Relaxation and Transient Processes in Disordered LC Systems

1. Introduction

As is well known, the problem of relaxation and transient processes in conducting media is closely connected with the problem of the conductivity of the media [1,2,3], because the conductivity of a conducting medium

$\sigma$ is determined by its relaxation time $\tau$:

$$\sigma ={\displaystyle \frac{q^{2}n\tau }{m}}$$

(1)

where $n$ and $m$ are the particle concentration and mass, respectively, and $q$ is an electric charge.

In the case of a disordered, multi-component LC system, which is composed of reactances from inductors and capacitors (non-dissipative elements) with random connections between them, it initially seems that no relaxation or transient processes are possible due to the non-dissipative character of the elements. Nevertheless, due to disorder in the system, caused by random placement and connections of reactances, finite dissipation arises at the percolation threshold. As a result, relaxation and transient processes appear in the LC systems.

To study this problem, the exact method developed in [4,5,6] was used. This method is based on the symmetry of two-dimensional equations relative to rotational or dual symmetry. Below we apply this method, which is described in the Appendix A in more detail, to calculate the effective conductivity of a discrete LC system consisting of randomly connected inductances and capacitances of two types with different values.

The paper is structured as follows. In Section 2 we calculate the effective conductivity of a disordered multi-component LC system. In Section 3 we study the transient currents at short and long times. Section 4 concludes the paper. The details of the calculations are provided in the Appendix A.

2. Effective Conductivity of Many-Phase System Consisting of Randomly Connected L and C at Percolation Threshold

Let us consider the conductivity of a four-component medium consisting of randomly placed inductors of two kinds and capacitors of two kinds (non-dissipative elements) with different values (a four-color chessboard) at equal concentrations of inductors $p_1$ and capacitors $p_2$: $p_1=p_2=1/4$.

At the percolation threshold $p_c=1/2$ we need to use the rotational transformations (A2) with coefficients $a=c=0$. Therefore, according to formulae (A7) and (A8) the following expression for the effective conductivity is obtained:

$$Y^e(0)=\sqrt{{\displaystyle \frac{Y_1{Y}_3(Y_2+Y_4)-Y_2{Y}_4(Y_1+Y_3)}{Y_1+Y_3-Y_2-Y_4}}}$$

(1)

The key feature of this solution is that the expression yields a real value. This result is unusual and interesting because a medium consisting of inductances and capacitances—elements with purely imaginary impedances—nevertheless exhibits a real effective conductivity at the percolation threshold. The analogous result was first obtained in [4], see also [6].

Let us analyze this formula (1) in the case when $Y_1=i\omega L_1$,$Y_3=i\omega L_2$ and $Y_2=1/i\omega C_1$, $Y_4=1/i\omega C_2$. After some transformations, the following expression is obtained:

$$Y^e(0)=\sqrt{\left({\displaystyle \frac{L_1+L_2}{C_1+C_2}}\right)\left({\displaystyle \frac{1+{\omega }^2/{\Omega }_1^2}{1+{\omega }^2/{\Omega }_2^2}}\right)}$$

(2)

Here we introduce the following notations: ${\Omega }_1^2=\left({\displaystyle \frac{C_1+C_2}{1/L_1+1/L_2}}\right)$ and ${\Omega }_2^2=\left({\displaystyle \frac{L_1+L_2}{1/C_1+1/C_2}}\right)$.

At low frequencies $\omega <<{\Omega }_1, {\Omega }_2$ the following formula is obtained:

$$Y^e(0,\omega \to 0)\approx \sqrt{{\displaystyle \frac{L_{eff}}{C_{eff}}}}=\sqrt{\left({\displaystyle \frac{L_1+L_2}{C_1+C_2}}\right)}$$

(3)

In the opposite case at high frequencies $\omega >>{\Omega }_1, {\Omega }_2$ another expression is obtained:

$$Y^e(0,\omega \to \infty )\approx \sqrt{{\displaystyle \frac{L_{eff}}{C_{eff}}}}=\sqrt{{\displaystyle \frac{1/C_1+1/C_2}{1/L_1+1/L_2}}}$$

(4)

And at intermediate frequencies the value depends on the relation between frequencies $\omega , {\Omega }_1, {\Omega }_2$.

In the case ${\Omega }_1<<\omega << {\Omega }_2$ we obtain from (2):

$$Y^e(0)\approx {\displaystyle \frac{1}{\omega C_{eff}}}={\displaystyle \frac{1}{\omega \sqrt{C_1{C}_2}}}$$

(5)

For clarity, these results are presented graphically in Figure 1.

