ATLC (Arbitrary Transmission Line Calculator) based on finite difference is used to find the characteristic impedance of a pair of parallel square conductors in air.
ATLC cannot calculate Zo for conductors in an unbounded space, so the conductors must be contained within a grounded bounding surface. In this case a rectangular form is used. Fig 1 shows a structure that ATLC can model, it is two conductors contained within a grounded bounding box. To obtain results that approach a pair of conductors in free space, the bounding box needs to be large wrt the conductor size and spacing. The bounding box is a square of 1000mm sides and the conductors are square of 20mm sides. Increasing the size of the bounding box increases Zo by less than 1% for the dimensions modelled.
For computing efficiency, the symmetry about the neutral plane (the vertical axis of symmetry in Fig 1) has been exploited to halve the problem, and a single square conductor is modelled within the rectangular bounding box to give half the differential Zo of a pair of such conductors with twice the spacing. Fig 2 shows an example of the reduced model that was computed.
The key outcome of the modelling is a relationship between Zo and the ratio of D/d, the ratio of centre to centre spacing to side length. Fig 3 shows the model points and a fitted curve. The curve fit is an exponential form, D/d=A+B*e^(C*Zo) where:
Alternatively, by rearranging the expression, Zo=ln((D/d-A)/B)/C (or Zo=105.2*ln(2.475*D/d-1.3359))).
For D/d>=1.2, the error between the exponential model and the ATLC simulation is less than 1% of D/d. Extrapolation beyond the range modelled may yield inaccurate results. The ATLC simulation probably does not account for proximity effect (since the model does not include conductivity of the conductors), so Zo figures below about 100Ω are likely to be underestimates, more so with lower Zo.
ATLC uses a lossless model, so Zo will always be real.
Transmission line sections can be used as part of an impedance matching arrangement such as quarter wave length lines, stubs etc.
Another application is the transmission line used to feed a log periodic array, where a pair of square tubes are used for the boom / transmission line, and element halves are mounted each side of the transmission line.
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