CROSSTALK
4.3 COUPLED LINE ANALYSIS
4.3.2 Coupled Noise
Recall from Section 4.1 that the elements on the matrix diagonals represent the self-inductance and total capacitance, respectively, while the off-diagonal elements represent the mutual terms. Once we obtain the effective inductance and capacitance, we can calculate the effective impedance and propagation velocity:
Z0,eff,n=
Leff,n
Ceff,n (4-44)
νp,eff,n= 1
Leff,nCeff,n (4-45)
We can use equations (4-44) and (4-45) along with the physical length of the transmission line to analyze or simulate the behavior of any line as a simple noncoupled line while accounting for the coupling to the other lines in the system.
This technique is best used during the early design phase of a bus, when I/O transceiver impedances and line-to-line spacing are being chosen. In addition, it is useful only for signals traveling in the same direction. For signals traveling in opposite directions, fully coupled simulations are required to comprehend the effects of crosstalk.
At this point it is important to note that although the SLEM method gives correct results for systems with two coupled lines, it is an approximation that will not exactly match the actual modal impedances and velocities for three or more coupled lines. As such, its use should be restricted to early design exploration aimed at narrowing down the solution space. Final simulations should always be done with fully coupled models. The accuracy of the SLEM model (for three lines) is reasonable for cross sections in which the spacing/height ratio is greater than 1. When this ratio is less than 1, the SLEM approximation should not be used. In Section 4.4 we introduce a technique for producing exact solutions to systems of three or more coupled lines.
vout
vin VCC vIH,min
VSS
Slope = −1 Slope = −1
vIH,max VCC
vOH,min
vOL,max VSS
Figure 4-14 Receiver transfer characteristic and switching thresholds.
on the signal that causes an excursion back below vI L will cause erroneous switching (Figure 4-15). It is the job of the system designer to make sure that this does not happen.
Refer to Chapter 11 for additional details on the operation of receivers for high-speed links.
Qualitative Description Before developing a quantitative treatment of crosstalk-induced noise, we describe the behavior qualitatively by looking at the propagation of the aggressor and coupled noise wave. From our discussion of mutual inductance and capacitance it is apparent that energy is coupled from one line to another only during signal transitions (i.e., the rising and falling edges).
Subsequently, we look at the propagation of a rising edge on the aggressor line.
Figure 4-16 shows a pair of coupled lines terminated at both ends that represent a typical system. As the incident wave on the aggressor line is launched, it imme- diately begins coupling over to the victim line through the mutual capacitance and inductance. Current that couples through the mutual capacitance (iC) splits into forward-traveling (if) and backward-traveling (ib) components in the victim line, as shown in Figure 4-17. Current that couples through the mutual inductance (iL) travels back toward the source end. As a result, we have a forward-coupled wave that is a function of the difference between the capacitively coupled current and the inductively coupled current. Since it is based on the difference between capacitive and inductive coupling, the amplitude may have the same polarity as
Time (ns)
Voltage (V)
vIH vIL
Threshold violation Clean signal
Figure 4-15 Receiver threshold violation.
z = 0 z = I
Z0 Z0
Z0 Z0
Near-end ×talk pulse at t = 0 Aggressor: Z0, td, I
Victim: Z0, td, I Far-end ×talk pulse at t = 0 t = 0
Figure 4-16 Initial launch of aggressor signal and coupling of noise.
Line 1
Line 2
CM dz
∆iC
∆if
∆vf
∆ib L0 dz
Cg dz
Cg dz
LM dz
iL Z0
Z0
∆vb Z0 Z0
v1 i1
L0 dz
Figure 4-17 Coupled circuit subsection for crosstalk noise analysis.
