# Electrical Network Functions of Common-Ground Uniform PassiveRLC Ladders and Their Elmore’s Delay and Rise Times

Authors: D. B. Kandić; B. D. Reljin; I. S. Reljin
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

In the paper are presented the expressions for all network functions of common-ground, uniform passive ladders having, in general, complex terminations at both their ends. The Elmore's delay and rise times calculated for selected types ofRLC ladders have indicated their slight deviation from delay and rise times obtained according to their classical definitions. For common-ground, integrating type RC ladder with voltage-step input, the Elmore's delay- and rise-times are produced in closed-form, both for ladder nodes and points. Furthermore, it is proposed a particular common-ground, uniform RLC ladder being amenable to application as delay line for pulsed and analog input signals. For this ladder, the Elmore's delay and rise times relating to their node voltages are produced in a closed-form, enabling thus with the realization of artificial (a) pulse delay line with arbitrarily and independently specified overall Elmore's delay and rise times and (b) true delay line with arbitrarily specified delay time for frequency bounded analog and/or pulsed input signals. In cases (a) and (b), precise procedures are formulated for calculation of ladder length and of all its RLC parameters. The obtained results are illustrated with several practical examples and are, also, verified through pspice simulation.


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## Body

## 1. Introduction

Modeling of digital MOS circuits byRC networks has become a well-accepted practice for estimating delays [1–6]. In digital integrated circuits, signal propagation delay through conducting paths with distributed resistance and capacitance is frequently a significant part of the total delay. These conducting paths, or “interconnections,” can be modeled quite accurately by nonuniform, branched RC ladder networks, also known as “RC trees” [3]. Computationally simple bounds for signal delay in linear RCtree networks were found in [3] and have been used in several practical MOS timing analyzers reported in [6], but certain circuits used in MOS logic cannot be modeled as RC trees since they contain one or more closed loops of resistors, and these general RCnetworks are being referred to as “RCmeshes”. In these networks, the time delay defined according to Elmore [7] is proved to be the valid estimate, and this fact has been used in [5] to advantage in an approach to MOS timing analysis of general RC networks containing RCmeshes. Simple closed-form bounds for signal propagation delay in linear RC tree models for MOS interconnections derived in [3] are, also, valid for the more general class of linear networks known as RC meshes, which are useful as models for portions of MOS logic circuits that cannot be represented as RCtrees [6].Elmore’s delay is an extremely popular timing-perfomance metric which is used at all levels of electronic circuit design automation, particularly forRC tree analysis. The widespread usage of this metric is mainly attributable to its property of being a simple analytical function of circuit parameters, and its drawbacks are the uncertainty of accuracy and restriction to being the estimate only for the step-response delay. An extension of Elmore’s delay definition has been proposed in [8] to accommodate the effect of nonunit-step (slow) excitations and to handle multiple sources of excitation, in order to show that delay estimation for slow excitations is no harder than for the unit-step input. In [9] it has been shown that this extension of Elmore’s delay time offers a provision to deal with slow varying excitations in timing analysis of MOS pass transistor networks. In addition, in [10] it has been reported that Elmore’s delay is an absolute upper bound on the actual 50% delay of an RC tree response. Also in [10], it has been proved that this bound holds for input signals other than steps and that actual delay asymptotically approaches to Elmore’s delay as the input signal rise-time increases. It has been emphasized in [11] that RC tree step responses always are monotonic, and this is why Elmore’s definitions of both delay and rise time [7] are applicable on complex RC tree networks.In this paper we will firstly derive the general, closed-form expressions for input and transfer functions of common-ground, uniform, and passive ladders with complex double terminations, making a distinction between the signal transfer to ladder nodes and to its points. Then, simplifications of the obtained results are produced for ladders with seven specific pairs of complex double terminations being interesting from the practical point of view. Thereafter, for a uniform, common-groundRLCladder with (a) a relation between its parameters resembling to analogous relation between per-unit-length parameters of distortionless transmission line and (b) symmetric, resistive double termination resembling to characteristic impedance of distortionless line [12], it will be shown that its Elmore’s delay and rise times for point voltage transmittances can be efficiently calculated (but not in the closed form) by using of the numerical scheme proposed herein. In this case, we will see that the obtained numerical values for Elmore’s times differ slightly from the delay  and rise times obtained according to their classical definitions and by using pspice simulation when ladder is excited by a step voltage.For the integrating type of distributed RC impedance, common-ground ladder as a model of two-wire line, it has been suggested that it might be used as a true pulse delay line [13], provided that, Schmitt triggers are used for reshaping the delayed and edge-distorted transmitted signals. This type of delay line with step-input excitation has already been thoroughly investigated in [14]—where Elmore’s delay time is given in closed form only for ladder nodes, and for Elmore’s rise time is offered a conjecture relating to its lower bound for overall network. Elmore’s rise times for all nodes of integrating, RCopen-circuited ladder are given in closed form in [15] and the obtaining of Elmore’s delay and rise times both for nodes and points is discussed to some extent in [16]—for other types of open-circuited ladders. In this paper, we are going to formulate the explicit closed-form expressions for Elmore’s delay and rise times both for the node and point voltages of integrating open circuited, common-ground RCladder and will conclude that this type of network is not recommendable for pulse delay line in its own right, since Elmore’s rise time of each point voltage is not less than the twice of its delay-time.And finally, we will propose a type of uniform, common-groundRLC ladder amenable for application as delay line both for pulsed and/or analog input signals. Elmore’s delay and rise times of this ladder relating to the node voltage transmittances are produced in closed form, opening thus with the following possibilities in ladder realization:(i)
for pulsed inputs, the overall Elmore delay and rise times may be specified arbitrarily,(ii)
for pulsed inputs, the minimum ladder length (i.e., the minimum necessary number of sections) is calculated straightforwardly by using only the overall Elmore delay and rise times,(iii)
the ladderRL parameters are calculated uniquely from the assumed nonminimal ladder length, overall Elmore delay time, and the assumed capacitance values,(iv)
for realization of true delay for pulsed and/or analog input signals with arbitrary variation in time, minimum ladder length is calculated from the specified true delay time (which asymptotically tends to Elmore’s delay time when the number of sections tends to infinity) and maximum frequency in the spectrum of the signal being transmitted along the ladder purporting to represent delay line. And again, as in (iii), the ladderRL parameters are determined from the ladder length, overall ladder true delay time (being approximately equal to Elmore’s delay time), and the assumed capacitance values. The previous approach and obtained results are illustrated and verified with realization examples of large true delay time (=5 [ms]) for pulsed, sine, distorted sine, CAM, FM; and chirp frequency and sweep amplitude input signals.Recall that in network synthesis the ladder topology is a preferable one, since it has very low sensitivity to variations ofRLCparameters [16, 17].


## 2. Network Functions of Common-Ground, Uniform, and PassiveRLCLadders

Consider a uniform, grounded, and passiveRLCladder in Figure 1 with N identical sections, whose impedances Z1=Z1(s) and Z2=Z2(s) are positive real rational functions in complex frequency s=σ+jω (j:=-1). There on are denoted the Laplace transforms E=E(s) of the voltage excitation e=e(t), Uk=Uk(s) and Ik=Ik(s) of the point voltagesuk=uk(t) and currents ik=ik(t)(k=0,N¯), and Um′=Um′(s) and Im′=Im′(s) of node voltagesum′=um′(t) and currents im′=im′(t)(m=0,N-1¯). The internal impedances of the voltage excitation (voltage generator) and the load are Zg=Zg(s) and ZL=ZL(s), respectively. All the initial conditions associated to the network reactive (LC) elements are assumed to be zero.Figure 1
The general, common-ground, uniform ladder withN sections.For ladder in Figure1, the mesh transform equations for its nth section (n=1,N¯) read
(1)(Z1+Z2)·In-1-Z2·In=Un-1,Z2·In-1-(Z1+Z2)·In=Un,orintherecurrenceform[UnIn]=[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2]·[Un-1In-1].The boundary values of voltages and currents in the set of matrix difference equations (1) are (U0,I0) and (UN,IN). For n=0,N¯, from (1) it immediately follows that
(2)[UnIn]=[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2]n·[U0I0],mmwhereU0=E-Zg·I0∧UN=ZL·IN.
