dalyb@godzilla.UUCP (Brian Daly) (01/17/90)
Subject: Re: Need Theoritical Antenna Info Newsgroups: rec.ham-radio,sci.electronics Keywords: Antennas John Moore (NJ7E) posted a request for information on theoritical formulas for effective aperture, input impedance, and radiation resistance for short/ long dipoles, short and long quarter wave ground planes, and magnetic dipole antennas. Here's a brief summary of equations: Reference: Balanis, "Antenna Theory: Analysis and Design", Harper and Row. This text is available from the Arizona State University bookstore. Kraus, "Antennas", McGraw Hill Short Dipole: (wavelength/50) <= length <= (wavelength/10) Effective Aperture Ae = 3(wavelength)**2 / ( 8 * pi) Radiation Resistance: Rr = (20 * pi**2) * (length/wavelength) Input Impedance: For a small dipole of length l and wire radius a: Reactance: Xin = -120 [ln(l/a)-1] / tan(kl) where k = (2 * pi) / wavelength Resistance: Rin = Rr as given above Note that if the antenna length is slightly less than 1/2 wavelength, the reactance will go to zero. For l of a halfwavelength, the reactance is approximately zero and the resistance is close to 50 ohms. (42.5 ohms for wire of small radius). Long Dipole: The analysis for a finite length dipole gets a little complicated. Balanis (reference above) provides a FORTRAN program in his text to compute radiation resistance, directivity, and input resistance for a finite length dipole based on the following equations: Radiation resistance Rr = 60 { C + ln(kl) - Ci(kl) + (1/2)sin(kl) * [Si(2kl) - 2Si(kl)] + (1/2)cos(kl) * [C + ln(kl/2) + Ci(2kl) - 2Ci(kl)] } where Ci,Si are cosine and sine integrals C = Euler's constant = 0.5772 Input resistance: Rin = Rr / (sin (kl/2) **2) Input reactance: Xin = Xm / (sin(kl/2) **2) where Xm = 30 { 2Si(kl) + cos(kl)[2Si(kl) - Si(2kl)] - sin(kl)[2Ci(kl) - Ci(2kl) - Ci(( 2k * {a**2})/l)]} Effective Aperture (maximum): Aem = {(wavelength **2) / (4 * pi)} * (directivity) Balanis figure 4.8 plots directivity versus dipole length for 0 to 3 wavelengths. The calculation for the directivity is rather complicated; refer to the reference for details of the calculations. Magnetic Dipole: By applying the principle of duality, a magnetic dipole of magnetic moment Iml is equivalent to a small electrical loop of radius a and constant electric current Io provided that: Iml = jSwuIo where S = area of the loop = pi * a**2 w = radian frequency u = permeability For a small electric loop of radius a: Radiation resistance: Rr = 20 * (pi)**2 * {C/wavelength}**4 where C = circumference of the loop Effective Aperture (maximum) Aem = 3 * (wavelength)**2 / (8 * pi) Short/Long (what is a short or long??) Quarter Wave Ground Plane: The analysis of a quarter wave ground plane antenna is essentially a vertical electrical dipole above a ground plane. When the length of the antenna is a quarter wave, and the height above the ground plane is zero, then: Input impedance = 36.5 + j21.25 (note that this is one half the input impedance of a half wave dipole) Effective aperture (maximum) Aem = 0.13 * (wavelength)**2 I hope this helps answer some of your questions. I refer you to the references I mentioned for more detailed information. The Balanis text does contain many FORTRAN programs, which may be of some use. Brian K. Daly WB7OML AG Communication Systems, P.O. Box 52179, Phoenix, Az. 85072-2179 UUCP: {...!ames!ncar!noao!asuvax|uunet!zardoz!hrc|att}!gtephx!dalyb Phone: (602) 582-7644 FAX: (602) 582-7111
feg@bonnie.ATT.COM (Forrest Gehrke,2C-119,7239,ATTBL) (01/18/90)
In article <481303aa.1423f@godzilla.UUCP> dalyb@godzilla.UUCP (Brian Daly) writes: > >Subject: Re: Need Theoritical Antenna Info >Newsgroups: rec.ham-radio,sci.electronics >Keywords: Antennas > > Note that if the antenna length is slightly less than 1/2 wavelength, > the reactance will go to zero. For l of a halfwavelength, the reactance > is approximately zero and the resistance is close to 50 ohms. (42.5 ohms > for wire of small radius). > I think this statement is a bit mixed up. The resistance is going to be more like 72 ohms. For an electrical 1/2 wavelength the reactance will be 42.5 ohms and the resistance will be 73 ohms. > > The analysis of a quarter wave ground plane antenna is essentially > a vertical electrical dipole above a ground plane. > > When the length of the antenna is a quarter wave, and the height above > the ground plane is zero, then: > > Input impedance = 36.5 + j21.25 (note that this is one half the input > impedance of a half wave dipole) > As I noted above for a halfwave, your parenthetic comment now agrees with the halfwave case. Again, if the monopole is slightly less than an electrical 1/4 wavelength the reactance will go to zero. BTW, I think most of these cases assume the current loop location, e.g. for the half wave dipole we are looking at the impedance at the dipole's center. If we were looking at the current node the impedance would be quite different. Forrest Gehrke feg@clyde.ATT.COM k2bt