran University of Science and Technology, Tehran, Islamic Republic of Iran COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 3 No. 2, 211 pp q Emerald Group Publishing Limited DOI 1.118/ Abstract Purpose The purpose of this paper is to analyze the effect of the white, colored and mixture noise perturbations as Gaussian process on the parameters of the RL electrical circuit including potential source and resistance. Design/methodology/approach By adding different noise terms in the voltage and resistance parameters of an RL electrical circuit, the deterministic model is replaced by a stochastic differential equation (SDE). Findings Owing to the application of multiple Ito s formula the analytical solutions of resulted SDEs have been obtained. Furthermore, based on a numerical method involving Euler-Maruyama scheme, the solution of the problem at the point of interest as a continuous time stochastic process has been obtained. Also shown is that the confidence interval for mean of solutions with colored and mixture noises is better than white noise. Practical implications Numerical tests via Matlab programming are performed in order to show the efficiency and accuracy of the present work. Numerical experiments show that an excellent estimation on the solution can be obtained within a couple of minutes time at Pentium IV-2.4 GHz PC. Originality/value It is believed that the stochastic model of an RL circuit with colored and mixture noises in potential source has not been studied before. Furthermore, according to latest information from the research works, two stochastic parameters in voltage and resistance of RL circuit including colored and mixture noise processes have been investigated for the first time in this paper. Keywords Circuits, Stochastic modelling, Numerical analysis, Simulation Paper type Research paper 1. Introduction The effects of intrinsic noise within physical phenomena are ignored when mathematical models of their behavior are constructed using deterministic differential equations. Such equations describe what is occurring on average to the system. However, more information is often required to obtain a full picture of the behavior of the model. In these circumstances, a stochastic model is used. In fact by adding random elements into the differential equations, a stochastic differential equation (SDE) arises. Every unwanted signal that adds to the information is called noise. Noise in dynamical

2 system is usually considered as a nuisance. SDEs include a random term which describes the intrinsic noise, or randomness, within dynamical systems. In recent decades, the mathematics of SDEs have found many applications in areas such as chemistry, physics, mathematical biology and engineering (Kloeden and Platen, 1995; Carletti et al., 24; Carletti, 26). However, in certain non-linear systems, including electronic circuits the presence of noise can enhance the detection of weak signals. Among other disturbing signals electronic noise in the electronic devices is not avoidable. The noise behavior of the devices disturbs the desired electric behavior of integrated circuits. Also, presence of the electronic noise in a circuit lowers the device dimensions. In the parameters of electric circuits such as potential source and resistance, we mainly can distinguish among white, colored and mixture noises. These stochastic processes describe different randomly disturbing phenomena in the RL circuit. Based on the theory of SDEs, randomly disturbed electric circuits can be modeled in the time domain. The theory of SDEs has proven to be the proper approach for modelling and solving ordinary differential equations disturbed by white, colored and mixture noise processes (Sections ). To date several works have been presented on the application of SDEs to the research of radar scattering and wireless communications. For example, Field and Tough (2) via an excellent research work have been applied SDEs to analyze K-distributed noise in electromagnetic scattering. Charalambous and Menemenlis (21) have been used SDEs to modelling multipath fading channels successfully. The effects of white noise in electrical circuits have been studied by many researchers (Kampowsky et al., 1992; Penski, 2; Rawat and Parthasarathy, 28; Ham and Hajimiri, 2; Kamarianakis and Frangos, 21). The literature reviews showed that Kampowsky et al. (1992) have been described classification and numerical simulation of electrical circuits with white noise. Penski (2) has been presented a numerical method for SDEs with white noise and its application in circuit simulation. It can be observed from a good work of Rawat and Parthasarathy (28) that an SDE with white noise has been used for modelling RC circuit. Up to now, very little work has been done on filed models with non-white noises. Since the path of a Wiener process are nowhere differentiable, a white noise cannot be considered a stochastic process in the usual way but it can be approximate by conventional stochastic processes with wide spectral bands which are commonly known as colored noise processes. The most famous example of this kind of noise is the Ornestein-Uhlenbeck process which is explained in Section 2. In the present work, we consider a specific numerical method for obtaining the solution of an ordinary SDE that arises in mathematical modelling of electrical circuits. This numerical method uses Euler-Maruyama (EM) scheme to represent the solution of the problem at the point of interest as a continuous time stochastic process. Furthermore, we are interested to obtain the analytical solution of voltage and resistance in the stochastic model of electrical circuit. For an RL circuit described by a SDE such solutions are delivered by the so-called Ito formulas (Oksendal, 2). The outline of this paper is as follows. In Section 2, the deterministic and stochastic model of an RL circuit is described. The stochastic model with white noise in a potential source is considered in Section 2.2. The SDEs of RL circuit with colored and mixture noises are established in Sections 2.3 and 2.4, respectively. Section 3 covers the stochastic model of RL circuit with two noise terms in voltage and resistance. A stochastic perspective of RL circuit 813