Since the roots (say λ1 and λ2) of the characteristic equation corresponding to 2×2 matrix appearing in (1),
(3)det{[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2]-λ·U2}mm=λ2-2·(Z1Z2+1)·λ+1=0mmm(U2is2×2identitymatrix),
satisfy the relations λ1+λ2=2·(Z1/Z2+1) and λ1·λ2=1, then by taking for convenience λ1:=exp(τ) and λ2:=exp(-τ), we obtain cosh(τ)=1+Z1/Z2. By using Cayley-Hamilton’s theorem we may put
(4)[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2]nm=A0·U2+A1·[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2],
where the “constants” A0 and A1 are determined from system of equations [exp(τ)]n=A0+A1·exp(τ) and [exp(-τ)]n=A0+A1·exp(-τ), whose solution is A0=-sinh[(n-1)·τ]/sinh(τ) and A1=sinh(n·τ)/sinh(τ).By substitutingA0 and A1 in (4) and bearing in mind that
(5)1+Z1Z2=cosh(τ),-(Z12Z2+2·Z1)=Z2-Z2·(Z1+Z2Z2)2=Z2·[1-cosh2(τ)]=-Z2·sinh2(τ),
we finally obtain from (2) and (4) for n=0,N¯ that
(6)[UnIn]=[1+Z1Z2-(Z12Z2+2·Z1)-1Z21+Z1Z2]n·[U0I0]=[cosh(n·τ)-Z2·sinh(τ)·sinh(n·τ)-sinh(n·τ)Z2·sinh(τ)cosh(n·τ)]mm·[U0I0].Since uniform ladder sections are electrically reciprocal, then their characteristic impedanceZc and the quantity Zc·sinh(τ) can easily be produced in the form
(7)Zc=Z2·sinh(τ)=Z1·(Z1+2·Z2),Zc·sinh(τ)=Z2·sinh2(τ)=Z2·[cosh2(τ)-1]=Z1·(Z1Z2+2).For the ladder depicted in Figure1, the complete set of the network immittance and voltage/current transmittance functions can be produced, by using the relations (2), (6), and (7):
(8)InE=(ZLZccosh[(N-n)·τ]m+ZLZc·sinh[(N-n)·τ])×(+(Zc+Zg·ZLZc)·sinh(N·τ))-1Zg·ZLZc(Zc+Zg·ZLZc)Zg·ZLZcZg·ZLZc(Zg+ZL)·cosh(N·τ)mm+(Zc+Zg·ZLZc)·sinh(N·τ))-1,InI0=(Zc·cosh[(N-n)·τ]m+ZL·sinh[(N-n)·τ])×(Zc·cosh(N·τ)+ZL·sinh(N·τ))-1,UnE=(ZL·cosh[(N-n)·τ]m+Zc·sinh[(N-n)·τ])×(+(Zc+Zg·ZLZc)·sinh(N·τ))(Zg+ZL)·cosh(N·τ)mm+(Zc+Zg·ZLZc)·sinh(N·τ))-1,UnU0=(ZL·cosh[(N-n)·τ]m+Zc·sinh[(N-n)·τ])×(ZL·cosh(N·τ)+Zc·sinh(N·τ))-1,UnIn=(ZL·cosh[(N-n)·τ]m+Zc·sinh[(N-n)·τ])×(Zc·cosh[(N-n)·τ]mmm+ZL·sinh[(N-n)·τ])-1·Zc,n=0,N¯.SinceZ1=Zc·(sinh(τ)/[1+cosh(τ)])=Zc·tanh(τ/2), then the voltages and currents of impedances Z2 connecting the middle nodes of ladder sections (Mm) to common-node O (Figure 1) are given as
(9)Um′=Um+Um+11+cosh(τ),Im′=Um′Z2=Um+Um+1Z2·[1+cosh(τ)]=Um+Um+1Z1+2·Z2,Um′E=(τ2(ZL-Z1)·cosh[(N-m)·τ]m+[Zc-ZL·tanh(τ2)]m·sinh[(N-m)·τ]τ2)×(Zg·ZLZc(Zg+ZL)·cosh(N·τ)mm+(Zc+Zg·ZLZc)·sinh(N·τ))-1,mmmmmmmmmmmmmm=0,N-1,-Um′U0=(τ2(ZL-Z1)·cosh[(N-m)·τ]m+[Zc-ZL·tanh(τ2)]mm·sinh[(N-m)·τ]τ2)×(ZL·cosh(N·τ)+Zc·sinh(N·τ))-1.Consider now the particular selection of{Zg,ZL} to investigate the generation of (i) finite length, frequency-selecting ladders with various input and transfer immittances and voltage and current transmittances and (ii) specific ladder which can take the role of finite pulse delay line without pulse attenuation and with independently controlled pulse delay and rise times in Elmore’s sense [7], calculable in closed form. We will assume that impedances Zg and ZL are rational positive real functions ins, so that they are realizable by passive transformerlessRLCnetworks [17]. All network functions in (8) and (9) are real rational functions ins, except Un/In, which must be rational,positive real function ins, since it is the input immittance of RLCnetwork.Case A (
Z
g
-arbitrary
∧
Z
L
→
∞
[
Ω
]) (open-circuited ladder). From (8) and (9) it follows,
(10)InE=sinh[(N-n)·τ]Zc·cosh(N·τ)+Zg·sinh(N·τ),InI0=sinh[(N-n)·τ]sinh(N·τ),UnE=cosh[(N-n)·τ]cosh(N·τ)+(Zg/Zc)·sinh(N·τ),UnU0=cosh[(N-n)·τ]cosh(N·τ),UnIn={cotanh[(N-n)·τ]}·Zc,n=0,N¯.(11)Um′E=cosh[(N-m)·τ]-(Z1/Zc)·sinh[(N-m)·τ]cosh(N·τ)+(Zg/Zc)·sinh(N·τ)=sinh[(N-m)·τ]-sinh[(N-m-1)·τ]sinh(τ)·[cosh(N·τ)+(Zg/Zc)·sinh(N·τ)],Um′U0=cosh[(N-m)·τ]-(Z1/Zc)·sinh[(N-m)·τ]cosh(N·τ)=sinh[(N-m)·τ]-sinh[(N-m-1)·τ]sinh(τ)·cosh(N·τ),mmmmmmmmmmmmmmmmmm=0,N-1.-Since we havecosh(τ)=1+Z1/Z2, then by Property 2 (Appendix A), Un/U0  (n=1,N¯) (10) can be easily converted into real rational function of Z1/Z2 or of s. Similarly, by Property 3 (Appendix A), Im/I0(m=1,N-1¯) (10) can be, also, easily converted into real rational function of Z1/Z2 or of s. When Z1 and Z2 are one-element-kind impedances, the zeros and poles of both Un/U0 and Im/I0 can be easily determined in the closed form. Since by Property 3  sinh[(N-m)·τ] and sinh[(N-m-1)·τ] contain the same factor sinh(τ)(m=0,N-1¯), then Um′/U0(m=0,N-1¯) (11) can be, also, easily converted into real rational function, either of Z1/Z2 or of the complex frequency s.Case B (Zg=Z2·cosh(τ)=Z1+Z2∧ZL→∞ [Ω]). In this case (10) and (11) simplify to
(12)InE=sinh[(N-n)·τ]Z2·sinh[(N+1)·τ],InI0=sinh[(N-n)·τ]sinh(N·τ),UnE=cosh[(N-n)·τ]·sinh(τ)sinh[(N-1)·τ],UnU0=cosh[(N-n)·τ]cosh(N·τ),UnIn={cotanh[(N-n)·τ]}·Zc,n=0,N¯,(13)Um′E=2·cosh[(N-m-(1/2))·τ]·sinh(τ/2)sinh[(N+1)·τ]=sinh[(N-m)·τ]-sinh[(N-m-1)·τ]sinh[(N+1)·τ],Um′U0=cosh[(N-m-(1/2))·τ]cosh(τ/2)·cosh(N·τ)=sinh[(N-m)·τ]-sinh[(N-m-1)·τ]sinh(τ)·cosh(N·τ),mmmmmmmmmmmmmmmmmm=0.N-1-.Case C (Zg-arbitrary∧ ZL=0[Ω] (short-circuited ladder)) From (8) and (9) it is obtained,
