A Transmission Line Power Paradox
and Its Resolution
Steve Stearns, K6OIK
Technical Fellow
Northrop Grumman
Electromagnetic Systems Laboratory
San Jose, California
stearns@ieee.org
k6oik@arrl.net
1
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
ARRL Pacificon Presentations by K6OIK
2001
Archived at
http://www.fars.k6ya.org
Jam-Resistant Repeater Technology: Signal Separation,
Identification, Routing, Control, and Automatic Logging
Mysteries of the Smith Chart

2002
How-to-Make Better RFI Filters Using Stubs
2003
Twin-Lead J-Pole Design
2004
Antenna Impedance Models – Old and New
2005
Novel and Strange Ideas in Antennas and Impedance Matching
2006
Novel and Strange Ideas in Antennas and Impedance Matching 2

2007
New Results on Antenna Impedance Models and Matching

2008
Antenna Modeling for Radio Amateurs

2009
(convention held in Reno)
2010

2011
Facts About SWR, Reflected Power, and Power Transfer on Real
Transmission Lines with Loss
Conjugate Match Myths
2012
Transmission Line Filters Beyond Stubs and Traps

2013
Bode, Chu, Fano, Wheeler: Antenna Q and Match Bandwidth

2014
A Transmission Line Power Paradox and Its Resolution

1999
2000
2
Mysteries of the Smith Chart
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA

October 10-12, 2014

Outline
 Antenna books
 Software
 Antenna modeling
 Circuit design
 Smith chart
 Transmission line theory – a brief review




Distributed parameters
Complex characteristic impedance and propagation constant
Attenuation constant and phase constant
Published formula for line loss with mismatched load
 The Loss Paradox
 The Resolution
 Power delivery myth
 Improved formulas for line loss with mismatched load
 References
3
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Favorite Antenna Books
 Books for antenna engineers and students








R.C. Hansen and R.E. Collin, Small Antenna Handbook, Wiley, 2011, ISBN
0470890835
Modern Antenna Handbook, C.A. Balanis editor, Wiley, 2008, ISBN
0470036346
Antenna Engineering Handbook, 4th ed., J.L. Volakis editor, McGraw-Hill,
2007, ISBN 0071475745. First published in 1961, Henry Jasik editor
C.A. Balanis, Antenna Theory, 3rd ed., Wiley, 2005, ISBN 047166782X.
First published in 1982 by Harper & Row
J.D. Kraus and R.J. Marhefka, Antennas, 3rd ed., McGraw-Hill, 2001, ISBN
0072321032. First published in 1950
S.J. Orfanidis, Electromagnetic Waves and Antennas, draft textbook online
at http://www.ece.rutgers.edu/~orfanidi/ewa/
E.A. Laport, Radio Antenna Engineering, McGraw-Hill, 1952
http://snulbug.mtview.ca.us/books/RadioAntennaEngineering
G.V. Ayzenberg, Shortwave Antennas, 1962, transl. from Russian, DTIC
AD0706545. http://www.dtic.mil/dtic/tr/fulltext/u2/706545.pdf
 Antenna research papers




4
IEEE AP-S Digital Archive, 2001-2009 (1 DVD), JD0307
IEEE AP-S Digital Archive, 2001-2006 (1 DVD), JD0304
IEEE AP-S Digital Archive, 2001-2003 (1 DVD), JD0301
IEEE AP-S Digital Archive, 1952-2000 (2 DVDs), JD0351
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Favorite Antenna Books continued
 Books for Radio Amateurs






