1. Theory

Definition: Crosstalk is the noise or delay change induced on a victim line by switching neighbours through coupling capacitance and mutual inductance; inductive effects are the overshoot, ringing and delay changes that appear when a line’s inductance is comparable to its resistance at the signal edge rate (Devgan; Ismail, Friedman and Neves).
Capacitive coupling: At tight pitch the coupling capacitance is a large share of each line’s capacitance, so a switching neighbour can inject a noise pulse on a quiet line or change the effective capacitance of a switching one by up to about plus or minus 30 per cent (Devgan; computed from the Miller factor, background knowledge).
Inductive range: Inductance matters only for lines within a range of lengths set by the rise time, the time of flight and the attenuation, mainly wide, low-resistance global lines with fast edges (Ismail, Friedman and Neves).
Design impact: Neglecting inductance in repeater design oversizes repeaters and adds delay, and inductive coupling between long buses can exceed capacitive coupling (Ismail and Friedman; Zhang and Friedman).

2. Working Principle

Capacitive injection: An aggressor edge drives a current $C_{c} d V / d t$ into the victim, which its driver must sink; a weak or floating victim follows the aggressor by the capacitive divider (Devgan; background knowledge).
Miller effect: The charge a line must supply through a coupling capacitor depends on whether the neighbour moves with it, stays still or moves opposite, which changes its effective capacitance by a factor of 0 to 2 on the coupling part (Zhang and Friedman; background knowledge).
Inductive behaviour: When the edge is fast compared with the time of flight and the line loss is small, the line behaves as a transmission line, with reflections and overshoot, and its delay approaches the time of flight rather than the RC diffusion time (Ismail, Friedman and Neves).
Inductive coupling: Return currents spread over a wide area, so mutual inductance between lines decays slowly with spacing and couples lines that are capacitively well separated; shields close to the signal provide nearby return paths (Zhang and Friedman; background knowledge for return paths).

3. Applications

Signal-integrity sign-off: Peak-noise screening of millions of nets with bounds such as Devgan’s metric before detailed simulation (Devgan).
Timing with coupling: Worst- and best-case delay with Miller factors and timing windows in static timing analysis (background knowledge).
Global wiring and clocks: RLC repeater sizing and shield insertion on wide upper-layer buses and clock lines (Ismail and Friedman; Zhang and Friedman).
Physical design fixes: Wider spacing, shielding, buffer sizing and staggered repeaters to reduce coupling (Zhang and Friedman; background knowledge).

