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

1 Department of Physics & Electrical Engineering, School of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia 2Department of General Electrical Engineering, School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia 3 Department of Telecommunications, School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia


Introduction
Modeling of digital MOS circuits by RC networks has become a well-accepted practice for estimating delays [1][2][3][4][5][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 RC tree 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 RC networks are being referred to as "RC meshes".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 RC meshes.Simple closedform 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 RC trees [6].
Elmore's delay is an extremely popular timingperfomance metric which is used at all levels of electronic circuit design automation, particularly for RC tree analysis.The widespread usage of this metric is mainly attributable to its property of being a simple analytical function of Mathematical Problems in Engineering 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 commonground, 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-ground RLC ladder with (a) a relation between its parameters resembling to analogous relation between perunit-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, RC open-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 RC ladder 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, commonground RLC 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 Recall that in network synthesis the ladder topology is a preferable one, since it has very low sensitivity to variations of RLC parameters [16,17].
By substituting  0 and  1 in (4) and bearing in mind that we finally obtain from ( 2) and ( 4) for  = 0,  that Since uniform ladder sections are electrically reciprocal, then their characteristic impedance   and the quantity   ⋅ sinh() can easily be produced in the form For the ladder depicted in Figure 1, the complete set of the network immittance and voltage/current transmittance functions can be produced, by using the relations (2), (6), and (7): Since  1 =   ⋅ (sinh()/[1 + cosh()]) =   ⋅ tanh(/2), then the voltages and currents of impedances  2 connecting the middle nodes of ladder sections (  ) to common-node  (Figure 1) are given as Consider now the particular selection of {  ,  } to investigate the generation of (i) finite length, frequencyselecting 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   and   are rational positive real functions in s, so that they are realizable by passive transformerless RLC networks [17].All network functions in (8) and ( 9) are real rational functions in s, except   /  , which must be rational, positive real function in s, since it is the input immittance of RLC network.(8) and ( 9) it follows,
It is clear that relations ( 7)-( 9) offer many other possibilities for the selection of   ,   ,  1 , and  2 that lead to generation of versatile ladders realizing different types of frequency selective networks.An interesting one seems to be the common-ground RLC ladder 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.

