# Localization Properties of Non-Periodic Electrical Transmission Lines

## Abstract

**:**

## 1. Introduction

## 2. Electric Transmission Lines

#### 2.1. Direct and Dual Transmission Lines

#### 2.2. Mixed Transmission Lines

#### 2.3. Relation with the Tight-Binding Model

#### 2.4. Spectrum of Allowed Frequencies

#### 2.5. Methods to Obtain the ${I}_{n}\left(\omega \right)$ Electric Current Function

#### 2.5.1. Recurrence Method

#### 2.5.2. Hamiltonian Map Method

#### 2.6. Diagnostic Tools

#### 2.6.1. Usual Diagnostic Tools

#### 2.6.2. The Average Overlap Amplitude ${C}_{\omega}$

#### 2.6.3. The Transmission Coefficient ${T}_{\omega}$

## 3. Disordered Transmission Lines

#### 3.1. Aperiodic Transmission Lines

#### 3.1.1. Generalized Fibonacci Sequence

#### 3.1.2. Generalized Thue–Morse Sequence

#### 3.1.3. Incommensurate Sequences

#### 3.2. Long-Range Correlated Disorder

#### 3.2.1. Discrete Sequences

#### 3.2.2. Continuous Sequences

#### 3.3. Diluted Disordered Systems

#### 4. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TL | Transmission Lines |

FFM | Transmission Lines |

FFT | Fast Fourier Transform |

## References

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**Figure 1.**A partial view of an ideal transmission line. ${Z}_{n}$ (${\gamma}_{n}$) represent horizontal (vertical) impedances, respectively. For direct TL, ${Z}_{n}$ is associated with inductances and ${\gamma}_{n}$ with capacitances. Conversely, for dual TL, ${Z}_{n}$ is associated with capacitances and ${\gamma}_{n}$ with inductances. The arrows indicate the direction of the electric current in each cell. We arbitrarily consider the initial flow from the left, because we are using open boundary conditions

**Figure 2.**A segment of a mixed transmission line formed by $p=2$ direct cells and $q=3$ dual cells. The full system is formed by the repetition of the basic unit formed by $d=\left(p+q\right)$ cells. Inductances are represented by rectangles and capacitances by circles. In addition, dual cells are marked with orange color-filled symbols. The arrows indicate the direction of the electric current in each cell.

**Figure 3.**(

**a**) Global sub-band structure of the Shannon entropy $S\left(\omega \right)={R}_{1}\left(\omega \right)$ versus $\omega ,$ for the Fibonacci quasi-periodic distribution of inductances ${L}_{n}$ discussed in Ref. [78]. (

**b**) Self-replication structure of the sub-band indicated by the vertical arrow in (

**a**).

**Figure 4.**$\left(N{C}_{\omega}\right)$ versus $\omega $ for the Fibonacci distribution of inductances ${L}_{n}$, for mixed TL with fixed $p=2$, considering two values of $q.$ (

**a**) $q=2$ ($d=4$ bands) and (

**b**) $q=3$ ($d=5$ bands). Vertical arrows indicate the bands to be studied in Figure 5.

**Figure 5.**$\left(N{C}_{\omega}\right)$ versus $\omega $ for the Fibonacci distribution of inductances ${L}_{n}$ for mixed TL, for $p=2,$ $q=3$. A detail of Figure 4. (

**a**) Third sub-band, (

**c**) fourth sub-band. Figures (

**b**,

**d**) show the self-replication of the sub-bands indicated by vertical arrows in (

**a**,

**c**), respectively.

**Figure 6.**$\xi \left(\omega \right)$ versus $\omega $ for the $m-$tupling distribution of inductances ${L}_{n}$ in the direct TL, for three values of $m,$ namely $m=\left\{2,3,13\right\}$ (

**a**–

**c**). (

**d**) $\mathsf{\Lambda}\left(\omega \right)$ versus $\omega .$ A short vertical bar indicates the existence of an extended state ($\mathsf{\Lambda}\left(\omega \right)\ge 1)$. The number of extended states for $m=3$ is very small compared to the case $m=2$. Conversely, for $m>>3$ ($m=8$ and $m=13$), the number of extended states increases and becomes comparable to the $m=2$ case.