Rothammel’s Antennenbuch, 13th ed., A. Krischke (DJ0TR) editor, DARC
Verlag, 2013
ARRL Antenna Book, 22nd ed., H.W. Silver (N0AX) editor, American Radio
Relay League, 2011, ISBN 087259694X
J. Devoldere (ON4UN), ON4UN’s Low-Band Dxing, 5th ed., American
Radio Relay League, 2011, ISBN 087259856X
Practical Wire Antennas 2, I. Poole (G3YWX) editor, Radio Society of Great
Britain, 2005, ISBN 1905086040
J. Sevick (W2FMI), The Short Vertical Antenna and Ground Radial, CQ
Communications, 2003, ISBN 0943016223
L. Moxon (G6XN), HF Antennas for All Locations, 2nd ed., Radio Society of
Great Britain, 1983, ISBN 1872309151
 ARRL Antenna Compendium series – eight volumes
 ARRL Antenna Classics series – seven titles
5
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Recent Antenna Books of Interest
R.C. Hansen and R.E.
Collin, Small Antenna
Handbook, Wiley, 2011
H.W. Silver, N0AX,
Antenna Modeling for
Beginners, ARRL, 2012
6
Y. Zhang et al., Higher
Order Basis Based Integral
Equation Solver
(HOBBIES), Wiley, 2012
A. Krischke, DJ0TR, ed.,
Rothammels Antennen
Buch, 13th ed, DARC,
2013
Steve Stearns, K6OIK
Elsherbeni et al., Antenna
Analysis and Design using FEKO
Electromagnetic Simulation
Software, SciTech, 2014
S. Nichols G0KYA, An
Even More Wire
Introduction to Antenna Antenna Classics vol. 3,
Modelling, RSGB, 2014 ARRL, 2014
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Software
7
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Antenna Modeling Programs for Radio Amateurs
 EZNEC http://www.eznec.com






EZNEC v.5 demo program Free
EZNEC-ARRL v.3 & v.4
$50 (on ARRL Antenna Book CD-ROM)
EZNEC v.5
$90
EZNEC+ v.5
$140
EZNEC Pro/2 v.5
$500
EZNEC Pro/4 v.5
$650 (sold only to NEC-4 licensees)
 4nec2 http://home.ict.nl/~arivoors/
 Free, 11,000 segments, two optimizers
 MMANA-GAL http://hamsoft.ca/pages/mmana-gal.php
 Free Basic version 8,192 segs. Pro version 32,000 segs, $130
 NEC-4 https://ipo.llnl.gov/data/assets/docs/nec.pdf
 Noncommercial user license $300
 FEKO Lite http://www.feko.info
 Free LITE version, no time limit
 WIPL-D Lite http://www.wipl-d.com
 Free 30-day full demo, or Lite version $449 (Artech House)
 HOBBIES http://em-hobbies.com
 Similar to WIPL-D Lite but more capability, $149 to $217 online
8
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Accessory Software for EZNEC
 AutoEZ 2.0.8 by Dan Maguire, AC6LA, http://www.ac6la.com
 Visual Basic program $79, or free demo version (30 segment limit)
 Controls EZNEC to make multiple runs
– A GUI for a GUI for NEC
 Requires Excel and EZNEC installed on computer
 Has an optimizer – Nelder-Mead algorithm
 Can read NEC, AO, and MMANA-GAL files
 Doesn’t work with EZNEC-ARRL
 Replaces MultiNEC which is no longer available
9
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Software for Smith Charting and Network Design
 SimSmith 9.9 by Ward Harriman AE6TY
 Free download from http://ae6ty.com/Smith_Charts.html
 Smith Chart Calculator 2.1 by Gorik Stevens
 Free download from http://sourceforge.net/projects/gnssmithchart/
 Smith 3.10 by Fritz Dellsperger HB9AJY
 Free download from http://fritz.dellsperger.net
 QuickSmith 4.5 by Nathan Iyer KJ6FOJ
 Free download from http://www.nathaniyer.com/qsdw.htm
 JJSmith 2.10 by J. Bromley K7JEB and J. Tonne W4ENE (SK)
 Free download from http://tonnesoftware.com/jjsmith.html
 linSmith by James Coppens ON6JC/LW3HAZ
 Free download from http://jcoppens.com/soft/linsmith/index.en.php
 XLZIZL by Dan Maguire AC6LA, 2005. No longer available.
 WinSMITH 2.0, Noble Publishing, 1995. No longer available.
 MicroSmith 2.3, ARRL, 1992. No longer available.
10
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Smith Chart Programs for Ladder Network Design
11
SimSmith 9.9
Smith 3.10
QuickSmith 4.5
winSMITH 2.0
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
General RF Circuit Design, Analysis, and Optimization
 Software for Radio Amateurs
 Quite Universal Circuit Simulator (QUCS) 0.0.18, 2014
– Free download from http://qucs.sourceforge.net
 Ansoft Serenade SV 8.5 (student version), Ansoft, 2000. No
longer available. See article by David Newkirk W9VES in QST
Jan. 2001.
 ARRL Radio Designer 1.5, ARRL, 1995. No longer available.
 Professional electronic design automation (EDA)
software
 Agilent Advanced Design System (ADS)
 Applied Wave Research Microwave Office (MWO)
 ANSYS Designer
12
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Transmission Line Theory
A Brief Review
13
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Oliver Heaviside, 1850-1925
14
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Heaviside Telegrapher’s Equations
Uniform transmission line
I(x)
Infinitesimal segment
Rx
V(x)
Lx
Gx
Cx
 d 2V
dV

