7/28/2019 Trask Tem Tutorial

1/5

1

A Tutorial on TEM Transmission Lines

by

Chris Trask / N7ZWYSonoran Radio Research

P.O. Box 25240Tempe, AZ 85285-5240

Email: christrask@earthlink.netExpanded and Revised 29 July 2005

Introduction

The concept of TEM transmission lineshas been a distinct element of electronics en-gineering for well over 70 years [1, 2]. Theterm TEM (TransverseElectroMagnetic [3],also known as TransverseElectric andMag-netic [4]) refers to a condition in which boththe electric and magnetic fields are parallelto a boundary plane [5] and there are nolongitudial components of either field.

Other terms such as transverse electric(TE) and transverse magnetic(TM) refer toconditions in which the electric field or mag-netic field, respectively, of a propagating wave

is parallel to a boundary plane, in this casebeing the surface of the conductors of a trans-mission line, while at the same time the ac-companying magnetic or electric fields, re-spectively, still have some longitudinal (oraxial) components [6]. Both of these termsare normally associated with wave guides [5].

Misconceptions

There are a number of misconceptions

regarding the theory and practical applica-tions of TEM transmission line, especiallycoaxial cable, which include:

It is quite impossible to build a current balun of

any ratio other than 1:1 using multiple transmis-sion line transformers on a single core unless flux

leakage between transmission lines is terrible. [7]

It is physically impossible to build a transmissionline current balun other than 1:1 on a single core

when the windings have mutual coupling throughthe core. [8]

I think the problem comes from the fact (that)

circuit engineers look at physical appearance, and

consider any two parallel conductors transmis-sion lines...[9]

Twisted pair, parallel lay, or coaxial...it only is

excited with TEM mode (transmission line modeenergy transfer) when fed with differential voltages

at the start of the pair, and currents are equaland opposite[10].

Coax does not always behave like a transmis-sion line, [11]

those who think any two parallel or concentric

conductors when fed start-to-finish or end-to-endon one conductor forces a TEM mode are not

viewing the system correctly. [12]

It is obvious from these remarks thatthere are a number of misconceptions regard-ing the general theory of TEM transmissionline, which are contrary to establishedelectromagnetics theory as well as engineer-ing practice. These misconceptions are sub-stantial obstacles in comprehending ad-

vanced applications of transmission lines,such as the design of transmission line trans-formers.

In order to avoid having these and othermisconceptions regarding the theory and ap-plication of TEM transmission line becomewidespread, Ill provide here a brief tutorialon the subject, beginning with an overview of

7/28/2019 Trask Tem Tutorial

2/5

2

is underscored by the fact that discussionson the electromagnetics of TEM mde trans-mission line do so in terms of the currents inthe conductors [14]. It only matters that theconditions of equal and opposite currents andequal voltages are met.

Generalised TEM Transmission Lines

Lets examine the parameters that arecommon to all forms of TEM mde transmis-sion line, and well later do a detailed studyof coaxial cable as it is the easiest to com-prehend since all of the electric and magneticfields are contained between the conductorswhen in TEM mde. For all types of TEMmde transmission lines (coaxial, parallel

wire, twisted wire, waveguide, etc.) the equa-tions have the same basic form. Referring toFig. 2, if R, L, G, and C are the total seriesresistance, series inductance, shunt conduct-ance, and shunt capacitance per unit lengthz, then the transmission line may be ex-pressed as [5]:

(1)

(2)

which are derived from Maxwells theoremsand equations [1, 5, 6]. Well dispense withthe usual two or more chapters of differentialcalculus that normally bring us to this point.

the fundaments of TEM theory as it applies toTEM mde transmission lines, a generalisa-tion of the parameters of TEM mde trans-mission lines, and then a detailed look at co-axial cable, which is a very familiar and read-

ily understood form of TEM mde transmis-sion line.

Essential Concepts

Fig. 1 illustrates the essential conceptof TEM mde transmission line in general-ised form, where the two lines represent thetwo conductors of the transmission line, re-gardless of whether it is made of parallelwires, twisted wires, coaxial cable, or othermeans. Here, the currents are equal in mag-nitude and opposite in phase, which is con-sistent with Lenzs Law [5, 6, 13], while thevoltages along the length of the two conduc-tors are equal in both magnitude and phase.This may also be seen to say that the voltageacross the two conductors is equal at all pointsalong the transmission line.

