1. Electromagnetics (ENGR 367)
Types of T-lines
and Their Applications

2. Outline of Lecture Identify Types of T-lines versus Waveguides
Detail construction and applications of each type
Provide formulas for T-line parameters of each type that depend on
LF or HF operation
Lossy or Lossless conditions
Show examples of T-line parameter calculation
Draw some conclusions

3. Types of T-lines Coaxial cable Two-wire line
Stripline and Microstrip Line
Other specialized variations

4. T-lines versus Waveguides
T-lines operate in the TEM mode only (Transverse ElectroMagnetic waves)
Waveguides carry EM waves that
May propagate in the TM or TE modes at higher frequencies and may perform more efficiently than T-lines in the microwave range
Take a zig-zag path as they reflect off the conducting boundaries and propagate along the main axis of the waveguide

5. Detailed Description of Individual T-line Types and Their Applications
Coaxial Cable: Basic Construction
Radio-grade flexible RG-59 (Z0 = 75 ) coaxial cable.
A: outer plastic sheath B: copper screen C: inner dielectric insulator D: copper core
Other standard types have similar construction.
from

6. Coaxial Cable Other aspects of basic construction
Stiffness options available on the market
Flexible type: has braided sheath
Rigid type: has a solid sheath
(Sheath in either case typically made of Cu)
Dielectric insulating layer
Thickness and permittivity determine
Characteristic Impedance (Z0)
Attenuation ()
May be either solid or perforated

7. Applications of Coaxial Cable
Short runs to connect
Home video equipment
Ham radio setups (transciever  antenna)
Satellite TV (dish Rx  set)
Cable modem & VSAT for Internet Access
Broadcast radio communication (Tx  ant.)

8. Applications of Coaxial Cable
Long runs to connect
Formerly radio and TV networks
(Now replaced by optical and satellite networks)
Presently cable TV signals

9. Specialized Variations of Coax
Triaxial Cable (triax): includes a 3rd layer of shielding, insulation and sheathing, the latter is grounded to further reduce outside interference
Twin-axial cable (twinax): balanced twisted pair within a cylindrical shield for shielded and balanced differential signals
Multi-conductor coax

10. Review Skin Depth
A measure of the depth electromagnetic waves penetrate into a conductor where the amplitude has decayed by e-1 = 0.368
Note that as frequency (f) increases, the skin depth becomes smaller and more significant!

11. Coax T-line Geometry

12. T-line Parameters for Coax (from Hayt & Buck, 7/e, pp. 483-484)
Assuming HF operation such that the skin depth  << a = radius of inner conductor
Lossless approx. (R << wL and G << wC):
Modifications for the lossy case:

13. T-line Parameters for Coax (from Hayt & Buck, 7/e, p. 485)
Modifications for LF operation where the skin effect is negligible ( >> a): current distributes uniformly throughout conductor cross-sections
Resistance of conductors increases

14. T-line Parameters for Coax (from Hayt & Buck, 7/e, p. 485)
Modifications for LF operation ( >> a):
Internal inductance of conductors becomes significant
For intermediate frequencies (  a):
Expressions for parameters become more complicated
One can refer to handbook values as needed

15. Example of Parameter Calculations for Coax
Exercise 1 (D14.2a, H&B, 7/e, p. 486)
Given: a coax T-line with a = 4 mm, b = 17.5 mm, and c = 20 mm. Each conductor has  = 2 x 107 S/m, and the dielectric has r = 1, r = 3, and / =
Find: L, C, R, G, and Z0 at 150 MHz.
Solution: 1st find the skin depth; compare to a

16. Example of Parameter Calculations for Coax
Exercise 1 (continued)
Solution: next calculate coax T-line parameters

17. Example of Parameter Calculations for Coax
Exercise 1 (continued)
Solution: check validity of lossless approx. for Z0

18. Two-wire Line Basic Construction
Two parallel circular conductors of equal radius and conductivity enclosed in a plastic insulating material
Dielectric insulator
Provides mechanical spacing and some rigidity
Affects Z0 and 

19. Applications of Two-wire Line
As a lead-in to carry low level signals from antenna over a short run to a TV or FM Rx
Connections in regular telephone networks
In the conceptual development of a more sophisticated waveguide systems (Fast Neutron Research Facility of CMU waveguide/Waveguide%20theory%202.html)