3 COMPEL 3,2 814 Numerical experiments are then carried out to simulate the confidence interval for the expectation of solution for resulting SDEs in Section 4. Section 5 offers conclusions and future directions. Some references are introduced at the end of this paper. 2. Mathematical modelling of an RL electrical circuit 2.1 The deterministic model Any electrical circuit consists of resistor (R), capacitor (C) and inductor (L). These circuit elements can be combined to form an electrical circuit in four distinct ways: the RC, RL, LC and RLC circuits. Then, an inductor-resistor circuit (RL) is an electrical circuit composed of resistor and inductor driven by a voltage or current source. The ordinary differential equation describing the behavior of the RL circuit is given by Kirchhoff s current law: L di dt þ RIðtÞ ¼VðtÞ; IðÞ ¼I ; ð1þ where the resistance R and the inductance L are constants and V(t) denotes the potential source at time t (Su and Strunz, 25). If V(t) is a piecewise continuous function, the solution of the first-order linear differential equation (1) is: IðtÞ ¼IðÞexp 2Rt þ 1 L L Rðt 2 sþ exp 2 VðsÞds: L 2.2 The stochastic model with white noise Now, let us allow some randomness in the potential source V(t). Then voltage may not be deterministic but of the form: V * ðtþ ¼VðtÞþ noise ¼ VðtÞþajðtÞ; ð2þ where j(t) is a white noise process of mean zero and variance one, and a is non-negative constant, known as the intensity of noise. To be able to substitute this into the equation of the circuit, we have to describe the noise mathematically. It is reasonable to look at it as a stochastic process j(t)dt by a term dw(t), where W(t) is the Wiener process. We get an SDE: diðtþ ¼ 1 L VðtÞ 2 R L IðtÞ dt þ a L dwðtþ: ð3þ In order to obtain analytic solution of equation (3), we use Ito formula. Based on Ito formula the derivative of the function: is as follows: gðt; IðtÞÞ ¼ e ðrt=lþ IðtÞ ðrt=lþ VðtÞ dgðt; IðtÞÞ ¼ e L dt þ e ðrt=lþ a L dwðtþ: Therefore, the solution of equation (3) is: IðtÞ ¼e 2ðRt=LÞ IðÞþ 1 L e ðrðs2tþ=lþ VðsÞds þ a L e ðrðs2tþ=lþ dwðsþ: ð4þ

4 Note that I(t) is a random process and it is expectation is as follows: mðtþ ¼EðIðtÞÞ ¼ e ð2rt=lþ EðI Þþ 1 L VðsÞe ðrðs2tþ=lþ ds; t. : ð5þ The second moment DðtÞ ¼EðIðtÞ 2 Þ can be computed as a solution of the following ordinary differential equation (Kolarova, 25): ddðtþ ¼ 2 2R DðtÞþ2mðtÞ VðtÞ dt L L þ a 2 L 2 : ð6þ A stochastic perspective of RL circuit 815 Furthermore, I(t) is a Gaussian process with distribution NðmðtÞ; s 2 ðtþþ, where s 2 ðtþ ¼EðIðtÞ 2 Þ 2 m 2 ðtþ. So based on the properties of the normal distribution, we obtain: while: PðjIðtÞ 2 mðtþj, 1:96sðtÞÞ ¼ 2fð1:96Þ 2 1 ¼ :95; fðxþ ¼ 1 Z x e 2ð1=2Þs 2 ds: 2p 21 ð7þ 2.3 The stochastic model with colored noise A white noise process can be consider as derivative of a Wiener process. Since the paths of a Wiener process are nowhere differentiable, a white noise cannot be considered as stochastic process in the usual way, so it must be interpreted in the sense of generalized functions. A white noise process cannot be physically realized but can be approximated by conventional stochastic processes with wide spectral bands which are commonly known as colored noise processes. Definition 1. The stochastic process X(t) is called colored noise if it is an Orstein-Uhlenbeck process that satisfies the linear additive SDE: dxðtþ ¼ mxðtþdt þ sdw ðtþ; where m and s are constants. The explicit solution of SDE (8) is given by: XðtÞ ¼e mt XðÞþs e 2ms dwðsþ : ð9þ Now, the noise term j(t) in equation (2) is considered as a colored noise process. Therefore: V * ðtþ ¼VðtÞþ colored noise With substituting dx(t) from equation (8) into equation (3) instead dw(t), we have: diðtþ ¼ 1 L VðtÞ 2 R L IðtÞ dt þ a L dxðtþ: ð1þ ð8þ