Capacitive Crosstalk

Name	Formula	Related Formulae	Variables/Constants
Capacitive crosstalk on a floating victim	$Δ V_{v} \approx V_{D D} \frac{C _{c}}{C _{c} + C _{g, v}} (victim not driven, aggressor step)$	$C_{g, v} = C_{g} + C_{c, o t h er}$	$Δ V_{v} : n o i se o n t h e v i c t im$ $V_{D D} : a g g r essor s w in g$ $C_{c} : co u pl in g c a p a c i t an ce t o t h e a g g r essor$ $C_{g, v} : a l l o t h er c a p a c i t an ce o f t h e v i c t im$ $C_{g} : c a p a c i t an ce t o t h e pl an es$ $C_{c, o t h er} : co u pl in g t o o t h er q u i e t n e i g hb o u r s (c ha r g e - s ha r in g b o u n d, ba c k g r o u n d k n o w l e d g e); a d r i v e n v i c t im r eco v er s t h r o ug h i t s d r i v er r es i s t an ce, so t h e a c t u a l p e ak i s l o w er; c a p a c i t an ce v a l u es o n t h e W i r e C a p a c i t an ce p a g e$ $(co m p u t e d : w i t h t h e w = t = s = h e x am pl e o f 3.95 ε t o g r o u n d an d 0.79 ε p er n e i g hb o u r, o n e a g g r essor g i v es 14 p er ce n t o f V_{D D} an d tw o a g g r essor s 29 p er ce n t)$
Peak noise bound for a driven RC victim (Devgan)	$V_{2, ma x} = - A_{22}^{- 1} C_{c} A_{11}^{- 1} B_{1} \overset{u}{˙}$	$I_{c} = C_{c} \dot{V}_{1, ss}, V_{2, ma x} (i) = V_{2, ma x} (parent) + R_{i} \sum I_{c, downstream}$	$V_{2, ma x} : p e ak n o i se v ec t or o n t h e v i c t im$ $A_{11}, A_{22} : co n d u c t an ce ma t r i ces o f a g g r essor an d v i c t im$ $C_{c} : co u pl in g c a p a c i t an ce ma t r i x$ $B_{1} : in p u t v ec t or$ $\overset{u}{˙} : s l o p e o f t h e a g g r essor in p u t r am p$ $I_{c} : co u pl in g c u r r e n t s f r o m t h e a g g r esso r^{'} s s t e a d y - s t a t e r am p s l o p e \dot{V}_{1, ss}$ $R_{i} : r es i s t an ce f r o m t h e p a r e n t n o d e t o n o d e i; D e v g an : t h e v i c t i m^{'} s p e ak e q u a l s a D C so l u t i o n w i t h co u pl in g c u r r e n t s inj ec t e d, an u pp er b o u n d f or R C c i r c u i t s co m p u t ab l e b y in s p ec t i o n o n t r ees; i t ma t c h e d s im u l a t i o n a t 200 ps r i se (0.0463 a g ain s t 0.0462 V) an d o v er es t ima t e d b y 7.73 t o 14.11 p er ce n t a t 100 ps r i se; m u l t i pl e a g g r essor s b y s u p er p os i t i o n w i t hin t imin g w in d o w s; r u n t im es o f 0.35 s a g ain s t 360.55 s b y ma t r i x f or 500, 000 e l e m e n t s$ $(n o co m p u t e d v a l u e)$
Miller coupling factor and delay variation	$C_{e f f} = C_{g} + 2 MCF C_{c}, MCF = 0, 1, 2$	$\frac{C _{e f f, ma x} - C _{e f f, min}}{C _{e f f, n o m}} = \frac{4 C _{c}}{C _{g} + 2 C _{c}}$	$C_{e f f} : e f f ec t i v e s w i t c h e d c a p a c i t an ce o f a l in e b e tw ee n tw o n e i g hb o u r s$ $C_{g} : c a p a c i t an ce t o t h e pl an es$ $C_{c} : co u pl in g t o e a c h n e i g hb o u r$ $M C F : M i l l er co u pl in g f a c t or (0 f or n e i g hb o u r s s w i t c hin g in t h e s am e d i r ec t i o n, 1 f or q u i e t n e i g hb o u r s, 2 f or o pp os i t e s w i t c hin g; v a l u es ab o v e 2 a r e p oss ib l e w i t h u n e q u a l s l e w s) (ba c k g r o u n d k n o w l e d g e); Z han g an d F r i e d man : t h e s am e - d i r ec t i o n an d o pp os i t e - d i r ec t i o n c a ses f or m t h e tw o d eco u pl e d m o d es w i t h g r o u n d c a p a c i t an ce c_{g} an d c_{g} + 2 c_{c}$ $(co m p u t e d : w i t h 3.95 ε an d 0.79 ε, C_{e f f} i s 3.95, 5.53 an d 7.11 ε, 71, 100 an d 129 p er ce n t o f t h e q u i e t v a l u e, a s p r e a d o f 57 p er ce n t)$

