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By extending the dc resistance equation (4.1), the frequency dependence of the resistance in a transmission line conductor can be approximated. Frequency-dependent resistance will sometimes be referred to ac resistance in this book. At low frequencies, the ac resistance will be identical to the dc resistance because the skin depth will be much greater than the thickness of the conductor. The ac resistance will remain approximately equal to the dc resistance until the frequency increases to a point where the skin depth is smaller than the conductor thickness. Figure 4.2 depicts the current distribution on a microstrip line at high frequencies. Notice that the current distribution is concentrated on the bottom edge of the transmission line. This is because the fields between the signal line and the ground plane pull the charge to the bottom edge. Also notice that the current distribution curves up the side of the conductor. This is because there is still significant field concentration along the thickness (the t dimension in Figure 4.2) of the conductor. The amount of cross-sectional area

Figure 4.2: Current distribution on a microstrip transmission line. 63% of the current is

in which the current is flowing will become smaller as the frequency increases [see equation (4.2)].

The losses in the conductor can be approximated using the dc resistance and the skin effect formulas by substituting the skin depth for the conductor thickness:

(4.3a)

Note that the approximation is valid only when the skin depth is smaller than the conductor thickness. Furthermore, equation (4.3a) is only an approximation because it assumes that all the current is flowing in the skin depth and is presented in this form for instructional purposes only. As this section progresses, more accurate methods of calculating the ac losses will be presented. Notice that when the skin depth equation (4.2) is inserted into the dc resistance equation, the ac resistance becomes directly proportional to the square root of the frequency

F and the resistivity ρ. Note that in equation (4.3a) the length term has been excluded to give the ac resistance units of resistance per unit length.

Figure 4.3 is a plot of the skin depth versus frequency for a copper conductor. Note that the skin depth is greater than the conductor thickness at frequencies below approximately 1.7 MHz. Figure 4.4 is a plot of resistance as a function of frequency for the example copper cross section. Note that the initial portion of the curve is constant at the dc resistance. This section

Figure 4.4: Ac resistance as a function of frequency.

corresponds to frequencies where the skin depth is greater than the conductor thickness. The curve begins to change with the square root of frequency when the skin depth becomes smaller than the conductor thickness. Although the curve shown in Figure 4.4 is not based on an exact model, it is well suited to help the reader understand the fundamental behavior of skin effect resistance. A good way to match measurements when simulating both the ac and dc resistance of a transmission line in a simulator such as SPICE is to combine Rac and

Rdc:

(4.3b)

The skin effect resistance of the conductor, however, is only one part of the total ac resistance. The portion that is not included in equation (4.3a) is the resistance of the return current on the reference plane. The return current will flow underneath the signal line in the reference plane and will be concentrated largely in one skin depth and will spread out perpendicular to the trace direction, with the highest amount of current concentrated directly beneath the signal conductor. An approximate current density distribution in the ground plane for a microstrip transmission line is [Johnson and Graham, 1993]

(4.4)

Figure 4.5: Current density distribution in the ground plane.

where Io is the total signal current, D the distance from the trace (see Figure 4.5), and H the

height above the ground plane. Figure 4.5 is a graphical representation of this current density distribution.

An approximation of the effective resistance of the ground plane can be derived using a technique similar to that used to find the ac resistance of the signal conductor. First, since 63% of the current will be confined to one skin depth (δ), then for the resistance calculation, the approximation may be made that the ground current flows entirely in one skin depth, as was approximated for the signal conductor ac resistance. Second, the equation

(4.5)

shows that 79.5% of the current is contained within a distance of ±3H (6H total width) away from the center of the conductor. Thus, the ground return path resistance can be

approximated by a conductor of cross section Aground = δ × 6H. Substituting this result into

equation (4.1) yields (4.6)

The total ac resistance is the sum of the conductor and ground plane resistance: (4.7)

(4.8)

Equation (4.8) should be considered a first-order approximation. However, since surface roughness can increase resistance by 10 to 50% (see "Effect of Conductor Surface Roughness" below), equation (4.8) will probably provide an adequate level of accuracy for most situations.

A more exact formula for the ac resistance of a microstrip can be derived through conformal mapping techniques.

(4.9)

Equation set (4.9) was derived using conformal mapping techniques and appears to have excellent agreement with experimental results [Collins, 1992]. These formulas are

significantly more cumbersome than (4.8) but should yield the most accurate results. Equation (4.8) will tend to yield resistance values that are larger then those in (4.9). Often, the slightly larger values given by (4.8) are used to roughly approximate the additional resistance gained from surface roughness.