5 COMPEL 3,2 816 Note that equation (1) is an ordinary SDE with colored noise. By using the Ito formula, it can be shown that: and: IðtÞ ¼e 2ðRt=LÞ IðÞþ 1 L e ðrðs2tþ=lþ VðsÞds þ a L e ðrðs2tþ=lþ dxðsþ; ð11þ EðIðtÞÞ ¼ e ð2rt=lþ EðI Þþ 1 L e ðrðs2tþ=lþ VðsÞds þ am L EðX Þ e ðrðs2tþ=lþ e ms ds: ð12þ 2.4 The stochastic model with mixture noise A mixture noise may be interpreted as any linear combination of Wiener processes. Definition 2. The process X(t) is a mixture noise if it satisfy the linear additive SDE: dxðtþ ¼ Xn k¼1 a k dw k ðtþ; X n k¼1 a k ¼ 1; ð13þ where W i are independent Wiener processes and a i are constants. In this case, the noise term j(t) in equation (2) is considered as a mixture noise process. Then the voltage with mixture noise term is introduced by: V * ðtþ ¼VðtÞþ mixture noise : With substituting dx(t) from equation (13) into equation (3), instead dw(t), we obtain: diðtþ ¼ 1 L VðtÞ 2 R L IðtÞ dt þ a L dxðtþ: ð14þ Equation (14) is an ordinary SDE with mixture noise. Again by using Ito formula the analytical solution of equation (14) is: IðtÞ ¼e 2ðRt=LÞ IðÞþ 1 L þ Xn a a k L k¼1 e ðrðs2tþ=lþ VðsÞds It is obvious that I(t) is a random process and its expectation is: e ðrðs2tþ=lþ dw k ðsþ : ð15þ EðIðtÞÞ ¼ EðI Þe ð2rt=lþ þ 1 L e ðrðs2tþ=lþ VðsÞds: ð16þ If the random variable IðÞ ¼I is constant, then the expectation of the stochastic solution is equal to the deterministic solution of the circuit.

6 3. RL circuit with two stochastic parameters This section is intended to provide some randomness in the electrical source V(t) as well as in the resistance R(t). For this aim, the voltage V(t) and resistance R(t) with white noise are introduced as: A stochastic perspective of RL circuit and: V * ¼ V þ aj 1 ; ð17þ 817 The SDE describing this situation is: R * ¼ R þ bj 2 : diðtþ ¼ 1 L ðvðtþ 2 RIðtÞÞdt 2 b L IðtÞdW 2ðtÞþ a L dw 1ðtÞ; ð18þ ð19þ where a and b are non-negative constants. Kolarova (25) has been shown that the solution of equation (19) is as the following form: IðtÞ ¼IðÞe ðð2rt=lþ2ð2b 2 =2L 2 Þt2ðb=LÞW 2 ðtþþ þ 1 L þ a L VðsÞe ððrðs2tþ=lþþðb 2 =2L 2 Þðs2tÞþðb=LÞðW 2 ðsþ2w 2 ðtþþþ ds e ððrðs2tþ=lþþðb 2 =2L 2 Þðs2tÞþðb=LÞðW 2 ðsþ2w 2 ðtþþþ dw 1 ðsþ: Now, we suppose dx 1 and dx 2 are two colored noises of the form: dx 1 ¼ mx 1 dt þ sdw 1 ðtþ and: dx 2 ¼ mx 2 dt þ sdw 2 ðtþ; ð2þ ð21þ where W 1 and W 2 are independent Wiener processes. With substituting dx 1 and dx 2 from equations (2) and (21) into equation (19) instead dw 1 and dw 2, respectively, we have: diðtþ ¼ 1 L ðvðtþ 2 RIðtÞÞdt 2 b L IðtÞdX 2ðtÞþ a L dx 1ðtÞ: ð22þ For the analytical solution of the SDE (22), first define the function: and then compute: F t ¼ e ððr=lþtþðb 2 s 2 =2L 2 Þtþðb=LÞX 2 ðtþþ ; dðfðtþiðtþþ ¼ dðe ððr=lþtþðb 2 s 2 =2L 2 Þtþðb=LÞX 2 ðtþþ IðtÞÞ: By using multidimensional Ito formula to the function: gðt; x; yþ : ð; 1Þ R 2! R; gðt; x; yþ ¼e ððr=lþtþðb 2 s 2 =2L 2 Þtþðb=LÞxÞ y;