Elmore's Delay and Rise Times for Selected Types of Common-Ground, Uniform RLC Ladders
Consider the ladder in Figure 1 having  1 =  +  ⋅ ,  2 = 1/ 2 =  +  ⋅ , and   =   = (/) 1/2 .Suppose that it holds the relation  := / = /, similar to the one associated with per-unit-length parameters of distortionless transmission line [18,19].Let us define the auxiliary parameter  := 1/( ⋅ ) 1/2 , and let us recast the network voltage transmittances describing the transfer of emf E to the ladder points with voltages  0 ,  1 , . . .,   (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: and then by using ( 8) and the Properties 2 and 3, let us express the point voltage transmittances   () =   /E ( = 0,  − 1) and   () =   / in the following form: 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  = 0 with the step-voltage e with amplitude  [V], we must do the following.
(i) Firstly, check that the point voltages   =   (t) ( = 0, ) 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  = 6 sections that E = 10 [V], then its voltage step-responses   =   ()( = 0, 6) and the node response   0 as well, obtained through pspice simulation and depicted in Figures 2 and 3  reveal that Elmore's definitions cannot be applied for responses u 0 and   0 , since the occurrence of overshoots, whereas for all other point voltages   =   (t) ( = 1, 6), those definitions are applicable.
(ii) Secondly, calculate the coefficients 2 ) Mathematical Problems in Engineering The times calculated according to Elmore's definitions.
All data are related to the ladder with zero initial conditions.
:=   (, )  0 (, ) [ = 0, (2 − 2 + 1) ;  = 0, ( − 1)] , Calculation of coefficients  0 (N, n), 1 (N, n),  2 (N, n),  0 (N),  1 (N), and  2 (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 Another interesting ladder in Figure 1 is the one with , and  := / = /.Again, let it be  := 1/( ⋅ ) 1/2 .By using of ( 8) and Properties 2 and 3, the point voltage transmittances   () =   /E ( = 0,  − 1) and   () =   /E are obtained as follows: 2 ) 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 {, , , , , } 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 with 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 transmittances   () =   /E ( = 0, ) and the node voltage transmittances    () =    /( = 0,  − 1), by using ( 8), (9), and Property 2 or Case A (10): Observe that if  → 0, then also  → 0, and from (30) and (31) it follows that   / → 1 ( = 1, ) and    / → 1 ( = 0,  − 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   ( = 1, ) and    ( = 0,  − 1) for  = 0.If excitation e(t) of the network with zero initial conditions is step voltage at  = 0 with amplitude , then  = () = L{()} = /.From (30), (31), and Remark 1 we can, also, easily see that In this case it can be shown that (i) the node and the point voltages are strictly monotone in  [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  For the ladder in Figure 4 the following system of state-space equations can be written in -domain: Since the real symmetric × regular tridiagonal matrix A is hyperdominant [17], then it is also positive definite and both similar and congruent to diagonal matrix D = diag( 1 ,  2 , . . .,   ) [20], where   > 0 ( = 1, ).So, there exists an orthogonal matrix Q [20], such that If we introduce the coordinate transformation () = Q ⋅ (), then the vector differential equation in () = [ 1 ()  2 () ⋅ ⋅ ⋅   ()] (32) takes on the following coordinate decoupled form: d () T ( × 1 column vector of the "transformed" state-space initial conditions).
For  = 8, the matrices Q and D with entries rounded up to 3 decimal places are being obtained as Now, if we suppose that the excitation () = () is the step voltage at  = 0 with amplitude  = 1 [V], then the unique solution of the vector differential equation (32) is produced in following form: we obtain that The point voltages are obtained according to relations   () = [  () +  +1 ()]/2 ( = 1,  − 1), and finally, for  = 8, we can produce by using (36) the closed form solutions of point and node voltages (in [V]), as "" and "" functions of normalized (i.e., dimensionless) "time"  = /: (37) All "" and "" functions of  are depicted in Figures 5  and 6 in the "time" interval  ∈ [0, 120].
Assume that Ω 0 = 4/( ⋅ ) is normalizing frequency and recast (30) and (31) in the following form by using Remark 1 (Appendix A):   () The coefficients  1 (, ),  2 (, ),  1 (), and  2 () (27) necessary for calculation of   (, ) and   (, ) according to Elmore's definitions (28) are obtained for   () ( = 1,  − 1) (38) in the following form (see Corollary 3 in Appendix A): whereas Elmore's delay times T   (N, m) and rise times T   (N, m) for node voltages u   ( = 0,  − 1) (Figure 1) are calculated by using ( 38) and (39): where From (43) it follows that   (N, N) ∝  2 and from (44) that  x 2 (t) = u 1  (t) u 3 (t) x 5 (t) = u 4  (t)  Finally, consider that the ladder in Figure 8 is analogue to the ladder in Figure 1.Assume  = / = 1/( ⋅ ) and In this case, from (7) we obtain Mathematical Problems in Engineering From (50) it is found that e − = /(s + ), and from (9) it follows that 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 / = 1/( ⋅ ), 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 and (c) u(t) is a monotone increasing function in t (i.e., u(t) has no overshoots) since we have From ( 52) we obtain the "a" and "b" coefficients necessary for calculation of Elmore's delay time   and rise time   (B.5) for the network in Figure 8: The procedure for determining the ladder parameters when both Elmore's times   and   are specified consists of the following three steps: (a) firstly, we calculate  = ant[2 ⋅ (  /  ) 2 ] + 1, and then from (54) we calculate the actual rise time  Ract =   ⋅ (2/) 1/2 <   (  remains unchanged), (b) assume C, and (c) calculate  = (  /)/ and  = (  /) 2 /C.When solely Elmore's delay time   is specified and   is left unspecified, we arbitrarily select some reasonably great N, so as to produce the sufficiently small   of the output u for the unit-step input and then follow the steps (b) and (c).
But in the case when N is small and we want to realize Elmore's delay time   for pulse train with period T and duty cycle , it can be shown that one of the following two conditions must be satisfied, in order to prevent severe distortion of transmitted "pulses": (i)   ≤ (1 − ) ⋅ /2, when  ≥ 1/2, or (ii)   ≤  ⋅ /2, when  < 1/2.For the ladder in Figure 8  Consider now the same ladder excited by the pulse train e(t) with amplitude 10 [V], provided that the condition (i) is violated by supposing that  = 2 [ms] and  = 0.6.The results of pspice simulation in the time interval  ∈ [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., 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 {, , , } parameters of common-ground, uniform RLC ladder (Figure 8).
In the previous examples, we have investigated Elmore's times of voltages at points and/or nodes in several characteristic types of common-ground uniform RLC ladders excited, either by step or by pulse-train emfs.In the most general case, consider the realization of physical delay time   for arbitrary excitation e(t) (i.e., true delay time), by using the RLC ladder in Figure 8 (true delay line).If we presume for this network that  → ∞, then from (52) and (54) it readily follows that lim so that the overall network response becomes , where Arg[f (z)] is expressed in radians and should be multiplyed by 180/ to be expressed in [deg].
The function |()| is depicted in Figure 13.From the set of its associated numerical data it can be seen that for Re{} = 0 we have |()| = 2.7182 . . .when 0 ≤ Im{} ≤ 2 ⋅ 10 −3 and |()| →  when Im{} → 0. In other words, for |()| to be close to e, it is sufficient in this case to provide that 2 ⋅  max ⋅   / ≤ 2 ⋅ 10 −3 , where  max is the maximum frequency in the spectrum of the signal e(t) being transmitted through the RLC ladder in Figure 8   In the paper [12] it is proved that by using distortionless transmission line with resistive load equal to the   ≈ 962.25 [km] from the line sending end, but when distortionless line is lossy and terminated with characteristic impedance (  /  ) 1/2 , then besides the time delay  ⋅ (  ⋅   ) 1/2 at distance d, the signal attenuation factor exp[ ⋅ (  ⋅   )] (  and   are the per-unit-length resistance and conductance of the lossy line, resp.) must be also taken into account [18].Realization of pulse delay   = 5 [ms] with lumped RLC network 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   for signal with maximum spectral frequency  max can be accomplished by using the uniform ladder in Figure 8 with length N selected according to the condition   / ≤ 1/(1000 ⋅  max ).The ladder parameters are calculated according to relations  = (  /)/ and  = (  /) 2 /, where  may be selected arbitrarily.

