Time-domain response bounds for linear time invariant systems with application to transmission line traveling waves

Abstract

A novel approach to deriving physical bounds on the time-domain (TD) response of a class of linear time-invariant systems is described. The approach is illustrated on determining the (worst-case) maximum voltage induced along a loaded transmission line (TL), to which previously established TD bounds are not directly applicable. Illustrative numerical examples are presented.

Introduction

According to Bode¹, Ch. XIII, physical characteristics of a linear time-invariant (LTI) and causal system in the real-frequency domain (FD) are restricted due to analytic properties of its response in the right half of the complex frequency plane. A similar conclusion has been recently drawn for its time-domain (TD) response, thus introducing TD physical bounds on the causal TD response of a LTI system^2,3,4. Envisaged applications of the TD bounds are in designing of time-varying electromagnetic (EM) structures (e.g., ^5,6,7,8,9) the performance of which can exceed the best achievable one that corresponds to their static counterparts^10,11,12.

The TD bounds as summarized in² have been derived under certain assumptions that limit their application to systems whose real-frequency response satisfies certain sign requirements. Indeed, the real and imaginary parts of the remainder function (see², Eq. (3) for its definition), as they relate to first-order systems², passive RLC loads and thin-sheet dielectric/magnetic layers³ – among other notable examples – are strictly either positive or negative along the imaginary axis of complex-frequency plane. This article demonstrates that this property does not hold true for the traveling-wave response on a transmission line (TL), highlighting the need to extend our previous work on the subject^2,3. Addressing this gap is the key goal of the present work, in which the proposed extension is both described and illustrated through representative examples.

Problem definition

Throughout this article, t denotes the (real-valued) time coordinate. The Heaviside unit-step function is denoted by $\textrm{H}(t)$, $\delta (t)$ denotes the Dirac-delta distribution and $\delta ^{(1)}(t)$ is its first derivative. The time convolution is denoted by ${*}$ and the time-integration operator is for a causal signal f(t) defined as $\partial _t^{-1} f(t) = \int _{\tau =0}^tf(\tau )\textrm{d}\tau$.

We shall analyze the response of a loss-free TL of finite length, $\ell > 0$, that extends along the x-axis (see Fig. 1). The TL consists of perfectly electrically conducting (PEC) thin-wire conductor that is located relatively close to the PEC ground plane. The TL is embedded in free space that is described by (scalar, real-valued and positive) electric permittivity, $\epsilon _0$, and magnetic permeability, $\mu _0$. The TL is then specified by its (real-valued and positive) characteristic impedance, $Z^\textrm{C}$, and wave speed $c_0 = (\epsilon _0\mu _0)^{-1/2}$.

Fig. 1

Transmission line with voltage source excitation and impedance load.

Specifically, we will derive upper bounds on the space-time voltage distribution, V(x, t), as excited by a causal, impulsive voltage source, $V_0(t)$. The property of causality is incorporated via a one-sided Laplace transformation, say

$$\begin{aligned} \hat{V}_0(s) = \int _{t = 0}^\infty \exp (-s t)V_0(t)\textrm{d}t, \end{aligned}$$

(1)

where $s\in \mathbb {C}$ is the complex frequency with $\Re (s) > 0$, since $V_0(t)$ is a bounded and causal function of time. The (real-valued and positive) internal resistance of the source is denoted by $R^\textrm{S}$. The TL is at its far end, $x = \ell$, terminated by the load impedance that is represented by its (causal) TD impedance operator $Z^\textrm{L}(t)$ (in ($\Omega /s$)).

If the load impedance is purely resistive, $Z^\textrm{L}(t) = R^\textrm{L}\delta (t)$ with $R^\textrm{L} > 0$ being the (real-valued and positive) load resistance, the (causal) space-time voltage distribution admits a straightforward closed form solution expressible in terms of traveling, voltage-wave constituents¹³, Ch. 8

$$\begin{aligned}&V(x,t) = \tfrac{1}{2}(1-\Gamma _1)\sum _{k=0}^\infty (\Gamma _1\Gamma _2)^k V_0\Big (t-\frac{x}{c_0}-\frac{2k\ell }{c_0}\Big ) \nonumber \\&+ \tfrac{1}{2}\Gamma _2(1-\Gamma _1)\sum _{k=0}^\infty (\Gamma _1\Gamma _2)^k V_0\Big (t-\frac{2\ell -x}{c_0}-\frac{2k\ell }{c_0}\Big ), \end{aligned}$$

