The use of printed-circuit differential transmission lines in modern digital systems is increasing. Formerly, printed-circuit differential transmission lines were used for RF distribution, directional couplers and filters. Now, differential interface standards like EIA/TIA-644 and BLVDS are spurring their use in pulse signal handling systems. But the theory and design approaches used for RF applications do not adequately address the design requirements for differential lines in digital applications; these require different design approaches and mathematical design tools. So, to more easily understand these transmission lines in the digital environment, let’s review a few definitions and concepts: Then we can explore a simplified method for design that gives sufficient accuracy for all but the most demanding uses.

Transmission lines on printed circuit boards (see Fig. 1) are a system consisting of two or more conducting paths: A conductor, or trace, and another trace and/or conducting plane in close proximity. Here, for the geometries of microstrip and stripline, conductor width is symbolized as w, conductor thickness as t, edge-to-edge conductor spacing as s and dielectric height or thickness as h. The dielectric height, h, is often symbolized as b when describing stripline. Both types of line have image planes, sometimes called virtual-ground planes, which may be either circuit ground or power planes. Microstrip has a surface conductor separated from a plane by a dielectric or insulator material. Stripline’s conductor is embedded in the dielectric and located, centered or otherwise, between two conducting planes.

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Fig. 1: Essential mechanical differences between surface microstrip and sandwiched stripline

The dielectric layer in these structures is described by its property, permittivity or dielectric constant (e_r), relative to that of free space. The permittivity of common dielectrics is greater than 1 (the permittivity of free space.) The effective permittivity for microstrip, a combination of that of air above the lines and the board insulator material below the lines, is greater than 1 but less than the e_r of the base material. The effective e_r for stripline is that of the dielectric embedding the conductor.

Differential transmission lines consist of two strip conductors spaced closely and forming a complete conducting loop path for the signal. To function as a coupled differential pair, the spacing between the conductors must normally be less than w or h. A conducting plane is not needed for there to be a complete transmission path. Naturally, printed-circuit differential lines may also retain their planes as image conductors thereby adding other conducting paths.

The basic electrical properties characterizing an individual microstrip or stripline are its distributed capacitance and inductance per unit length, symbolized as C₀ and L₀. (Other factors such as conductor losses can come into play for long lines but are usually negligible for the majority of circuit board applications.) These properties exist for the conductor itself (L₀) or in concert with its image conductor formed by a virtual-ground plane as in microstrip or the two virtual-ground planes for stripline (C₀).

From these properties, the characteristic impedance is defined as . The property impedance results from a dynamic signal condition: the ratio of dynamic voltage to current at a corresponding point on the line system. Though often thought of as a static value, impedance exists and has practical importance only when a dynamically changing signal is present on the transmission line system.

Traditionally, the properties of transmission lines are evaluated assuming that the applied excitation produces an ideal propagating transverse-electromagnetic (TEM) field. This condition is valid for stripline but not for microstrip because of its composite dielectric interface with air. Impedance relationships are derived based on the line’s properties, usually using a quasi-static signal approach. However, the frequency dependent nature of the effective-dielectric constant and characteristic impedance are better handled by applying full-wave analysis techniques.

Capacitance is the easiest line property to measure or calculate. One common method assumes excitation is purely TEM and that a quasi-static charge is present on the conductors. The region around the conductor is sub-divided into areas and the electrostatic capacitance is found for each area from the line’s geometrical properties. The area capacitances are summed into an overall C₀. Inductance, on the other hand, is difficult to measure directly and so is normally computed from conductor geometry, or indirectly from a measurement of the line’s propagation delay. Once these intrinsic properties are determined, then the line’s impedance can be computed. However, pulse-excitation, as used in digital signaling, does not result in TEM-mode propagation. This does affect the application and accuracy of impedance relationships based on these assumptions, which are the majority in current literature and usage.

The design and evaluation of differential lines presents not only the difficulties just mentioned but also own unique problems. Until recently, most of the interest in differential-line structures concerned their use in microwave directional-couplers and filters. But adaptation of computational design methods for these structures to differential-transmission lines is not simple, particularly in the case of coupled striplines.

