S.C. Dutta Roy, Sudarshan Rao Nelatury
IMAGE LICENSED BY INGRAM PUBLISHING
Distributed RC networks arise naturally in metallic interconnections between any two elements in integrated circuits (ICs). In the early stages of ICs, they were considered as parasitics and were avoided. Then Kaufman’s article [1] appeared in 1960 and showed its useful application in a null network. Since then, many research efforts have been directed toward applying them, either with a uniform distribution or by shaping them, for various useful purposes [2], [3], [4], [5], [6], [7].
In this article, it is shown that a uniformly distributed RC (UDRC) bandpass network can give a gain greater than unity. While it is known that lumped RC networks in any configuration can give more than unity gain under suitable conditions [8], the case of distributed RC networks has never been considered in the literature. The aim of this article is to make teachers and students of electrical engineering aware of this fact.
The basic circuit configuration, called ${N}_{b},$ considered here is shown in Figure 1. The symbols used in this figure and in the text are as follows. With reference to the distributed network,
Figure 1. The basic circuit configuration considered here.
The other symbols are
Figure 2. The source network, ${N}_{a},$ from which ${N}_{b}$ is derived.
Note that ${\omega}_{0} = {1}{/(}{R}_{T}{/}{C}_{T}{)}$ is the 3-dB cutoff frequency of a single-section RC low-pass filter with series resistance ${R}_{T}$ and shunt capacitance ${C}_{T}$. This is the simplest rough model of the UDRC network. The model is rough because the actual bandwidth, calculated from (1), which follows, is about 2.5 times ${\omega}_{0}$.
Actually, the network of Figure 1 is a reorientation of the null network shown in Figure 2 [1], [2], henceforth referred to as ${N}_{a}$.
The transfer function (TF) of an exponentially tapered distributed RC network is given in [2] and can be easily adapted to the present case by setting the tapering factor equal to zero. After some simplifications, the TF of ${N}_{a},$ denoted by ${T}_{a},$ becomes, in terms of x, the following: \[{T}_{a} = \frac{\left[{{k}{\left({jx}\right)}^{1/2} + {jx}\,{\sinh}{\left({jx}\right)}^{1/2}}\right]}{\left[{{k}{\left({jx}\right)}^{1/2}\,{\cosh}{\left({jx}\right)}^{1/2} + {jx}\,{\sinh}{\left({jx}\right)}^{1/2}}\right]}{.} \tag{1} \]
It is obvious that the TF of ${N}_{b},$ ${T}_{b},$ will be \[{T}_{b} = {1}{-}{T}_{a}{.} \tag{2} \]
Simulations were carried out for ${\left\vert{{T}_{b}}\right\vert}$ only for various values of k. The results are shown in Figure 3.
Figure 3. Simulation results for ${\left\vert{{T}_{b}}\right\vert}$.
Clearly, for k > 15, the response shows a gain greater than unity, the maximum value of which increases with increasing k. However, this increase does not go on for ever; simulations beyond k = 15 up to k = 1,000 show that with increasing k, the response curve shows an increase in ${x}_{peak},$ the value of x at which the peak occurs, with a corresponding increase in the peak value, ${{\left\vert{{T}_{b}}\right\vert}}_{\max},$ to be denoted by ${T}_{b,\text{peak}}$. Both saturate at ${x}_{peak}\cong{9.2}$ and ${{\left\vert{{T}_{b}}\right\vert}}_{\max}\cong{1.148},$ as shown in Figure 4. The variation of the bandwidth over which the gain exceeds unity with k is shown in Figure 5.
Figure 4. The variation of ${x}_{peak}$ and ${T}_{b,\text{peak}}$ with k.
Figure 5. The variation of the more than unity gain bandwidth with k.
It is desirable to have closed formulas for the 3-dB bandwidth and ${x}_{peak}$. However, it has been found that this leads to a highly involved transcendental equation, which can be solved only by numerical methods. Instead of that, we simply continued the simulations beyond k = 50, noted the desired data, and plotted them in the design graphs, shown in Figures 4 and 5. Regarding the 3-dB bandwidth, our simulations showed that, for higher values of k, the frequency response shape was more of a highpass filter, with an overunity gain in the left side, just like overshoot in the step response. Therefore, except for lower values of k, the upper 3-dB point is not meaningful.
It has been shown that a UDRC network is able to achieve a maximum gain of 1.148 for high values of k. This has not been known in the existing literature, and it is hoped that teachers and students of electrical engineering will take notice of this fact. Distributed RC modeling of interconnects is essential in microwave circuits as the energy transmission through them is not instantaneous, and the transmission delays play an important part in the characteristics of the designed circuit.
We thank the editor-in-chief of IEEE Microwave Magazine for his valuable suggestions, which helped to improve the manuscript.
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[8] S. C. Dutta Roy, “Revisiting passive RC networks with over unity gain,” submitted for publication.
[9] W. C. Elmore, “The transient response of damped linear networks with particular regard to wideband amplifiers,” J. Appl. Phys., vol. 19, no. 1, pp. 55–63, Jan. 1948, doi: 10.1063/1.1697872.
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Digital Object Identifier 10.1109/MMM.2022.3218177