(2)

where

$$\begin{aligned} \Gamma _1 = \frac{R^\textrm{S}-Z^\textrm{C}}{R^\textrm{S}+Z^\textrm{C}} \text { and } \Gamma _2 = \frac{R^\textrm{L}-Z^\textrm{C}}{R^\textrm{L}+Z^\textrm{C}}, \end{aligned}$$

(3)

denote the voltage-type, source and load reflection coefficients, respectively. Since (2) represents merely a superposition of progressively time-shifted and attenuated copies of the excitation pulse shape $V_0(t)$, we can readily find its upper bound as

$$\begin{aligned} |V(x,t)|&\le \tfrac{1}{2}V_m(1-\Gamma _1)(1+|\Gamma _2|)\sum _{k=0}^\infty |\Gamma _1\Gamma _2|^k \nonumber \\&= \tfrac{1}{2}V_m(1-\Gamma _1)\frac{1+|\Gamma _2|}{1-|\Gamma _1\Gamma _2|}, \end{aligned}$$

(4)

where we used the sum of a (convergent) geometric series and $V_m > 0$ denotes the amplitude of the excitation pulse. In practice, however, the load impedance is rarely purely resistive due to its either dominant or parasitic capacitive or/and inductive behavior. In such circumstances, the TD solution (2) and hence the corresponding bound (4) cease to be valid. This highlights the need for a general framework to derive TD physical bounds on traveling-wave solutions. Section "Problem solution" addresses this by introducing such an approach.

Since the voltage distribution V(x, t) along a TL that is terminated by the causal excitation and passive load is a causal and bounded function of time, we can in part follow the lines of reasoning presented in^2,3,4, and derive TD physical bounds using its analytic behavior in the right half of the complex-frequency plane. The procedure is demonstrated for both purely resistive load and for a serial RL load.

Problem solution

In this section, we shall derive upper bounds on the voltage distribution along loaded TLs. For the sake of clarity, the methodology is first illustrated on the resistively loaded TL for which a simple bound is readily achievable (see (4)). Second, the methodology is applied to a TL loaded by a serial RL load, for which an upper bound is not immediately available.

Transmission line with purely resistive load

The s-domain solution that corresponds to (2) can be expressed as

$$\begin{aligned} \hat{V}(x,s)&= \tfrac{1}{2}\hat{V}_0(s)\hat{\Lambda }(s)\exp (-sx/c_0) \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\Gamma _2\hat{\Lambda }(s)\exp [-s(2\ell -x)/c_0], \end{aligned}$$

(5)

where

$$\begin{aligned} \hat{\Lambda }(s)&= \frac{1-\Gamma _1}{1-\Gamma _1\Gamma _2\exp (-2s\ell /c_0)}, \end{aligned}$$

(6)

Upon extracting the instantaneous response, (5) can be cast into the following form

$$\begin{aligned} \hat{V}(x,s)&= \tfrac{1}{2}\hat{V}_0(s)\Lambda _\infty \exp (-sx/c_0) \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\Gamma _2\Lambda _\infty \exp [-s(2\ell -x)/c_0] \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\hat{\Omega }(s)\exp (-sx/c_0) \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\Gamma _2\hat{\Omega }(s)\exp [-s(2\ell -x)/c_0], \end{aligned}$$

(7)

where $\Lambda _\infty = \lim _{s\rightarrow \infty }\hat{\Lambda }(s) = 1-\Gamma _1$, and

$$\begin{aligned} \hat{\Omega }(s) = (1-\Gamma _1)\frac{\Gamma _1\Gamma _2\exp (-2s\ell /c_0)}{1-\Gamma _1\Gamma _2\exp (-2s\ell /c_0)} \end{aligned}$$

(8)

is an analytic function in $\Re (s) > 0$ that tends to zero as $s\rightarrow \infty$ and

$$\begin{aligned} \hat{\Omega }(s) = (1-\Gamma _1)\frac{\Gamma _1\Gamma _2}{1-\Gamma _1\Gamma _2} + o(1) \text { as } s\rightarrow 0. \end{aligned}$$

(9)

It is further noted that both $\Re [\hat{\Omega }(\mathrm{i} \omega )]$ and $\Im [\hat{\Omega }(\mathrm{i} \omega )]$ are oscillating functions for $\omega \in \mathbb {R}$, which violates the necessary conditions for the TD bounds as introduced in Ref. ². Fig. 2 shows the oscillatory behavior of the real and imaginary parts along $\{s = \mathrm{i} \omega , -10c_0/\ell \le \omega \le 10c_0/\ell \}$ for $\Gamma _1 = -2/3$ and $\Gamma _2 = 3/5$.

