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| 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 |

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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

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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 Out1 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 Ladder line ~ 0.04 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. Return to: Beginning of this file

Return to: KE3FL home pageSWR Calculation ExampleFor 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. Return to: Beginning of this file

Return to: KE3FL home pageSome Notes on Antenna BandwidthL. B. Cebik, W4RNL, QRPARCI #2572 1434 High Mesa Drive Knoxville, Tennessee 37938-4443 cebik@utk.edu Return to: Beginning of this file

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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. 6. Antenna shortening and loading: Although antenna loading for 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 about 70 kHz for most common loading schemes--just about half 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 Canada. Return to: Beginning of this file

Return to: KE3FL home pageTable 1 Selected 2:1 SWR Bandwidths for Wire AntennasReturn to: Beginning of this file

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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 Return to: Beginning of this file

Return to: KE3FL home pageBASIC Program ListingReturn to: Beginning of this file

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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 "Anotherrequency or 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. 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 $10 for this program are greatly accepted :) Please indicate which program you are making a donation for, thanks. Please make checks payable to: Circle Software and send them to: P.O. Box 74 Mt Airy MD 21771 (end of this file) Return to: Beginning of this fileuit" 280 A$=INKEY$ 290 IF A$="f" OR A$="F" THEN 10 ELSE IF A$="q" OR A$="Q" THEN 300 ELSE 280 300 END Return to: Beginning of this file

Return to: KE3FL home pagePower Output/Dipole Bandwidth Program[Click Here] to download the Zipped program and its associated files. The file name is:PWROUT.ZIP

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