Inductive Effects

Name	Formula	Related Formulae	Variables/Constants
When inductance matters	$\frac{t _{r}}{2 L C} < l < \frac{2}{R} \frac{L}{C}$	$ζ = \frac{R _{l}}{2} \frac{C _{l}}{L _{l}}, T_{0} = l L C, t_{r} > \frac{4 L}{R} \Rightarrow RC model valid at all lengths$	$t_{r} : in p u t r i se t im e$ $L, C, R : in d u c t an ce, c a p a c i t an ce an d r es i s t an ce p er u ni t l e n g t h$ $l : l in e l e n g t h$ $ζ : d am p in g f a c t or o f a s in g l e - sec t i o n l u m p e d R L C m o d e l$ $R_{l}, C_{l}, L_{l} : t o t a l l in e v a l u es$ $T_{0} : t im e o f f l i g h t; I s mai l, F r i e d man an d N e v es : in d u c t an ce i s s i g ni f i c an t o n l y w i t hin t hi s l e n g t h r an g e, t h e l imi t s a r e n o t s ha r p, osc i l l a t i o n a pp e a r s f or ζ b e l o w 1, an d in d u c t an ce i s u nim p or t an t w h e n t h e r i se t im e e x cee d s ab o u t tw i ce t h e t im e o f f l i g h t; f or a 0.25 μ m I B M C M O S l in e w i t h L = 1 \times 1 0^{- 7} H/cm, R = 400 Ω/ cm, C = 1 \times 1 0^{- 12} F/cm an d t_{r} = 0.25 ns t h e r an g e i s 0.3259 t o 1.58 cm, an d w i t h L = 1 \times 1 0^{- 8} H/cm i t i s e m pt y (1.25 ab o v e 0.5 cm)$ $(co m p u t e d : t h e f i r s t c a se g i v es 0.395 t o 1.58 cm, n o t 0.3259 (see t h e Q / A); 4 L / R i s 1 ns an d 0.1 ns f or t h e tw o c a ses, so o n l y t h e f i r s t c an b e in d u c t i v e a t t_{r} = 0.25 ns; ζ = 0.63 f or a 1 cm l in e o f t h e f i r s t c a se)$
Repeater insertion with inductance	$h_{o pt} = h_{o pt, R C} [1 + 0.16 T_{L / R}^{3}]^{- 0.24}, k_{o pt} = k_{o pt, R C} [1 + 0.18 T_{L / R}^{3}]^{- 0.3}$	$T_{L / R} = \frac{L _{t} / R _{t}}{R _{0} C _{0}} (my reading of the garbled definition)$	$h_{o pt}, k_{o pt} : o pt im u m r e p e a t er s i z e an d n u mb er w i t h in d u c t an ce$ $h_{o pt, R C}, k_{o pt, R C} : B ak o g l u R C v a l u es (R C D e l a y p a g e)$ $T_{L / R} : r a t i o o f t h e l in e^{'} s L / R t im e co n s t an t t o t h e r e p e a t e r^{'} s in t r in s i c t im e co n s t an t$ $L_{t}, R_{t} : t o t a l l in e in d u c t an ce an d r es i s t an ce$ $R_{0}, C_{0} : minim u m in v er t er o u tp u t r es i s t an ce an d in p u t c a p a c i t an ce; I s mai l an d F r i e d man : t h e R L C d e l a y m o d e l i s w i t hin 5 p er ce n t o f A S / X o v er a w i d e r an g e, b o t h o pt ima f a l l a s in d u c t an ce r i ses, u s in g R C r e p e a t er s o n in d u c t i v e l in es in cr e a ses t o t a l d e l a y b y 10, 20 an d 30 p er ce n t a t T_{L / R} o f 3, 5 an d 10, an d in cr e a ses r e p e a t er a r e a b y 154 an d 435 p er ce n t a t 3 an d 5, w h er e 5 i s co mm o n f or 0.25 μ m t ec hn o l o g y$ $(co m p u t e d : t h e cor r ec t i o n f a c t or s g i v e h k = 0.39 a t T_{L / R} = 3 an d 0.19 a t 5, so R C s i z in g u ses 2.54 an d 5.35 t im es t h e a r e a, 154 an d 435 p er ce n t m or e, r e p r o d u c in g t h e p a p e r^{'} s a r e a p e na l t i es an d co n f i r min g t h e c u bi c e x p o n e n t)$
Coupled RLC lines and shielding	$t_{f 1} \approx h (l + l_{m}) c_{g}, t_{f 2} \approx h (l - l_{m}) (c_{g} + 2 c_{c})$	$V_{p e ak} = max {V_{n o i se} (t_{f ma x}), V_{n o i se} (3 t_{f ma x})}, \frac{L _{m}}{L} \frac{C _{c}}{C} < 0.1 (loose coupling)$	$t_{f 1}, t_{f 2} : t im es o f f l i g h t o f t h e e v e n (s am e - d i r ec t i o n) an d o dd (o pp os i t e - d i r ec t i o n) m o d es$ $h : l in e l e n g t h$ $l, l_{m} : se l f an d m u t u a l in d u c t an ce p er l e n g t h$ $c_{g}, c_{c} : g r o u n d an d co u pl in g c a p a c i t an ce p er l e n g t h$ $V_{p e ak} : p e ak cr oss t a l k n o i se$ $t_{f ma x} : l a r g er t im e o f f l i g h t, L_{m} / L an d C_{c} / C : in d u c t i v e an d c a p a c i t i v e co u pl in g r a t i os; Z han g an d F r i e d man : t h e m o d e l i s w i t hin 6.8 p er ce n t o f S P I C E o n a v er a g e f or i d e n t i c a l l in es an d 3.1 p er ce n t f or n o ni d e n t i c a l l in es, t h e n o i se i s n o t m o n o t o ni c in e i t h er co u pl in g, a d i s t r ib u t e d R C m o d e l s u f f i ces f or ζ ab o v e ab o u t 1.5, a s hi e l d b e tw ee n a g g r essor an d v i c t im c u t s m u t u a l in d u c t an ce m os t, tw o s hi e l d s s t i l l l e a v e c a p a c i t i v e co u pl in g, t h r ee g i v e t h e l o w es t n o i se, an d o n e s hi e l d p er tw o g l o ba l l in es i s a p r a c t i c a l co m p r o mi se$ $(co m p u t e d : t h e i r i d e n t i c a l - l in e e x am pl e w i t h K_{C} = 0.769 an d K_{L} = 0.217 ha s a p r o d u c t o f 0.17, ab o v e t h e 0.1 l oose - co u pl in g l imi t)$

