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3-wire transmission line analysis

When a transmission line has three conductors (e.g. a directional coupler, a microstrip with both signal traces and a ground bridge), there is no single \(Z_0\): instead there are three pair-wise impedances.

Y-decomposition

Following atlc2 (and standard microwave-engineering practice), model the 3-conductor line as three impedances in a "Y" configuration:

        ZoR
   +1 ───────●
              \
               ●─── center
              /
   −1 ───────●
        ZoB
   ───── ZoG ────── ground

A single atlc2/lineforge run with only two of the three conductors active at a time (the third floating) gives one of the three pair-wise impedances:

Run Active conductors Result
1 red (+1), blue (−1) only; green floats \(Z_{0,\text{GCZ}}\) ("green current zero")
2 red, green; blue floats \(Z_{0,\text{BCZ}}\)
3 green, blue; red floats \(Z_{0,\text{RCZ}}\)

These satisfy:

\[ \begin{aligned} Z_{0,\text{RCZ}} &= Z_{0,B} + Z_{0,G} \\ Z_{0,\text{GCZ}} &= Z_{0,R} + Z_{0,B} \\ Z_{0,\text{BCZ}} &= Z_{0,R} + Z_{0,G} \end{aligned} \]

Solve for the Y-impedances:

\[ Z_{0,R} = \tfrac{1}{2}(Z_{0,\text{GCZ}} + Z_{0,\text{BCZ}} - Z_{0,\text{RCZ}}) \]

(and analogous for \(Z_{0,G}\), \(Z_{0,B}\)).

Coupler odd/even modes

For a quarter-wave directional coupler (a common 3-wire geometry. Two parallel signal traces over a ground plane), it's standard to characterize the odd mode (push-pull drive, \(V_R = -V_B\)) and even mode (in-phase drive, \(V_R = V_B\)). atlc2 provides a shortcut: a single run with the green ground floating, where the floating voltage equals the center voltage \(V_C\):

\[ \begin{aligned} Z_{0,B} &= Z_{0,\text{GCZ}} \cdot (V_\text{float} + 1) / 2 \\ Z_{0,R} &= Z_{0,\text{GCZ}} - Z_{0,B} \\ Z_{0,G} &= Z_{0,R} \\ Z_\text{ODD} &= 2 \cdot Z_{0,R} \\ Z_\text{EVEN} &= Z_{0,R}/2 + Z_{0,B} \end{aligned} \]

Caveat from atlc2 docs: this single-run approach is convenient but often

5% off due to the asymmetric charge distribution producing radiation that the single-mode solver doesn't capture correctly. For accurate coupler characterization, run two solves directly with the right boundary conditions for each mode (Phase 4's solve_differential API).

Net current and ZoG warning

If the grounded conductor carries a non-zero net current \(I_\text{gnd}\) (reported as Ignd in atlc2 / result.ignd_pct in lineforge), the line is radiating and the reported \(Z_0\) is not a true characteristic impedance.

Rule of thumb: if \(|I_\text{gnd}/I_R| > 4\%\), reconsider the geometry. Something is wrong with the assumed boundary conditions, or the line is unsuitable as a transmission line as defined.

References

  • atlc2 docs §"A discussion of 3-wire transmission lines": http://www.hdtvprimer.com/kq6qv/atlc2.html.
  • E. M. T. Jones et al., Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech 1980, Chapter 5 (directional couplers).
  • D. Pozar, Microwave Engineering, 4th ed., §7.6 (coupled-line couplers).