(14)InE=cosh[(N-n)·τ]Zg·cosh(N·τ)+Zc·sinh(N·τ),InI0=cosh[(N-n)·τ]cosh(N·τ),UnE=sinh[(N-n)·τ]sinh(N·τ)+(Zg/Zc)·cosh(N·τ),UnU0=sinh[(N-n)·τ]sinh(N·τ),UnIn=tanh[(N-n)·τ]·Zc,n=0,N¯,(15)Um′E=Zc·sinh[(N-m)·τ]-Z1·cosh[(N-m)·τ]Zc·sinh(N·τ)+Zg·cosh(N·τ)=cosh[(N-m)·τ]-cosh[(N-m-1)·τ]sinh(τ)·[sinh(N·τ)+(Zg/Zc)·cosh(N·τ)],Um′U0=sinh[(N-m)·τ]-(Z1/Zc)·cosh[(N-m)·τ]sinh(N·τ)=cosh[(N-m)·τ]-cosh[(N-m-1)·τ]sinh(τ)·sinh(N·τ),mmmmmmmmmmmmmmmmmm=0,N-1.-
According to Properties 2 and 3, respectively, In/I0(n=1,N¯) and Um/U0(m=1,N-1¯) (14) can be easily converted into real rational functions, either of Z1/Z2 or of the complex frequency s. If Z1 and Z2 are one-element-kind impedances, the zeros and poles of both In/I0 and Um/U0 can be determined straightforwardly in closed form. Now, since Zc·sinh(τ)/Z1=2+Z1/Z2 and/or (1+Z1/Z2)2-sinh2(τ)=1, it can be seen from (15) that Um′/U0(m=0,N-1¯) is, also, convertible into real rational function, either of Z1/Z2 or of the complex frequency s, by using of both Properties 2 and 3.Case D (Zg=Z2·cosh(τ)=Z1+Z2∧ZL=0[Ω]). From (14) and (15) it readily follows that
(16)InE=cosh[(N-n)·τ]Z2·cosh[(N+1)·τ],InI0=cosh[(N-n)·τ]cosh(N·τ),UnE=sinh[(N-n)·τ]·sinh(τ)cosh[(N+1)·τ],UnU0=sinh[(N-n)·τ]sinh(N·τ),UnIn=tanh[(N-n)·τ]·Zc,n=0,N,¯(17)Um′E=cosh[(N-m)·τ]-cosh[(N-m-1)·τ]cosh[(N+1)·τ],Um′U0=cosh[(N-m)·τ]-cosh[(N-m-1)·τ]sinh(τ)·sinh(N·τ),mmmmmmmmmmmmmmmmmm=0,N-1-
By using of Properties 2 and 3 and sinh2(τ)=(1+Z1/Z2)2-1, it can be seen from (16) and (17) that the network functions Ik/I0(k=1,N¯), Um/E(m=0,N-1¯), Un/U0(n=1,N-1¯), Up′/E(p=0,N-1¯), and Uq′/U0(q=0,N-1¯) can be easily converted into real rational functions, either of Z1/Z2 or of the complex frequency s.Case E (Zg=ZL=Z1). From (8) and (9), after using the relations Zc=Z2·sinh(τ),Z1=Zc·tanh(τ/2), and cosh(τ)=1+Z1/Z2, it can be easily obtained that
(18)InE=(+Z1Zc·sinh[(N-n)·τ])cosh[(N-n)·τ]m+Z1Zc·sinh[(N-n)·τ])×(+Z1+Z2Zc·sinh(N·τ)]2·Z1·[Z1+Z2Zccosh(N·τ)mmmmmm+Z1+Z2Zc·sinh(N·τ)])-1=sinh[(N-n+1)·τ]-sinh[(N-n)·τ]2·Z1·sinh[(N+1)·τ],InI0=cosh[(N-n)·τ]+(Z1/Zc)·sinh[(N-n)·τ]cosh(N·τ)+(Z1/Zc)·sinh(N·τ)=cosh[(N-n+1)·τ]+cosh[(N-n)·τ]cosh[(N+1)·τ]+cosh(N·τ),UnE=cosh[(N-n)·τ]+(Zc/Z1)·sinh[(N-n)·τ]2·[cosh(N·τ)+((Z1+Z2)/Zc)·sinh(N·τ)]=sinh[(N-n+1)·τ]+sinh[(N-n)·τ]2·sinh[(N+1)·τ],UnU0=cosh[(N-n)·τ]+(Zc/Z1)·sinh[(N-n)·τ]cosh(N·τ)+(Zc/Z1)·sinh(N·τ)=cosh[(N-n+1)·τ]-cosh[(N-n)·τ]cosh[(N+1)·τ]-cosh(N·τ),UnIn=cosh[(N-n)·τ]+(Zc/Z1)·sinh[(N-n)·τ]sinh[(N-n)·τ]+(Zc/Z1)·cosh[(N-n)·τ]mm·Zc=sinh[(N-n+1)·τ]+sinh[(N-n)·τ]cosh[(N-n+1)·τ]+cosh[(N-n)·τ]mm·Zc,n=0,N¯,(19)Um′E=(sinh[(N-m)·τ])×((1+Z1Z2)cosh(N·τ)·sinh(τ)mm+(1+Z1Z2)·sinh(N·τ))-1=sinh[(N-m)·τ]sinh[(N+1)·τ],Um′U0=(2·sinh[(N-m)·τ])×((2+Z1Z2)cosh(N·τ)·sinh(τ)mm+(2+Z1Z2)·sinh(N·τ))-1=2·sinh[(N-m)·τ]sinh[(N+1)·τ]+sinh(N·τ),mmmmmmmmmmmmmmm=0,N-1-.Case F (Zg=ZL=Z1+2·Z2). From (8) and (9), after using relations Zc=Z2·sinh(τ),Z1=Zc·tanh(τ/2),cosh(τ)=1+Z1/Z2, and Zc·sinh(τ)/(Z1+2Z2)=Z1/Z2, it readily follows that
(20)InE=(Z1+2·Z2Zccosh[(N-n)·τ]m+Z1+2·Z2Zc·sinh[(N-n)·τ])×([Z1+Z2Zccosh(N·τ)+Z1+Z2Zc·sinh(N·τ)]2·(Z1+2·Z2)mm·[Z1+Z2Zccosh(N·τ)+Z1+Z2Zc·sinh(N·τ)])-1=sinh[(N-n+1)·τ]+sinh[(N-n)·τ]2·(Z1+2·Z2)·sinh[(N+1)·τ],InI0=(Z1+2·Z2Zccosh[(N-n)·τ]m+Z1+2·Z2Zc·sinh[(N-n)·τ])×(cosh(N·τ)+Z1+2·Z2Zc·sinh(N·τ))-1=sinh[(N-n+1)·τ]+sinh[(N-n)·τ]sinh[(N+1)·τ]+sinh(N·τ),UnE=(ZcZ1+2·Z2cosh[(N-n)·τ]m+ZcZ1+2·Z2·sinh[(N-n)·τ])×(2·[cosh(N·τ)+Z1+Z2Zc·sinh(N·τ)])-1=sinh[(N-n+1)·τ]-sinh[(N-n)·τ]2·sinh[(N+1)·τ],UnU0=(ZcZ1+2·Z2cosh[(N-n)·τ]m+ZcZ1+2·Z2·sinh[(N-n)·τ])×(cosh(N·τ)+Zc(Z1+2·Z2)·sinh(N·τ))-1=cosh[(N-n+1)·τ]+cosh[(N-n)·τ]cosh[(N+1)·τ]+cosh(N·τ),UnIn=(ZcZ1+2·Z2cosh[(N-n)·τ]m+ZcZ1+2·Z2·sinh[(N-n)·τ])×(Zc(Z1+2·Z2)sinh[(N-n)·τ]mm+Zc(Z1+2·Z2)·cosh[(N-n)·τ])-1·Zc=cosh[(N-n+1)·τ]+cosh[(N-n)·τ]sinh[(N-n+1)·τ]+sinh[(N-n)·τ]·Zc,mmmmmmmmmmmmmmmmmn=0,N¯.(21)Um′E=(cosh[(N-m)·τ])×((Z1Z2+2)mm·[cosh(N·τ)+Z1+Z2Zc·sinh(N·τ)])-1=sinh(τ)·cosh[(N-m)·τ][1+cosh(τ)]·sinh[(N+1)·τ],Um′U0=2(Z1/Z2)+2m·cosh[(N-m)·τ]cosh(N·τ)+(Zc/(Z1+2·Z2))·sinh(N·τ)=2·cosh[(N-m)·τ]cosh[(N+1)·τ]+cosh(N·τ),mmmmmmmmmmmm=0,N-1-.Case G (Zg=ZL=Zc=k·Z1(k>1)). From (7) it is easily obtained that Z2=[(k2-1)/2]·Z1,sinh(τ)=2·k/(k2-1),cosh(τ)=(k2+1)/(k2-1),exp(τ)=(k+1)/(k-1), and tanh(τ/2)=1/k. And from (8) and (9) it follows that
(22)InE=12·k·Z1·(k-1k+1)n,InI0=UnU0=(k-1k+1)n,UnE=12·(k-1k+1)n,UnIn=Zc=k·Z1,n=0,N¯,(23)Um′E=k-12·k·(k-1k+1)m,Um′U0=k-1k·(k-1k+1)m,m=0,N-1-.From (22) and (23) it can be seen that even when emf e [with Laplace transform E=E(s)] is pulsed, the node and point voltages with respect to common-ground O (Figure 1) are reproduced faithfully and without delay but with geometrically progressive amplitude attenuation, regardless of the impedance Z1.It is clear that relations (7)–(9) offer many other possibilities for the selection of Zg, ZL, Z1, and Z2  that lead to generation of versatile ladders realizing different types of frequency selective networks. An interesting one seems to be the common-ground RLCladder realizing (a) pulse delay with independently selected Elmore’s delay and rise times and/or (b) true delay for either pulsed or analog frequency limited input signals. These topics will be considered in the following section.