Important Questions

Ques: How much noise can a switching neighbour put on a quiet line?

An upper estimate for a victim that is not held by its driver is the capacitive divider $V_{D D} C_{c} / (C_{c} + C_{g, v})$ (background knowledge). Using the Sakurai-Tamaru values for equal width, thickness, spacing and height, 3.95 times the permittivity to ground and 0.79 per neighbour, one aggressor gives about 14 per cent of the supply and two aggressors switching together 29 per cent (computed). A driven victim recovers through its driver, so the real peak is lower. Devgan’s metric bounds this case for RC networks by injecting the coupling currents of the aggressor’s steady ramp and solving the victim at DC: it matched simulation at 200 $ps$ rise and overestimated by 7.73 to 14.11 per cent at 100 $ps$ , so it is safe but slightly pessimistic for fast edges. Reference: A. Devgan, ‘Efficient Coupled Noise Estimation for On-Chip Interconnects’, ICCAD 1997 (IBM Austin) (equation reconstructed by the fetch tool) (fetch-tool summary); Wire Capacitance and Low-k Dielectrics page of this vault.

Ques: How much can switching neighbours change a wire’s delay?

Through the Miller effect on the coupling capacitance. With neighbours switching in the same direction the coupling capacitors carry no charge, with quiet neighbours they count once, and with opposite switching twice, so the effective capacitance is $C_{g} + 2 MCF C_{c}$ with factors of 0, 1 and 2 (background knowledge). For the example of 3.95 and 0.79 times the permittivity, this gives 71, 100 and 129 per cent of the quiet value, a spread of 57 per cent (computed), and the wire part of the delay changes in proportion. Unequal slews can push the effective factor above 2 (background knowledge). Zhang and Friedman treat the same and opposite cases as decoupled modes with ground capacitance $c_{g}$ and $c_{g} + 2 c_{c}$ , whose different times of flight shape the crosstalk waveform on coupled RLC lines. Reference: J. Zhang and E. G. Friedman, ‘Crosstalk Modeling for Coupled RLC Interconnects With Application to Shield Insertion’, IEEE Transactions on VLSI Systems 14(6), 641-646 (2006) (fetch-tool summary); Wire Capacitance and Low-k Dielectrics page of this vault.

Ques: Is the lower length limit for significant inductance in Ismail et al.’s first example 0.326 cm or 0.395 cm?

