FUNDAMENTALS FOR SIGNAL INTEGRITY
2.3 WAVE PROPAGATION
2.3.4 Propagation of Time-Harmonic Plane Waves
As will be demonstrated in subsequent chapters, the propagation of time-harmonic plane waves is of particular importance for the study of transmission-line or other guided-wave structures. This allows us to study a simplified subset of Maxwell’s equations where propagation is restricted to one direction (usually along the z-axis) and time is removed as described in Section 2.3.3. A plane wave is defined so that propagation occurs in only one direction (z) and the fields do not vary with time in the x- and y-directions. If the fields were observed at an instant in time, they would be constant in thex–y plane for any given pointz and would change for different values ofzort. Figure 2-11 depicts a plane wave propagating in thez-direction.
To study the behavior of time-harmonic plane waves, it is necessary to re-derive the wave equation from the time-harmonic form of Maxwell’s
x
y
z Direction of propagation
Figure 2-11 Plane wave propagating in thez-direction.
equations using the procedure employed in Section 2.3.1. Again, assume a source-free, linear, homogeneous medium:
∇ ×(∇ × E)= −j ωµ(∇ ×H )
The formula can be further simplified by using the following vector identity (Appendix A):
∇ ×(∇ × E)= ∇(∇ · E)− ∇2E
Since we have assumed a source-free medium, the charge density is zero (ρ=0) and Gauss’s law reduces to∇ · E =0, yielding
∇2E+j2ω2µεE= ∇2E−ω2µεE=0 (2-38) Substitutingγ2=ω2µε yields
∇2E−γ2E=0 (2-39)
which is thetime-harmonic plane-wave equation for the electricfield, where γ is known as the propagation constant.
If the solution is limited to plane waves propagating in thez-direction that have an electricfield component only in thex-direction, the wave equation becomes (see Appendix A)
∇2E= ∂2Ex
∂x2 +∂2Ex
∂y2 +∂2Ex
∂z2 = ∂2Ex
∂z2 −γ2Ex=0
∂2Ex
∂z2 −γ2Ex=0
(2-40)
which is a second-order ordinary differential equation with the general solution Ex=C1e−γ z+C2eγ z (2-41) where C1 and C2 are determined by the boundary conditions of the particular problem.
As discussed in Chapter 3, equation (2-41) and its magnetic field equiva- lent will prove to be particularly important for signal integrity because they describe the propagation of a signal on a transmission line. The first term, C1e−γ z, describes completely the forward-traveling part of the wave propagating in thez-direction (i.e., down the length of the transmission line), and the second term, C2e+γ z, describes the propagation of the rearward-traveling wave in the
−z-direction. Observing equation (2-41) allows the definition of an important term, thepropagation constant:
γ =α+jβ (2-42)
The terms in (2-42) have special meanings used throughout the book to describe the medium where the electromagnetic wave is propagating, whether it is in free space, is an infinite dielectric, or is a transmission line. Specifically,αis theloss term, which describes signal attenuation as it propagates through the medium. The loss term accounts for the fact that real-world metals are not infinitely conductive (except superconductors) and dielectrics are not perfect insulators (except free space), both of which are discussed in detail in Chapters 5 and 6. The imaginary portion of (2-42), β, called the phase constant, essentially dictates the speed at which the electromagnetic wave will travel in the medium. To visualize these waves propagating as described in (2-41), it is necessaryfirst to recover the time dependency removed in Section 2.3.3. Considering only the forward-propagating component of a wave in a vacuum, replacingC1with the magnitude of the electric field, restoring the time dependency as in (2-37), and applying the identity of equation (2-31) yields
E(z, t)=Re(Ex+e−γ zej ωt)=Re(E+xe−αze−jβzej ωt)=e−αzEx+cos(ωt−βz) (2-43) Assuming that the loss term is zero (α=0), Figure 2-12 depicts successive snapshots of a wave propagating though space. To determine how fast the wave is propagating, it is necessary to observe the cosine term for a small duration of time t. Since the wave is propagating, a small change in time will be proportional to a small change in distancez, which means that an observer moving with the wave will experience no phase change because she is moving at the phase velocity (νp).