## 3. Elmore’s Delay and Rise Times for Selected Types of Common-Ground, UniformRLCLadders

Consider the ladder in Figure1 having Z1=R+L·s, Y2=1/Z2=G+C·s, and Zg=ZL=(L/C)1/2. Suppose that it holds the relation α:=R/L=G/C, similar to the one associated with per-unit-length parameters of distortionless transmission line [18, 19]. Let us define the auxiliary parameter β:=1/(L·C)1/2, and let us recast the network voltage transmittances describing the transfer of emf E to the ladder points with voltages  U0,U1,…,UN (Figure 1)—in the form suitable for calculating of Elmore’s delay and rise times [7] (Appendix B). Firstly, we should observe that the following holds:
(24)Z1Z2=(s+αβ)2,Z1ZL=ZLZc·sinh(τ)=s+αβ,ZcZL·sinh(τ)=s+αβ·[(s+αβ)2+2],1Zg+ZL·(Zc+Zg·ZLZc)·sinh(τ)mm=s+α2·β·[(s+αβ)2+3],
and then by using (8) and the Properties 2 and 3, let us express the point voltage transmittancesTn(s)=Un/E(n=0,N-1¯) and TN(s)=UN/E in the following form:
(25)Tn(s)=(β22)n·((s+α)·∏i=1N-n{[i·π2·(N-n)](s+α)2mmmmmmmmm+2·β2·sin2[i·π2·(N-n)]}mm+β·∏j=1N-n{[2·j-14·(N-n)·π](s+α)2mmmmmmm+2·β2·sin2[2·j-14·(N-n)·π]}∏j=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)][(s+α)2+2·β2·sin2(2·j-14·N·π)][(s+α)2+2·β2·sin2(2·j-14·N·π)])×(∏j=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)]β2(s+α)·[(s+α)2+3·β2]mm·∏i=1N-1[(s+α)2+2·β2·sin2(i·π2·N)]+2·βmm·∏j=1N[(2·j-14·N·π)(s+α)2mmmmmm+2·β2·sin2(2·j-14·N·π)(2·j-14·N·π)]∏i=1N-1[(s+α)2+2·β2·sin2(i·π2·N)])-1,TN(s)=(β22)N·(2·β)×([(s+α)2+2·β2·sin2(2·j-14·N·π)]∏j=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)](s+α)·[(s+α)2+3·β2]mm·∏i=1N-1[(s+α)2+2·β2·sin2(i·π2·N)]+2·βmm·∏j=1N[(i·π2·N)(s+α)2mmmmmm+2·β2·sin2(2·j-14·N·π)]∏i=1N-1[(s+α)2+2·β2·sin2(i·π2·N)]+2·β)-1.
As the calculation example of Elmore’s times, let us suppose for this type of ladder that the specified parameters are N=6, R=50 [Ω], L=10 [mH], G=5 [mS], and C=1 [μF]. We firstly calculate ZL=Zg=(L/C)1/2=100[Ω],α=5·103 [1/s], and β=104 [1/s], and then we recast Tn(s)=Un/E(n=0,N-1¯) and TN(s)=UN/E (25) in the following form:
(26)Tn(s)=(β22)nm·(A0(N,n)+A1(N,n)·s+A2(N,n)·s2mmm+⋯+A2N-2n+1(N,n)·s2N-2n+1)×(B2N+1(N)B0(N)+B1(N)·s+B2(N)·s2mmm+⋯+B2N+1(N)·s2N+1)-1,TN(s)=(β22)Nm·(2·β)((N)s2N+1B0(N)+B1(N)·s+B2(N)mmmmmm·s2+⋯+B2N+1(N)·s2N+1)-1,
where all “A” and “B” coefficients are positive. To apply Elmore’s definitions of delay and rise times (Appendix B) for all points in the ladder with zero initial conditions (Figure 1) and excited at t=0 with the step-voltage e with amplitude E[V], we must do the following.(i)
Firstly, check that the point voltagesuk=uk(t)(k=0,N¯) have no overshoots, or eventually if they are present, overshoots must be less than 5% of those voltages steady state values [7]. For example, if we have assumed for ladder with N=6 sections that E=10 [V], then its voltage step-responses uk=uk(t)(k=0,6¯) and the node response u0′ as well, obtained through pspice simulation and depicted in Figures 2 and 3 in the interval t∈[0,750][μs], reveal that Elmore’s definitions cannot be applied for responses u0 and u0′, since the occurrence of overshoots, whereas for all other point voltages uk=uk(t)(k=1,6¯), those definitions are applicable.(ii)
Secondly, calculate the coefficientsA0(N,n), A1(N,n), A2(N,n), B0(N), B1(N), and B2(N), so as to determine the parameters a1(N,n), a2(N,n), b1(N), and b2(N) of the normalized transfer functions Tn(s)/Tn(0)(n=0,N-1¯) and TN(s)/TN(0) obtained from (26) according to the relations
(27)Tn(s)Tn(0)=(1+a1(N,n)·s+a2(N,n)·s2m+⋯+a2N-2n+1(N,n)·s2N-2n+1)×(1+b1(N)·s+b2(N)·s2mm+⋯+b2N+1(N)·s2N+1)-1,Tn(0):=(β22)n·A0(N,n)B0(N),ai:=Ai(N,n)A0(N,n)[i=0,(2N-2n+1);¯n=0,(N-1)¯],bj:=Bj(N)B0(N)[j=0,(2N+1)¯],TN(s)TN(0)=(1)(1+b1(N)·smmm+b2(N)·s2+…+b2N+1(N)·s2N+1)-1,TN(0):=β2N+12N-1·B0(N).
Calculation of coefficients A0(N,n),A1(N,n), A2(N,n), B0(N), B1(N), and B2(N) in (26) might be a tedious task, especially when N is large, since these coefficients are produced from (25) as cumbersome expressions which cannot be put in the closed form. So, we must resort to making of a numerical application (say in MATHCAD) for automatic calculation of A0(N,n), A1(N,n), A2(N,n), B0(N), B1(N), B2(N), a1(N,n), a2(N,n), b1(N), b2(N), and Elmore’s times, for any given set of input parameters {N,n,R,L,C,G}, provided that the condition R/L=G/C is satisfied.(iii)
Calculate Elmore’s delay time (TD) and rise time (TR) of any point step-voltage response having no overshoot, according to definitions (B.5) in Appendix B and by using (27)
(28)TD(N,n)=b1(N)-a1(N,n),TR(N,n)=2·π·{b12(N)-a12(N,n)+2·[a2(N,n)-b2(N)]}.For RLC ladder having N=6, R=50[Ω],L=10 [mH], G=5 [mS], C=1 [μF]—excited at t=0 by step emf e with amplitude E=10 [V], the obtained results are summarized in Table 1.Table 1
Delay and rise times of the point voltages in the considered common-ground, uniformRLCladder.
n
0
1
2
3
4
5
6
a
1 [ms]
0.813
0.680
0.547
0.415
0.286
0.158
0
a
2 [μs]
0.323
0.225
0.145
0.082
0.037
0.010
0
b
1 [ms]
0.804
0.804
0.804
0.804
0.804
0.804
0.804
b
2 [μs]
0.315
0.315
0.315
0.315
0.315
0.315
0.315
T
D [ms]1
Not applicable
0.124
0.257
0.388
0.518
0.647
0.804
T
R [ms]1
Not applicable
0.157
0.200
0.229
0.250
0.286
0.326
T
D
c [ms]2
Not applicable
0.115
0.257
0.395
0.528
0.658
0.817
T
R
c [ms]2
Not applicable
0.190
0.252
0.295
0.324
0.349
0.373
u
n
(
∞
) [V]3
5.999
2.999
1.498
0.747
0.370
0.178
0.075
1The times calculated according to Elmore's definitions.