Aggressor: Z0, td, I Near-end ×talk pulse at t = ½ td ⋅I t = ½ td ⋅I
Far-end ×talk pulse at t = ½ td ⋅I Victim: Z0, td, I
Figure 4-18 Propagation of incident aggressor signal and coupled noise pulses.
the aggressor signal, or it may have opposite polarity. The forward coupled wave then begins to propagate toward the far end of the victim line (z=l). We also have a backward coupled wave that is a function of the sum of the capacitively and inductively coupled currents. The backward traveling noise will have the same polarity as the aggressor signal, which is a direct result of the summing of the inductive and capacitive coupling. The backward crosstalk noise propagates toward the near end of the victim line, where it is immediately detectable.
As the incident wave on the aggressor propagates toward the far end, it contin- ues to couple energy over to the victim line. As Figure 4-18 shows, the forward crosstalk pulse on the victim line propagates alongside the aggressor signal. Since coupling continues along the length of the line, the amplitude of the far-end noise pulse grows as it propagates along the length of the coupled pair. The backward crosstalk pulse propagates back toward the near end (z=0).
The coupling of energy to the victim line continues as the aggressor propagates along the line until it reaches the far end at timet =τdl, whereτd is the signal propagation delay per unit length and l is the line length. Alternatively, we could define the time as t=l/νp, where νp is the propagation velocity of the signal. At that point, assuming that we have a matched termination, the coupling ceases since the aggressor does not generate reflected waves. As Figure 4-19 demonstrates, arrival of the far-end crosstalk noise occurs simultaneously with arrival of the aggressor signal. Since the far-end crosstalk travels along with the aggressor signal, the crosstalk noise pulse grows in amplitude but does not grow in width. Coupling occurs only during the signal transition, so the width of the forward coupled pulse will be approximately equal to the rise (or fall) time of
Z0, td, I t = td ⋅I
Z0, td, I
Near-end ×talk pulse at t = td ⋅I
Far-end ×talk pulse at t = td ⋅I
Figure 4-19 Propagation of coupled noise pulses as the aggressor reaches the far end (z=l).
t = 2td ⋅I z = 0 z = l Z0
Z0 Z0
Z0, td, l
Z0, td, l
Near-end ×talk pulse at t = 2 td ⋅I
Figure 4-20 Completion of the noise pulse at the near end(z=0).
the aggressor signal, as thefigure illustrates. Even though additional energy is no longer being coupled at this point, the backward crosstalk wave must still travel back to the near end. This takes a full propagation delay (td=τdl) to complete and is shown in Figure 4-20.
The shape of the near-end pulse depends on the electrical length of the coupled lines relative to the transition time of the aggressor signal. Consider the case for which the coupled length is less than one-half of the signal rise time. The beginning of the signal transition will reach the far end before the rising edge at the driving end has completed one-half of the transition. As the coupled noise propagates back to the near end, the signal at the far end continues to change, and therefore to couple more energy to the victim. Not until the rising-edge transition hasfinished propagating all the way to the far end does coupling stop.
In that case, the near-end crosstalk pulse will have a similar shape (but different amplitude) to the far-end pulse, as shown in Figure 4-21a.
On the other hand, if the coupled length is greater than one-half the signal rise time, the near-end crosstalk pulse will reach maximum amplitude and will then begin to spread in time, a phenomenon known as saturation, which is shown in Figure 4-21b. To understand this effect, consider a case where the coupled electrical length is equal to several rise times. For this situation, we can visualize the signal edge as a traveling wave on the aggressor line. Recalling that the lines couple only during the transition, we can imagine the backward crosstalk as a train of pulses of equal magnitude with a width equal to the rise time that propagate back to the near end. As a result, the backward noise does not grow in magnitude, but instead, spreads out in time. Figure 4-22 illustrates the coupled pulse propagation as just described.
