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Prof. David R. Jackson Dept. of ECE

Adapted from notes by Prof. Jeffery T. Williams. ECE 5317-6351 Microwave Engineering. Fall 2019. Prof. David R. Jackson Dept. of ECE. Notes 3. Transmission Lines Part 2: TL Formulas. Overview. In this set of notes we develop some general formulas that hold for any transmission line.

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Prof. David R. Jackson Dept. of ECE

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  1. Adapted from notes by Prof. Jeffery T. Williams ECE 5317-6351 Microwave Engineering Fall 2019 Prof. David R. Jackson Dept. of ECE Notes 3 Transmission Lines Part 2: TL Formulas

  2. Overview • In this set of notes we develop some general formulas that hold for any transmission line. • We first examine the coaxial cable as an example.

  3. Coaxial Cable Here we present a “case study” of one particular transmission line, the coaxial cable. Find C, L, G, R For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation of C and L using electrostatics and magnetostatics. We will assume no variation in the z direction, and take a length of one meter in the z direction in order to calculate the per-unit-length parameters.

  4. h = 1[m] Coaxial cable a l0 -l0 b Coaxial Cable (cont.) Find C (capacitance / length) From Gauss’s law:

  5. Coaxial Cable (cont.) Coaxialcable Hence We then have:

  6. Coaxial Cable (cont.) Find L(inductance / length) From Ampere’s law: Coaxial cable Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency. Magnetic flux: Center conductor S

  7. Coaxial Cable (cont.) Hence

  8. Coaxial Cable (cont.) Observations: (independent of frequency) (independent of frequency) This result actually holds for anytransmission line that is homogenously filled* (proof omitted). *This result assumes that the permittivity is real. To be more general, for a lossy line, we replace the permittivity with the real part of the permittivity in this result.

  9. Coaxial Cable (cont.) For a lossless (or low loss) cable:

  10. Coaxial Cable (cont.) Find G(conductance / length) Coaxial cable From Gauss’s law:

  11. Coaxial Cable (cont.) We then have or

  12. Coaxial Cable (cont.) Observation: This result actually holds for anytransmission line that is homogenously filled* (proof omitted). *This result assumes that the G term arises only from conductivity, and not polarization loss.

  13. Coaxial Cable (cont.) As just derived, Hence: This is the loss tangent that would arise from conductivity. or This result actually holds for anytransmission line that is homogenously filled* (proof omitted). *This result is very general, and allows the G term to come from either conductivity or polarization loss.

  14. Accounting for Dielectric Loss Complex Permittivity The permittivity becomes complex when there is polarization (molecular friction) loss. Loss term due to polarization (molecular friction) Example: Distilled water heats up in a microwave oven, even though there is essentially no conductivity!

  15. Accounting for Dielectric Loss (cont.) Effective Complex Permittivity The effective permittivity accounts for conductive loss. Effective permittivity that accounts for conductivity We then have Hence

  16. Accounting for Dielectric Loss (cont.) Most general expression for loss tangent: Loss due to conductivity Loss due to molecular friction The loss tangent accounts for both molecular friction and conductivity.

  17. Accounting for Dielectric Loss (cont.) For most practical insulators (e.g., Teflon), we have Note:The loss tangent is usually (approximately) constant for practical insulating materials, over a wide range of frequencies. Typical microwave insulating material (e.g., Teflon): tan = 0.001.

  18. Accounting for Dielectric Loss (cont.) Important point about notation: (We will adopt this convention also.) In this case we then write: The effective complex relative permittivity

  19. Coaxial Cable (cont.) Find R(resistance / length) Coaxial cable Rs= surface resistance of metal (This is discussed later.) Outer conductor Inner conductor = skin depth of metal

  20. General Formulas for (L,G,C) The three per-unit-length parameters (L, G, C) can be found from These values are usually known from the manufacturer. These formulas hold for any homogeneously-filled transmission line.

  21. General Formulas for (L,G,C) (cont.) The derivation of the previous results follows from: Multiply and divide

  22. General Formulas for (L,G,C) (cont.) Example: A transmission line has the following properties: (frequency is only needed for G) Results: Note: We cannot determine R without knowing the type of transmission line and the dimensions (and the conductivity of the metal).

  23. Wavenumber Formulas General case (R,L,G,C): Lossless case (L,C): Dielectric loss only (L,C,G): (please see next slide)

  24. Wavenumber (cont.) Dielectric loss only (L,C,G): Note: The mode stays a perfect TEMzmode if R = 0. kz = kfor anyTEMz mode.

  25. Common Transmission Lines Coax (surface resistance of metal for inner or outer conductors) (skin depth of metal for inner or outer conductors)

  26. Common Transmission Lines Twin-lead Two identical conductors (surface resistance of metal) (skin depth of metal)

  27. Common Transmission Lines (cont.) Microstrip Approximate CAD formula er

  28. Common Transmission Lines (cont.) Microstrip Approximate CAD formula Note: Usually w/h > 1 for a 50  line. er

  29. Common Transmission Lines (cont.) Approximate CAD formula Microstrip er

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