Reactance of Small AntennasKirk T. McDonaldJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544(June 3, 2009; updated September 11, 2012)1ProblemEstimate the capacitance and inductance of a short, center-fed, linear dipole antenna whosearms each have length h and radius a. Also estimate the inductance of a small loop antennaof major radius b and minor radius a.For completeness, consider also the real part, its so-called radiation resistance, of theantenna impedance in the approximation of perfect conductors.2Solution2.1 Short, Center-Fed, Linear Dipole An tennaThis solution follows sec. 10.3 of [1].2.1.1 CapacitanceThe key assumption is that the electric field lines from one arm of the dipole antenna to theother follow semicircular paths (the principal mode), as shown in the figure below.1If so, all the field lines emanating from charge dQ in interval dr at distance r from thecenter of the antenna cross a surface of area 2πr dr sin θ that lies on a cone of half angle θ,so the electric field strength at (r, θ)isE =dQ/dr2π0r sin θ. (1)1On the right is Fig. 86 from [2].1The voltage difference between the two arms of the antenna is2ΔV =2π/2θminEr dθ =dQ/drπ0π/2a/rdθsin θ=dQ/drπ0ln[tan(θ/2)]π/2a/r=dQ/drπ0ln(2r/a). (2)This voltage difference should be independent of position along the antenna.3The chargedistribution dQ/dr is indeed constant to a good approximation for short dipole antennas,but the factor ln(2r/a)=−ln(θmin/2) is constant only for a biconical dipole antenna (asmuch favored theoretically by Schelkunoff). A reasonable approximation for a linear dipoleantenna is to use r = h/2 as a representative length in eq. (2), which leads to the estimateΔV ≈dQ/drπ0ln(h/a). (3)The corresponding capacitance per unit length along the antenna isdCdr≈π0ln(h/a), (4)and the total capacitance isC ≈π0hln(h/a). (5)This estimate ignores the contribution to the capacitance of roughly π0a2/d associated withthe electric field in the gap d between the terminals of the antenna, as is reasonable whend ≈ a since then ln(h/a) h/a ≈ dh/a2.2.1.2 InductanceFor a quick estimate of the inductance of the antenna we note when the arms carry currentI the magnetic field near the conductors varies with distance asB ≈μ0I2πr. (6)The magnetic flux associated with the linear antenna isΦ=LI ≈ hhaBdr≈μ0hI2πlnha, (7)where we note that the current drops from I to0overlengthh of each arm. Then, ourrough estimate of the inductance L isL ≈μ0h2πlnha. (8)2In general the electric field is related to the scalar and vector potentials by E = −∇V − ∂A/∂t =−∇V − iωA, assuming a time dependence of the form eiωt. Then, 21E · dl = V1− V2− iω 21A · dl.However, close to a small linear dipole antenna the electric field is much larger than the magnetic field (see,for example, [3]), and the contribution of the vector potential to the electric field in negligible in this region.3The vanishing of the tangential component of the electric field along the (ideal) conductor implies thatthis conductor is an equipotential only if the vector potential can be neglected. For examples where thisdoes not hold, see [4, 5].22.1.3 ReactanceThe reactance of a short linear antenna (h λ) is largely due to its capacitance,Xsmall linear= ωL −1ωC≈−1ckC≈−ln(h/a)π0ckh= −Z0πln(h/a)kh= −Z0π2λ2hln(h/a), (9)where (c =1/√0μ0being the speed of light in vacuum)Z0=μ00= μ0c =10c= 377 Ω. (10)The reactance (9) falls with increasing length h of the arms of the antenna, and vanisheswhenω =1√LC≈2πμ0h1π0h=√2ch= kc =2πcλ, i.e., h ≈√2λ2π=λ4.44. (11)Thus, the rough estimates (5) and (8) of C and L for a linear antenna give a fairly goodprediction of “resonance” when half-length h ≈ λ/4.2.1.4 Relation between Reactance and “Free Oscillation”As an aside, we note that frequencies at which the terminal reactance vanishes correspondto those of “free oscillation of the antenna (with its terminals shorted).In a “free oscillation”4radiation is ignored and the (near) fields are standing waves thatobey the Helmholtz equation, (∇2+ k2)ψ =0,whereψ is any scalar component of theelectric and magnetic fields. Electromagnetic energy is stored in the (near) fields, whichoscillates between “electric” and “magnetic” terms, there being no exchange of energy withthe perfect conductor.For driven oscillations of a conductor, a nonzero terminal reactance implies an exchangeof energy between the energy/voltage source and the (near) electromagnetic fields.Thus, if the reactance in nonzero at some frequency, that frequency cannot correspond toa “free oscillation” (for which there is no exchange of energy between fields and conductors).2.2 Small Loop Antenna2.2.1 InductanceOne definition of a small loop antenna is that the spatial variation of the current around theloop can be neglected. In this case the inductance is essentially that of a circular loop/torusof,say,majorradiusb and minor radius a, supposing that all the current in on the surfacebecause of the skin effect.ForaquickestimatewenotewhentheloopcarriescurrentI the magnetic field near theconductor varies with distance asB ≈μ0I2πr, (12)4“Free oscillations” of (perfect) conductors were perhaps first discussed in [6]. See also, [7].3so the magnetic flux linked by the loop isΦ=LI ≈ 2πbbaBdr≈ μ0bI lnba, (13)and the inductance L isL ≈ μ0b lnba= μ0bln8ba− 2.08. (14)A more exact calculation using toroidal coordinates [8] shows that the number 2.08 = ln 8in eq. (14) is actually 2 when b  a.2.2.2 CapacitanceA loop antenna has a small capacitance C associated with the gap between its terminal.However, the capacitive reactance 1/iωC is negligible in practice, so we skip estimating thecapacitance C.52.2.3 ReactanceThe reactance of a small loop antenna is essentially that due to its inductance,Xsmall l oop≈ ωL ≈ μ0ωb lnba= μ0ckb lnba= Z02πbλlnba. (15)A Appendix: Radiation Resistance of Small AntennasFor completeness, we include the well-known calculations of the radiation resistance Rradofsmall antennas, noting that the time-average radiated power P is related to the peak currentI0at the antenna terminals byP =I20Rrad2=μ0|¨p|212πc=μ0ω4|p0|212πc, i.e., Rrad=μ0ω4|p0|26πcI20, (16)where p0is the peak electric dipole moment of the antenna (or p0= m0/c in case the