|The Liberal Arts Major and Antenna Theory|
Email Question on TowerTalk Reflector
Can someone direct me to a web site that
explains antenna input impedance for a Liberal Arts Major with a smattering
of college algebra some 35 years ago? I've never understood imaginary
numbers. I am working on yagi design here and the program defines input
inpedance in imaginary numbers...such as 41.3 -j .9 ohms. OK. I am trying
to interpret that stuff.
Gene Smar AD3F answered my question on TowerTalk with this elegant email:
I can't point you to anything definitive on the Web right now, but I'll keep looking for you. In the meantime, let me pass along a few rules of thumb that might help you understand complex numbers (numbers with j's in them) and antenna impedance.
1. A complex antenna impedance like R + jX or R - jX means that the power you create in your transmitter's final amp(s) will not ALL be sent out over the air. Only the stuff that can gets burned up by the R part is real (R) power. The j part (or i for imaginary in most math texts) is energy that you CAN'T get at. It's the energy that's stored in the antenna's equivalent inductance (+j part) or capacitance (-j part).
Think of this energy as a minimum balance in your checking account. You can't spend it - it has to stay there untouched. Also, it's not just a fixed minimum balance; it's a given fraction of ALL the money (energy) you put into the antenna system. The +/-j energy (stored in the inductive or capacitive field of the equivalent antenna) is NOT available for use (transmission).
2. However, you CAN get at this +/-j energy if you cancel the +j or -j with a -j or a +j. That is, if your antenna is 35 +j10, then if you put in a -j10 in series with this impedance, you'll end up with 35 +j10 -j10 or 35 REAL Ohms! (The two 10's cancel. That's called matching the antenna.) So if your rig puts out 100 Watts into this antenna-plus-matching-system, your antenn will be able to actually get at (transmit) the full 100 Watts.
In a gamma matching system for a Yagi, for example, that's exactly what you do with the series capacitor. The gamma rod represents an inductive (+j) reactance to the coax. You move the gamma rod tap around on the driven element until you hit the 50 Ohm tap point, or close to it. At this tap point the coax might see 50 +j15 Ohms, for example. You adjust the series capacitor (you add more and more -j to the impedance) until you've added in exactly -j15 Ohms of capacitive reactance. Here your SWR bridge or other measuring device will show R = 50, j = 0 Ohms. Your SWR is 1:1 (50 Ohms of the coax is connected to 50 REAL Ohms of the antenna.)
But the -j part will be exactly 15 Ohms at only ONE FREQUENCY.
When you move above or below that frequency (the resonant frequency),
you'll change the sum of the + and - parts. Your SWR will increase above
1:1. How far you can go in frequency before SWR gets too high (we hams
pick 2:1) depends on what the +j was to start with and how the -j
3. The Yagi example you used says that at some frequency the Yagi's feedpoint impedance is 41.3 REAL OHMS and 0.9 capacitive reactance Ohms (-j means capacitive). To get a better match to 50 Ohm coax, you'd have to introduce about 1 Ohm of INDUCTIVE reactance in series with the coax and the antenna's feedpoint. That would eliminate the almost 1 Ohm of capacitive reactance in your example.
OR - you could shorten the driven element a bit. This shortening DECREASES the capacitive reactance (makes it more inductive) of the element at that frequency (whatever that freq is.)
4. Antenna elements that are LONGER than their quarter-WL size are inductive; SHORTER elements are capacitive. That's why you need a loading COIL (inductance) to make short mobile HF whips seem LONGER - the antenna element itself is SHORTER than a quarter-WL on the lower freqs. The loading coil actually is cancelling the capacitive reactance of a short whip. If you were to model a LONG inverted L antenna for 160M (make it 170 feet long), then you would see its feedpoint impedance would be 50 +j 100 Ohms or so. It's Longer than a quarter WL at 160 M (133 feet is about 1/4 WL.) You'd then have to insert a series capacitance to knock down the inductive reactance of this LONGER wire element. And W1BB's (SK) design for a long inverted L for Topband included just such a series capacitor.
Good for you if you've been able to read this far and not get totally bored or confused. I guess the main thing to remember about +/-j is that you can't get at the energy that is stored in that component of the impedance. You have to eliminate or minimize it to make all the REAL energy available to the REAL impedance of the antenna. (No minimum balance required.)
George T. Daughters, K6GT wrote this to me about this question:
That "stuff" is telling you that your impedance (at the given frequency) looks like a 41.3 Ohm resistor in series with a capacitor that has 0.9 Ohms of reactance at that frequency. You can tell it's a capacitor, because it has a minus sign. If it had inductive reactance, the sign on the imaginary term would be positive. That's easy enough, huh? Now, if you can contrive to "match" the 0.9 Ohms of capacitive reactance with 0.9 ohms of inductive reactance, then the series combination would be 41.3 +0.9j -0.9j = 41.3 Ohms resistive. This would be pretty close to 50 Ohms, and would probably make you and your transceiver very happy. Remember, this only works for one frequency. As the frequency changes, one of the reactances goes UP, and one goes DOWN. Hence we build antenna tuners. Then we can match pretty much anything at pretty much any frequency.
Does this help at all?
George T. Daughters, K6GT