As afinal note, we must realize that the crosstalk pulse magnitudes and shapes that we just described, and for which we will develop quantitative models in the next section, are specific to the matched termination case. Terminating the lines simplifies the analysis by eliminating the need to deal with reflected crosstalk and crosstalk from reflected aggressor signals. In general, the characteristics of the crosstalk noise are heavily dependent on the amount of coupling and the termination. For cases with imperfect termination and/or complex topologies, we recommend using a simulator to analyze the behavior of the system. Hall et al. [2000] describe crosstalk shapes for some general nonperfect termination schemes.
(a)
Driven signal tr
tr
tr
Near-end crosstalk Far-end crosstalk td ⋅I
(b) Driven signal
Near-end crosstalk
Far-end crosstalk td ⋅I
2 td ⋅I tr
tr
Figure 4-21 Forward- and backward-coupled noise pulses: (a) nonsaturated; (b) sat- urated.
Quantitative Development Having developed an intuitive understanding of crosstalk and the characteristics of crosstalk noise, we now derive the equations for noise at each end of a quiet transmission line (line 2 in Figure 4-17) that is induced by coupling from an adjacent coupled line (line 1) that is actively driven. Noise will be coupled from line 1 to line 2 through the mutual inductance LMdz and mutual capacitance CMdz. Line 1, whose characteristic impedance is Z0, has an incident pulse of magnitude v1 (voltage) and i1 (current), and is terminated in its characteristic impedance at both ends. We want to come up with expressions for the backward (vb) and forward (vf) noise pulses on line 2, which also has a characteristic impedance ofZ0with termination at both ends. Our derivation follows the method presented by Seraphim et al. [1989].
z = I
z = 0
Aggressor: Z0, td, I
Victim: Z0, td, I Near-end ×talk pulse at t = 0
Far-end ×talk pulse at t = 0 t = 0
Z0
Z0
Z0
Z0
(a)
Aggressor: Z0, td, I
Victim: Z0, td, I Far-end ×talk pulse at t = ½td ⋅I
(b)
Near-end ×talk pulse at t = ½td ⋅I
t = ½td ⋅I z = 0 z = I
Z0
Z0 Z0
Z0, td, I
Z0, td, I
Far-end ×talk pulse at t = td ⋅I (c)
Near-end ×talk pulse at t = td ⋅I t = td ⋅I
(d) Z0, td, I
Z0, td, I
Z0 Z0
Z0
z = 0 z = I
Near-end ×talk pulse at t = 2 td ⋅I t = 2 td ⋅I
Figure 4-22 Summary of propagation of forward- and backward-coupled noise: (a) ini- tial wave launch; (b) halfway down the line; (c) one full trip down the line; (d) round trip.
We begin the development by applying Ohm’s law at each end of line 2 to get the amplitudes of the noise pulses at the near end (vb) and far end (vf):
vb=ibZ0 (4-46)
vf =ifZ0 (4-47)
Current is coupled from line to line through the mutual capacitance:
iC=CM dzdv1
dt (4-48)
The coupled current splits into separate branches on line 2 that flow in both directions:
iC=ib+if (4-49)
Combining equations (4-46) through (4-49) gives an expression for the voltage pulses created by the coupling through the mutual capacitance,
vb+vf =Z0CM dzdv1
dt (4-50)
Next we define the capacitive coupling coefficient as the ratio of the mutual capacitance between lines to the total capacitance of the line:
KC ≡ CM
Cg+CM
(4-51) Along with the capacitive coupling coefficient definition, we apply expressions for the characteristic impedance of the line, Z0=
L0/Cg+CM, to equation (4-50), resulting in
vb+vf =
L0CM2 Cg+CM
dzdv1
dt (4-52)
By substitutingνp=1/
L0(Cg+CM)and performing some algebra, we arrive at the following expression for the sum of forward and backward crosstalk induced by the mutual capacitance:
vb+vf = 1
νpKC dzdv1
dt (4-53)
Turning now to the inductance, we note that the mutual inductance acts as a coupling transformer. Current on line 1 induces a voltage on line 2 that travels in the direction opposite that of the incident signal on line 1. As a result, it creates a
