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Abstract BbJs_”õ ø_Éõíù¼ƒbV¡N RLC cÕÚ˜Ÿƒbí‡ú_sŸjá[b, ©_õísŸj á[b, ªà KVL F[ýíä³HpÀít l|V; Bb?T|püít Vl¥¼à@íôbvÈ ,¯vÈ |×¾· · ·; â_Òí!‹)ø, BbT|íj¶ªcà s_”õV¡N, y?üí,lÍ$í¡bWe propose two-pole one-zero second-order approx-imations for transfer function in RLC trees. The ap-proximation matches the first three moments of the original transfer function. Fundamental loop matrix formulation of circuit equations allows efficient and si-multaneous computations of moments, and thus ap-proximations, at all the nodes. Explicit formulas for step response parameters such as delay time, rise time, overshoot, etc., are given. Simulations show improved accuracy over existing second-order approximations.
I INTRODUCTION
RLC trees are useful in modelling interconnect lines in VLSI circuits [2]. Step response parameters, such as de-lay time, at capacitor nodes in the tree are important for routing and wire sizing optimization. Low-order approximation for the corresponding transfer functions are required for estimating the parameters without solv-ing the complete RLC tree equations.
Ismail, et.al. [2] proposed a second-order approx-imation that matches the first two moments of the original transfer function. The second-order transfer function, with unit DC gain, is completely character-ized by the damping ratio ζ and undamped natural frequency ωn. Estimates of various step response
pa-rameters such as delay time, rise time, overshoot, etc., are proposed. In an effort to improve the accuracy of the second-order approximation, we propose a more general two-pole and one-zero second-order approxima-tion. The three parameters of transfer function are de-termined by matching the first three moments of the original transfer function. Simulation results show that the additional degree of freedom in the second-order transfer function indeed improves the accuracy of the approximation, in term of frequency response and step response, and thus also improves the accuracy of
esti-mates of step response parameters.
The fundamental loop matrix formulation of circuit equations is ideal for RLC trees [1]. The matrix for-mulation is simple. It involves only diagonal matrices and matrices with zeros and ones. The computation of moment matrices of the transfer matrix from source to all the capacitor voltage is simple and very efficient. Since the moments for each capacitor node are com-puted simultaneously, the proposed method constructs approximate for every source-to-node transfer function. The paper is organized as follows. In Section II, we compute the transfer matrix of the RLC tree using fundamental loop matrix. In Section III, we give a recursive formula for computing the moment matrices of the transfer matrix. The formula for damping ratio, undamped natural frequency, and zero location of the second-order approximation are given in Section IV. Explicit formulas for delay time, rise time, overshoot etc, are given in Section V. Simulation examples and comparisons are given in Section VI. Finally, Section VII is a brief conclusion.
II TRANSFER MATRIX OF RLC TREE
For an RLC tree, the tree graph is uniquely defined and it consists of the voltage source branch and the R and L branches. The capacitor branches are the links that defines the fundamental loops [1]. KVL equations of these fundamental loops and KCL equations at the capacitor nodes completely specify the interconnection of the tree.
To write the circuit equations, let’s consider the simple RLC tree shown in Figure 1, where the input voltage source is vs, the tree branch voltages are vt1, vt2, and vt3, and the link voltages are vl1, vl2, and vl3, respectively; the branch currents it1, it2, it3, and the link currents il1, il2, and il3 are defined accordingly in associated reference directions [1]. Note that we treat each series RL connection as a single branch. The KVL equations for the three fundamental loops are
vt1+ vl1= vs (loop 1) vt1+ vt2+ vl2 = vs (loop 2) vt1+ vt3+ vl3 = vs (loop 3)
In matrix form, the equations become 1 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 | {z } B vt vl = 1 1 1 vs (1) where vt = [vt1 vt2 vt3] T, v l= [vl1 vl2 vl3] T, and B ∈
R3×6 is the fundamental loop matrix associated with the tree. The matrix B is partitioned as
B =h F ... I i (2)
where F, I ∈ R3×3. The ijth entry of F are either 1 or
0 depending on whether the jth tree branch is in loop i or not. Thus (1) becomes
F vt+ vl= 1 1 1 vs (3)
The KCL equations at capacitor nodes are it1 = il1+ il2+ il3 it2 = il2
it3 = il3
In matrix form, the equations become
it= FTil (4)
where it = [it1 it2 it3]
T and i
l = [il1 il2 il3]
T. The
branch equations for the tree branches and the link branches are respectively
vti = Riiti+ Li diti dt i = 1, 2, 3 (5) and ili = Ci dvli dt i = 1, 2, 3 (6)
The equation (3), (4), (5), and (6) thus completely describe the RLC tree in Figure 1.