2The times obtained through PSPICE simulation according to the classical definitions of TDc (“50%” delay time) and TRc (“10%–90%” rise time).
3All data are related to the ladder with zero initial conditions.Figure 2
A set of voltage responses at the selected points of the considered ladder (and at theM0 node).Figure 3
Another set of voltage responses at the selected points of the considered ladder.Another interesting ladder in Figure1 is the one with Z1=R+L·s, Y2=1/Z2=G+C·s, Zg=0[Ω],ZL=(L/C)1/2, and α:=R/L=G/C. Again, let it be β:=1/(L·C)1/2. By using of (8) and Properties 2 and 3, the point voltage transmittancesTn(s)=Un/E  (n=0,N-1¯) and TN(s)=UN/E are obtained as follows:
(29)Tn(s)=(β22)n·((s+α)·∏i=1N-n{[i·π2·(N-n)](s+α)2+2·β2mmmmmmmmmm·sin2[i·π2·(N-n)]}mm+β·∏j=1N-n{sin2[2·j-14·(N-n)·π](s+α)2+2·β2mmmmmmmm·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]∏j=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)])×((s+α)·∏i=1N[sin2(2·j-14·N·π)(s+α)2+2·β2mmmmmmmmm·sin2(i·π2·N)]mm+β·∏j=1N[sin2(2·j-14·N·π)(s+α)2+2·β2mmmmmmmm·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]∏j=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)])·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)])-1,TN(s)=(β22)N·(2·β)×((s+α)·∏i=1N[sin2(i·π2·N)(s+α)2+2·β2mmmmmmmmm·sin2(i·π2·N)]m+β·∏j=1N[·sin2(2·j-14·N·π)](s+α)2+2·β2mmmmmm·sin2(2·j-14·N·π)]∏i=1N[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]∏i=1N[(s+α)2+2·β2·sin2[2·j-14·(N-n)·π]}·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)]·sin2[2·j-14·(N-n)·π]}·sin2[2·j-14·(N-n)·π]}[(s+α)2+2·β2·sin2(2·j-14·N·π)])-1.In this case, Elmore’s delay and rise times can be calculated in the similar way as it has been done in (25). The specified set of parameters {R,L,C,G,N,n} provided that the voltages of selected ladder points and/or nodes satisfy the condition (i), Elmore’s times can be calculated by using (26)–(29), but further consideration of this point will be left to the reader.Also, an interesting ladder is the one withZ1=R/2,Y2=1/Z2=C·s,Zg=0[Ω],andZL→∞[Ω](Figure 1), which resembles to a delay line [13], but it does not truly behave like it and rather may be used for generation of delayed time markers with Elmore’s times expressible in the closed form. To see this, let us produce the point voltage transmittancesTn(s)=Un/E(n=0,N¯) and the node voltage transmittancesTm′(s)=Um′/E(m=0,N-1¯), by using (8), (9), and Property 2 or Case A (10):
(30)Tn(s)=cosh[(N-n)·τ]cosh(N·τ)=(1R·C)n·(∏i=1N-n{[2·i-14·(N-n)·π]s+4R·Cmmmmm·sin2[2·i-14·(N-n)·π]})×(∏j=1N[s+4R·Cmmmmm·sin2(2·j-14·N·π)]∏j=1N[s+(4(R·C))),-1mmmmmmmmmmmn=0,N-1,-(31)TN(s)=(1R·C)N·(2)(∏j=1N[s+4R·Cmmmmmm·sin2(2·j-14·N·π)])-1,Tm′(s)=2/(R·C)s+(4/(R·C))·(Tm+Tm+1),m=0,N-1-.
Observe that if s→0, then also τ→0, and from (30) and (31) it follows that  Un/E→1  (n=1,N¯) and Um′/E→1  (m=0,N-1¯). This is in obvious physical agreement with the network behaviour in DC operating regime. The previous conclusions could be, also, formally verified by using Remark 1 (Appendix A) in calculation of Tn(n=1,N¯) and Tm′(m=0,N-1¯) for s=0. If excitation e(t) of the network with zero initial conditions is step voltage at t=0 with amplitude E, then E=E(s)=ℒ{e(t)}=E/s. From (30), (31), and Remark 1 we can, also, easily see that uk(0)=lims→∞[E·Tk(s)]=0[V] and uk(∞)=lims→0[E·Tk(s)]=E[V](k=1,N¯). In this case it can be shown that (i) the node and the point voltages are strictly monotone in t [11], so that Elmore’s definitions can be applied leading to (ii)   closed-form expressions of delay and rise times for node voltages [15]. To illustrate the point (i) let us consider the network in Figure 4 whereon capacitance voltages are denoted with xi=xi(t)(i=1,N¯). Let us introduce notation χ:=t/RC, xi(t):=ξi(χ),  x(t)=[x1(t)x2(t)⋯xN(t)]T, ξ(χ)=[ξ1(χ)ξ2(χ)⋯ξN(χ)]T (“T”-operation of matrix transposition), and e(t)=ε(χ). For the ladder in Figure 4 the following system of state-space equations can be written in χ-domain:
(32)dξ(χ)dχ=A·ξ(χ)+B·ε(χ),A:=[-310⋯01-21⋯0⋯⋯⋯⋯⋯0⋯1-210⋯01-1](N×Nreal,symmetric,tridiagonalmatrix),B:=[20⋯0]T (N×1 column vector), ξ(0)=[00⋯0]T [V] (N×1 column vector of state initial conditions).Figure 4
The common-ground, uniform, integratingRCladder with N sections.Since the real symmetricN×N regular tridiagonal matrix A is hyperdominant [17], then it is also positive definite and both similar and congruent to diagonal matrix D=diag(d1,d2,…,dN) [20], where di>0  (i=1,N¯). So, there exists an orthogonal matrix Q [20], such that -Q-1·A·Q=-QT·A·Q==D. If we introduce the coordinate transformation ξ(χ)=Q·ζ(χ), then the vector differential equation in ξ(χ)=[ξ1(χ)ξ2(χ)⋯ξN(χ)] (32) takes on the following coordinate decoupledform:
(33)dζ(χ)dχ=Q-1·A·Q·ζ(χ)+Q-1·B·ε(χ)=-D·ζ(χ)+Q-1·B·ε(χ),ζ(0)=[00⋯0]T (N×1 column vector of the “transformed” state-space initial conditions).ForN=8, the matrices Q and D with entries rounded up to 3 decimal places are being obtained as(34)Q=[-0.4970.478-0.4410.386-0.3170.235-0.1450.0490.478-0.3170.0490.235-0.4410.497-0.3860.145-0.4410.0490.386-0.4780.1450.317-0.4970.2350.3860.235-0.478-0.0490.497-0.145-0.4410.317-0.317-0.4410.1450.4970.049-0.478-0.2350.3860.2350.4970.317-0.145-0.478-0.3860.0490.441-0.145-0.386-0.497-0.441-0.2350.0490.3170.4780.0490.1450.2350.3170.3860.4410.4780.497],(35)D=[3.961000000003.663000000003.111000000002.390000000001.610000000000.888000000000.337000000000.038].Now, if we suppose that the excitation e(t)=ε(χ) is the step voltage at t=0 with amplitude E=1 [V], then the unique solution of the vector differential equation (32) is produced in following form:
(36)ξ(χ)=Q·[1-e-d1·χd10⋯001-e-d2·χd2⋯00⋯⋱00⋯01-e-dN·χdN]·Q-1·B,wherefrom(asitwasexpected)mmmmmmmmmmmmmmweobtainthatx(∞)=ξ(∞)=-A-1·B=[11⋯1]T[V](N×1columnvector).
