Perturbation-Based Stochastic Modeling of Nonlinear Circuits

Abstract

This paper presents a general model for a nonlinear circuit, in which, the circuit parameters (e.g. resistance and capacitance) are subject to random fluctuations due to noise, which vary with time. The fluctuating amplitudes of these parameters are assumed to be Ornstein–Uhlenbeck (O.U.) processes and not the white noise owing to temporal correlations. The nonlinear circuit is represented by a system of nonlinear differential equations depending upon a set of parameters that fluctuate slowly with time. To model these fluctuations, we use the theory of Ito’s stochastic differential equations (SDEs). Then the driving force of the circuit dynamics is in accordance with the general perturbation theory decomposed into the sum of a strong linear component and a weak nonlinear component by the introduction of a small perturbation parameter. The circuit states are expanded in the powers of this small perturbation parameter and recursive solutions to the various approximates obtained. Finally, the approximate expressions for the output states are obtained as stochastic integrals with respect to Brownian motion processes. The proposed method is applied to a half-wave rectifier circuit which is built out of a diode, a resistor and a capacitor. The diode is represented by nonlinear voltage–current equation, and resistance and capacitance are subject to random fluctuations due to noise, which vary slowly with time. The results, obtained using the proposed method, are compared with those obtained via the conventional perturbation-based deterministic differential equations model for a nonlinear circuit. Hence, the noise process component, present at the output, is obtained.

Appendices

Appendix A

Let B(t) be a Brownian motion process. Then the O.U. process is given as follows:

$$dx(t)=-\gamma_{1}x\,dt+\sigma_{1}\,dB(t) $$

We approximate the process x(t) using the Euler–Maruyama scheme in the following way:

Appendix B: Perturbation-Based Deterministic Differential Equations Modeling of Nonlinear Circuits

Consider a system whose input x(t) and output y(t) are related by the differential equation as follows:

$$ \frac{d\mathbf{y}(t)}{dt}=\mathbf{A}\mathbf{Y}(t)+\epsilon\mathbf{f}\bigl(\mathbf{Y}(t) \bigr)+\mathbf{B}_{o}\mathbf{x}(t) $$

(71)

Here, $\mathbf{x}(t)\in\Bbb{R}^{n}$, $\mathbf{y}(t)\in\Bbb{R}^{p}$, $\mathbf{A}\in\Bbb{R}^{p \times p}$

$$\mathbf{f}:\Bbb{R}^p \rightarrow\Bbb{R}^p\mbox{ is a smooth nonlinear map and }\mathbf{B}_{o}\in R^{p \times n}. $$

ϵ is a small perturbation parameter that signifies the fact that the nonlinear effect is small. This state equation is not, in general, solvable in close form and hence, we try an expansion of y(t)=y(t,ϵ) in powers of ϵ:

$$ y(t,\epsilon)=\sum_{m=0}^{\infty}y^{(m)}(t) \epsilon^{m} $$

(72)

Substituting (72) into (71) and equating coefficients of ϵ ^m, m=0,1,2,3,… successively, we get a sequence of linear differential equations for y ^(m) in terms of y ^(m−1),…,y ⁽⁰⁾. For example, equating coefficients of ϵ ⁰ gives

$$ \frac{d\mathbf{y}^{(0)}}{dt}=\mathbf{A}\mathbf{y}^{(0)}+\mathbf{B}\mathbf{x}(t) $$

(73)

This has the solution

$$ \mathbf{y}^{(0)}(t)=\int_{0}^{t}e^{(t-\tau)\mathbf{A}} \mathbf{B}\mathbf{x}(\tau)\,d\tau $$

(74)

assuming zero initial conditions. Then equating coefficients of ϵ ¹ gives

$$ \frac{d\mathbf{y}^{(1)}(t)}{dt}=A\mathbf{y}^{(1)}(t)+f\bigl(Y_{0}(t) \bigr) $$

(75)

The solution to it is as follows:

(76)

Equating coefficients of ϵ ², we get

$$ \frac{d\mathbf{y}^{(2)}(t)}{dt}=\mathbf{A}\mathbf{y}^{(2)}(t)+\sum _{j=1}^{p}\frac{\partial\mathbf{f}(\mathbf{y}^{(0)}(t))}{\partial y_{j}}y_{j}^{(1)}(t) $$

(77)

The solution is given as follows:

$$ y^{(2)}(t)=\int_{0}^{t}e^{(t-\tau)\mathbf{A}} \sum_{j=1}^{p}\biggl(\frac {\partial\mathbf{f}(\mathbf{y}^{(0)}(\tau))}{\partial y_{j}}y_{j}^{(1)}( \tau)\biggr)\,d\tau $$

(78)

In general, equating coefficients of ϵ ^m, m≥1, we get

$$\frac{d\mathbf{y}^{m}(t)}{dt}=\mathbf{A}\mathbf{y}^{(m)}(t)+ \mbox{Coefficients of }\epsilon^{m-1}\mbox{ in } \mathbf{f}\bigl(y^{(0)}(t)+\epsilon y^{(1)}(t)+\epsilon^{2}y^{(2)}(t)+\cdots\bigr) $$

We note that the coefficient of ϵ ^m−1 in f(y(t)) will be expressible in terms of y ^(k)(t), 0≤k≤m−1 and will not involve y ^(m)(t).