A differential-line structure can simultaneously support two different modes of propagation, depending upon the polarities of the signals present on the two conductors. These modes are usually referred to as even - with both excitation signals having the same instantaneous polarity - and odd - having opposite polarities. These excitation modes cause the differential-line system to exhibit different characteristic impedances depending upon the instantaneous polarity of the propagating signal. For microstrip the two modes even have different velocities of propagation.

If the lines can be considered as operating in the TEM mode, a quasi-static analysis can approximate both the self- and mutual-inductance and capacitance characteristics accurately enough for practical design use. Even so, analysis and synthesis of differential-line structures can be difficult and computationally intensive. Not everyone has the computational power, or software, needed to effect such designs. Perhaps fortunately, absolute precision in the manufacture of printed-circuit board transmission lines is rarely achieved in practice. So, a simple, practical, method for designing differential lines with acceptable accuracy is most welcome.

A simplified method for designing differential lines does exist. This is based on the relationship of the constants used to express crosstalk voltages between conductors, and the self-impedances exhibited by those conductors. These approximations are sufficiently accurate for practical designs. The key concept of this approach is the relationship that exists between the crosstalk constants and the mutual-inductance and capacitance of the coupled-line pair. Moreover, this approach allows the computation of differential impedances and intrinsic parameters directly from the more easily found self-impedances of the individual lines. These self-impedances for microstrip and stripline can be found with simplified, closed-form expressions based on line geometry (see reference 7.) When teamed with expressions or graphs of the backward (or reverse) crosstalk-constant, a simple method of designing differential lines results.

The backward crosstalk that exists for both microstrip and stripline structures has been extensively investigated and documented (see references.) The backward crosstalk-constant is expressed (assuming the most common case where both lines are the same size) as:

,

where, L_m and C_m are the mutual-inductance and capacitance of the line pair,

T_d is the propagation velocity of the individual lines ,

and Z₀ is the characteristic self-impedance of the individual lines.

For actual design of differential-line geometries, published tables or graphs can provide the needed backward crosstalk-constants. The differential impedance, Z_D, can then be expressed simply as: . This approach makes differential-line design quite a bit easier.

Though tables and graphs of the backward crosstalk-constant are available for various geometries, K_B can be approximated with sufficient accuracy using these relationships:

for microstrip, and,

for stripline, .

Combining these expressions for K_B with that for Z_D gives two simple relationships by which differential-line geometries can be synthesized or analyzed. For microstrip the relationship is:

.

For stripline the relationship is:

.

The simplicity of these relationships allows implementation of a slide-rule-type calculator. National Semiconductor’s Transmission Line RAPIDESIGNER slide-rule is a simple, powerful tool for designing differential and other types of transmission lines. Its use in the design of many differential-line systems on circuit cards and backplanes has demonstrated the tool’s practicality and accuracy. Perhaps the nicest thing about the slide-rule is its cost . . . free. The Transmission Line RAPIDESIGNER is available through any of National Semiconductor’s offices or customer response centers worldwide. Ordering information is also available on National’s Web site.

Bibliography:

1. Ivor Catt, "Crosstalk (Noise) in Digital Systems," IEEE Transactions on Electronic Computers, Vol. EC-16, No. 6, Dec. 1967, pp. 743-763.
2. H.R. Kaupp, "Pulse Crosstalk Between Microstrip Transmission Lines," Proceedings - International Electronic Circuit Packaging Symposium, Aug. 1966, Vol. 2/5, pp. 1-12.
3. A. Feller, H.R. Kaupp and J.J. DiGiacomo, "Crosstalk and Reflections in High-Speed Digital Systems," Proceedings - Fall Joint Computer Conference, 1965, pp. 511-525.
4. John A. DeFalco, "Predicting Crosstalk in Digital Systems," Computer Design, June 1973, pp. 69-75.
5. N.C. Arvanitakis, J.T. Kolias and W. Radzelovage, "Coupled Noise Prediction in Printed Circuit Boards for a High-Speed Computer System," Proceedings - International Electronic Circuit Packaging Symposium, Aug. 1966, Vol. 2/6, pp. 1-11.
6. John B. Connolly, "Cross Coupling in High Speed Digital Systems," IEEE Transactions on Electronic Computers, Vol. EC-15, No. 3, June 1966, pp. 323-327.
7. James A. Mears, "Transmission Line RAPIDESIGNER Operation and Applications Guide", AN-905, National Semiconductor Corp., 1996, http://www.national.com/an/AN/AN-905.pdf.