**Figure 7.**$\xi \left(\omega \right)$ versus $\omega $ for $m-$tupling distribution of inductances ${L}_{n}$ in direct TL in a restricted region of frequencies of the Figure 6. (

**a**) $m=2,$ (

**b**) $m=13.$ We can see that the sub-bands of extended states ($\xi \approx 0.667$) for $m=2$ is much wider than the sub-bands of extended states of case $m=13.$

**Figure 8.**$\left(N{C}_{\omega}\right)$ versus $\omega $ for the $m-$tupling distribution of amplitudes ${\epsilon}_{n}$ of non-linear capacitances of the dual TL with ${\epsilon}_{A}=0.1$ and ${\epsilon}_{B}=0.03.$ (

**a**) Case $m=2$ and (

**b**) case $m=3.$ The horizontal dashed lines indicate the value $\left(N{C}_{\omega}\right)\approx 1.62,$ for the periodic linear case, i.e., ${\epsilon}_{A}={\epsilon}_{B}=0.0.$ In addition, at the top of each figure, we indicate with a vertical bar the presence of extended states which meet condition $\mathsf{\Lambda}\left(\omega \right)\ge 1$.

**Figure 9.**The integrated density of states $IDOS\left(\omega \right)$ versus $\omega $ for the $m-$tupling distribution of amplitudes ${\epsilon}_{n}$ of the non-linear capacitances of the dual TL for ${\epsilon}_{A}=0.1$ and ${\epsilon}_{B}=0.03$. (

**a**) $IDOS\phantom{\rule{4pt}{0ex}}$for four m values, i.e., $m=\left\{2,3,8,13\right\}$. For each $m,$ we used a fixed ${N}_{m},$ i.e., ${N}_{m}=\left\{{2}^{21},{3}^{13},{8}^{7},{13}^{6}\right\}.$ The $IDOS$ for $m=2$ is the greatest of all, and the $IDOS$ for $m=3$ is the smallest of all. For increasing values of m ($m=8$ y $m=13$), the $IDOS$ tends to the value corresponding to $m=2$. (

**b**) Fixed $m=3,$ as a function of the $N={m}^{k},$ with $k=\left\{10,12,13\right\}$. The $IDOS$ corresponding to $N={3}^{10}$ is the largest of all. When N increases, the $IDOS$ tends to zero, $IDOS\to 0.$

**Figure 10.**Map $\left(b,\omega \right)$ for the Aubry–André distribution of inductances with ${L}_{0}=4.0$. Each point of the map indicates an extended state, because $\mathsf{\Lambda}\left(\omega \right)\ge 1.0.$ For increasing values of the amplitude $b,$ the number of sub-bands of extended states diminishes, and for b close to $b={L}_{0},$ there is only a small sub-band where almost all states are extended states, namely $0\le \omega \le 0.45$.

**Figure 11.**(

**a**) Lyapunov exponent $\lambda \left(\omega \right)$ versus $\omega $ for two values of the b amplitude $b=\left\{1.5,3.99\right\}$. For $b=3.99\approx {L}_{0}$ (thick red line), only one band of extended states ($\lambda \left(\omega \right)\to 0$) can be observed for $\omega \le 0.45.$ Conversely, for $b=1.5$ (thin black line), we can see several sub-bands of extended states ($\lambda \left(\omega \right)\to 0$) separated by gaps. Within these sub-bands, we can find more localized states and gaps. (

**b**) The spectrum of the extended states, $\mathsf{\Lambda}\left(\omega \right)\ge 1$ versus $\omega $ for fixed $b=1.5$ and $N={10}^{6}.$ The vertical dashed arrows that cross both images indicate the edge of the gaps, in which phase transitions occur.

**Figure 12.**(

**a**) Lyapunov exponent $\lambda \left(\omega \right)$ versus $\omega $ for the case $b=2.0$ and $N=2\times {10}^{5}.$ The vertical arrows indicate the frequencies ${\omega}_{1}=0.5006231,$ ${\omega}_{2}=0.6336884,$ and ${\omega}_{3}=0.7577136,$ to be studied in (

**b**–

**d**). In these last three images, we show the scaling behavior of $\left(N{C}_{\omega}\right)$ for three values of $N,$ namely $N=\left\{8,12,16\right\}\times {10}^{4}.$ For each frequency ${\omega}_{1},$ ${\omega}_{2}$ and ${\omega}_{3},$ we can see a phase transition from extended states to localized states at the critical value ${b}_{c}=2.0.$ For $b\le {b}_{c}$ we find only extended states, because for almost every amplitude b, all $\left(N{C}_{\omega}\right)$ values coalesce into a single one, i.e., $\left(N{C}_{\omega}\right)\to const.>0.$ On the contrary, for $b>{b}_{c},$ $\left(N{C}_{\omega}\right)$ grows as system size N grows, indicating a localized behavior.