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 of RLC ladders indicated slight deviation from delay and rise times values obtained according to their classical definitions.
For the common-ground, integrating RC 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, uniform RLC 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 RLC parameters 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 RLC parameters.The obtained results are illustrated with several practical examples and are verified through pspice simulation.systems), applicable only when its step response is monotonic (nonovershooting), originate from Elmore [7]: In many practical situations it has been noticed that, when the unit-step response of network, or system, hasaovershoot less than 5% of its steady-state unity value, (B.2) will still be holding and the obtained results will be in close agreement with those produced by using the conventional definitions of delay and rise times [7].Since  ] () = L{ℎ ] ()}, then from Elmore's definitions (B.2) it is obtained that  ] () = ∫

Figure 1 :
Figure 1: The general, common-ground, uniform ladder with N sections.

Figure 2 : 6 Figure 3 :
Figure 2: A set of voltage responses at the selected points of the considered ladder (and at the  0 node).

Figure 4 :
Figure 4: The common-ground, uniform, integrating RC ladder with N sections.

Figure 5 :Figure 6 :
Figure 5: () and () functions for the specified nodes and points of ladder in Figure 4.

Figure 8 :
Figure 8: Common-ground, uniform ladder, which may play the role of artificial delay line (ADL) for sufficiently large N.
in the time interval  ∈ [0, 40] [ms].Therefrom, we find the classical delay and rise times of the output voltage u(t),   ≈ 5 [ms] and   ≈ 575 [s]

Figure 14 :Figure 15 :
Figure 14: The plot of function Arg[()] in the z-region of interest.

Table 1 :
Delay and rise times of the point voltages in the considered common-ground, uniform RLC ladder.
and Elmore's times, for any given set of input parameters {, , , , , }, provided that the condition / = / is satisfied.
Figure 13: The plot of function |()| in the -region of interest.(z) becomes close to the number e.It is obvious from (55) that these conditions relate to certain restrictions in the span of N, , and .Observe that () →  when  → 0. After some manipulation, from (56) we easily obtain      ()     =  (Im{}/(Re 2 {}+Im 2 {}))⋅ tan[Im{}/(1+Re{})] having almost true time delay equal to Elmore's delay time   , or  ≥  max ⋅ [  ], where [  ] is delay time   expressed in [ms].The function Arg[()] [deg] is depicted in Figure14.From the set of its associated numerical data it can be seen