The paper’s figure of merit is $t_{r} / (2 L C) < l < (2/ R) L / C$ , and for $L = 1 0^{- 7}$ $H/cm$ , $R = 400$ $Ω/ cm$ , $C = 1 0^{- 12}$ $F/cm$ and $t_{r} = 0.25$ $ns$ it quotes 0.3259 to 1.58 $cm$ . The upper limit follows exactly, but the lower one gives 0.395 $cm$ , and 0.3259 would need a rise time of about 0.206 $ns$ (computed). The second example, with $L = 1 0^{- 8}$ $H/cm$ , reproduces exactly: 1.25 above 0.5 $cm$ , so the range is empty (computed). The difference probably comes from a rise-time definition or a typing error in the first case (my reading). Since the paper stresses that the limits are not sharp, 0.395 $cm$ , which follows from the stated formula and values, is the more likely intended value. Reference: Y. I. Ismail, E. G. Friedman and J. L. Neves, ‘Figures of Merit to Characterize the Importance of On-Chip Inductance’, DAC 1998 (University of Rochester copy) (fetch-tool summary).

Ques: How large is the error if repeaters are designed with an RC model on an inductive line?

Ismail and Friedman correct Bakoglu’s optimum size and number by $[1 + 0.16 T_{L / R}^{3}]^{- 0.24}$ and $[1 + 0.18 T_{L / R}^{3}]^{- 0.3}$ , where $T_{L / R}$ compares the line’s $L / R$ time constant with the repeater’s intrinsic delay. They report that RC-designed repeaters raise the total delay by 10, 20 and 30 per cent at $T_{L / R}$ of 3, 5 and 10, and the repeater area by 154 and 435 per cent at 3 and 5, with 5 common in 0.25 $μ m$ technology. The correction factors reproduce the area penalties exactly: the product of size and number falls to 0.39 and 0.19, so RC sizing uses 2.54 and 5.35 times the area (computed). Inductive lines therefore need fewer, smaller repeaters, which also saves power. Reference: Y. I. Ismail and E. G. Friedman, ‘Effects of Inductance on the Propagation Delay and Repeater Insertion in VLSI Circuits’, DAC 1999 (University of Rochester copy; equations reconstructed from a garbled PDF) (fetch-tool summary).

Ques: Why do global buses need shields?

Because inductive coupling reaches farther than capacitive coupling. Zhang and Friedman show that peak crosstalk on coupled RLC lines is not monotonic in either the capacitive or the inductive coupling, and that a distributed RC model suffices only when the damping parameter exceeds about 1.5. Their identical-line example, with capacitive and inductive coupling ratios of 0.769 and 0.217, has a product of 0.17, above the 0.1 limit for loose coupling (computed). A shield placed between aggressor and victim lowers the mutual inductance more than shields placed outside, two shields still leave capacitive coupling, and three give the lowest noise at the most area. They recommend about one shield per two global signal lines. Their model is within 6.8 and 3.1 per cent of SPICE for identical and nonidentical lines. Reference: J. Zhang and E. G. Friedman, ‘Crosstalk Modeling for Coupled RLC Interconnects With Application to Shield Insertion’, IEEE Transactions on VLSI Systems 14(6), 641-646 (2006) (fetch-tool summary).

Ques: Which wires need RLC rather than RC models?

Wires inside the length range $t_{r} / (2 L C) < l < (2/ R) L / C$ of Ismail, Friedman and Neves. Short lines charge faster than the wave crosses them, and long lines are damped by their resistance, so both behave as RC. For the paper’s line with $R = 400$ $Ω/ cm$ , $L = 1 0^{- 7}$ $H/cm$ and $C = 1 0^{- 12}$ $F/cm$ , a 1 $cm$ section has a damping factor of 0.63, below 1, so it can ring (computed). If the rise time exceeds $4 L / R$ , 1 $ns$ for that line, no length is inductive (computed). This makes wide, low-resistance global lines and clock lines with fast edges the main RLC candidates, while narrow local wires with hundreds of ohms per micrometre remain RC. The authors propose the criterion as a CAD screen for which nets need RLC extraction. Reference: Y. I. Ismail, E. G. Friedman and J. L. Neves, ‘Figures of Merit to Characterize the Importance of On-Chip Inductance’, DAC 1998 (University of Rochester copy) (fetch-tool summary).

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Crosstalk and Inductive Effects