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Negative inductance and negative capacitance of bend TL

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McFlasher

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Hello everyone,

during my study I came across this forum and wanted to post a question to the community, despite having read the already existing threads.

I am looking for a formulation of the inductance and capacitance for the bend of a two-wire Transmission Line (TL). I found some papers where these quantities are derived analytically using the vector potential.

All of these papers indicate a negative value for the inductance and the capacitance of the bend, which seems to me mathematically correct. However, I have a problem in understanding the physical meaning of such an expression, since there are no active elements at all and the derivation is (at least in one paper) not related to time-varying currents.

Maybe some of you have an idea on how to interpret such a negative quantitiy?

Sources:
SCIENCE APPLICATIONS INC BERKELEY CALIF, "Equivalent Lumped Parameters for a Bend in a Two-Wire Transmission Line.", Jul 1977
Abstract:
"This report calculates the equivalent lumped parameters of a bend in a parallel-wire transmission line. The equivalent lumped inductance of the bend is calculated exactly within the thin-wire assumption. The inductance for an abrupt bend is obtained explicity in simple closed form. The inductance for a smooth bend modeled by a circular arc is reduced to one-dimensional integrals that can readily be computed. The equivalent lumped capacitance of the bend is calculated from a variational principle. The capacitance of an abrupt bend is obtained explicity in simple form. The capacitance of a circular bend is expressed in terms of one-dimensional integrals to be computed numerically. (Author)"
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K. Tomiyasu "The Effect of a Bend and Other Discontinuities on a Two-Wire Transmission Line"; Proceedings of the IRE, Vol. 38, Issue 6, June 1950
Abstract:
The theoretical problem of a bend on a two-wire transmission line is analyzed by the vector-potential method. The equivalent circuit elements are obtained by comparing the variableline parameters near the bend with the conventional line parameters found on an infinite line. The effect of the nonrotational symmetry of the axial vector potential on the conductor surfaces is treated by an approximate method to obtain one of the equivalent series elements. Experimental values of the circuit elements for the bend were found to be in good agreement with the theoretical. Good agreement was also found between experimental and theoretical values for open-ended and bridged-end lines.
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J. Kasper ; R. Vick "The Effect of a Bend on the Stochastic-Field Coupling to a Single Wire Transmission Line Over a Conductive Ground Plane"; EMC Europe 2019
Abstract:
The classical transmission line theory does not take not into account the variations in the parameters of a transmission line or any change in its uniform properties as well as the influence on coupling of not straight, e.g. bent wires. In this paper, a theory is derived which uses simple position dependent per-unit-length parameters to account to these effects. The theoretical problem of a bend on a single wire transmission line is analyzed based on the wave equations and the solution of the Green's functions. The model is validated against a simulation model based on the method of moments. Moreover, the new model is compared to an already existing simulation model based on the BLT equations (named after Baum-Liu-Tesche) for the field-to-wire coupling of a stochastic field to transmission line networks, which does not take into account the coupling between the lines.
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The quantities Ld and Cd represent the inductance and the capacitance of the bend alone. There isn't a physical meaning they just indicate the difference in inductance and capacitance of a cable with and without bend. So a negative value indicate that the inductance (or the capacitance) of a bent cable is less than that of a straight cable.

Ltotal = Lstraight + Ld
Ctotal = Cstraight + Cd
 
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    FvM

    FvM

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Thank you for your answer!

I understand your point but:
In general the bend should be able to be modeled in an equivalent circuit, like it is done in the field of microstrip arrays (T-Network). Even if the value of the bend (Ld, Cd) is negativ, the total value of the respective quantity should be positive nevertheless. Otherwise it would be some kind of perpetuum mobile.
If the derived p.u.l quantities of the bend from Tomiyasu and Kasper were to be used in such an equivalent circuit Ctotal and Ltotal would be negative (in certain ranges).
 

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