 ( R  j L) I ( x ) 
 ( R  j L) (G  j C )V ( x )

2
dx

 dx
  

 d 2I
dI
 (G  j C )V ( x ) 
 ( R  j L) (G  j C ) I ( x )

2
dx

 dx
15
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Transmission Line Solution: Waves
 Two waves traveling in opposite directions
V ( x )  V0 e x
 V0 e  x
V0  x
I ( x) 
e
Z0
V0  x

e
Z0
Attenuation per
unit length
 Propagation constant
  ( R  j L) (G  j C )    j
 Characteristic impedance
R  j L
Z0 
 R0  jX 0
G  j C
16
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
Phase per
unit length
October 10-12, 2014
Distortionless Line (Oliver Heaviside, 1887)
 A line is distortionless if R, L, G, and C are constants
independent of frequency and
R G

L C
or
RC  GL
or
R L

G C
 The first and second equations make clear that dissipationless or
lossless line (R = G = 0) is also distortionless
 In most practical lines the insulation is so good that
R G

L C
or
RC  GL
or
R L

G C
Oliver Heaviside, “Simple Properties of the Ideally Perfect Telegraph Circuit,” Section XLI of the
series “Electromagnetic Induction and Its Propagation,” The Electrician, p. 124, June 16, 1887
17
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Seven Properties of Distortionless Line
 Characteristic impedance is purely real
R  j L
Z0 

G  j C
L
 R0  j 0
C
 Characteristic impedance is independent of frequency
 Attenuation is independent of frequency
 Attenuation is minimum
 

0
L C
 Velocity of propagation is independent of frequency
 Velocity of propagation is maximum
   



0
R L G C

18
Traveling waves account for power delivery along the line
PDelivered  PForward  PReverse
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Transmission Line Distributed Parameters from
Physical Dimensions and Material Properties
s
dielectric
(m, e, s)
a
dielectric
(m, e, s)
b
c
 Parameter
G S/m
C F/m
 m
19
Steve Stearns, K6OIK
a
1 1
  
2 s c  a b 
1
R /m
L H/m
conductor
(mc, ec, sc)
m
2
 b   1 1 
ln a  2  a  b 



a
1
 as c
m
1 s 

cosh
  2a
2a 
2s
b
ln
a
2 e
b
ln
a
1
 f mc s c
ARRL Pacificon Antenna Seminar, Santa Clara, CA
s
cosh 1
e
cosh 1
s
2a
s
2a
1
 f mc s c
October 10-12, 2014
Round Open-Wire Transmission Line (PEC in Air)
d
s
 Accurate formula
 For high frequencies (assumes that skin depth is small  << a)
s
Z 0  119.917 cosh 1  
d 
 Approximate formula
 Assumes, in addition, large spacings: s/d > 3
or large impedances: Z0 > several hundred
 2s 
 2s 
Z 0  120 ln    276 log10  
d 
d 
20
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Characteristic Impedance
 Exact formula
1
R
R  j L
L
j L


G
G  j C
C
1
j C
Z 0  R0  jX 0 
 High-frequency, low-loss approximation
Z0 
L
e
C
| Z0 | 
R0 
X0  
 j R
G 



