 Power Output Program Information by Phil Karras, KE3FL

| Introduction | Power Output Calculation Example | SWR Calculation Example | Some Notes on Antenna Bandwidth | Table 1 Selected 2:1 SWR Bandwidths for Wire Antennas | BASIC Program Listing | Power Output/Bandwidth Program |

Introduction

The equations used for the SWR and power output calculations came from "THE ARRL HANDBOOK FOR RADIO AMATEURS 1992"

Look for SWR and dB to understand how SWR is calculated and what it is, and to find the equations for dB and how it relates to %losses.

The equations for the dipole antenna bandwidth calculation section came from: L. B. Cebik, W4RNL who can be contacted at QRP-L@lehigh.edu These calculations are for the estimated 2:1 SWR bandwidth points.

Programmed by: Philip Karras/KE3FL for Circle Software

73, Phil/KE3FL
KE3FL@juno.com
Circle Software
P.O. Box 74
Mt. Airy, MD 21771

Power Output Calculation Example

For this example I'm going to use the reflected power from the antenna due to the impedance mismatch as 50 watts (which would come from a mismatch SWR of about 5.8:1- see below) for a 100 watt transmitter. This is to make things a bit easier to visualize. If we were to loose all 50 watts we would be down by 3dB of power going out the antenna. Our second assumption is that the feed-line losses are 1% (about 0.04dB - ladder line) per 100 feet, and we have 100 feet of feed-line.
```
I hope a table will make this clear:

Trip   direction    power   Pwr Out
1        F        100.0    49.37
1        B         49.5      -
2        F         49.0    24.5
2        B         24.5      -
3        F         24.3    12.1
3        B         12.1      -
4        F         12.0     6.0
4        B          6.0      -
5        F          5.9     3.0
5        B          3.0      -
6        F          2.9     1.5
6        B          1.5      -
7        F          1.46    0.73
7        B          0.73     -
8        F          0.72    0.36
8        B          0.36     -
9        F          0.36    0.18
9        B          0.18     -
10        F          0.177   0.08
10        B          0.088    -
11        F          0.087   0.04
11        B          0.044    -
12        F          0.043   0.02
12        B          0.022    -
13        F          0.021   0.011
13        B          0.011    -
14        F          0.011   0.005
14        B          0.005    -
15        F          0.005   0.003
15        B          0.003    -     OK enough already!

Lets add up how much power got out: ~98 Watts.  So, there you have
it.  It took 15 round trips to get the power out, but we got 98
watts out of 100 with 1% loss per 100 ft.  So, how long did this
take?

15 X 200 ft = 3000 ft of feed-line total length.  Signals travel at
186,000 mi/s 3000 ft (well make it easy and say 1 mile, 5280 ft)
AND since I've increase the distance a bit I will ignore the
decrease speed of the RF in the feed-line, so this comes out to
about 0.000005 sec.  This would not only not be heard on Morse
code, but we wouldn't be able to hear such alterations on an SSB
signal either.

So we see that for extremely low loss feed-line, even an SWR of
about 6:1 causes only a 2% loss in transmitted power.

If we increase the feed-line loss to say 1.2 dB (about 25%) we get:

Trip   direction    power   Pwr Ant   Pwr Out
1        F        100.0    75.0      37.50
1        B         37.5      -
2        F         28.13   21.09     10.55
2        B         10.55     -
3        F          7.91    5.93      2.97
3        B          2.97     -
4        F          2.22    1.67      0.83
4        B          0.83     -
5        F          0.63    0.47      0.23
5        B          0.23     -
6        F          0.18    0.13      0.07
6        B          0.07     -
7        F          0.05    0.04      0.02
7        B          0.02     -
8        F          0.02    0.01      0.01 - That's All!

Lets add up how much power got out: ~52 Watts.  So, there you have
it.  If we have a high SWR we'd better have a very low loss
feed-line, and if we have even one dB of loss in the line, we'd
better have a good SWR match to the antenna!

NOTE: For a PERFECT feed-line to antenna match (1:1 SWR) and a
1.2dB feed-line loss we would only see 75% of our transmitted power
getting to the antenna! So, better feed-line is the bottom line!

dB loss per 100 ft at 7.04 MHz  (Ref: ARRL Handbook 1992 Pg 16-14)
RG-58 ~ 0.98
RG-8  ~ 0.42

Equations used:  dB = 10 log(P2/P1) for dB loss P2 < P1
SWR = (1+r)/(1-r)
ro = sqrt(Pr/Pf) = (Pr/Pf)^1/2