voltage difference across the differential segment (dz) which travels back toward the source:
vb−vf =LM dzdi1
dt (4-54)
In the same way that we defined a capacitive coupling coefficient, we define the inductive coupling coefficient as the ratio of the mutual inductance between lines to the self-inductance of the line:
KL≡ LM
L0 (4-55)
Application of Ohm’s law at the driven end of line 1 (i1=v1/Z0) yields
vb−vf =dzLM
Z0
dv1 dt =dz
L2M(Cg+CM) L0
L0 L0
dv1 dt
=dz
L0(Cg+CM)L2M L20
dv1
dt
which reduces to another expression relating the forward and backward coupled noise to a coupling coefficient, in this case the inductive coupling coefficient:
vb−vf =dzKL
νp
dv1
dt (4-56)
Since (4-53) and (4-56) give us two expressions with two unknowns, we can solve them forvbandvf and takedz→0 in the limit, to get
dvf
dz = KC−KL
2νp
dv1
dt (4-57)
dvb
dz = KC+KL
2νp
dv1
dt (4-58)
Forward Crosstalk Integrating (4-57) fromz=0 to z=l gives an expression for forward crosstalk:
vf = 1
2(KC−KL) l νp
dv1
dt (4-59)
By approximating dv1/dt as the ratio of the voltage swing v and a 10 to 90%
rise timetr, we have our final expression for the forward crosstalk:
vf = 1
2(KC−KL) l νp
v tr
(4-60)
Noting again that the forward crosstalk is a function of the difference between capacitive and inductive coupling, we see that the coupled pulse will have the same polarity as the aggressor signal if there is more capacitive coupling than inductive coupling in the system, and vice versa for the case where inductive coupling dominates. Equation (4-60) also suggests that it is possible to have no forward crosstalk if we canfind a case whereKC=KL. In fact, this is always true for coupled lines in a homogeneous dielectric (proof is left for the reader as Problem 4-9). On the other hand, for typical transmission lines in inhomogeneous dielectrics, such as microstrip PCB traces, the inductive coupling is generally greater than the capacitive coupling, so that the forward crosstalk pulse has the opposite magnitude from that of the aggressor signal.
As we described earlier, the width of the forward crosstalk pulse is
tpw,f ∼=tr (4-61)
where tr is the rise time of the signal. Note that (4-60) and (4-61) are equally applicable for a falling-edge transition.
Reverse Crosstalk To get an expression for the reverse (near-end) crosstalk, we must take into account the fact that the coupling region travels in the direction opposite to the coupled waved. The output wave at the left in Figure 4-17 is a superposition of the waves coupled at earlier times that propagate and sum at the near end. This requires that we integrate fromz=0 to z=l while accounting for the travel time of the wave:
vb= KC+KL
2νp l
z=0
dv(t−2z/νp)
dt dz (4-62)
After integration we have an expression for the coupled noise at the near end.
vb(t)= KC+KL
4
v1(t)−v1 t−2l νp
(4-63) The apparent reduction of the effect of the coupling coefficient after the inte- gration of (4-62) is caused by the fact that the energy coupling of the backward crosstalk is spread out over a pulse width of 2l/vp. The width of the backward coupled pulse is calculated with
tpw,b=2τdl (4-64)
where τd is the propagation delay per unit length and l is the coupled length.
Equations (4-63) and (4-64) assume that the backward crosstalk has saturated, which is realistic for multi-Gb/s links.
Finally, we note that the equations that we derived in this section apply to situation in which both lines are terminated at each end. Other configurations, such as when the near end is not terminated, will have different equations to
describe the amplitudes and shapes of the crosstalk pulses. The modified crosstalk equations can be derived by considering the effect of reflections as described in Section 3.5.