2   L  C 
L
C
 R
L
G 

cos

C
 2 L 2 C 
 R
L
G 

sin

C
 2 L 2 C 
E.W. Kimbark, Electrical Transmission of Power and Signals, Wiley, 1949
21
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Propagation Constant
 Exact formula
    j  ( R  j L) (G  j C )  j LC  1 
R
j L
 1
G
j C
 High-frequency, low-loss approximation
   LC e
  R
G 
 
j   

 2  2 L 2 C  

 R
G  1 R

  
   LC sin

 G | Z 0 |
 2 L 2 C  2  | Z 0 |

   LC

1
vp  

LC
E.W. Kimbark, Electrical Transmission of Power and Signals, Wiley, 1949
22
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Attenuation per 100 feet (dB)
Attenuation Constant of Common Transmission Lines
  k0  k1 f  k2 f
Source: ARRL Antenna Book, 21st ed., p. 24-20, 2007
23
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Voltage and Current Standing Waves
Source: R.A. Chipman, Schaum’s Theory and Problems
of Transmission Lines, p. 170, McGraw Hill, 1968
24
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Ohmic and Dielectric Heating Alternate Along a
Mismatched Line
For distortionless line, heating is uniform
I2R
25
Steve Stearns, K6OIK
V2G
I2R
ARRL Pacificon Antenna Seminar, Santa Clara, CA
V2G
October 10-12, 2014
Total Line Loss
26
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Published Formula
Why SWR < 20:1?
Source: ARRL Antenna Book, 22nd ed., p. 23-11, 2011
27
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Matched Loss and Attenuation Constant
 Power at load end in terms of power at transmitter end of line
PF,Tx
PR,Tx
Transmission
Line
Length l
PF,Load
PR,Load
1
PF ,Load   PF ,Tx
a
PR ,Load  a  PR ,Tx
 a is the matched loss or power attenuation ratio in linear units
 A real number greater than one
 Related to the line’s attenuation constant  or scattering parameter s21
Don’t confuse
Latin a and Greek 
 e 2 l

a   or
10  l / 1000

a
28
Steve Stearns, K6OIK
for  in nepers/meter and l in meters
for  in dB /100 feet and l in feet
1
| s21 |2
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Derivation of Published Formula for Total Line Loss
Pin
Total Loss (dB)  10 log10
Pout
 10 log10
PF ,Tx  PR ,Tx
PF , Load  PR , Load
 Key Assumption
P

 1  R ,Tx
 PF ,Tx  
PF ,Tx


 10 log10 

P
F
,
Load

  1  PR , Load

PF , Load

1 | in |2
 10 log10 a
1 | Load |2
Matched Loss






1 | in |2
 10 log10 a  10 log10
1 | Load |2
Additional
Loss
1 | in |2
  l (dB)  10 log10
1 | Load |2
29
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Total Line Loss in Terms of SWR at Load
Total Loss (dB)   l (dB)  10 log10
1
2
(
SWR

1
)
Load
a2
 1) 2  ( SWRLoad  1) 2
( SWRLoad  1) 2 
( SWRLoad
Matched Loss
Additional
Loss
Published line loss formula depends on matched loss and SWR.
30
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Transmission Line Total Loss
VSWR at Load
J. Reisert W1JR , Ham Radio, p. 27, Oct. 1987
31
Steve Stearns, K6OIK
ARRL Handbook, 91st ed., p. 20.5, 2014
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Additional Loss Due to Mismatch
 The additional loss can be expressed in terms of the SWR at the
line’s input or its output
1  | in |2
( SWRTx  1) 2  ( SWRTx  1) 2
10 log10
 10 log10
2
2
1  a | in |
( SWRTx  1) 2  a 2 ( SWRTx  1) 2
Additional Loss (dB) 
1
1
2
2
|

|
(
SWR

1
)

