CROSSTALK
4.2 COUPLED WAVE EQUATIONS
Before proceeding with the analysis of coupled systems, wefirst derive the wave equations, to reinforce the notion of wave propagation and to allow us to study the effects of mutual inductance and capacitance. The wave equation is central to our analysis of transmission lines, and extension to coupled systems will lend insight into the effects of crosstalk on the propagation of signals in a coupled system.
4.2.1 Wave Equation Revisited
We start our derivation of the transmission-line equations by focusing on the isolated line case shown in Figure 4-7. The circuit must satisfy Kirchhoff’s laws.
L0
+ +
C0
dz v(z)
i(z) i(z + dz)
v(z + dz)
− −
Figure 4-7 Differential circuit subsection for a lossless transmission line.
Application of Kirchhoff’s voltage law (KVL) gives the voltage drop across the incremental inductance:
v(z)−v(z+dz)= −j ωL0i(z)dz (4-19) where j ωL0dzis the frequency-dependent impedance of the inductor. Equation (4-19) contains a frequency term, ω, implying a sinusoidal input which has the form v(t)=V0ej ωt. A sinusoidal signal has the property that its derivative is a scaled version of the original signal, dv(t)/dt=d(V0ej ωt)/dt =j ωV0ej ωt= j ωv(t). Note that the current,i(z,t), is also a sinusoid, so thatdi(t)/dt =j ωi(t).
Equation (4-19) is equivalent to Faraday’s law for the response of an induc- tor to a transient current [v=L(di/dt)]. The analysis that follows is equally applicable for a digital input, since such a signal is composed of the superposition of multiple sinusoids, although the wave equation requires a separate solution for each frequency in the envelope of the driven signal. The next steps are to divide (4-19) through by dz, followed by differentiation with respect toz:
dv
dz = −j ωL0i (4-20)
d2v
dz2 = −j ωL0di
dz (4-21)
Looking now at the incremental capacitance of the subsection, we apply Kirch- hoff’s current law to find the change in current, as shown in equation (4-22), where (j ωC0dz)−1is the impedance of the capacitor:
i(z+dz)−i(z)= −j ωC0v(z)dz (4-22) Equation (4-22) is equivalent toi=C0(dv/dt) in the time domain. Dividing through bydz gives
di
dz = −j ωC0v(z) (4-23)
Substituting (4-23) into (4-21) to get an expression in terms of v yields the voltage wave equation:
d2v
dz2 +ω2L0C0v=0 (4-24)
The same approach will give the wave equation for the current:
d2i
dz2+ω2L0C0i=0 (4-25)
Equations (4-24) and (4-25) should be familiar as the wave equations for a uniform lossless transmission line.
4.2.2 Coupled Wave Equations
Our goal now is to generalize equations (4-24) and (4-25) to then-coupled-line case. Figure 4-8 describes the circuit for a subsection of a pair of coupled lines.
Our derivation follows the same method that we used above for the isolated transmission line. We start by applying KVL to develop equation (4-26). Note that the voltage drop on line 1 depends on the mutual inductance LM and the current on the adjacent line i2(z), in addition to the line self-inductance L and driving current i1(z). We can also create a corresponding expression (4-27) for the voltage drop on line 2:
v1(z)−v1(z+dz)= −j ωL0i1(z) dz−j ωLMi2(z) dz
= −j ω[L0i1(z)+LMi2(z)]dz (4-26) v2(z)−v2(z+dz)= −j ω[L0i2(z)+LMi1(z)]dz (4-27) We can write the equations for the voltage drop across the coupled subcircuit in compact matrix form, where the boldface symbols represent the compact matrix:
dv
dz = −j ωLi (4-28)
where dv dz = 1
dz
v1(z)−v1(z+dz) v2(z)−v2(z+dz)
taken in the limit asdz→0 L=
L0 LM
LM L0
i=
i1(z, t ) i2(z, t )
Applying the isolated line method to the current change caused by the capaci- tances of the coupled line yields
i1(z)−i1(z+dz)= −j ω(Cg+CM)v1(z)dz+j ωCMv2(z) dz
= −j ω[(Cg+CM)v1(z)−CMv2(z)] dz (4-29)
L0
L0 LM CM Cg
Cg
+ i1(z)
i2(z)
i1(z + dz)
i2(z + dz)
dz
v1(z + dz) v2(z + dz)
v1(z) v2(z)
− − +
− − + +
Figure 4-8 Differential circuit subsection for two lossless coupled transmission lines.
i2(z)−i2(z+dz)= −j ω[(Cg+CM)v2(z)−CMv1(z)] dz (4-30) di
dz = −j ωCv (4-31)
where di dz= 1
dz
i1(z)−i1(z+dz) i2(z)−i2(z+dz)
taken in the limit as dz→0 C=
Cg+CM −CM
−CM Cg+CM
v= v1(z)
v2(z)
Notice in equations (4-29) and (4-30) that the current change on line 1 caused byv1is proportional to the sum of the capacitance to groundCg and the mutual capacitance between lines CM. This is consistent with our earlier discussion of mutual capacitance. We can also reassure ourselves that this is correct by con- sidering the response to a potential stimulus. Let usfirst assume that a potential, v, is applied to line 1 (relative to ground), while no potential is applied to line 2.
In this case, we must charge up both the capacitance to ground and the mutual capacitance between lines, a result that (4-29) predicts. In the second case we assume that the same potential is applied to both lines. In this situation, lines 1 and 2 remain at the same potential, so that no charge is stored in the electric field between the lines. Therefore, line 1 need charge up only the capacitance to ground, a result that is also predicted by (4-29).
Returning to our derivation, we differentiate (4-28) with respect to zto get d2v
dz2 = −j ωLdi
dz (4-32)
Substituting fordi/dzfrom (4-31) results in the coupled voltage wave equation d2v
dz2 =ω2LCv (4-33)
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.