CROSSTALK
4.3 COUPLED LINE ANALYSIS
4.3.1 Impedance and Velocity
Following our method, we can also derive the current wave equation for the coupled lines:
d2i
dz2 =ω2CLi (4-34)
Equations (4-33) and (4-34) comprehend wave propagation on each line in the system due to the source on the line itself and to sources being coupled from other lines through the electromagneticfields. Notice also that they bear a striking similarity to equations (4-24) and (4-25). In fact, (4-24) and (4-25) reduce to (4-33) and (4-34) for the casen=1.
A couple of additional observations about (4-33) and (4-34) are worth pointing out. First, in equation (4-34) the order of multiplication of theLandCmatrices is reversed from that of (4-33). Since they are matrices, the product LC is not necessarily equal to CL. Second, the compact matrix equations are extensible to an arbitrary number of coupled transmission lines, providing the means for analyzing practical systems, as we demonstrate in the sections that follow.
TABLE 4-1. Summary of Effective Capacitance and Inductance for a Couple Pair
Mode Line 1 Line 2 Ceffective Leffective
Even
hi
hi lo
lo
hi
hi lo
lo
Cg L0+LM
Odd
hi
hi lo
lo
hi
hi lo
lo
Cg+2CM L0−LM
Quiet
hi
hi
hi lo
lo
lo
lo hi hi
hi hi lo lo lo lo
Cg+CM L0
Z0,odd=
L0−LM
Cg+2CM (4-37)
νp,isolated= 1
L0(Cg+CM) (4-38)
νp,even= 1
√(L0+LM)C0
(4-39)
νp,odd= 1
(L0−LM)(Cg+2CM) (4-40) Equations (4-35) through (4-40) are exact for the two-line case, and they give us a simple way to analyze a coupled pair via a lattice diagram or simulation of a single line using the effective characteristic impedance and effective propagation velocity. We call the models created using this method single-line equivalent models (SLEMs) [Hall et al., 2000].
It is interesting to note that the mutual inductance is always added or sub- tracted in the opposite manner as the mutual capacitance for odd- and even-mode propagation. The fields in Figure 4-9 help us understand why this is true. Con- sidering odd-mode propagation as an example, the effect of mutual capacitance must be added because the conductors are at different potentials. Additionally,
Electric field: Odd mode Electric field: Even mode
Magnetic field: Even mode Magnetic field: Odd mode
Figure 4-9 Odd- and even-mode electric and magnetic field patterns for a simple two-conductor system.
since the current in the two conductorsflows in opposite directions, the currents induced on each line due to the coupling of the magneticfields always oppose each other and cancel the effect of the mutual inductance. Therefore, the mutual inductance must be subtracted and the mutual capacitance must be added to cal- culate the odd-mode characteristics. These characteristics of even- and odd-mode propagation are due to the assumption that the signals are propagating only in transverse electromagnetic (TEM) mode, so that the electric and magneticfields are always orthogonal to each other.
With homogeneous dielectrics, the product ofLandCremains constant, since fields are confined within the uniform dielectric:
LC= 1
νp2I (4-41)
where I is the identity matrix. Thus, in a multiconductor homogeneous system such as a stripline array, if L is increased by the mutual inductance, C must be decreased by the mutual capacitance such thatLC remains constant. Subse- quently, a stripline, or buried microstrip, which is embedded in a homogeneous dielectric, should not exhibit velocity variations due to different switching modes.
It will, however, exhibit pattern-dependent impedance variation.
In a nonhomogeneous dielectric where the electricfields fringe through more than one dielectric material, such as an array of microstrip lines,LC is not con- stant for different propagation modes because the electromagnetic fields travel partially in air and partially in the board’s dielectric material. In a microstrip, the effective dielectric constant is a weighted average between air and the dielectric material of the board. Because thefield patterns change with different propaga- tion modes, the effective dielectric constant will change depending on thefield densities contained within the board’s dielectric material and the air. Thus, the
0.005 in 0.005 in 0.005 in
0.005 in εr = 4.0
0.002 in
Figure 4-10 Cross section of the PCB-based coupled transmission-line pair for Example 4-2.