Setting the term inside the cosine of (2-43) to a constant (ωt−βz=constant) and differentiating allows the definition of the phase velocity from the cosine term in (2-43):
νp = dz dt = ω
β m/s (2-44)
at t = t1 at t = t2 E+x cos (wt − bz)
at t = t0
∆t
∆t
∆z ∆z= νp
l
Figure 2-12 Snapshots in time of a plane wave propagating along thez-axis, showing the definition of phase velocity.
The relationship between the frequency and its wavelengths is calculated based on the speed of light, which is the phase velocity (νp) in a vacuum:
f = c
λ hertz (2-45)
Sinceω=2πf andcis the speed of light in a vacuum (ca. 3×108m/s), equation (2-45) can be substituted into (2-44) to obtain a useful formula forβ in terms of the wavelengthλ:
c= ω β = 2π c
βλ →β= 2π
λ rad/m (2-46)
The speed of light in a vacuum is defined as the inverse of the square root of the product of the permeability and the permittivity of free space:
c≡ 1
√µ0ε0 m/s (2-47)
Calculation ofλ in terms of (2-47) allows the phase constant β to be rewritten in terms of the properties of free space:
β=2πf√
µ0ε0=ω√
µ0ε0 rad/m (2-48)
This is expanded on later in this chapter to include propagation of a wave in a dielectric medium.
Now that the propagation constant has been defined, (2-43) can be rewritten in physical terms, assuming free space (which is lossless, soα=0):
E(z, t)=Re(Ex+e−j ωz√µ0ε0ej ωt)=Ex+cos(ωt−ωz√
µ0ε0) (2-49) Since (2-49) is a solution to the wave equation, the magneticfield is found simply by using Faraday’s law (∇ × E+j ωB =0):
∂
∂z(Ex+e−j ωz√µ0ε0)ej ωt = −j ωµ0Hy+ Hy+=
√µ0ε0
µ0 Ex+e−j ωz√µ0ε0ej ωt = 1
η0Ex+e−j ωz√µ0ε0ej ωt (2-50)
= 1
η0Ex+cos(ωt−ωz√ µ0ε0) whereη0 is theintrinsic impedance of free spaceand has a value of 377:
η0≡ µ0
ε0 =377 (2-51)
Equations (2-49) and (2-50) describe how a plane wave propagates in free space.
The intrinsic impedance and the speed of light are constants that describe how the electromagnetic wave will propagate through the medium. The speed of light defines the phase delay of the wave, and the intrinsic impedance describes the relationship between the electric and magnetic fields. However, for wave prop- agation in other media, such as the dielectric of a printed circuit board (PCB), the speed of light and the intrinsic impedance are calculated using the relative permittivityεr and relative permeabilityµr, which simply describe the properties of the material relative to free-space values. Note that bothµr andεr are unitless values that are real numbers for loss-free media but become complex for lossy media, as described in Chapters 5 and 6. Thespeed of light (referred to as the phase velocity for media other than free space) and the intrinsic impedance in a medium is calculated as
νp= 1
√µrµ0εrε0 = c
√µrεr
m/s (2-52)
η ≡
µrµ0
εrε0 = µ
ε = E
H ohms (2-53)
Note that for free space,µr and εr are both defined to be unity.
Equations (2-54) and (2-55) summarize the TEM plane waves of both the electric and magneticfields in general form, with the time dependency removed:
Ex(z)=Ex+e−γ z+Ex−eγ z (2-54) Hy(z)= 1
η(E+xe−γ z−E−xeγ z) (2-55) where γ =α+jβ is the propagation constant; α describes how the signal is attenuated by conductor and dielectric losses, described in full detail in Chapters 5 and 6; and β is the phase constant, as defined by (2-46) when the phase velocity in (2-52) is substituted for the speed of light in a vacuum(c). Note that the second term in (2-55) is negative. This is because the sign of the exponent for the reverse traveling wave does not cancel the negative sign in Faraday’s law as it did for the forward-traveling wave in equation (2-50) when the derivative with respect to zwas calculated. The termsEx+ and E−x describe the directions of each component of a propagating wave. For example, the total propagating wave could have a portion of the electric field propagating in the+z-direction and another propagating in the−z-direction. Figure 2-13 depicts a time-harmonic TEM plane wave propagating along thez-axis.