Pr = reflected power
Pf = forward power

ro = reflection coefficient, rho

(Ref. ARRL Handbook 1992 for SWR: Pg 16-2 and for dB: Pg 2-7)

I hope this helps clarify why you can use ladder line with a good
ATU and not worry too much about feed-line-antenna SWR.

SWR Calculation Example

For this example I will use 450 mW of forward power and 100 mW of
reflected power.

Pf = 450 mW    and    Pr = 100 mW

ro = SQRT(Pr/Pf) = .4714      (ro = reflection coefficient, rho)

SWR = (1 + ro)/(1 - ro)  =  2.78:1

I hope this helps.  Just use your calculator and see how it works.

Some Notes on Antenna Bandwidth

L. B. Cebik, W4RNL, QRPARCI #2572
1434 High Mesa Drive
Knoxville, Tennessee 37938-4443
cebik@utk.edu

For most of us, the antenna's bandwidth is the number of Hz for which the
antenna will exhibit a less than 2:1 SWR.  We usually measure bandwidth
at the transmitter output, and hence put a large pile of variables on top
of the basic idea of SWR bandwidth.  So let's begin again and see how the
concept actually works.

An antenna--for example, a resonant half-wavelength dipole operated on
its fundamental frequency--has a natural feedpoint impedance.  For a
lossless wire dipole in free space, that figure is just about 72 ohms.
In fact, NEC-2 models of just such an antenna using wire diameters from
#30 to over 2.5" show less than 1 ohm variation in the 72-ohm feedpoint
impedance.

Relative to that impedance, a 2:1 SWR will occur as the feedpoint
impedance (off resonance, a complex of resistance and reactance) reaches
about 144 ohms at points higher or lower than resonance.  The number of Hz
(of KHz or MHz) between those frequencies is the 2:1 SWR bandwidth of the
antenna.  The bandwidth will vary with the diameter of the antenna
element in a regular but nonlinear manner.

2:1 SWR bandwidth is approximately (but again, nonlinearly) proportional
to frequency.  For a given wire size, a resonant dipole at 28 MHz will
have (about) twice the bandwidth of a resonant dipole at 14 MHz.

To help you gain a reasonable expectation of the 2:1 SWR bandwidth of
resonant half-wavelength dipoles, I am attaching a small BASIC utility
program that will produce bandwidth tables for any HF frequency for wires
from #30 (0.01" diameter) to 2.5" diameter.  It is roughly calibrated to
NEC- 2 models for lossless wire resonant dipoles in free space and to 72
ohms.  The algorithms are generally accurate to about 5%, with some
matrix-center variations reaching about 10%.  The figures are roughly
applicable also to resonant quarter-wavelength vertical antennas.

Table 1 summarizes a few data points for thin, medium, and thick antenna
elements on 80, 40, 20, and 10 meters.  The increase of bandwidth with
frequency for a given wire size is evident.  Notice also that it takes
nearly a 100:1 wire size increase to double the bandwidth of the antenna
on any given frequency.

The degree of error in the program is of no concern, since real antennas
and antenna systems will introduce larger variations that no table can
account for in advance.  Hence, the program is only for getting some
reasonable expectations, not for predicting bandwidth with precision.  The
bandwidth you actually measure will vary with the following variables:

1.  Antenna type: Low impedance antenna types will generally (but
not always) have wider bandwidths than high impedance
antennas.

2.  Antenna material: Copper and aluminum have losses that affect
antenna bandwidth, especially with small diameter wires (less
than #20).

3.  Antenna environment: Placing an antenna some height above
ground less than about 2 wavelengths will alter both the
natural feedpoint impedance and the bandwidth at that
impedance.  Ground clutter in the near field of the antenna
will affect both in ways that are for practical purposes
unpredictable.

4.  Feedline mismatch: Feeding a 72-ohm antenna with our common
50-ohm coax starts us out at 1.4:1 SWR, hence decreasing the
2:1 SWR bandwidth.  The reduction of SWR bandwidth is a
function of a complex curve that begins with a shallow
decrease, narrows to the inverse of the SWR at the 1.4 SWR
point and then decreases rapidly toward zero as the basic
mismatch SWR grows to 2.  Hence, for the case of the dipole
fed with 50-ohm coax, we should expect about 70% of the
program's estimated bandwidth.  (This fact explains why
some claim a slightly wider band width for inverted Vee
configurations: being closer to 50 ohm natural feedpoint
impedance, Vees introduce less bandwidth narrowing due to the