The point voltages are obtained according to relations uj(χ)=[ξj(χ)+ξj+1(χ)]/2(j=1,N-1¯), and finally, for N=8, we can produce by using (36) the closed form solutions of point and node voltages (in [V]), as “ξ” and “u” functions of normalized (i.e. dimensionless) “time” χ=t/RC:
(37)ξ1(χ)=1-0.125·e-3.961·χ-0.125·e-3.663·χ-0.125·e-3.111·χ-0.125·e-2.390·χ-0.125·e-1.610·χ-0.125·e-0.888·χ-0.125·e-0.337·χ-0.125·e-0.038·χ,u1(χ)=1-0.002·e-3.961·χ-0.021·e-3.663·χ-0.555·e-3.111·χ-0.100·e-2.390·χ-0.149·e-1.610·χ-0.194·e-0.888·χ-0.229·e-0.337·χ-0.247·e-0.038·χ,ξ2(χ)=1+0.120·e-3.961·χ+0.083·e-3.663·χ+0.0139·e-3.111·χ-0.076·e-2.390·χ-0.174·e-1.610·χ-0.264·e-0.888·χ-0.333·e-0.337·χ-0.370·e-0.038·χ,u2(χ)=1+0.005·e-3.961·χ+0.035·e-3.663·χ+0.061·e-3.111·χ+0.039·e-2.390·χ-0.0584·e-1.610·χ-0.216·e-0.888·χ-0.380·e-0.337·χ-0.485-e-0.038·χ,ξ3(χ)=1-0.110·e-3.961·χ-0.0128·e-3.663·χ+0.109·e-3.111·χ+0.155·e-2.390·χ+0.0572·e-1.610·χ-0.168·e-0.888·χ-0.428·e-0.337·χ-0.601·e-0.038·χ,u3(χ)=1-0.006·e-3.961·χ-0.0372·e-3.663·χ-0.013·e-3.111·χ+0.085·e-2.390·χ+0.126·e-1.610·χ-0.045·e-0.888·χ-0.404·e-0.337·χ-0.705·e-0.038·χ,ξ4(χ)=1+0.097·e-3.961·χ-0.0616·e-3.663·χ-0.135·e-3.111·χ+0.016·e-2.390·χ+0.196·e-1.610·χ+0.0770·e-0.888·χ-0.380·e-0.337·χ-0.809·e-0.038·χ,u4(χ)=1+0.009·e-3.961·χ+0.026·e-3.663·χ-0.047·e-3.111·χ-0.072·e-2.390·χ+0.107·e-1.610·χ+0.165·e-0.888·χ-0.291·e-0.337·χ-0.897·e-0.038·χ,ξ5(χ)=1-0.0780·e-3.961·χ+0.115·e-3.663·χ+0.041·e-3.111·χ-0.160·e-2.390·χ+0.0193·e-1.610·χ+0.253·e-0.888·χ-0.203·e-0.337·χ-0.986·e-0.038·χ,u5(χ)=1-0.009·e-3.961·χ-0.004·e-3.663·χ+0.065·e-3.111·χ-0.056·e-2.390·χ-0.084·e-1.610·χ+0.229·e-0.888·χ-0.080·e-0.337·χ-1.055·e-0.038·χ,ξ6(χ)=1+0.059·e-3.961·χ-0.123·e-3.663·χ+0.0890·e-3.111·χ+0.0470·e-2.390·χ-0.188·e-1.610·χ+0.205·e-0.888·χ+0.042·e-0.337·χ-1.125·e-0.038·χ,u6(χ)=1+0.011·e-3.961·χ-0.011·e-3.663·χ-0.026·e-3.111·χ+0.094·e-2.390·χ-0.140·e-1.610·χ+0.089·e-0.888·χ+0.157·e-0.337·χ-1.172·e-0.038·χ,ξ7(χ)=1-0.036·e-3.961·χ+0.101·e-3.663·χ-0.141·e-3.111·χ+0.142·e-2.390·χ-0.0929·e-1.610·χ-0.0260·e-0.888·χ+0.273·e-0.337·χ-1.220·e-0.038·χ,u7(χ)=1-0.012·e-3.961·χ+0.031·e-3.663·χ-0.037·e-3.111·χ+0.020·e-2.390·χ+0.029·e-1.610·χ-0.130·e-0.888·χ+0.342·e-0.337·χ-1.245·e-0.038·χ,ξ8(χ)=71+0.012·e-3.961·χ-0.0380·e-3.663·χ+0.067·e-3.111·χ-0.102·e-2.390·χ+0.152·e-1.610·χ-0.234·e-0.888·χ+0.412·e-0.337·χ-1.270·e-0.038·χ.All “ξ” and “u” functions of χ are depicted in Figures 5 and 6 in the “time” interval χ∈[0,120].Figure 5
ξ
(
χ
) and u(χ) functions for the specified nodes and points of ladder in Figure 4.Figure 6
ξ
(
χ
) and u(χ)  functions for the specified nodes and points of ladder in Figure 4.Assume thatΩ0=4/(R·C) is normalizing frequency and recast (30) and (31) in the following form by using Remark 1 (Appendix A):
(38)Tn(s)=(∏i=1N-n{1+sΩ0·sin2[((2·i-1)/(4·(N-n)))·π]})×(∏j=1N[1+sΩ0·sin2(((2·j-1)/(4·N))·π)])-1,mmmmmmmmmmmmmmmmmmmmmn=0,N-1¯,(39)TN(s)=(1)×(∏j=1N[1+sΩ0·sin2(((2·j-1)/(4·N))·π)])-1,Tm′(s)=Tm+Tm+12·(1+(s/Ω0)),m=0,N-1.¯The coefficientsa1(N,n),a2(N,n), b1(N), and b2(N)  (27) necessary for calculation of TD(N,n) and TR(N,n) according to Elmore's definitions (28) are obtained for Tn(s)(n=1,N-1¯) (38) in the following form (see Corollary 3 in Appendix A):
(40)a1(N,n)=1Ω0m·∑i=1N-n1sin2[((2·i-1)/(4·(N-n)))·π]=R·C2·(N-n)2,b1(N)=1Ω0m·∑j=1N1sin2(((2·j-1)/(4·N))·π)=R·C2·N2,(41)a2(N,n)=12·Ω02m·{∑i=1N-n1sin4[((2·i-1)(4·(N-n)))·π][∑i=1N-n1sin2[((2·i-1)/(4·(N-n)))·π]]2mm-∑i=1N-n1sin4[((2·i-1)/(4·(N-n)))·π]∑i=1N-n1sin4[((2·i-1)(4·(N-n)))·π]}=(R·C)224·(N-n)2·[(N-n)2-1],(42)b2(N)=12·Ω02m·{∑i=1N-n1sin4[((2·i-1)(4·(N-n)))·π][∑i=1N1sin2(((2·i-1)/(4·N))·π)]2mmm-∑i=1N1sin4(((2·i-1)/(4·N))·π)[∑i=1N1sin2(((2·i-1)(4·N))·π)]}=(R·C)224·N2·(N2-1),
provided that  a1(N,N)=a2(N,N)=0, and finaly, Elmore’s delay and rise times for point voltages un(n=1,N¯) (Figure 4) are calculated in the closed form according to the relations
(43)TD(N,n)=b1(N)-a1(N,n)TD(N,n)=R·C2·[N2-(N-n)2],(44)TR(N,n)=2·π·{b12(N)-a12(N,n)+2·[a2(N,n)-b2(N)]}=R·C·π6m·[N2-(N-n)2]·{2·[N2+(N-n)2]+1},
whereas Elmore’s delay times TD′(N,m) and rise times TR′(N,m) for node voltages um′(m=0,N-1¯) (Figure 1) are calculated by using (38) and (39):
(45)TD′(N,m)=b1′(N)-a1′(N,m),TR′(N,m)=2·π·{b1′2(N)-a1′2(N,m)+2·[a2′(N,m)-b2′(N)]},where(46)a1′(N,m)=R·C4·[(N-m)2+(N-m-1)2],b1′(N)=1Ω0m·[1+∑j=1N1sin2(((2·j-1)/(4·N))·π)]=R·C4·(2N2+1),(47)a2′(N,m)=(R·C)248m×{(N-m)2·[(N-m)2-1]mm+(N-m-1)2·[(N-m-1)2-1]},(48)b2′(N)=12·Ω02m·{[1+∑i=1N1sin2(((2·i-1)/(4·N))·π)]2mmm-1-∑i=1N1sin4(((2·i-1)/(4·N))·π)[1+∑i=1N1sin2(((2·i-1)/(4·N))·π)]2[1+∑i=1N1sin2(((2·i-1)/(4·N))·π)]}=(R·C)224·N2·(N2+2).