Fig. 2

Oscillating remainder function along a section of $\Re (s) = 0$.

This difficulty, however, can be circumvented using the inequality (A6) as presented in Appendix Appendix A. Indeed, as the TD counterpart of (8) is readily available, it can be used to express the corresponding $\Theta (t)$ (see (A4)) as

$$\begin{aligned} \Theta (t) = (1-\Gamma _1)\sum _{k=1}^\infty (\Gamma _1\Gamma _2)^k[1-\textrm{H}(t-2k\ell /c_0)], \end{aligned}$$

(10)

which shows that $|\Theta (t)|$ is a monotonically decreasing function of time. Indeed, the kth constituent of $\Theta (t)$ can be expressed as $\Theta (0^+)(\Gamma _1\Gamma _2)^k$ and has a bounded support in $\{2(k-1)\ell /c_0< t <2k\ell /c_0\}$. Consequently, it has two properties: (a) its amplitude decays and (b) its “time of arrival” increases as k grows. These properties remain valid for any passive internal and load impedances. Consequently, the use of (A6) with (9) leads to the desired bound

$$\begin{aligned} |\partial _t^{-1}\Omega (t)|\le 2(1-\Gamma _1)\frac{|\Gamma _1\Gamma _2|}{|1-\Gamma _1\Gamma _2|}. \end{aligned}$$

(11)

In the TD counterpart of (7) we consider the “worst-case” step pulse $V_0(t) = V_m\textrm{H}(t)$ and get

$$\begin{aligned} |V(x,t)| \le \tfrac{1}{2}V_m(1-\Gamma _1)(1+|\Gamma _2|)\left[ 1+\frac{2|\Gamma _1\Gamma _2|}{|1-\Gamma _1\Gamma _2|}\right] . \end{aligned}$$

(12)

We can further use $|1-\Gamma _1\Gamma _2|\ge 1-|\Gamma _1\Gamma _2|$ in (12) and obtain

$$\begin{aligned} |V(x,t)| \le \tfrac{1}{2}V_m(1-\Gamma _1)(1+|\Gamma _2|)\frac{1+|\Gamma _1\Gamma _2|}{1-|\Gamma _1\Gamma _2|}, \end{aligned}$$

(13)

which gives apparently a $(1+|\Gamma _1\Gamma _2|)$-times higher bound than (4). Note, for example, that if the resistive load is perfectly matched, i.e., $\Gamma _2 = 0$, then bounds (4), (12) and (13) give $|V(x,t)|\le \tfrac{1}{2}V_m(1-\Gamma _1)$. The latter is, in fact, equal to $V_m Z^\textrm{C}/(R^\textrm{S}+Z^\textrm{C})$, which corresponds to the amplitude of incident voltage pulse traveling from the source to the load. Therefore, if there is non-vanishing load capacitance or/and inductance, all bounds (4), (12) and (13) will fail due to an (unavoidable) reflection against the load. A way out of the difficulty is presented in the ensuing Section "Transmission line with RL load".

Transmission line with RL load

For a series RL load, the TD load impedance operator is given by $Z^\textrm{L}(t) = R^\textrm{L}\delta (t) + L^\textrm{L}\delta ^{(1)}(t)$, which corresponds to $\hat{Z}^\textrm{L}(s) = R^\textrm{L} + sL^\textrm{L}$. The pertaining s-domain voltage distribution can be cast into the following form (cf., (7))

$$\begin{aligned} \hat{V}(x,s)&= \tfrac{1}{2}\hat{V}_0(s)\Lambda _\infty \exp (-sx/c_0) \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\Lambda _\infty \exp [-s(2\ell -x)/c_0] \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\hat{\Omega }^+(s)\exp (-sx/c_0) \nonumber \\&+ \tfrac{1}{2}\hat{V}_0(s)\hat{\Omega }^-(s)\exp [-s(2\ell -x)/c_0], \end{aligned}$$

(14)

where $\Lambda _\infty = 1-\Gamma _1$ and (cf., (8))

$$\begin{aligned} \hat{\Omega }^+(s)&= (1-\Gamma _1)\frac{\Gamma _1\hat{\Gamma }_2(s)\exp (-2s\ell /c_0)}{1-\Gamma _1\hat{\Gamma }_2(s)\exp (-2s\ell /c_0)}, \end{aligned}$$