( SWRLoad  1) 2
Load
Load
2
2
a
a

10
log
10
1  | Load |2
( SWRLoad  1) 2  ( SWRLoad  1) 2
1
10 log10
 Next slide shows the additional loss graphed both ways
32
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Additional Loss versus Matched Loss
8
10
SWR at Load
20
15
Additional Loss Due to Mismatch dB
Additional Loss Due to SWR dB
7
6
SWR at Transmitter
7
5
5
4
3
4
2
3
1.5
2
10
7
5
4
1
3
2
1
1.5
0
0.1
0
1
2
3
4
5
Matched Loss dB
33
Steve Stearns, K6OIK
6
7
0.1
1
Matched Loss dB
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
10
The Loss Paradox
34
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Which Case has Greater Loss?
Case A
Case B
Z0 = 50 
Z0 = 50 
l = /8
5
l = /8
Typical lossy line
SWR = 10
35
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
500 
A Clue
Case A
Case B
Z0 = 50 
Z0 = 50 
5
l = /8
500 
l = /8
I2R
V2G
Typical lossy line
Ohmic loss dominates
SWR = 10
Dielectric loss negligible
36
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Total Line Loss According to Published Formula
Total Loss (dB)   l (dB)  10 log10
1 2
9
2
a
112  92
112 
 Does not depend on RL except through SWR !
 For a given SWR, does not depend on whether RL > Z0 or RL < Z0 !
 For a given SWR, does not depend on where ZL lies on Smith
Chart’s SWR circle !
Published line loss formula depends on matched loss and SWR.
37
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
The Paradox
 Assume a lossy 50- line in which copper loss dominates
dielectric loss
 Case A
 A short lossy 50- line terminated with a 5- resistor
 Line current is high at the load
 Case B
 Same line terminated with a 500- resistor
 Line current is low at the load
 SWR at load is 10:1 in both cases
 By published formula, line loss is the same in both cases
 By physical reasoning, line loss is greater in Case A than Case B
A contradiction !
38
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
The Resolution
39
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
The Problem
 The total line loss formula depends for its proof on the condition
 Delivered power is forward power minus reverse power
PDelivered  PForward  PReverse
 While true for distortionless line, this statement is not true for
general transmission lines
 Certain kinds of power meters make a similar assumption
40
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Delivered Power Along a General Transmission Line
PDelivered


1
1
*
 Re{ V I }  V I *  V * I
2
4
1
 (VF  VR ) ( I F*  I R* )  (VF*  VR* ) ( I F  I R )
4
1
 VF I F*  VR I R*  VR I F*  VF I R*  VF* I F  VR* I R  VR* I F  VF* I R
4




 

 
1
1
1
VF I F*  VF* I F  VR I R*  VR* I R  VR I F*  VF I R*  VR* I F  VF* I R
4
4
4
1
 PForward  PReverse  VR I F*  VF I R*  VR* I F  VF* I R
4



Cross terms !
41
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014

Cross Power Terms


1
VR I F*  VF I R*  VR* I F  VF* I R
4
1 VF*VR VFVR* VFVR* VF*VR 
  *  * 

4  Z0
Z0
Z0
Z 0 
Cross Power Terms 
 1
1
1 
*
*

 (VFVR  VFVR )   * 
4
 Z0 Z0 
 0 if VFVR* is real or Z 0 is real
 Cross terms vanish if either of two conditions is met
 Real Z0, e.g. distortionless line
 Real reflection coefficient, i.e. real VR /VF or real ZL /Z0
42
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Formulas for Total Loss of Terminated General
Transmission Lines
 Loss expressed in terms of the load impedance
ZL
Total Loss (dB)  20 log10 cosh γ L 
sinh γ L
Z0

RIN
10 log10
RL
 Loss expressed in terms of line input impedance
Z IN
Total Loss (dB)  20 log10 cosh γ L 
sinh γ L
Z0
RIN
 10 log10
RL
 where Z0 is complex characteristic impedance taken as
Z 0  R0  jX 0

 R0  j R0

43
Steve Stearns, K6OIK
Other useful forms are
obtained from equations
on Slides 20 and 21
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Test Case for Numerical Validation of Loss Formulas
 Input parameters (N6BV example)