7 COMPEL 3,2 818 we obtain: dgðt; X 2 ðtþ; IðtÞÞ ¼ dðe ððr=lþtþðb 2 s 2 =2L 2 Þtþðb=LÞX 2 ðtþþ IðtÞÞ ¼ dðfðtþiðtþþ þ 1 2 It can be easily shown that: R ¼ FðtÞ L þ b 2 s 2 2L 2 IðtÞdt þ FðtÞ b L IðtÞdX 2ðtÞþFðtÞdIðtÞ b 2 s 2 L 2 FðtÞIðtÞðdX 2 ðtþþ 2 þ FðtÞ b L ðdx 2ðtÞdIðtÞÞ: ðdx i ðtþþ 2 ¼ s 2 dt; dx i ðtþdt ¼ dtdx i ðtþ ¼; and: dx 1 ðtþdx 2 ðtþ ¼: Therefore: finally: VðtÞ dðfðtþi ðtþþ ¼ FðtÞ L dt þ a L dx 1ðtÞ ; IðtÞ ¼IðÞe ðð2rt=lþ2ð2b 2 s 2 =2L 2 Þt2ðb=LÞX 2 ðtþþ þ 1 L þ a L VðsÞe ðrðs2tþ=lþþðb 2 s 2 =2L 2 Þðs2tÞþðb=LÞðX 2 ðsþ2x 2 ðtþþ ds e ðrðs2tþ=lþþðb 2 s 2 =2L 2 Þðs2tÞþðb=LÞðX 2 ðsþ2x2ðtþþ dx 1 ðsþ: 4. Numerical simulation The main part of stochastic calculus is the Ito and Stratonovich calculus. In this paper, we consider SDE of kind Ito. A general SDE is given by: dxðtþ ¼ f ðt; XðtÞÞdt þ gðt; XðtÞÞdW ðtþ; where f(t, X(t)) and g(t, X(t)) are drift and diffusion term, respectively, and W(t) is a Wiener process with the property WðtÞ 2 WðsÞ, Nð; t 2 sþ; t $ ; s $. Generally, SDEs cannot be solved using traditional mathematics for the steps of the transformation because the Wiener process is non-differentiable, instead we need special techniques such as Ito and Stratonovich calculus (Oksendal, 2). However, there is not always a closed form solution for SDEs, hence researchers have looked for solving them numerically. The methods based on numerical analysis are reported in Oksendal (2), Penski (2) and Rawat and Parthasarathy (28), which involve ð23þ

8 discrete time approximation in a finite time interval over the sample paths. Neglecting the errors due to numerical approximation, the simplest time discretization approach is based on EM approximation which we adopt in this paper. There are similar relationships between the numerical methods for ordinary differential equations and those for SDEs. 4.1 Euler-Maruyama method Let X ¼ {XðtÞ : t [ ½t ; TŠ} satisfying the SDE (23) on t # t # T, with the initial value Xðt Þ¼x and for a given partition t ¼ t # t 1 # # t n # # t N ¼ T; of the time interval [t,t ]. EM approximation is a continuous time stochastic process Y ¼ {YðtÞ : t # t # T} satisfying the iterative scheme equation: A stochastic perspective of RL circuit 819 Y nþ1 ¼ Y n þ f ðt n ; Y n Þðt nþ1 2 t n Þþgðt n ; Y n ÞðWðt nþ1 Þ 2 Wðt n ÞÞ; for n ¼ ; 1;...; N 2 1 with initial value Y ¼ x, where we have Y n ¼ Yðt n Þ, Dt n ¼ððt N 2 t Þ=nÞ ¼ððT 2 t Þ=nÞ, t n ¼ t þ nd, D ¼ max{dt 1 ;...; Dt N } and DW n ¼Wðt nþ1 Þ2Wðt n Þ, Nð; DnÞ. The sequence Y ¼ {Y n : n ¼ ; 1;...; N 2 1} is the value of the EM approximation at the instants t n. The EM method has a strong order of accuracy and it is numerically stable. Furthermore, the EM approximation that is resulted from iterative scheme (24) converges to the Ito solution of SDE (23) (Kolarova, 25). 4.2 Discussion We will focus our attention on EM scheme. Let us consider an RL electrical circuit, when R, L and V(t) are constants, I ¼, X ¼, (the initial value of colored noise). In that case, the expectation of the stochastic solutions with white, colored and mixture noises is equal to the deterministic solution of the circuit. Using the EM scheme and Matlab programming, we simulate and plot the confidence interval for the mean of solutions. As shown in Figure 1, the confidence intervals for mean of solutions with ð24þ Current (I(t)) Confidence interval with color noise 2 Confidence interval with white noise Expectation of the solutions Time (t) Figure 1. Comparison confidence intervals with white and colored noises