slight mismatch).

5.  Feedline losses: Even well-matched
transmitter-feedline-antenna systems introduce some losses in
the feedline.  The effect of these losses is to reduce the SWR
at the transmitter end of the line, thus giving a wider 2:1
SWR bandwidth.  This wider bandwidth is usable, so long as we
understand and evaluate the acceptability of the power losses
involved.

the sake of shortening reduces the feedpoint impedance, it
introduces components that raise antenna Q and narrow the
bandwidth.  As a rule of thumb, bandwidth is reduced by the
percentage of shortening of the antenna.  For example, a 33'
vertical on 80 meters is about half size, and its bandwidth is
the bandwidth of a full size quarter-wave vertical.

Understanding these bandwidth-altering factors along with the basic output
of the program can give us reasonable expectations for antenna bandwidth
for the various bands.  If our antenna system is more than about 20% off
the mark, then we begin to search for possible problems.

Remember that these notes do not apply to antennas fed with parallel
feedline and an ATU: those we always tune for 1:1 SWR and maximum power
output to the line and antenna.

Finally, if you do not like typing BASIC programs or converting them to C,
a version of the program will appear in VE3ERP's HAMCALC collection,
available in the Lehigh.edu archives or directly from Murph.  Address: Mr.
George Murphy, VE3ERP; 77 McKenzie Street; Orillia, Ontario; L3V 6A6

Table 1  Selected 2:1 SWR Bandwidths for Wire Antennas

Frequency      3.5 MHz        7 MHz          14 MHz         28 MHz

Wire Size
(diameter)                    2:1 SWR Bandwidth in MHz

#28 AWG
(0.013")       0.17           0.35           0.73           1.63

#12 AWG
(0.081")       0.19           0.40           0.86           1.91

#4 AWG
(0.204")       0.22           0.46           0.98           2.18

(1")           0.30           0.63           1.35           3.06

BASIC Program Listing

10 ' BW.BAS
20 CLS:SCREEN 0: COLOR 2,,4:CLS
30 ER\$=STRING\$(70,32):BW\$="###.###":WIRE\$="#.###":S\$=STRING\$(10,32):
T\$=STRING\$(6,32)                                                                        
40 ' Estimates 2:1 SWR bandwidth of halfwavelength dipoles for a range of
common wire and tubing sizes.  Algorithm is based on NEC models of
lossless wire dipoles in free space and is based on a feedpoint
50 ' impedance of 72 ohms.  Program does not account for material losses,
feedline losses, mismatches, or the antenna environment.  Accuracy
averages 5%.
60 PRINT "   Estimated 2:1 SWR bandwidth of half-wavelength dipoles at
any HF frequency"
70 LOCATE 2,25:PRINT "by L. B. Cebik, W4RNL"
80 LOCATE 3,15:INPUT "Enter any frequency from 3 - 30 MHz:   ",F
90 IF F>30 OR F<3 THEN LOCATE 3,5:PRINT ER\$:GOTO 80
100 PRINT "Wire size","Wire dia.","Bandwidth";S\$;"Wire
dia.";T\$;"Bandwidth"::PRINT "  AWG  ","inches","   MHz  ";S\$;"
inches",T\$;" MHz   "
110 FOR J=30 TO 2 STEP -2
120 AWG\$=MKS\$(J):N=J:AWG=J
130 K#=(.46/.005)^(1/39):WIRE=.46/K#^(N+3):DIA=WIRE
140 DIA2=DIA-((.4343*LOG(30/F))*(DIA/(2*(2.56/DIA))))
150 BWBASE=(.0469+(((F/3)-1)*(.0116/9)))*F
160 BW=((SQR(DIA2))+.9)*BWBASE
170 PRINT AWG,:PRINT USING WIRE\$;WIRE,:PRINT" ",:PRINT USING BW\$;BW
180 NEXT
190 FOR J=.375 TO 2.5 STEP .125
200 DIA=J
210 DIA2=DIA-((.4343*LOG(30/F))*(DIA/(2*(2.56/DIA))))
220 BWBASE=(.0469+(((F/3)-1)*(.0116/9)))*F
230 BW=((SQR(DIA2))+.9)*BWBASE
240 K=(J*8)+3:LOCATE K,50
250 PRINT USING WIRE\$;J,:PRINT S\$;:PRINT USING BW\$;BW
260 NEXT
270 LOCATE 23,5:PRINT "Another requency or uit"
280 A\$=INKEY\$
290 IF A\$="f" OR A\$="F" THEN 10 ELSE IF A\$="q" OR A\$="Q" THEN 300 ELSE 280
300 END

Power Output/Dipole Bandwidth Program

associated files.

The file name is:  PWROUT.ZIP

You will need to use PKUNZIP to extract the needed files.  I suggest
doing so in a new sub-directory.  Let me know how it works for you by
sending me an e-mail message.   <!--
BuildIt('ke3fl', 'l', 'qs', '1');
//-->
to ke3fl.

If you enjoy this program and would like to support my efforts in this
area you may do so by sending a small donation.

No donations are expected from non-profit teaching institutions or
individual casual users.

Donations are expected from for-profit organizations and individuals.

Donations of \$10 for this program are greatly accepted :)

Please indicate which program you are making a donation for,
thanks.

Please make checks payable to: Circle Software
and send them to:              P.O. Box 74
Mt Airy  MD  21771

(end of this file)