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Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 15 ECE 6340 Intermediate EM Waves 1

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At z = 0 : At z = z : Attenuation Formula Waveguiding system (WG or TL): S Waveguiding system 2

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Attenuation Formula (cont.) Hence If so 3

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Attenuation Formula (cont.) From conservation of energy: where = power dissipated per length at point z so 4 S

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Attenuation Formula (cont.) Hence As z 0 : Note: The point z = 0 is arbitrary. 5

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Attenuation Formula (cont.) General formula: z0z0 This is a perturbational formula for the conductor attenuation. 6 The power flow and power dissipation are usually calculated assuming the fields are those of the mode with PEC conductors.

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Attenuation on Transmission Line Attenuation due to Conductor Loss The current of the TEM mode flows in the z direction. 7

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Attenuation on Line (cont.) C= C A + C B Power dissipation due to conductor loss: Power flowing on line: zz S A B CACA CBCB I ( Z 0 is assumed to be approximately real.) 8

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Hence Attenuation on Line (cont.) 9

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R on Transmission Line Ignore G for the R calculation ( = c ): R z L z CzCz G z z z I 10

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R on Transmission Line (cont.) so Hence Substituting for c 11

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Total Attenuation on Line Method #1 so Hence, If we ignore conductor loss, we have a TEM mode. 12

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Total Attenuation on Line (cont.) Method #2 The two methods give approximately the same results. where 13

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Example: Coax Coax I I z a b A B 14

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Example (cont.) Hence Also, Hence (nepers/m) 15

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Example (cont.) Calculate R: 16

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Example (cont.) This agrees with the formula obtained from the “DC equivalent model.” (The DC equivalent model assumes that the current is uniform around the boundary, so it is a less general method.) b a DC equivalent model of coax 17

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Skin Inductance R z C zC z G z L z L 0 z This extra (internal) inductance consumes imaginary (reactive) power. The “external inductance” L 0 accounts for magnetic energy only in the external region (between the conductors). This is what we get by assuming PEC conductors. An extra inductance per unit length L is added to the TL model in order to account for the internal inductance of the conductors. 18

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Skin Inductance (cont.) R z C zC z G z L z L 0 z Imaginary (reactive) power per meter consumed by the extra inductance: Skin-effect formula: I Circuit model: Equate 19

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Skin Inductance (cont.) Hence: 20

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Skin Inductance (cont.) Hence or 21

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Summary of High-Frequency Formulas for Coax Assumption: << a 22

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Low Frequency (DC) Coax Model At low frequency (DC) we have: a b c t = c - b Derivation omitted 23

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Tesche Model This empirical model combines the low-frequency (DC) and the high-frequency (HF) skin-effect results together into one result by using an approximate circuit model to get R( ) and L( ). F. M. Tesche, “A Simple model for the line parameters of a lossy coaxial cable filled with a nondispersive dielectric,” IEEE Trans. EMC, vol. 49, no. 1, pp. 12-17, Feb. 2007. Note: The method was applied in the above reference for a coaxial cable, but it should work for any type of transmission line. 24 (Please see the Appendix for a discussion of the Tesche model.)

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Twin Lead a x y h a x y h DC equivalent model Twin Lead Assume uniform current density on each conductor ( h >> a ). 25

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Twin Lead a x y h or 26 (A more accurate formula will come later.)

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Wheeler Incremental Inductance Rule x y A B L 0 is the external inductance (calculated assuming PEC conductors) and n is an increase in the dimension of the conductors (expanding the surface into the active field region). Wheeler showed that R could be expressed in a way that is easy to calculate (provided we have a formula for L 0 ): H. Wheeler, "Formulas for the skin-effect," Proc. IRE, vol. 30, pp. 412-424, 1942. 27

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The boundaries are incremented a small amount n into the field region Wheeler Incremental Inductance Rule (cont.) PEC conductors x y A B S ext nn 28 L 0 = external inductance (assuming perfect conductors).

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Derivation of Wheeler Incremental Inductance rule Wheeler Incremental Inductance Rule (cont.) Hence We then have PEC conductors x y A B S ext nn 29

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Wheeler Incremental Inductance Rule (cont.) PEC conductors x y A B S ext nn 30 From the last slide, Hence

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Wheeler Incremental Inductance Rule (cont.) Example 1: Coax a b c 31

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Example 2: Twin Lead a x y h Wheeler Incremental Inductance Rule (cont.) 32 From image theory:

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Example 2: Twin Lead (cont.) a xy h Wheeler Incremental Inductance Rule (cont.) Note: By incrementing a, we increment both conductors simultaneously. 33

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Example 2: Twin Lead (cont.) a xy h Wheeler Incremental Inductance Rule (cont.) 34 Summary

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Attenuation in Waveguide A waveguide mode is traveling in the positive z direction. 35 We consider here conductor loss for a waveguide mode.

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Attenuation in Waveguide (cont.) or Hence Power flow: Next, use 36

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Assume Z 0 WG = real ( f > f c and no dielectric loss) Hence Attenuation in Waveguide (cont.) Vector identity: 37

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Then we have Attenuation in Waveguide (cont.) C S x y 38

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Total Attenuation: Attenuation in Waveguide (cont.) Calculate d ( assume PEC wall ) : where 39 so

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Attenuation in dB Use z = 0 z S Waveguiding system (WG or TL) 40

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so Hence Attenuation in dB (cont.) 41

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or Attenuation in dB (cont.) 42

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Appendix: Tesche Model C ZaZa ZbZb G zz L0L0 The series elements Z a and Z b account for the finite conductivity, and give us R and L for each conductor at any frequency. 43

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Appendix: Tesche Model (cont.) Inner conductor of coax Outer conductor of coax The impedance of this circuit is denoted as 44

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Inner conductor of coax At low frequency the HF resistance gets small and the HF inductance gets large. 45 Appendix: Tesche Model (cont.)

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Inner conductor of coax At high frequency the DC inductance gets very large compared to the HF inductance, and the DC resistance is small compared with the HF resistance. 46 Appendix: Tesche Model (cont.)

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The formulas are summarized as follows: 47 Appendix: Tesche Model (cont.)

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