44
Line type:
Length:
Frequency:
Alpha:
Velocity factor:
R0 = real part of Z0 :
Load resistance RL :
Load reactance XL :
Steve Stearns, K6OIK
Wireman #551, 450- window line
100 feet
1.83 MHz
0.095 dB/100 feet
0.915
402.75 
4.5 
−1,673 
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Z = 4.5 –j 1,673  – What Antenna Is This?
 Electrically small dipole – three step calculation
 f L 
RRad  5(  L)  20 

c


c
RRad
 L
 24.735 m  81.15 ft
2 f
5
2
2
L ≈  /6.6
 f L   L 
L  L 

 ln  1
X Ant  120 cot 
ln

1


120
cot
 

 2  d


 2c   d
L /d ≈ 30
L
24.735
 d

 0.8224 m  32.38 in
 X Ant
1  29.078
 f L
Cage dipole?

120 cot 
 2c 
1 e
Efficiency if a single
wire/tube, not a cage
45
Steve Stearns, K6OIK
2  m0 f 1
4  107 f
RLoss 

 91 m 
3
s  d 3 5 .8 d
100 RRad
4 .5
 Efficiency (%) 

 99.998 %
RRad  RLoss 4.500091
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Calculated Values from TLW v3.24
46
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Validation Results
R0 = (given)
Published
Formula
402.75
X0 =
TLW
New Formula 1 New Formula 2
402.7
402.75
402.75
-3.446913256
-3.46
-3.446913256
-3.446913256
RL = (given)
4.5
4.5
4.5
4.5
XL = (given)
1773
1773
1773
1773
|| at load
0.994894069
0.99488
0.994894069
0.994894069
SWR at load =
390.7013472
389.99
390.7013472
390.7013472
Rin =
5.255606806
5.26
5.255606806
5.255606806
Xin =
-22.99763828
-23.00
-22.99763828
-22.99763828
|Zin| =
23.59052287
23.60
23.59052287
23.59052287
-77.12735721
-77.13
-77.12735721
-77.12735721
SWR at input =
74.11147916
74.07
74.11147916
74.11147916
Total Loss (dB) =
Additional Loss Due
to Mismatch (dB) =
7.220380739
13.277
13.27435696
13.27435696
7.125380739
13.182
13.17935696
13.17935696
Angle(Zin) =
47
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Resistance and Reactance Along the Transmission Line
6
Resistance in ohms
5
4
3
2
1
0
0
10
20
0
30
40
50
60
70
80
90
100
80
90
100
Distance from Transmitter in feet
-200
Reactance in ohms
-400
-600
-800
-1,000
-1,200
-1,400
-1,600
-1,800
0
10
20
30
40
50
60
70
Distance from Transmitter in feet
48
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Voltage and Current Along Transmission Line
7,000
6,000
Average Input Power 1500 W
Maximum RMS Voltage 6625.5 V
RMS Voltage volts
5,000
4,000
3,000
2,000
1,000
0
0
10
20
VLoad
RMS Current in amperes
TLW load voltage is wrong
PLoad
1500  101.3274357
70.5758 W



 6625.5 VRMS
4.5
GLoad
1.60776 mS
4.52  16732
30
40
50
60
70
80
90
100
80
90
100
Distance from Transmitter in feet
20
15
Average Input Power 1500 W
Maximum RMS Current 16.894 A
10
5
0
0
10
20
30
40
50
60
70
Distance from Transmitter in feet
49
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Summary of Test Example
 The published formula underestimates line loss by 6.05 dB
 Assuming 1500 watts input to line
 Published formula predicts incorrectly
 Total line loss = 7.22 dB
 Power delivered to load = 284.5 W
 Power dissipated in line = 1215.5 W
 New formulas and TLW predict correctly
 Total line loss = 13.27 dB
 Power delivered to load = 70.6 W
 Power dissipated in line = 1429.4 W
Ignoring the seemingly small imaginary part of Z0 of practical
transmission lines can give large error in predicted delivered power.
50
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
Summary and Conclusion
 Showed the published formula for total line loss has a
fundamental problem – stemming from a hidden assumption
PDelivered  PForward  PReverse
 Showed the published formula and graphs for mismatched line
loss are correct only in two special cases
 Real Z0, e.g. distortionless line
 Real reflection coefficient, i.e. real VR /VF or real ZL/Z0
 “SWR < 20” is not the correct condition to assure accuracy
 Gave improved formulas for the total loss of general lines
 Showed the new formulas agree with Transmission Lines for
Windows (TLW)
Ignoring the seemingly small imaginary part of Z0 of practical
transmission lines can give large error in predicted delivered power.
51
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
References – Papers and Books
 Papers
 T.M. Papazoglou, “Review of the Maximum Efficiency of Transmission Lines,”
IEEE Trans. Education, Nov. 1992
 R.J. Vernon and S.R. Seshadri, “Reflection Coefficient and Reflected Power on a