LCproduct will be mode dependent in a nonhomogeneous system. TheLCprod- uct, however, will remain constant for a given mode. Therefore, a microstrip will exhibit both a velocity and an impedance change, due to different switching pat- terns. It should be noted that the description above holds for a single frequency.
The product ofLCvaries with frequency but remains constant at each frequency point for a given mode.
Example 4-2 The PCB transmission lines depicted in Figure 4-10 have the following inductances and capacitances:
L=
3.592×10−7 3.218×10−8 3.218×10−8 3.592×10−7
H/m C=
8.501×10−11 −2.173×10−12
−2.173×10−12 8.501×10−11
F/m Assume that the waveform is driven into the line att =1 ns.
We have designed the PCB traces to have a typical (isolated) characteristic impedance of approximately 65 with a length of 0.2794 m (11 in.). They are driven by a 1-V 65-source, and are terminated to ground in 65 at the far end. The rise and fall times are 0.1 ns. Compare the analytical results with those from a fully coupled simulation for even- and odd-mode propagation.
SOLUTION
Step 1: Calculate the impedances and velocities for all of the switching pat- terns.
Z0,even=
3.592×10−7+3.218×10−8H/m
8.501×10−11−2.173×10−12 F/m =68.7 Z0,odd=
3.592×10−7−3.218×10−8 H/m
8.501×10−11+2.173×10−12)F/m =61.2
Z0,isolated=
3.592×10−7H/m
8.501×10−11 F/m =65.0
νp,even= 1
[(35.92+3.218)×10−8H/m][(85.01−2.173)×10−12 F/m]
=1.756×108 m/s
νp,odd= 1
[(35.92−3.218)×10−8H/m][(85.01+2.173)×10−12 F/m]
=1.873×108 m/s
νp,isolated= 1
3.592×10−7 H/m 8.501×10−11 F/m
=1.810×108 m/s
Step 2: Calculate the even-mode waveform. Calculate the values for the initial voltage and current waves, reflection coefficients,final voltage and current levels, and propagation delay as preparation for a lattice diagram analysis.
v(t =0, z=0)= Z0,even
RS+Z0,evenVS= 68.7
65 +68.7(1 V)=0.514 V i(t=0, z=0)= v(t =0, z=0)
Z0,even = 0.514 V
68.7 =7.48 mA (z=0)= RS−Z0,even
RS+Z0,even = 65−68.7
65−68.7 = −0.028 (z=l)= RT −Z0,even
RT +Z0,even = 65 −68.7
65 +68.7= −0.028 v(t = ∞)= Rt
RS+Rt
VS= 65
65 +65 (1 V)=0.500 V i(t = ∞)= vS
RS+Rt = 1.000
65+65 =7.69 mA td,even= l
νp,even = 11 in 1.756×108m/s
m
39.37 in. =1.592 ns The corresponding lattice diagram is shown in Figure 4-11.
Step 3: Calculate the odd-mode waveform. Repeat the analysis for the odd-mode propagation, with the lattice diagrams shown in Figure 4-12.
v(t =0, z=0)= Z0,odd
RS+Z0,oddVS= 61.2
65 +61.2(1 V)=0.485 V
v(z = 0) i(z = 0) v(z = l) i(z = l)
1.000 ns
Γ = −0.028 Γ = −0.028
0 l
z
t
0.000 V 0.00 mA
0.000 V 0.00 mA 0.514 V
−0.014 V 0.000 V
0.000 V 7.48 mA
−0.21 mA 0.01 mA
0.00 mA 0.514 V 7.14 mA
0.500 V 7.69 mA 0.500 V
0.500 V
0.500 V 7.68 mA
7.69 mA
7.69 mA 2.592 ns
4.182 ns 5.773 ns 7.364 ns
Figure 4-11 Lattice diagram for even-mode propagation of the coupled line pair for Example 4-2.