(15)

$$\begin{aligned} \hat{\Omega }^-(s)&= (1-\Gamma _1)\frac{\Gamma _1\hat{\Gamma }_2(s)\exp (-2s\ell /c_0)-[1-\hat{\Gamma }_2(s)]}{1-\Gamma _1\hat{\Gamma }_2(s)\exp (-2s\ell /c_0)}, \end{aligned}$$

(16)

with $\Gamma _1$ given by (3), while the load reflection coefficient is now given by

$$\begin{aligned} \hat{\Gamma }_2(s) = \frac{\hat{Z}^\textrm{L}(s)-Z^\textrm{C}}{\hat{Z}^\textrm{L}(s)+Z^\textrm{C}}. \end{aligned}$$

(17)

Note that at $s = 0$, we have $\hat{\Gamma }_2(0) = \Gamma _2$ (see (3)). Both $\hat{\Omega }^\pm (s)$ are analytic functions in $\Re (s) > 0$ including its boundary and tend to zero as $s\rightarrow \infty$. Consequently, their dominant terms around $s = 0$, i.e.,

$$\begin{aligned} \hat{\Omega }^+(0)&= (1-\Gamma _1)\frac{\Gamma _1\Gamma _2}{1-\Gamma _1\Gamma _2}, \end{aligned}$$

(18)

$$\begin{aligned} \hat{\Omega }^-(0)&= (1-\Gamma _1)\frac{\Gamma _1\Gamma _2-(1-\Gamma _2)}{1-\Gamma _1\Gamma _2}, \end{aligned}$$

(19)

can be used to establish the following late-time bounds (cf., (11))

$$\begin{aligned}&|\partial _t^{-1}\Omega ^+(t)|\le 2(1-\Gamma _1)\frac{|\Gamma _1\Gamma _2|}{|1-\Gamma _1\Gamma _2|}, \end{aligned}$$

(20)

$$\begin{aligned}&|\partial _t^{-1}\Omega ^-(t)|\le 2(1-\Gamma _1)\frac{|\Gamma _1\Gamma _2-(1-\Gamma _2)|}{|1-\Gamma _1\Gamma _2|}. \end{aligned}$$

(21)

Despite $\Theta ^{\pm }(t)$ (see (A4)) are not readily achievable in closed form, we have surmised that they share the properties stated below (10). Indeed, in line with (A4), $\Theta ^{\pm }(t)$ can be found by subtracting $\partial _t^{-1}\Omega ^\pm (t)$ from their terminal values given by (18) and (19), respectively. The former follows by carrying out the inverse Laplace transform of $\hat{\Omega }^\pm (s)/s$. This can be done via the term-by-term inversion of the geometric-series expansions of (15) and (16). It is easily seen that each s-dependent term of the series shows a simple pole singularity at $s = 0$ and a kth-order pole at $s = -1/\tau$ with $\tau = L/(R+Z^\textrm{C}) > 0$. Owing to the exponential factor $\exp (-2ks\ell /c_0)$, the subtraction of the simple-pole ($s = 0$) contributions from the terminal value leads to a series of the form (10), while the contributions of the pole at $s = -1/\tau$ lead to exponential decaying functions proportional to $\exp [-(t-2k\ell /c_0)/\tau ]\textrm{H}(t-2k\ell /c_0)$. In total, we again obtain monotonically decaying functions as was assumed for establishing (20) and (21). Using these inequalities and following the lines of reasoning of Section "Transmission line with purely resistive load", we end up with (cf., (12))

$$\begin{aligned} |V(x,t)| \le V_m(1-\Gamma _1)\left[ 1+ \frac{|\Gamma _1\Gamma _2|+|\Gamma _1\Gamma _2-(1-\Gamma _2)|}{|1-\Gamma _1\Gamma _2|}\right] . \end{aligned}$$

(22)

Note that even if the load resistor is matched to the TL, i.e., $R^\textrm{L} = Z^\textrm{C}$ and hence $\Gamma _2 = 0$, we can expect (possibly multiple) reflections due to the non-zero load inductance. In this case, from (22) we obtain $|V(x,t)| \le 2V_m(1-\Gamma _1)$, which is equal to $4 V_m Z^\textrm{C}/(R^\textrm{S}+Z^\textrm{C})$.