20. Parameters of Two-wire Line (from Hayt & Buck, 7/e, pp. 486-487)
For HF operation ( << a):
Lossless approximation

21. Parameters of Two-wire Line (from Hayt & Buck, 7/e, pp. 486-487)
For HF operation ( << a):
Modifications for Lossy conditions
In LF operation: must modify above by including R and L over entire cross-section of conductors as for coax

22. Parameters of Two-wire Line (from Hayt & Buck, 7/e, p. 487)
For LF operation ( >> a):
Inductance per unit length increases by twice the internal inductance of a straight round wire
Resistance per unit length becomes twice the dc resistance of a wire of radius a and conductivity c

23. Example of Parameter Calculation for Two-wire Line
Exercise 2 (D14.3, H&B, 7/e, p. 487)
Given: a two-wire T-line with conductors each of radius 0.8 mm and conductivity 3 x 107 S/m, separated by 0.8 cm in a dielectric where r = 2.5, r = 1, and d = 4x10-9 S/m
Find: , C, G, L & R at 60 Hz
Solution: validate LF model by comparing  to a

24. Example of Parameter Calculation for Two-wire Line
Exercise 2 (D14.3, H&B, 7/e, p. 487)
Solution: calculate the LF two-wire T-line circuit param’s

25. Striplines and Microstrip Lines
Basic Construction: Stripline
Single or double track strip of Cu imbedded in dielectric material sandwiched between conducting ground planes on top and bottom

26. Striplines and Microstrip Lines
Basic Construction: Microstrip
Single or double track strip of Cu on top of a dielectric substrate material above a single conducting ground plane
In practice, superstrate material may be other than air

27. Striplines and Microstrip Lines
Basic Construction: dielectric material
For HF applications, special substrate and superstrate (if present) materials must be used with high uniformity and low loss tangent (tan = ’/’) such as Rogers RT Duroid or FR4 as manufactured for this application (common PCB will not work!)

28. Applications of Stripline & Microstrip Line
Trace connections between devices in PCB microelectronic circuits
T-line connections between HF devices easily integrated with surface mount, distributed elements and microstrip antennas.
HF communication systems and devices with compact, flat, lightweight, constraints and short run requirements such as cell phones, portable PCs and and other wireless mobile devices

29. Parameters for Microstrip Line (Single Track)
If the strip width is large (w >> d), then the structure acts like a parallel plate line where for the low loss case

30. Parameters for Microstrip Line (Single Track)
If w  d or w < d (as typical for microstrip) then a quasi TEM mode may be assumed to account for the propagation of waves through the two different materials (e.g., air or superstrate and substrate dielectrics)
At low frequencies (f < 1.5 GHz) assuming negligible losses over a short run the propagation velocity is

31. Parameters for Microstrip Line (Single Track)
Definition of Effective Dielectric Constant (r,eff):
acts as a weighted average of the air (or superstrate)
and substrate dielectric constants with a proportion
determined by the field filling factor (q)

32. Parameters for Microstrip Line (Single Track)
Empirical Formulas for r,eff assuming w/d > 1.3
in terms of w/d for application to the dimensions of a pre-fabricated line
In terms of r and Z0 for finding the necessary r,eff based on a desired Z0

33. Parameters for Microstrip Line (Single Track)
Characteristic Impedance (Z0):
Based on the air-filled equivalent microstrip line
For w/d < 3.3

34. Example of Parameter Calculation for Microstrip Line
Exercise 3 (D14.4, H&B, 7/e, p. )
Given: a 2 mm wide microstrip line is fabricated on a mm thick substrate of lithium niobate (r = 4.8).
Find: r,eff, Z0 and vp
Solution: since w/d = 2 > 1.3:

35. Conclusions
Types of T-lines such as coax, two-wire, microstrip lines and other variations have
relative advantages in certain applications
a unique construction that determines their circuit parameters
Circuit parameters of T-lines depend on LF or HF operation as indicated by skin depth versus conductor size; parameters may be found by calculation or in a handbook

36. Conclusions
While coax has shielding to minimize interference in low signal applications, it may be too large and bulky for PCB and microelectronic circuit applications
While microstrip lines are easy to fabricate and compatible with microelectronic systems, they are analytically complex requiring numerical methods for accuracy

37. Conclusions
For higher frequency applications, waveguides (including rectangular or cylindrical “pipes,” parallel plates, dielectric slabs and optical fiber types) become more efficient than T-lines
The analysis of waveguides becomes a more advanced task with fully EM field wave equations

38. References and Other Resources
Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, 2006.
Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.