Lossy Transmission Line,” Proc. IEEE, Jan. 1969
 G.E. Stannard, Calculation of Power on a Transmission Line, Proc. IEEE, Jan.

1967
 K. Kurokawa, “Power Waves and the Scattering Matrix,” IEEE Trans. Microwave
Theory and Techniques, Mar. 1965
 W.W. Macalpine, “Computation of Impedance and Efficiency of Transmission
Line with High Standing-Wave Ratio,” Trans. AIEE, July 1953
 S.G. Lutz, “Transmission Line Load Impedance for Maximum Efficiency,” Trans.
AIEE, July 1951
 W.W. Macalpine, “Heating of Radio-Frequency Cables,” Trans. AIEE, July 1949
 Books
 M.E. Van Valkenburg and W.M. Middleton, eds., Reference Data for Engineers,
9th ed., Newnes/Butterworth-Heineman, 2002
 R.A. Chipman, Theory and Problems of Transmission Lines, McGraw-Hill, 1968

 R.W.P. King, Transmission Line Theory, McGraw-Hill, 1955. Republished by
Dover 1965
 H.H. Skilling, Electric Transmission Lines, McGraw-Hill, 1951

 W.C. Johnson, Transmission Lines and Networks, McGraw-Hill, 1950





52
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
References – Books continued


















53
J.J. Karakash, Transmission Lines and Filter Networks, Macmillan, 1950
E.W. Kimbark, Electrical Transmission of Power and Signals, Wiley, 1949
J.D. Ryder, Networks, Lines, and Fields, Prentice-Hall, 1949
S. Ramo and J.R. Whinnery, Fields and Waves in Modern Radio, Wiley, 1949
W. Jackson, High Frequency Transmission Lines, Methuen, 1945.
R.I. Sarbacher and W.A. Edson, Hyper and Ultrahigh Frequency Engineering,
Wiley, 1943, Chapter 9.
F.E. Terman, Radio Engineering, McGraw-Hill, 1937
E.A. Guillemin, Communication Networks, Vol. II: The Classical Theory of Long
Lines, Filters and Related Networks, Wiley, 1935
W.L. Everitt, Communication Engineering, McGraw-Hill, 1932
A.E. Kennelly, The Application of Hyperbolic Functions to Electrical Engineering
Problems, McGraw-Hill and Univ. London Press, 1916; Tables of Complex
Hyperbolic and Circular Functions, Harvard Univ. Press, 1914; Chart Atlas of
Complex Hyperbolic and Circular Functions, Harvard Univ. Press, 1914
C.P. Steinmetz, Elementary Lectures on Electric Discharges, Waves and
Impulses, and Other Transients, McGraw-Hill, 1914
H.B. Dwight, Transmission Line Formulas for Electrical Engineers and
Engineering Students, Constable, 1913
J.A. Fleming, The Propagation of Electrical Currents in Telephone and Telegraph
Conductors, Constable, 1911
O. Heaviside, Electromagnetic Theory, London, 1893-1912
O. Heaviside, Electrical Papers, London, 1872-1892
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014
The End
This presentation will be archived at
http://www.fars.k6ya.org/docs/k6oik
54
Steve Stearns, K6OIK
ARRL Pacificon Antenna Seminar, Santa Clara, CA
October 10-12, 2014