Example 4-3 We now analyze the coupling from an active line to a quiet line for the PCB transmission lines from Example 4-2. Recall that the lines had the following inductances and capacitances:
L=
3.592×10−7 3218×10−8 3218×10−8 3.592×10−7
H/m C=
8.501×10−11 −2.173×10−12
−2.173×10−12 8.533×10−11
F/m
The 0.2794-m-long traces have a typical (isolated) characteristic impedance of approximately 65 and are terminated to ground in 65 at the far end. They are driven by a 1-V 65-source with a 100-ps rise time. Compare the analytical results with those from a fully coupled simulation.
SOLUTION
Step 1: We start by calculating the impedance and propagation velocity:
Z0,isolated=
3.592×10−7H/m
8.501×10−11 F/m =65.0
νp,isolated= 1
3.592×10−7H/m 8.501×10−11 F/m
=1.810×108 m/s
Step 2: Since we plan to analyze the coupled noise, we need the coupling coefficients.
KC= 2.173×10−11 F/m
8.501×10−11 F/m =0.0256 KL= 3.218×10−8 H/m
3.593×10−7 H/m =0.0896 Step 3: Analysis for a rising edge:
v(t=0, z=0)= Z0
RS+Z0VS = 65.0
65 +65.0(1 V)=0.500 V i(t=0, z=0)=v(t =0, z=0)
Z0 = 0.500 V
65.0 =7.69 mA (z=0)=RS−Z0
RS+Z0 = 65 −65.0
65 +65.0 =0.000
(z=0.2794 m)= RT −Z0
RT +Z0 = 65−65.0
65+65.0 =0.000 td= l
νp = 0.2794 m 1.810×108 m/s
109ns s
=1.544 ns vf = 1
2(KC−KL) l νp
v tr
= 1
2(0.0256−0.0896) 0.2794 m 1.820×108m/s
× 0.500 V 100 ps
1012 ps s
= −0.247 V vb= KC+KL
4
v(t)−v t−2 l νp
= 0.0256+0.0896 4
(0.500 V)=0.014 V tpw,f =tr =100 ps
tpw,b=2τdl =2 11 in.
1.810×108 m/s
m 39.37 in.
109ns s
=3.088 ns
Since the reflection coefficients are zero, a lattice diagram is not necessary, as we can construct the waveform directly from our calculations.
Step 4: Comparison to simulation. Figure 4-23 compares our calculated results with those from SPICE time-domain simulations. We see that although waveforms nearly match, they are not identical. In particular, the rising edge of the active signal in the SPICE simulation has grown to approximately 200 ps at the receiver end of the transmission line. The degradation of the rising edge of the active signal can be attributed to the crosstalk mechanism using an energy conservation argument. To conserve energy, the active line must give up an amount of energy that is equal to the amount coupled to the quiet line. Recall that the reactive nature of the coupling mechanism means that energy is coupled to the quiet line only during signal transition, so that the rising edge is degraded as a direct result of the coupling. In addition, the increase in rise time at the receiver causes the width of the crosstalk pulse at the receiver end to be approximately 200 ps rather than the 100 ps predicted by our calculation.
As the example demonstrates, the crosstalk model presented here is an approxi- mation. This model begins to break down when the coupled length is long enough such that the difference in propagation delay between the even and odd modes
−0.3
−0.2
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time (ns)
Voltage (V)
vAggressor(z = 0)
vAggressor(z = I)
vvictim(z = 0)
vvictim(z = I)
−0.3
−0.2
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time (ns)
Voltage (V)
vAggressor(z = 0)
vAggressor(z = I)
vvictim(z = 0)
vvictim(z = I)
Figure 4-23 Comparison of (a) calculated and (b) simulated results for Example 4-3.
exceeds the rise time:
td,even−td,odd> tr
When this occurs, the forward crosstalk saturates. At that point the amplitude ceases to grow; instead, the pulse width increases. In the next section we show how modal analysis allows us to explain the noise coupling mechanism in terms of modal propagation velocities and gives us the means to calculate accurately the behavior of the system.