From (43) it follows that TD(N,N)∝N2 and from (44) that TR(N,n)>2·TD(N,n)(n=1,N¯), which does not qualify this type of ladder for using as pulse delay line in its own right. If in the ladder on Figure 4N=6, R=50[Ω], and C=2 [nF] and e(t) is periodic pulse train with amplitude 10 [V], duty-cycle 50%, and period 100 [μs], then excitation e(t) and node voltages x2(t)=u1′ (t), x5(t)=u4′(t), and point voltage u3(t)—obtained by pspicesimulation, are depicted on Figure 7 in interval t∈[0,120][μs].Figure 7
Some of thepspice simulation results for six segment integrating ladder in Figure 4 with R=50[Ω],C=2  [nF], and pulse-train excitation e(t) with amplitude 10 [V], period T=100[μs], and duty-cycle 50%.Finally, consider that the ladder in Figure8 is analogue to the ladder in Figure 1. Assume α=R/L=1/(R·C) and
(49)Zg=Z1=12·R·L·sR+L·s=R2·ss+α,Z2=1C·s,ZL=Z1+11+R·C·s=R2·s+2αs+α(Zg+ZL=R).Figure 8
Common-ground, uniform ladder, which may play the role of artificial delay line (ADL) for sufficiently largeN.In this case, from (7) we obtain
(50)Zc=Z1·(Z1+2·Z2)=R2·s+2αs+α=ZL,sinh(τ)=ZcZ2=s·(s+2α)2α·(s+α),cosh(τ)=1+Z1Z2=(s+α)2+α22α·(s+α).
From (50) it is found that e-τ=α/(s+α), and from (9) it follows that
(51)Tm′(s)=Um′E=ZL-Z1ZL+Z1·e-m·τ=αs+α·e-m·τ=(αs+α)m+1=1(1+(s/α))m+1(m=0,N-2¯),(52)T(s)=UE=UN-2′E·UUN-2′=TN-2′·αs+α=1(1+(s/α))N-1·αs+α=1(1+(s/α))N.Relation (52) could have been produced in a less formal way by observing that the ladder in Figure 8 is a constant resistance network when R/L=1/(R·C), since the input impedances (denoted by dashed arrows) seen from the pertinent pairs of nodes are all equal to R. If the ladder has no initial conditions and if it is excited by unit-step voltage e(t)att=0{E=E(s)=ℒ[e(t)]=1/s}, then for overall step response u(t)=ℒ-1[U(s)] it holds that (a) u(0)=lims→∞s·U(s)=0[V], (b) u(∞)=lims→0s·U(s)=1[V], and (c) u(t) is a monotone increasing function in t (i.e., u(t) has no overshoots) since we have
(53)0≤u(t)=ℒ-1[(αs+α)N·1s]=1-e-α·t·∑k=0N-1(α·t)kk!≤1,du(t)dt=αN·e-α·t·tN-1(N-1)!>0,t≥0.
From (52) we obtain the “a” and “b” coefficients necessary for calculation of Elmore’s delay time TD and rise time TR(B.5) for the network in Figure 8:
(54)a1=a2=0,b1=Nα,b2=N·(N-1)2·α2,TD=b1=Nα=LR·N=N·L·C,TR=1α·2π·N=TD·2πN=2π·N·L·C.The procedure for determining the ladder parameters when both Elmore’s timesTD and TR are specified consists of the following three steps: (a) firstly, we calculate N=ant[2π·(TD/TR)2]+1, and then from (54) we calculate the actual rise time TRact=TD·(2π/N)1/2<TR (TD remains unchanged), (b) assume C, and (c) calculate R=(TD/N)/C and L=(TD/N)2/C. When solely Elmore’s delay time TD is specified and TR is left unspecified, we arbitrarily select some reasonably great N, so as to produce the sufficiently small TR of the output u for the unit-step input and then follow the steps (b) and (c).But in the case whenN is small and we want to realize Elmore’s delay time TD for pulse train with period Tand duty cycle δ, it can be shown that one of the following two conditions must be satisfied, in order to prevent severe distortion of transmitted “pulses”: (i)TD≤(1-δ)·T/2, when δ≥1/2, or (ii)TD≤δ·T/2, when δ<1/2. For the ladder in Figure 8 with N=20, R=10[Ω],L=0.5 [mH], and C=5[μF], excited by the pulse train e(t) with amplitude 10 [V], period T=10 [ms], and duty-cycle δ=0.6, the obtained results of pspice simulation in the time interval t∈[0,20] [ms] are depicted in Figure 9, whereon are plotted the excitation e(t), voltage u9′(t) of the 10th capacitor, and the voltage u(t) of the 20th capacitor. Elmore’s delay and rise times for voltage um-1′ of the mth capacitor (m=1,N-1¯) (Figure 8) are, respectively, m·L·C and 2π·m·L·C, according to (54). For example, Elmore’s delay time of u9′(t) is 10·L·C=0.5[ms] and of u(t) is 20·L·C=1[ms]. Both these values are slightly different from those obtained according to the classical definition of delay time and they are denoted in Figure 9 for comparison. In this case, Elmore’s rise time for u obtained from (54) is TR=π/10 [ms] ≈ 560.5 [μs], whereas the classical rise time obtained from Figure 9 is slightly different, TRc≈568.9[μs].Figure 9
Realization of Elmore’s delay timeTD=1 [ms] for pulse-train input e(t) with amplitude 10 [V], period T=10 [ms], and duty-cycle δ=0.6, by ladder in Figure 8 with N=20, R=10[Ω],L=0.5[mH],andC=5[μF]. The output is u(t).Consider now the same ladder excited by the pulse traine(t) with amplitude 10 [V], provided that the condition (i) is violated by supposing that T=2 [ms] and δ=0.6. The results of pspice simulation in the time interval t∈[0,5] [ms] are depicted in Figure 10 and obviously exhibit a severe distortion of “pulses” u being transmitted to the end of the ladder.Figure 10
Realization of Elmore’s delay timeTD=1 [ms] for pulse-train input e(t) with amplitude 10 [V], period T=2 [ms], and duty-cycle δ=0.6, by ladder in Figure 8 with N=20, R=10[Ω],L=0.5 [mH], and C=5[μF]. The output is u(t).Consider again the same ladder excited by the pulse traine(t) with amplitude 10 [V], provided that condition (ii) is violated by supposing that T=2 [ms] and δ=0.4. The results of pspice simulation are depicted in Figure 11 in the time interval t∈[0,5] [ms] and obviously exhibit a severe distortion of “pulses” u being transmitted to the end of the ladder.Figure 11
Realization of Elmore’s delay timeTD=1 [ms] for pulse-train input e(t) with amplitude 10 [V], period T=2 [ms], and duty-cycle δ=0.4, by ladder in Figure 8 with N=20, R=10[Ω],L=0.5 [mH], and  C=5[μF]. The output is u(t).And finally, consider the ladder in Figure8 which should realize Elmore’s delay time TD=5 [ms] and rise time TR=560.5[μs]. Let the excitation e(t) be the pulse-train with amplitude 10 [V], period T=20 [ms], and duty-cycle δ=0.5. To avoid a severe distortion of output pulses u(t), it is sufficient to follow the previously formulated steps (a)÷(c). So, according to the step (a) we firstly calculate N=500. Then, in the step (b) we assume that, say, C=1 [μF] and finally in the step (c) we calculate R=10 [Ω] and L=100[μH]. The time variations of e(t)and u(t) obtained by pspice simulation are depicted in Figure 12 in the time interval t∈[0,40] [ms]. Therefrom, we find the classical delay and rise times of the output voltage u(t),TDc≈ 5 [ms] and TRc≈ 575 [μs], respectively. Both these times are slightly different from the pertinent Elmore’s times (54), whose obvious advantage is that are explicitly calculable as functions of {N,R,L,C} parameters of common-ground, uniform RLC ladder (Figure 8).Figure 12
Realization of Elmore’s delay timeTD=5 [ms] and rise-time TR=560.5[μs] for pulse-train input e(t) with amplitude 10 [V], period T=20 [ms], and duty-cycle δ=0.5, by ladder in Figure 8 with N=500, R=10[Ω],L=100[μH], and C=1[μF]. The output is u(t).In the previous examples, we have investigated Elmore’s times of voltages at points and/or nodes in several characteristic types of common-ground uniformRLCladders excited, either by step or by pulse-train emfs. In the most general case, consider the realization of physical delay timeTD for arbitrary excitatione(t) (i.e., true delay time), by using the RLCladder in Figure 8 (true delay line). If we presume for this network that N→∞, then from (52) and (54) it readily follows that
(55)limN→∞T(s)=limN→∞1(1+(s/α))N=limN→∞[(1+s·TDN)N/(s·TD)]-s·TD=e-s·TD,
so that the overall network response becomes u(t)=ℒ-1[U(s)]=ℒ-1[e-s·TD·E(s)]=e(t-TD) (i.e., u(t) is the exact replica of the excitation e(t) delayed by the time TD). Let us firstly introduce the notation
(56)z:=s·TDN=(σ+j·ω)·TDN=σ·TDN+j·ω·TDN,Re{z}:=σ·TDN,Im{z}:=ω·TDN,f(z):=(1+z)1/z
