IDEAL TRANSMISSION-LINE FUNDAMENTALS
3.3 TRANSMISSION-LINE PROPERTIES
3.3.5 Validity of the TEM Approximation
The assumptions of how the electric and magnetic fields are related for a propagating electromagnetic wave were discussed in Section 2.3.2. One of
t = 0 t = t1 t = t2
er > 1 er = 1
Ex_dielectric
Ex_air
Ex
Ez
Direction of signal propagation (+z)
Figure 3-13 The electric field develops a z-component when propagating down a microstrip transmission line, due to the nonhomogeneous dielectric.
the fundamental concepts used throughout signal integrity analysis is that the electric and magneticfields are orthogonal and there are no components in the z-direction. When waves propagate in this manner, it is called the transverse electromagnetic mode (TEM). However, when discussing the concept of an effective dielectric permittivity in Section 3.3.3, where the component of the electric field propagating through the air travels faster than the component propagating in the board material, it becomes obvious that the electric field is no longer restricted to a single component. For example, consider Figure 3-13, which depicts the side view of an electricfield established in thex-direction at t =0 on a microstrip transmission line between the signal conductor and the reference plane. As the signal begins to propagate down the line, the electricfield lines in the air will travel at a faster speed than those in the board, effectively tilting the electricfield in thez-direction. Consequently, the electricfield devel- ops a component in thez-direction which violates the assumption of the TEM approximation.
Furthermore, as the frequencies increase, the electricfield will become more confined to the region between the microstrip and the reference plane, result- ing in less fringing through the air, causing the effective dielectric permittivity to increase. To understand this, refer to Figure 3-14. When a dc voltage is applied between the signal conductor and the reference plane, the charge will be distributed uniformly across the cross section of the signal conductor. As the frequency of the signal is increased, the charge will tend to concentrate at the bot- tom of the signal conductor closest to the reference plane because that is the area of highestfield concentration. This means that for a microstrip transmission line, the electricfield will tend to concentrate in the board material, which increases the effective dielectric permittivity with increasing frequency. The charge distribution in transmission lines is discussed in more detail in Sections 3.4.4 and 5.1.2.
er= 1
Increased charge density
er> 1
Figure 3-14 At high frequencies, the proximity of thefields concentrates the charge in the bottom of the strip nearest the reference plane.
The frequency-dependent nature of the effective permittivity in a microstrip will cause the spectral components of the digital waveform (as calculated with a Fourier transform) to travel at different speeds, which will distort the waveform.
This is known as dispersion. A relatively simple formula for calculating how the effective dielectric permittivity for a microstrip changes with frequency due to the nonhomogeneous nature of the dielectric was developed empirically by [Collins 1992], and is given by
εeff(f )=εr−εr −εeff(f =0)
1+(f/fa)m (3-37)
where
fa= fb
0.75+(0.75−0.332ε−1.73r )(w/ h) fb= 47.746
h√
εr−εeff(f =0)tan−1
εr
εeff(f =0)−1 εr−εeff(f =0)
m=m0mc≤2.32 m0=1+ 1
1+√
w/ h+0.32
1+ w
h −3
mc=
1+ 1.4 1+w/ h
0.15−0.235e−0.45(f/fa) w h ≤0.7
1 w
h >0.7
whereεeff(f =0) is calculated with (3-35),f is in gigahertz, and the units ofw andhare millimeters.∗
∗Fortunately, a frequency-dependent effective dielectric permittivity does not pose significant obsta- cles to modeling transmission lines. In Chapter 10, techniques that employ frequency-dependent equivalent circuits using tabular SPICE models are described that allow a unique value of the transmission-line parameters to be described at every frequency point.
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
1 10 100
Frequency (GHz) er,eff
h = 2.5 mils h = 25 mils
Quasi-TEM approximation er= 1
w = 2h
er= 4.5 h
Figure 3-15 Effective dielectric permittivity compared to the quasi-TEM approximation for a 12-in. microstrip.
For most practical applications encountered in high-speed digital design, the TEM approximations are valid and the frequency-dependent nature described by (3-37) is ignored. For microstrip lines and other nonhomogeneous struc- tures, when TEM propagation is assumed, it is referred to as the quasi-TEM approximation. To demonstrate the conditions when the quasi-TEM approxima- tion breaks down, equation (3-37) was used to calculate the frequency variation of the relative effective dielectric permittivityεefffor two cases, a thick and a thin dielectric, as shown in Figure 3-15. The thin dielectric example (h=2.5 mils) represents the transmission-line dimensions typically used to design buses on conventional motherboards for personal computers. The thick dielectric example (h=25 mils) is an exaggerated case chosen to demonstrate when the TEM approximation breaks down (however, similar dimensions are sometimes used in radio-frequency (RF) applications where density is not as much of a concern).
Equation (3-35) was used to calculate the frequency-invariant quasistatic TEM value forεeff, and equation (3-37) was used to estimate the variation ofεeff with frequency. Note that for the thin case, the deviation from the quasi-TEM approx- imation is small up to very high frequencies, but the thick case begins to deviate much earlier.
To evaluate when the quasi-TEM approximation breaks down, we choose a metric of 1% error in total delay. Since the changes inεeffwill alter the velocity as shown in equation (2-52), the errors will accumulate for longer line lengths. If we choose a line length, the valid frequency range of the TEM approximation can be calculated. Figure 3-16 shows the percent error in propagation delay caused by the quasi-TEM approximation for a 12-in. microstrip. Note that for the thick case, the quasi-TEM will induce a 1% error in delay at about 8 GHz, and the thin case remains accurate to about 80 GHz. Consequently, for typical transmission-line
0 1 2
2 20 40 60 80 100
3 4 5 6 7 8 9 10
1% error threshold h = 25 mils h = 2.5 mils
Frequency (GHz)
Quasi-TEM error (%)
Figure 3-16 Bandwidth where the quasi-TEM approximation induces a 1% error in the delay for a 12-in. microstrip.
dimensions used in contemporary digital design, the quasi-TEM approximation for the transmission-line parameters is valid.
3.4 TRANSMISSION-LINE PARAMETERS FOR THE LOSS-FREE CASE