These are the conditions that excite TEM

mde, which again is equal and opposite cur-rents in the two conductors, which results inno net magnetic field, and equal voltages ei-ther along or across the two conductors, whichresults on no net electric field. It is of no con-sequence how the external exciting voltageor current is applied, either across the termi-nals at one end or along the length of oneside from one end to the other, and this point

( )

V

zR j L I= +

( ) Iz

G j C V= +

Figure 1 - Transmission Line inTransverse (TEM) Mde

Figure 2 - Incremental Model ofTEM Transmission Line

7/28/2019 Trask Tem Tutorial

3/5

3

and the propagation constant of (4) can beapproximated as [5]:

(8)

The reciprocal of the square root of theproduct of L and C provides us with the ve-locity of propagationor phase velocity[5, 14]:

(9)

where is the permeability of the transmis-sion line medium in Henries/meter (H/m) and is the permitivity of the transmission line me-dium in Farads/meter (F/m) [5].

Coaxial Cable

Lets now turn to the specific case ofcoaxial cable, using the illustration of Fig. 3as a guide. Here, the inner conductor has aradius of r

1and the outer conductor has an

inner radius of r2

and a thickness of t. Thespace between the inner and outer conduc-tors is filled with an insulating dielectric ma-terial such as PTFE that has a relativepermitivity(or dielectric constant)

rand a rela-

tive permeabilityr. If the conductors are con-

sidered to be lossless as per our earlierappoximation, the unit shunt capacitance Cis [14]:

(10)

Make careful note of the voltage con-ventions between Fig. 1 and Fig. 2. The lat-ter is always used when examining TEM trans-mission line, however the former is very of-ten, but not exclusively used when mappingthe voltages in systems of transmission line,such as in the design of transmission linetransformers (TLTs). For all intents and pur-poses, they are both the same as the actualvoltages are unaffected regardless as to howthey are visualised.

From (1) and (2) we can derive the char-acteristic impedanceZ

oof the transmission

line as related to the primary constants R, L,G, and C by [5]:

(3)

and the complex propagation constantcanbe approximated as [5]:

(4)

where is called the attenuation constantand is called the phase constant. The primaryconstants R, L, G, and C are directly relatedto the physical properties of the materialsused in the transmission line and remain un-affected by the application of the transmis-sion line.

For low-loss transmission line such asgood quality coaxial cable [5]:

(5)

(6)

and the characteristic impedance Zoof (3) can

be approximated as [5]:

(7)

R L

G C

ZL

Co

j L C

C2

rr

r

r

pF /m2

1

r

21

=

=

ln

.

ln

55 6

Figure 3 - Coaxial Transmission Line

p

1

L C

1

= = =

Z R j LG j Co

= ++

( ) ( )

j

R j L G j C

= + =

= + +

7/28/2019 Trask Tem Tutorial

4/5

4

from which we can derive the loss tangentofthe transmission line, which is [6]:

(15)

where and are often referred to as di-electric dispersion formulas[5, 16], which arevery rigorous and very much beyond thescope of this discussion. It is sufficient forour purposes to use the loss tangent dataprovided by the manufacturer of the cable weare using and from that derive the surface re-sistance R

s.

Going back to (9), we can define a quan-tity that is well known to radio amateurs, whichis the velocity factor of the cable:

(16)

where c is the speed of light, vis the perme-

ability of free space (4 x 107 H/m). and v

isthe permitiviity of free space (8.854x10-12 F/m). In general the relative permeability ofmost, if not all insulating materials is close to

unity, so (16) can be comfortably approxi-mated for coaxial cable as:

(17)

Coaxial cable is a good practical exam-ple of TEM transmission line as the skin ef-fect of the inner surface of the outer conduc-tor causes the current of the outer conductorto be concentrated on the inside surface. The

magnetic fields generated by the equal andopposite currents of the concentric inner con-ductor and the inner surface of the outer con-duct cancel outside the outer conductor inboth the near and far fields, leaving no netmagnetic field outside of the outer conductorthat would couple to nearby objects, such asthe magnetic core of a transmission line trans-former, which would cause additional losses

and the unit series inductance L is [14]:

(11)

The unit series resistance R is relatedto the conductivity of the conductors, and isfrequency dependent by way of a phenom-enon known as skin effect. We begin by firstdefining a quantity known as the 1/e depth ofpenetration[6, 14, 15]:

(12)

where is the conductivity of the material.

The current in a conductor will always con-centrate on the surface that is nearest thewave that creates the current [14], and in thecase of coaxial cable this is the electromag-netic field that exists between the inner sur-face of the outer conductor and the inner con-ductor. At high frequencies the skin effectcauses the current to flow only on the outersurface of the inner conductor and the innersurface of the outer conductor [14], and thiscondition persists as long as the thickness t

of the outer conductor (see Fig. 2) is appreci-ably greater than the skin depth.