9 COMPEL 3,2 82 colored and mixture noises (red graphs) are better than white noise (black graph) in comparison with the expectation of solutions (blue graph). The comparison between colored and mixture noises is shown in Figure 2. Figure 3 shows the deterministic solution and three sample paths of stochastic solution with two stochastic parameters, when s ¼ m ¼ 1 in equations (2) and (21), a ¼ 2 and b ¼ 2 in equation (22). 5. Conclusion The present study successfully applied different random terms involving white, colored and mixture noises as Gaussian processes to derive a stochastic model for Current (I(t)) Figure 2. Comparison confidence intervals with white and mixture noises 2 Confidence interval with mixture noise Confidence interval with white noise Expectation of the solutions Time (t) Current (I(t)) Figure 3. The deterministic solution and three sample paths of stochastic solution with two stochastic parameters Time (t)

10 an RL electrical circuit. Colored and mixture noises were added in the parameters of model since in many of real world applications the behavior of noises is not Wiener process. Some applications of the Ito calculus in conjunction with EM method to find the analytical and numerical solutions of stochastic models of RL electrical circuit are introduced. The effects of colored and mixture noise perturbations were analyzed in comparison with white noise. Numerical simulation is performed based on EM method via Matlab programming and the following result is obtained. The confidence interval for expectation of solutions with colored and mixture noises is better than white noise. In the future work, we plan to find further statistical properties of the stochastic solution with two parameters. Furthermore, we will focus on the application of colored and mixture noises for other engineering problems including RLC circuit and slow-drift motion. A stochastic perspective of RL circuit 821 References Carletti, M. (26), Numerical solution of stochastic differential problems in the biosciences, Journal of Computational and Applied Mathematics, Vol. 185 No. 2, pp Carletti, M., Burrage, K. and Burrage, P.M. (24), Numerical simulation of stochastic ordinary differential equations in biomathematical modeling, Mathematics and Computers in Simulation, Vol. 34 No. 2, pp Charalambous, C.D. and Menemenlis, N. (21), A state-space approach in modeling multipath fading channels via stochastic differential equations, IEEE International Conference on Communications, Helsinki, pp Field, T.R. and Tough, R.J.A. (2), Stochastic dynamics of the scattering amplitude generating K-distributed noise, Journal of Mathematical Physics, Vol. 44 No. 11, pp Ham, D. and Hajimiri, A. (2), Complete noise analysis for CMOS switching mixtures via SDEs, IEEE Custom Integrated Circuits Conference, Lake Buena Vista, FL, pp Kamarianakis, Y. and Frangos, N. (21), Deterministic and stochastic differential equations modeling for electrical networks, paper presented at Hellenic and European Research in Computational Mathematics Conference, Athens University of Economics Business, Athens. Kampowsky, W., Rentrop, P. and Schmidt, W. (1992), Classification and numerical simulation of electric circuits, Surveys Mathematics Industry, Vol. 2, pp Kloeden, P.E. and Platen, E. (1995), Numerical Solution of Stochastic Differential Equations, Springer, Berlin. Kolarova, E. (25), Modeling RL electrical circuits by SDEs, International Conference on Computer as a Tool, pp Oksendal, B. (2), Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin. Penski, C. (2), A new numerical method for SDEs and its application in circuit simulation, Journal of Computational and Applied Mathematics, Vol. 115, pp Rawat, T.K. and Parthasarathy, H. (28), Modeling of an RC circuit using a SDEs, Int. J. SC. Tech, Vol. 13 No. 2, pp Su, Q. and Strunz, K. (25), Stochastic circuit modeling with Hermite polynomial chaos, Electronic Letters, Vol. 41 No. 21, pp

11 COMPEL 3,2 822 Further reading Burrage, K. and Platen, E. (1994), Rung-Kutta methods for stochastic differential equations, Ann. Numer. Math, Vol. 1, pp Kim, C. and Lee, E.K. (26), Numerical method for solving differential equations with dichotomous noise, Physical Review, Vol. E73, p Corresponding author Rahman Farnoosh can be contacted at: To purchase reprints of this article please Or visit our web site for further details: