On 8/24/22 11:39 AM, TG Frerichs wrote:
I keep seeing the statement that a resonant halfwave center fed dipole has zero reactance at the feed point, and I don't think this is always correct.
For a center-fed dipole that is 1/2 wavelength and where the length/diameter ratio is greater than fifteen (i.e. a wire antenna) in free space, (e.g., no mutual impedance from a ground plane), I found a couple of references that indicate Xl <> -Xc. They agree as to values.
Quoting Blake:
The radiation resistance for an exactly half-wavelength dipole is found ... to be 73.1 ohms, referred to the maximum current point (dipole center). Therefore this is also the resistive component of the input impedance when the dipole is fed at the center. There is also a small reactive component of 42.5 ohms, inductive. This small inductive reactance may be eliminated by shortening the dipole to about 95% of a half-wavelength (i.e., to 0.475 {lamda}) The radiation resistance (and input impedance) is then 67 ohms. The pattern (beamwidth and gain) is not significantly affected by this slight shortening. ---- Blake, L. V., Antennas (Second), Artech House, Dedham, MA, 1984, pp. 175-176
And in the Antenna Engineering Handbook (Third) in the chapter on dipoles and monopoles I found a third-order polynomial that gave the same results given the above limitations. The claimed accuracy of this approximation is 0.5 ohms, and in the table entitled "Functions R(kl) and X(kl) Contained in the Formula of the Input Impedance of a Center-Driven Cylindrical Antenna" the values for pi/2 length (in radians) are R=73.12 and X=42.46. ---- Chen, To Tai and Long, Stuart A.,"Dipoles and Monopoles," Antenna Engineering Handbook (Third), Johnson, Richard C., Editor, McGraw-Hill, New York, NY, 1993, pp. 4-4, 4-5
Of course, if you start to include a ground plane, ideal or otherwise, these numbers go all to hell. But I think it does show that assuming that a resonant dipole has zero input reactance is not necessarily accurate.
In general: resonance => no reactive component (by definition)
In general: an exact half wavelength (or multiple thereof) has a reactive component for the feedpoint impedance (no matter where along the antenna the feedpoint is located).
In general: resonance is NOT an exact multiple of lambda/2 (at any multiple)
None of these are contradicted by the references above, nor any standard references (Kraus, Balanis, Orfanidis), all of which give explanations and derivations (multiple ones in most cases).
The subject of "analytical formulas" for feed point impedance (and mutual coupling) for various configurations (straight, angled, over a ground plane, next to other antennas in echelon, lined up, parallel, skew, non coplanar, you name it) - has been the subject of countless papers. There are formulas which attempt to approximate these things from first principles. There are also formulas which are a "best fit approximation" to measurements or models, and do not claim to be "representative of the actual theory" - just efficient and fast. A lot of these were used in the past for things like Yagi and Phased Array design. (See, e.g., King three term approximation)
Back before computers got fast, these equations were quite useful in design work. Now, though, it's probably easier to run a numerical model. Rather than write a function that calculates a series expansion for Sine or Cosine Integrals, and then figure out how to incorporate skew or images or whatever, it's easier to write a function that takes the parameter of interest, builds a model file, runs the model, and returns the desired parameter parsed from the output.