From (12) we can now define the sur-face resistance of the conducting material [5]:

(13)

The unit shunt conductance is relatedto the resistivity of the dielectric insulating ma-

terial, due to a phenomenon known as dielec-tric hysteresis, which is analagous to the mag-netic hysteresis in magnetic materials [6]. Itis convenient to describe the total losses ofthe transmission line as the equivalent con-ductivity[6]:

(14)

VFc

1

p v v

v v

r v r v r r

= = =

= =

VF1

r

Tan

=

Rs =1

= +

2

v r

L2

r

r

0.2r

r H / m

2

1

r2

1

= =

=

ln

ln

7/28/2019 Trask Tem Tutorial

5/5

5

References

1. King, R.W.P., Transmission Line Theory, McGraw-Hill, 1955.

2. Schelkunoff, S.A., The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields,Bell System Technical Journal, Vol. 13, No. 10, October 1934, pp. 532-579 (reprinted as Monograph B-816).3. Gannet, E.K. (ed), IEEE Standard Dictionary of Electrical and Electronics Terms, 1st ed., IEEE Standard

100-1972, Wiley, 1972.4. -----, Glossary of Telecommunications Terms, Federal Standard 1037C, Information Technology Serv-

ice (ITS), 7 August 1996.

5. Jordan, E.C. and K.G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd ed., Prentice-Hall,1968.6. Kraus, J.D. and K.R. Carver, Electromagnetics, 2nd ed., McGraw-Hill, 1973.7. Tom Rauch, 22 June 2005, http://mailman.qth.net/pipermail/qrp-l/20050622/021408.html

8. Tom Rauch, 22 June 2005, http://mailman.qth.net/pipermail/qrp-l/20050622/021442.html9. Tom Rauch, 25 June 2005, http://mailman.qth.net/pipermail/qrp-l/20050625/021618.html

10. Tom Rauch, 25 June 2005, http://mailman.qth.net/pipermail/qrp-l/20050625/021626.html11. Tom Rauch, 25 June 2005, http://mailman.qth.net/pipermail/qrp-l/20050625/021626.html

12. Tom Rauch, 27 June 2005,http://www.ibiblio.org/pub/academic/agriculture/agronomy/ham/ANTENNAS/20050630.ant

13. Marshall, S.V. and G.G. Skitek, Electromagnetic Concepts & Applications, 2nd ed., Prentice-Hall, 1987.

14. Matick, R.E., Transmission Lines for Digital and Communication Networks, McGraw-Hill, 1969.15. Ramo, S., J.R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, 2nd ed.,

Wiley, 1984.16. Debye, P., Polar Molecules, Dover, 1945.

17. Hamilton, N., RF Transformers Part 2: The Core,RF Design, January 1996, pp. 64-73.18. Lefferson, P., Twisted Magnet Wire TransmissionLine,IEEE Transactions on Parts, Hybrids, and Pack-

aging, Vol. 7, No. 4, December 1971, pp. 148-154.19. Trask, C., Designing Wide-band Transformers for HF and VHF Power Amplifiers,QEX, Mar/Apr 2005,pp. 3-15.

20. Breed, G.A., Transmission Line Transformer Basics,Applied Microwave & Wireless, Vol. 10, No. 4, May1998, p. 60.

21. Rotholz, E., Transmission-Line Transformers,IEEE Transactions on Microwave Theory and Techniques,Vol. 29, No. 4, April 1981, pp. 327-331.

beyond those of the cable itself [14, 17].

Similar equations may be developed forparallel wire transmission line [1, 5, 6] as wellas twisted wire transmission line [18], the lat-ter of which is an important element in thedesign of wideband transformers for RF ap-plications [19].

Advanced Applications

A thorough understanding of the basictheory of TEM transmission line is essentialin comprehending advanced applicationssuch as transmission line transformers (TLTs),which operate by transmitting energy by wayof the TEM transmission line mde rather than

on the more familiar coupling of magnetic flux

as with a conventional transformer [20, 21].

Closing Remarks

The theory of TEM transmission lines,especially coaxial cable, is a well- establishedelement of RF design and has been part ofthe technology for over 70 years. The equa-tions are both well established and easilyunderstood, and have their foundations inbasic electromagnetic theory such asMaxwells theorems and equations and LenzsLaw. A basic background in the theory of TEMtransmission line is an essential prerequisitein understanding advanced topics such astransmission line transformers, but not so dif-ficult as to be beyond the understanding of

those having entry level experience in the pro-fession of RF circuit design.