In the opposite case ${\Omega }_2<<\omega << {\Omega }_1$ we obtain from (2) – see Figure 2 as well:

$$Y^e(0)\approx \omega L_{eff}=\omega \sqrt{L_1 L_2}$$

(6)

3. Transient Currents in LC System

As is well known, in an LC system consisting of inductors (L) and capacitors (C), only oscillatory phenomena are possible. However, because a finite resistance exists in the investigated system at the percolation threshold – see formula (2), one can expect that transient currents and relaxation processes will appear:

$$I(t)=I_0 \exp \left(-{\displaystyle \frac{t}{{\tau }_{eff}}}\right)$$

(7)

Thus, the problem reduces to finding the value of the effective relaxation time ${\tau }_{eff}$. Let us study these currents in two limiting cases: at low and high frequencies, as described by the formulae (3) and (5).

In the system with inductance L and resistivity $R=\sqrt{L/C}$ the relaxation time is equal to: ${\tau }_L=L/R=L/\sqrt{L/C}=\sqrt{LC}$; in the system with capacitor C and the same effective resistivity R the relaxation time is equal to: ${\tau }_C=RC=C\sqrt{L/C}=\sqrt{LC}$. We therefore assume these quantities are equal within the system under investigation:

$${\tau }_{eff}=L_{eff}/R=R{C}_{eff}$$

(8)

The exact relation between two effective characteristics $L_{eff}$ and $C_{eff}$ is obtained:

$$L_{eff}/C_{eff}=R^2$$

(9)

As a result, in the case of low frequencies $\omega \to 0$ we obtain:

$${\displaystyle \frac{L_{eff}}{C_{eff}}}={\displaystyle \frac{L_1+L_2}{C_1+C_2}}$$

(10)

If we assume that at low frequencies the effective inductor $L_{eff}$ is equal to $L_{eff}=L_1+L_2$, which means that the main current paths in the disordered system are provided by the series-connected inductors and the parallel-connected capacitors $C_{eff}=C_1+C_2$ - see Figure 3

Thus, the effective relaxation time at low frequencies is:

$${\tau }_{eff}(\omega \to 0)=\sqrt{(L_1+L_2)(C_1+C_2)}$$

(11)

And in the case of high frequencies $\omega \to \infty$, we obtain the following relation:

$${\displaystyle \frac{L_{eff}}{C_{eff}}}={\displaystyle \frac{1/C_1+1/C_2}{1/L_1+1/L_2}}$$

(12)

If we assume that at high frequencies the effective inductor $L_{eff}$ is equal to $1/L_{eff}=1/L_1+1/L_2$, this means that in this limiting case (high frequencies) the main current paths in the disordered system are provided by the parallel-connected inductors and series-connected capacitors $1/C_{eff}=1/C_1+1/C_2$.

Figure 4. The current paths are provided by inductors with parallel connection and by capacitors with series connection.

Figure 4. The current paths are provided by inductors with parallel connection and by capacitors with series connection.

Thus, the effective relaxation time at high frequencies is:

$$1/{\tau }_{eff}(\omega \to \infty )=\sqrt{(1/L_1+1/L_2)(1/C_1+1/C_2)}$$

(13)

It is easy to see that in the case of the two-component LC system, when $L_1=L_2$ and $C_1=C_2$, the formulae (11) and (13) reduce to a single expression [3]:

$${\tau }_{eff}(\omega \to 0)={\tau }_{eff}(\omega \to \infty )=\sqrt{LC}$$

(14)

4. Discussion

In this work, we have studied current flow in a two-dimensional system of randomly connected circuit elements. It has been shown that the effective conductivity is constant, i.e., independent of phase concentration, and is equal to either the capacitive reactance or the inductive reactance—see formula (3). However, at the point of the percolation transition, a finite effective resistance appears in a system of randomly connected circuits $R=\sqrt{{\displaystyle \frac{L}{C}}}$. The emergence of a dissipative state with finite resistance can be understood as follows. Suppose the initial phase through which the current flows consists of inductors; consequently, all the energy is concentrated in the inductive elements. In order for current to flow through another phase, the capacitive phase, it is necessary to transfer this energy to the capacitances, and for this it is necessary to excite local oscillations in the circuits, during which energy will be transferred from the inductors to the capacitances. In other words, the transition from one non-dissipative state of the system, described by conduction, to another non-dissipative state, described by a different value, is possible only through the excitation of the circuit, i.e., through the dissipative state at the point of topological transition. In our opinion, the obtained results can be used to study the properties of multi-component metamaterials.