Note that the rising edge wave will be 0.485 V, starting from 0.000 V, while the falling edge wave will be−0.485 V, starting from 0.500 V.
i(t=0, z=0)= v(t=0, z=0)
Zodd =0.485 V
61.2 =7.92 mA (z=0)= RS−Z0,odd
RS+Z0,odd = 65−61.2
65+61.2 =0.030 (z=l)= RT −Z0,odd
RT +Z0,odd = 65 −61.2
65 +61.2 =0.030 Rising edge
v(t = ∞)= Rt
RS+RtVS = 65
65+65(1 V)=0.500 V i(t= ∞)= vS
RS+Rt = 1.000
65 +65 =7.69 mA Falling edge
v(t= ∞)=0.000 V i(t= ∞)=0.00 mA td,odd= l
νp,odd = 0.2794 m 1.873×108m/s
109ns s
=1.492 ns
Step 4: Compare the calculated results to simulated results. From Figure 4-13 we see that the results from SLEM analysis match results from fully coupled SPICE time-domain simulations using theL andCmatrices.
Having developed the SLEM approach for the two-line case, we want to gen- eralize it to deal with an arbitrary number of coupled lines, since real systems
v(z = 0) i(z = 0) v(z = l) i(z = l) Γ = 0.030
Γ = 0.030
0 l
z
t
0.000 V 0.00 mA
0.000 V 0.00 mA 0.485 V
0.014 V 0.001 V
0.000 V 7.92 mA
0.24 mA 0.01 mA
0.01 mA 0.485 V 8.58 mA
0.499 V 7.68 mA 0.500 V
0.500 V
0.500 V 7.69 mA
7.69 mA
(a)
7.69 mA 0.000 V
0.00 mA 1.000 ns
2.492 ns 3.984 ns 5.476 ns 6.976 ns
v(z = 0) i(z = 0) v(z = l) i(z = l)
1.000 ns
0 l
z
t
0.500 V 7.69 mA
0.500 V 7.69 mA
−0.485 V
−0.014 V
−0.001 V 0.000 V
−7.92 mA
−0.24 mA
−0.01 mA 0.00 mA 0.015 V −0.23 mA
0.001 V 0.01 mA 0.000 V
0.000 V
0.000 V 0.00 mA
0.00 mA
(b)
0.00 mA 2.492 ns
3.984 ns 5.476 ns
6.976 ns 0.000 V
0.00 mA
Γ = 0.030 Γ = 0.030
Figure 4-12 (a) Rising- and (b) falling-edge lattice diagrams for odd-mode propagation of the coupled line pair for Example 4-2.
typically contain larger numbers of coupled lines. In our earlier discussion we noted that for even-mode switching, the effective inductance is increased by the mutual inductances between lines, while the effective capacitance is decreased by the mutual capacitance. We also know that for odd-mode switching, the effec- tive inductance is decreased by the mutual inductances between lines, while the effective capacitance is increased by the mutual capacitances. Finally, the mutual inductances and capacitances for quiet lines do not change the effective
(a) 0.0
0.1 0.2 0.3 0.4 0.5 0.6
0.0 1.0 2.0 3.0 4.0 5.0
Time [ns]
Voltage [V]
v(z = 0) & v(z = 0)
v(z = l) & v(z = l)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0 0.5 1.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time [ns]
Voltage [V]
v(z = 0) & v(z = 0)
v(z = l) & v(z = l)
1.5 4.5
3.5 2.5 1.5 0.5
(b)
−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 4.5 Time [ns]
Voltage [V]
v(z = 0)
v(z = l) v(z = 0)
v(z = l)
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [ns]
Voltage [V]
v(z = 0)
v(z = l) v(z = 0)
v(z = l)
0.5
0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.0
Figure 4-13 Comparison of (a) even- and (b) odd-mode calculated (left) and simulated (right) results for Example 4-2.
inductance and capacitance. With this knowledge, we can write generalized approximations for the effective inductance and capacitance, adopting the matrix notation that we developed in Section 4.1 .
Leff,n=Lnn+
Lne−
Lno (4-42)
Ceff,n=Cnn− Cne+ Cno (4-43)
whereLeff,n =effective inductance of line n Lnn =self-inductance of line n
Lne =sum of the mutual inductances for the lines that switch in phase with line n(which we approximate as “even” mode)
Lno =sum of the mutual inductances for the lines that switch out of phase with linen (which we approximate as “odd” mode) Ceff,n =effective capacitance of line n
Cnn =total capacitance of linen
Cne=sum of the absolute values of the mutual capacitances for the lines that switch in phase with linen
Cno=sum of the absolute values of the mutual capacitances for the lines that switch out of phase with linen
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.