**Figure 13.**Transmission coefficient $T\left(\omega \right)$ versus $\omega ,$ for mixed transmission line with $p=2,$ $q=1$, for (

**a**) $b=0.3$, (

**b**) $b=0.7$, (

**c**) $b=1.1$ and (

**d**) $b=1.5$. For (

**a**) $b=0.3$ we find $d=\left(p+q\right)=3$ bands containing extended states, localized states and gaps. For increasing values of $b,$ the number of extended states within each band decreases. Therefore, for (

**c**) $b=1.1$, the leftmost band has already disappeared, and for (

**d**) $b=1.5,$ the rightmost band is about to disappear.

**Figure 14.**Schematic phase diagram $\left(b,\alpha \right),$ for a fixed frequency $\omega $, when ${C}_{n}$ and ${L}_{n}$ vary in phase in dual transmission lines. For this transmission line with long-range correlated distribution of ${C}_{n}$ and ${L}_{n}$, extended states can only be found for $\alpha \ge {\alpha}_{c}$ and $b\le {b}_{c}$.

**Figure 15.**(

**a**) $\mathsf{\Lambda}\left(\omega \right)$ versus $\omega ,$ for mixed TL with $p=1$ and $q=3,$ for $b=0.1.$ We consider two fixed values of the correlation exponents, $\alpha =2.3$ and $\beta =2.5.$ Only three bands are visible, because $\mathsf{\Lambda}\left(\omega \right)\to 0$ for the leftmost band (localized states). (

**b**) Phase diagram $\left(\alpha ,\beta \right)$ for $\omega =1.462121$ (indicated by the vertical arrow in (

**a**)). Only for $\alpha \ge {\alpha}_{c}=1.81$ and $\beta \ge {\beta}_{c}=1.68$, with ${\alpha}_{c}>{\beta}_{c}$ is it possible to find extended states.

**Figure 16.**Scaling behavior of $ln\left({C}_{\omega}\right)$ versus $ln\left(N\right)$ for mixed TL with $p=1,$ $q=3,$ $\omega =1.462121,$ and $b=0.1$. (

**a**) For fixed $\beta =3.6$, only for $\alpha \ge {\alpha}_{c}=1.81,$ we find straight lines with the same slope $m=-1.0$ ($R=1.0$), which indicates an extended behavior. (

**b**) For fixed $\alpha =2.0$, only for $\beta \ge {\beta}_{c}=1.68,$ we can obtain linear relationships with $m=-1.0.$

**Figure 17.**$\left(N{C}_{\omega}\right)$ versus $\omega ,$ for the Thue–Morse distribution of inductances, with ${L}_{A}=1.6,$ ${L}_{B}=1.5.$ Four values of the period P are considered, namely $P=\left\{1,2,3,4\right\}.$ The case $P=1$ corresponds to the usual Thue–Morse sequence without dilution. For $P\ge 2,$ the sequences are diluted with ${L}_{0}=1.8,$ with fixed ${C}_{n}=0.5.$ The resonances coincide with the left edge of each of the $(P-1)$ gaps generated by the dilution process.

**Figure 18.**Density of states $DOS\left(\omega \right)$ versus $\omega .$ To the left of each gap, the density of states does not fluctuate, which is an indication of the extended nature of the resonance located there. The same does not happen on the right side of each gap. In addition, we can see that the localization behavior of the $DOS\left(\omega \right)$, is identical to the localization behavior shown by $\left(N{C}_{\omega}\right)$ in Figure 17.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lazo, E.
Localization Properties of Non-Periodic Electrical Transmission Lines. *Symmetry* **2019**, *11*, 1257.
https://doi.org/10.3390/sym11101257

**AMA Style**

Lazo E.
Localization Properties of Non-Periodic Electrical Transmission Lines. *Symmetry*. 2019; 11(10):1257.
https://doi.org/10.3390/sym11101257

**Chicago/Turabian Style**

Lazo, Edmundo.
2019. "Localization Properties of Non-Periodic Electrical Transmission Lines" *Symmetry* 11, no. 10: 1257.
https://doi.org/10.3390/sym11101257