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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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Presentation on theme: "Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 15 ECE 6340 Intermediate EM Waves 1."— Presentation transcript:

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

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

3 Attenuation Formula (cont.) Hence If so 3

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

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

6 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.

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

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

9 Hence Attenuation on Line (cont.) 9

10 R on Transmission Line Ignore G for the R calculation (  =  c ): R  z L  z CzCz G  z z z I 10

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

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

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

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

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

16 Example (cont.) Calculate R: 16

17 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

18 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

19 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

20 Skin Inductance (cont.) Hence: 20

21 Skin Inductance (cont.) Hence or 21

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

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

24 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 , Feb 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.)

25 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

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

27 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 ,

28 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 nn 28 L 0 = external inductance (assuming perfect conductors).

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

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

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

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

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

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

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

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

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

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

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

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

41 so Hence Attenuation in dB (cont.) 41

42 or Attenuation in dB (cont.) 42

43 Appendix: Tesche Model C ZaZa ZbZb G zz 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

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

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

46 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.)

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


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