For a general RLC tree with n sections, the equa-tions are similar. The vectors it, il, vt, and vl now
all have n components; the fundamental loop matrix B ∈ Rn×2nis partitioned similarly as
B =h F ... I i
where the ikth element, fik, of F ∈ Rn×n is
fik=
1 if the kth tree branch is in the ith loop
0 if the kth tree branch is not in the ith loop
The KVL equations and KCL equations in matrix form are respectively
F vt+ vl= Evs (7)
and
FTi
l= it (8)
where E = [1 · · · 1]T ∈ Rn×1. The branch equations
in matrix form are
vt= Rit+ L dit dt (9) and il= C dvl dt (10)
where R = diag(R1, R2, · · · , Rn), L = diag(L1, L2, · · · , Ln),
and C = diag(C1, C2, · · · , Cn) are diagonal matrices
with element values on the diagonal entries.
To write the equations relating vl, the capacitor
voltage, to vs, the input voltage source, we substitute
(8) and (9) into (7) to get F (RFTi
l+ LFT
dil
dt) + vl= Evs (11) and substitute (10) into (11) to get
F RFTCdvl
dt + F LF
TCd2vl
dt2 + vl= Evs (12)
Taking Laplace transform of (12) with zero initial con-ditions to get
(F LFTCs2+ F RFTCs + I)V
l(s) = EVs(s)
where Vl(s) = L(vl(t)) and Vs(s) = L(vs(t)). Thus the
transfer matrix from vsto the capacitor node voltages
vlis
H(s) = Vl(s) Vs(s)
= (F LFTCs2+ F RFTCs + I)−1E
Note that H(s) is a rational matrix of dimension n × 1.
III MOMENT COMPUTATION
The kth moment mk of the transfer matrix H(s) is
defined as the coefficient of the term sk in the power
series expansion, at s = 0, of H(s). Since H(s) = H(0) + H0(0)s + 1
2!H
00(0)s2+ 1
3!H
(3)(0)s3+ · · ·
the moment of H(s) are mk=
1 k!H
(k)(0) k = 0, 1, 2, · · ·
To compute the moments mk, let’s writes
where A(s) = F LFTCs2+F RFTCs+I. Since A(0) = I, we have m0= H(0) = E. By computation, dA−1 ds |s=0 = −A −1dA dsA −1| s=0= −F RF T C d2A−1 d2s |s=0 = −[−2A −1dA dsA −1dA dsA −1+ A−1d 2 A ds2A −1]| s=0 = 2F RFTCF RFTC− 2F LFTC d3 A−1 d3s |s=0 = −[6A −1dA dsA −1dA dsA −1dA dsA −1 −3A−1d 2 A ds2A −1dA dsA −1 −3A−1dA dsA −1d 2 A ds2A −1]| s=0 = −6F RFT CF RFTCF RFTC+ 6F LFT CF RFTC+ 6F RFT CF LFTC
Note that A(s) is a matrix of polynomials of degree 2, therefore d3A(s)/ds3 = 0. From the above result, the first three moment matrices are:
m1 = −F RFTCm0 (13)
m2 = −F RFTCm1−F LFTCm0 (14)
m3 = −F RFTCm2−F LFTCm1 (15)
Note that the second and third moments depend on the previous two moments. It can be shown that the recur-sive formula holds true for other high-order moments as well, that is, for k ≥ 2,
mk = −F RFTCmk−1−F LFTCmk−2 (16)
The formula (16) computes the moments of the transfer matrix H(s) and thus the moments of the transfer functions from the source to all capacitor node are simultaneously computed. It is clear that simi-lar recursive formula for each source-to-capacitor node transfer function can be obtained from (16). To see this, let’s consider the simple 3-section RLC tree in Figure 1. Let mi
j be the jth moment of the transfer
function to node i, where i, j = 1, 2, 3. The first mo-ment is m1 = −F RFTCm0 = − R1C1+ R1C2+ R1C3 R1C1+ (R1+ R2)C2+ R1C3 R1C1+ R1C2+ (R1+ R3)C3 = m1 1 m2 1 m3 1 (17)
where F ∈ R3×3is defined in (2). The second moment
is m2 = −F RFTCm1−F LFTCm0 = − R1C1m11+ R1C2m21+ R1C3m31 R1C1m11+ (R1+ R2)C2m21+ R1C3m31 R1C1m11+ R1C2m21+ (R1+ R3)C3m31 − L1C1+ L1C2+ L1C3 L1C1+ (L1+ L2)C2+ L1C3 L1C1+ L1C2+ (L1+ L3)C3
The third moment is
m3 = −F RFTCm2−F LFTCm1 = − R1C1m12+ R1C2m22+ R1C3m32 R1C1m12+ (R1+ R2)C2m22+ R1C3m32 R1C1m12+ R1C2m22+ (R1+ R3)C3m32 − L1C1m12+ L1C2m22+ L1C3m32 L1C1m12+ (L1+ L2)C2m22+ L1C3m32 L1C1m12+ L1C2m22+ (L1+ L3)C3m32
From above the recursive formula, the moments of the transfer function to each node can also be computed recursively: mi 1 = − 3 X k=1 CkRik mi2 = − 3 X k=1 CkRikmk1− 3 X k=1 CkLik mi 3 = − 3 X k=1 CkRikmk2− 3 X k=1 CkLikmk1
where Rik is the sum of common resistance from the
input to node i and k and Lik is the sum of common
inductance from the input to node i and k.