Illustrative numerical examples

The analyzed TLs will be excited by a symmetrical trapezoidal pulse

$$\begin{aligned}&V_0(t) = (V_\textrm{m}/t_\textrm{r})[t\textrm{H}(t)-(t-t_\textrm{r})\textrm{H}(t-t_\textrm{r}) \nonumber \\&- (t-t_\textrm{w})\textrm{H}(t-t_\textrm{w}) + (t-t_\textrm{w}-t_\textrm{r})\textrm{H}(t-t_\textrm{w}-t_\textrm{r})], \end{aligned}$$

(23)

where $V_\textrm{m}$ denotes its amplitude, $t_\textrm{w}$ is the pulse time width (at half the pulse height), and $t_\textrm{r}$ is the pulse rise time (= pulse fall time $t_\textrm{f}$). In the examples that follow, we take the unit amplitude $V_\textrm{m} = 1.0\,\textrm{V}$. The pulse time width is taken to be a half of the travel time along the TL, $c_0t_\textrm{w} = \ell /2$, and, finally, the pulse rise/fall time is a tenth of the pulse time width, i.e., $t_\textrm{r} = t_\textrm{w}/10$. Fig. 3 shows the excitation pulse shape in the chosen time window of observation $\{0\le t \le 6\ell /c_0\}$. The internal resistance of the voltage source is $R^\textrm{S} = 10\,\Omega$ and the characteristic impedance of the analyzed TL is chosen as $Z^\textrm{C} = 50\,\Omega$. Consequently, the input reflection coefficient is $\Gamma _1 = -2/3$ (see (3)).

Fig. 3

Trapezoidal excitation pulse shape.

First, the TL is loaded by a resistor of $R^\textrm{L} = 200\,\Omega$, which leads to $\Gamma _2 = 3/5$. The corresponding voltage pulse shapes as evaluated at three observation points along the TL at $x = \{0,\ell /2,\ell \}$ are shown in Fig. 4. As can be seen, all the derived bounds (4), (12) and (13) are greater than |V(x, t)| from Fig. 4. The tightest bound in this case is the one given by (12) that returns the upper limit at about $2.10\,\textrm{V}$. Next, (4) and (13) lead to approximately $2.22\,\textrm{V}$ and $3.11\,\textrm{V}$, respectively.

Fig. 4

Voltage pulse shapes at three spatial points along a TL loaded by $R^\textrm{L} = 200\,\Omega$ and the bounds given by (4), (12) and (13).

In the second example, we add a small serial inductance of $L^\textrm{L} = 10^{-8}\,\textrm{H}$ to the resistor $R^\textrm{L} = 200\,\Omega$. The other parameters remain the same as in the first example. Fig. 5(a) shows the effect of the additional inductance on the shape of induced voltage. It seen that bounds (4) and (12) applying to a purely resistive load satisfy, in this particular case, the desired inequality, even in a tighter manner than (22). If, however, the load resistance gets closer to the characteristic impedance, both (4) and (12) will fail to predict the upper limit. This is illustrated in Fig. 5(b) that shows the voltage distribution for the RL load with $R^\textrm{L} = 60\,\Omega$ with $L^\textrm{L} = 10^{-8}\,\textrm{H}$. Owing to a relatively small static load reflection coefficient, $\Gamma _2 = 1/11$, both bounds (4) and (12) are relatively close to the voltage amplitude on the perfectly matched TL, $\tfrac{1}{2}V_m(1-\Gamma _1) = 5/6\,\textrm{V}$. The non-vanishing load inductance, however, gives rise to overshoot exceeding the maximum voltage level on the resistively loaded TL. Equation (22) leads approximately to $3.29\,\textrm{V}$, which safely bounds the induced voltage response.

Fig. 5

Voltage pulse shapes at three spatial points along a TL loaded by a serial RL load and the bounds given by (22), (4) and (12). (a) $R^\textrm{L} = 200\,\Omega$ with $L^\textrm{L} = 10^{-8}\,\textrm{H}$; (b) $R^\textrm{L} = 60\,\Omega$ with $L^\textrm{L} = 10^{-8}\,\textrm{H}$.

Conclusion

We have investigated upper bounds on the space-time voltage distribution induced along a loaded TL. Since this voltage response does not satisfy the conditions required by the previously published TD physical bounds (see ^2,3), we have extended the methodology to incorporate TD responses consisting of (a superposition of) causal, traveling wave constituents. Illustrative numerical examples have validated the proposed approach.