and then investigate the conditions under which the function f(z) becomes close to the number e. It is obvious from (55) that these conditions relate to certain restrictions in the span of N, σ, and ω. Observe that f(z)→e when z→0. After some manipulation, from (56) we easily obtain
(57)|f(z)|=e(Im{z}/(Re2{z}+Im2{z}))·atan[Im{z}/(1+Re{z})]m·([1+Re{z}]2mm+Im2{z})Re{z}/(2·[Re2{z}+Im2{z}]),Arg[f(z)]=Re{z}Re2{z}+Im2{z}·atan[Im{z}1+Re{z}]-Im(z)2·[Re2{z}+Im2{z}]·ln([1+Re{z}]2+Im2{z}),
where Arg[f(z)] is expressed in radians and should be multiplyed by 180/π to be expressed in [deg].The function|f(z)| is depicted in Figure 13. From the set of its associated numerical data it can be seen that for Re{z}=0_ we have |f(z)|=2.7182… when 0≤Im{z}≤2π·10-3 and |f(z)|→e when Im{z}→0. In other words, for |f(z)| to be close to e, it is sufficient inthis case to provide that 2π·fmax·TD/N≤2π·10-3, where fmax is the maximum frequency in the spectrum of the signal e(t) being transmitted through the RLC ladder in Figure 8 having almost true time delay equal to Elmore’s delay time TD, or N≥fmax·[TD], where [TD] is delay time TD expressed in [ms].Figure 13
The plot of function|f(z)| in the z-region of interest.The functionArg[f(z)] [deg] is depicted in Figure 14. From the set of its associated numerical data it can be seen that for Re{z}=0_ we have -0.1799964<Arg[f(z)] [deg] ≤0when0≤Im{z}≤2π·10-3andArg[f(z)]→0whenIm{z}→0.For  Arg[f(z)]≤0 to be close to 0, say to be |Arg[f(z)]|<0.18[deg], it is sufficient inthis case also to provide that 2π·fmax·TD/N≤2π·10-3orN≥fmax·[TD].Figure 14
The plot of functionArg[f(z)] in the z-region of interest.For properly selectedN (i.e., TD/N≤1/(1000·fmax)), the ladder parameters are found according to the previously formulated steps (b)÷(c): arbitrarily select C and calculate R=(TD/N)/C and L=(TD/N)2/C.As a final example, consider the ladder in Figure8 with parameters: N=500, C=1 [μF], R=10[Ω], L=100[μH], and the Elmore’s delay time TD=N·(L/R)=N·L·C=5 [ms]. Since condition fmax·TD/N≤10-3 is satisfied for fmax≤100 [Hz], then this ladder may be used as true delay line for every input signal e(t) with maximum spectral frequency up to fmax, with output u(t) being almost the replica of e(t) with no attenuation and virtually constant delay TD equal to Elmore’s delay time. The results for this ladder excited by various input signals e(t), produced by using of PSPICE simulation, are depicted in Figures 15, 16, 17, 18 and 19. There on it may be noticed that the overall output voltage of this ladder as of artificial delay-line (ADL) is represented by u(t)≈e(t-TD), with absolute error aer(t)=u(t)-e(t-TD).Figure 15
PSPICE verification of delay timeTD=5 [ms] for ±10 [V]/20 [Hz] sine-wave input e(t).Figure 16
PSPICE verification of delay timeTD=5 [ms] for input wave e(t) with period T=0.1 [s],  linear frequency chirp 0÷100 [Hz], and linearly increasing amplitude 0÷10 [V].Figure 17
PSPICE results for transfer with delayTD=5 [ms] of CAM input wave e(t) with carrier ±10 [V]/100 [Hz], modulation index m=0.5, and the intelligence frequency 50 [Hz] within ADL in Figure 8.Figure 18
PSPICE results for transfer with delayTD=5 [ms] of FM input wave e(t) with carrier ±10 [V]/50 [Hz], modulation index m=2 and intelligence frequency 25 [Hz] within ADL in Figure 8.Figure 19
PSPICE results for transfer with delayTD=5 [ms] of input wave e(t) with period T=0.1 [s], linear frequency chirp 0÷100 [Hz], and linearly increasing amplitude 0÷10 [V] within ADL in Figure 8.In the paper [12] it is proved that by using distortionless transmission line with resistive load equal to the line characteristic impedance L′/C′  (L′ and C′ are per-unit-length inductance and capacitance of line, resp.), even a relativelly small signal delay TD cannot be realistically produced. For example, signal delay TD=5 [ms] can be realized with lossless line having the constant parameters L′=1.5[μH/m],C′=18 [pF/m], and load resistance (L′/C′)1/2=500/3 [Ω], at the distance d=TD·(L′·C′)-1/2≈ 962.25 [km] from the line sending end, but when distortionless line is lossy and terminated with characteristic impedance (L′/C′)1/2, then besides the time delay d·(L′·C′)1/2 at distance d, the signal attenuation factor exp[d·(R′·G′)] (R′ and G′ are the per-unit-length resistance and conductance of the lossy line, resp.) must be also taken into account [18]. Realization of pulse delay TD=5 [ms] with lumped RLCnetwork in Figure 8 is verified in Figure 12 and seems to be the more comfortable approach than using lengthy distributed parameter networks. In general, realization of any time delay TD for signal with maximum spectral frequency fmax can be accomplished by using the uniform ladder in Figure 8 with length N selected according to the condition TD/N≤1/(1000·fmax). The ladder parameters are calculated according to relations R=(TD/N)/C and L=(TD/N)2/C, where C may be selected arbitrarily.Finally, in Figures20 and 21 the results of PSPICE AC-analysis are depicted with 1 [V] amplitude AC-input in the entire frequency range [0,100] [Hz] for network in Figure 8 with parameters N=500, C=1[μF],R=10[Ω], and L=100[μH]. In Figure 20 we see that the amplitude-frequency characteristics of the network are almost flat, and in Figure 21 we see that its phase-frequency characteristics are virtually linear with slope -1.8 [deg/Hz]. Hence we conclude that the network in Figure 8 may satisfactorily realize the time delay TD for any input signal e(t) with spectrum in range 0÷100 [Hz].Figure 20
The amplitude-frequency characteristics of the considered ladder in the frequency rangef∈[0,100] [Hz].Figure 21
The phase-frequency characteristics of the considered ladder in the frequency rangef∈[0,100][Hz].


## 4. Conclusions

In the paper, are derived general, closed-form expressions for network functions of common-ground, uniform, and passive ladders with complex double terminations are derived, making distinction in analysis between the signal transfer to ladder nodes and to its points. The obtained results are then simplified for ladders with seven specific pairs of complex double terminations. Elmore’s delay and rise times calculated for specific, important types ofRLCladders indicated slight deviation from delay and rise times values obtained according to their classical definitions.For the common-ground, integratingRC ladder with step input, Elmore’s delay  and rise times are produced in closed form, both for ladder nodes and ladder points. It has been shown that this type of ladder could not be recommended as pulse delay line in its own right, since its Elmore’s rise-times of point voltages are not less than the twice of their delay times.And finally, in this paper we have proposed a specific type of common-ground, uniformRLC ladder amenable for application as delay line, both for pulsed and analog input signals. For this type of ladder, Elmore’s delay  and rise times relating to the node voltages are produced in the closed form, which offers a possibility for realization of (a) the pulse delay line with arbitrarily specified Elmore’s delay and rise times and (b) the true delay line, both for pulsed and analog input signals, with arbitrarily selected delay time. For both cases (a) and (b), in the paper a procedure for the determination of ladder length and calculation of all its RLCparameters is specified. Recall that the ladder network topology is preferable one in the network synthesis, since it has very low sensitivity with respect to variations of RLCparameters. The obtained results are illustrated with several practical examples and are verified through pspice simulation.



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*Source: 691740-2013-07-30.xml*