It can be shown that the same formula holds for general tree with n section and for moments of any order. That is mi j= − n X k=1 CkRikmkj−1− n X k=1 CkLikmkj−2 for i = 1, 2, · · · , n, and j ≥ 2. IV SECOND-ORDER APPROXIMATION
We now consider matching the first three moments to obtain a second-order approximation. The three pa-rameters of the second-order approximation we should determine are the damping ratio ζ, undamped natu-ral frequency ωn, and zero location −z. We consider
scalar transfer function in this section, since an approx-imation of a transfer matrix is obtained component by component. Suppose the first three moments of an RLC tree transfer function m1, m2, and m3 are given.
The transfer function of the two-pole one-zero second-order approximation, with unit DC-gain, has the form
h(s) = ω 2 n(s + z) z(s2+ 2ζω ns + ωn2) = 1 + 1z s 1 +ω2ζns + 1 ωn2 s2 (18)
The power series expansion of h(s) can be obtained by long division: h(s) = 1 +ωn−2ζz zωn s +4ζ 2z − 2ζω n−z zω2 n s2 +−ωn+ 4ζz + 4ζ 2ω n−8ζ3z zω3 n s3+ · · · The moment matching equation are
m1 = ωn−2ζz zωn (19) m2 = 4ζ2z − 2ζω n−z zω2 n (20) m3 = −ωn+ 4ζz + 4ζ2ωn−8ζ3z zω3 n (21) Equations (19), (20), and (21) uniquely determine ωn,
ζ, and z. The solutions are ωn = s m2 1−m2 m2 2−m1m3 (22) ζ = − 1 2m1ωn (m2ωn2+ 1) (23) z = ωn m1ωn+ 2ζ (24) Hence given the moments m1, m2, and m3the formulas
(22), (23), and (24) determine the undamped natural frequency ωn , the damping ratio ζ, and zero location
−z.
V Step Response Parameters
The unit-step response of the second-order transfer func-tion h(s), in (18), is s(t) = 1 − e− σt p 1 − ζ2[(ζ − r) sin ω dt+ p 1 − ζ2cos ω dt] (25) where σ = ζωn, r = ωn/z, ωd = ωn p 1 − ζ2. The
delay time td is defined as the time it takes the signal
to rise to 50% of its final value. The rise time tr is
defined as the time it takes the signal to rise from 10% to 90% of its final value. To simplify expressions, let t0= ωnt, and g(t0) = f (t), hence (25) becomes,
g(t0) = 1 − e−ζt 0 p 1 − ζ2[(ζ − r) sin ( p 1 − ζ2t0) +p1 − ζ2cos (p1 − ζ2t0)] (26)
We note that (26) has two variables, ζ and r, only and that exact explicit formulas for delay time and rise time do not exist.
To find an approximate explicit formula for delay time, we determined the normalized delay time t0
d for
different values of ζ and r via simulations. Least squares curve fitting is then used. First for each r we fit the
normalized delay time t0
das a function ζ by 1/(aζ + b),
where a and b depend on the value r. We then fit the parameter a and b by second-order polynomials. The result is t0 d(r, ζ) = 1 pd(r)ζ + qd(r) (27) The same curve fitting method is used in estimating the rise time. The result is
t0 r(r, ζ) = 1 pr(r)ζ + qr(r) (28) where pd(r) = −0.0051r2−0.5989r − 0.3652, qd(r) = 0.5355r2+0.9136r+0.9542, pr(r) = −0.3886r2−0.1123r− 0.6959, and qr(r) = 0.6064r2+ 0.0762r + 0.9707.
Con-sequently, the delay time, td and rise time trare
td= t0 d ωn (29) tr= t0 r ωn (30) The peak time can be found by differentiating s(t) in (25) and then set s0(t) = 0. The peak time that the
maximum overshoot occurs is tp=
π − θ ωd
(31) where θ = tan−1(rp1 − ζ2/1 − rζ), and ω
d= ωn
p 1 − ζ2.
The maximum overshoot, Moobtained by substituting
(31) into (25) is Mo= p 1 − rζ + r2e− ζ(π−θ) √ 1−ζ2 (32) VI An Example
To examine the effectiveness of the proposed second-order approximation, we consider the RLC tree shown in Figure 2. The RLC tree has 6 sections and is con-sidered in [2].
Step response of each capacitor node for the origi-nal transfer function, the proposed approximation, and the approximation reported in [2] are computed using Matlab. The results are shown in Figure 3. In each plot, the solid line is the exact response, the dot line is the response of the proposed approximation, and the dot-dashed line is the response of the approximation reported in [2]. The plots show improved approxima-tion over that reported in [2]. We also note that for this example the proposed approximation gives step responses very close to the original.
The delay time tdand rise time trfor step response
of each node are shown in Table 1 and Table 2, respec-tively. In general, the proposed method and the ex-plicit formulas (29) and (30) give better estimate over that proposed in [2]. Over all, the formula (29) gives estimates of delay time with error less than 20%; the
formula (30) gives estimates of rise time with error less than 30%. Figure 4 shows the frequency magnitude re-sponse of the approximation error transfer function at each node. Again, the proposed approximations show improved accuracy over [2].
Table 1. delay time, in ns
node exact proposed formula (29) [2]
node 1 0.1235 0.1470 0.1473 0.1751 node 2 0.1953 0.2111 0.2212 0.2143 node 3 0.2021 0.2065 0.2066 0.2124 node 4 0.2608 0.2647 0.2658 0.2553 node 5 0.2720 0.2705 0.2717 0.2613 node 6 0.2575 0.2612 0.2624 0.2526
Table 2. rise time, in ns
node exact proposed formula (30)
node 1 0.2561 0.2343 0.2296 node 2 0.2134 0.2608 0.2545 node 3 0.2575 0.2668 0.2601 node 4 0.1943 0.2558 0.2521 node 5 0.2385 0.2720 0.2590 node 6 0.2295 0.2633 0.2592 VII CONCLUSION
We propose a method to obtain second-order approxi-mations for transfer functions in RLC trees. The two-pole one-zero approximation is shown to give improved accuracy over the existing second-order approximations in terms of step response, frequency response, esti-mated delay time and rise time. The results can be used to quickly estimate signal delay and other param-eters. In view of the accuracy obtained, the second-order model can also be used in dynamic simulation to replace the original tree.
References
[1] L. Chua, C. Desoer, and E Kuh, Linear and non-linear circuit, McGraw Hill 1987.
[2] Y. Ismail, E. Friedman, and J. Neves, “Equiva-lent elmore delay for RLC trees”, in IEEE Trans. Computer-Aided Design of Intergrated Circuits and System, vol. 19, pp. 83-97, Jan. 2000. [3] L. T. Pillage, and R. A. Rohrer, “Asymptotic
waveform evaluation for timing analysis”,IEEE Trans. Computer-Aided Design, vol. 9, pp. 352-366, APR. 1990.
Figure 1: A simple RLC tree
Figure 2: An example of an RLC tree
0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 1
time (ns) voltage (v) exact with zero without zero 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 2
time (ns) voltage (v) exact with zero without zero 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 3
time (ns)
voltage (v)
exact with zero without zero
0 0.5 1 1.5 2 2.5 3 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 4
time (ns) voltage (v) exact with zero without zero 0 0.5 1 1.5 2 2.5 3 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 5
time (ns) voltage (v) exact with zero without zero 0 0.5 1 1.5 2 2.5 3 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
1.4 step response of node 6
time (ns)
voltage (v)
exact with zero without zero
Figure 3: The step response of each node
0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9 frequency response of node 1
frequency (GH) |E(j ω )| without zero with zero 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2
0.25 frequency response of node 2
frequency (GH) |E(j ω )| without zero with zero 0 1 2 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.16 frequency response of node 3
frequency (GH) |E(j ω )| without zero with zero 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
frequency response of node 4
frequency (GH) |E(j ω )| without zero with zero 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.45 frequency response of node 5
frequency (GH) |E(j ω )| without zero with zero 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.45 frequency response of node 6
frequency (GH) |E(j ω )| without zero with zero
