ECEn 665: Antennas and Propagation for Wireless Communications 48 3.3 Loop Antenna An electric dipole antenna radiates an electric field that is aligned with the dipole and a magnetic field that radiates outward as circular field lines around the axis of the dipole. The wire loop antenna is in some sense the opposite of the dipole. A loop antenna radiates a magnetic field that is orthogonal to the plane of the loop, and the electric field radiates outward as circular field lines in the plane of the loop. The dipole antenna and loop antenna are said to be duals, because the electric and magnetic field pictures of the radiated electric field can essentially be exchanged. A thin wire loop antenna can be modeled with the equivalent current source J(r) ˆϕI δ(ρ a)δ(z) (3.38) The vector current moment is N dr e jkˆr r ˆϕ I δ(ρ a)δ(z ) (3.39) In order to evaluate the integral, we have to expand ˆϕ into fixed components that do not depend on the integration point r, using ˆϕ ˆx sin ϕ + ŷ cos ϕ (3.4) The phase in the integrand can be expressed as ˆr r (ˆx sin θ cos ϕ + ŷ sin θ sin ϕ + ẑ cos θ) (ˆxa cos ϕ + ŷa sin ϕ ) a sin θ(cos ϕ cos ϕ + sin ϕ sin ϕ ) a sin θ cos(ϕ ϕ ) Combining these results and evaluating the integrals over ρ and z leads to N adϕ e jka sin θ cos(ϕ ϕ ) I ( ˆx sin ϕ + ŷ cos ϕ ) (3.4) for the vector current moment. In the far field, we only need the transverse components N θ and N ϕ. N θ is zero since the current direction is confined to the x-y plane. We can form the ϕ component before integrating using N ϕ N x sin ϕ + N y cos ϕ Making the substitution u ϕ ϕ, adϕ e jka sin θ cos(ϕ ϕ ) I (sin ϕ sin ϕ + cos ϕ cos ϕ ) adϕ e jka sin θ cos(ϕ ϕ ) I cos(ϕ ϕ ) N ϕ I a ϕ ϕ cos u e jka sin θ cos u du Since the integrand is periodic, we can change the integration limits to N ϕ I a cos u e jka sin θ cos u du
ECEn 665: Antennas and Propagation for Wireless Communications 49 We now use the integral identity J n (z) j n 2π with which the vector current moment becomes where J (x) is the first order Bessel function. The far electric field is cos(nϕ)e jz cos ϕ dϕ (3.42) N ϕ I a2πjj (ka sin θ) (3.43) E ϕ jωµ e jkr 4πr I a2πjj (ka sin θ) aωµi e jkr 2r J (ka sin θ) (3.44) From the far field, we can see that the radiation pattern of the loop antenna is f(θ) J (ka sin θ) 2 (3.45) The first order Bessel function is oscillatory with respect to its argument. When ka is small, the argument of the Bessel function changes only over a small range as θ varies from zero to π, and the radiation pattern only has one broad lobe. When ka is large, the argument passes zeros of the Bessel function as θ varies, and the pattern has multiple lobes. 3.3. Electrically Small Loop (ka ) In the case of an electrically small loop with ka, using the small argument approximation J (x) x/2 the electric field can be approximated as E ϕ aωµi e jkr 2r ka sin θ 2 η(ka) 2 e jkr I sin θ (3.46) 4r which shows that the radiation pattern of a small loop antenna is the same as that of a Hertzian dipole. The polarization is different, because the Hertzian dipole radiates a ˆθ-polarized electric field, which is orthogonal to the ˆϕ polarization of the loop antenna. The similarity between the radiation patterns of the dipole and loop antennas is a reflection of the duality principle of electromagnetics. The magnetic field radiated by the electric dipole have the same form as the electric field radiated by the loop antenna. Because of this duality of the two antenna types, a small electric loop antenna is also sometimes called a magnetic dipole. To understand why, consider the magnetic Hertzian dipole current model M(r) ẑi m lδ(r) (3.47) This magnetic current is sometimes drawn as an arrow with two heads to distinguish it from an electric current. The far electric field radiated by the magnetic dipole current is identical to (3.46) if I m l jωµπa 2 I (3.48) In general, we can consider four related types of antennas: the electric dipole, electric loop, magnetic dipole, and magnetic loop (a fictitious magnetic current flowing in a loop). The electric dipole and electric loop are a dual pair of antennas, as are the magnetic dipole and magnetic loop, whereas the electric loop and magnetic dipole radiate identical far fields (as do the electric dipole and magnetic loop) and are said to be equivalent antennas.
ECEn 665: Antennas and Propagation for Wireless Communications 5 Since the fields radiated by a small loop antenna are the same as that of a Hertzian dipole (except for the polarization), the directivity is the same, but the radiation resistance is smaller: D(θ) 3 2 sin2 θ (3.49) P rad η π(ka)4 I 2 (3.5) 2 R rad η π(ka)4 6 (3.5) For an multi-turn loop antenna with N turns, the radiation resistance increases by a factor of N 2. The radiation resistance can be increased roughly by the magnetic permeability by adding a ferrite core through the loop antenna. The ferrite core loop antenna is sometimes called a loop-stick antenna, and was commonly used in AM radio receivers. At high frequencies, the ferrite adds a substantial additional loss term to the antenna input resistance and reduces the radiation efficiency, making ferrite loop antennas less useful for microwave applications. 3.3.2 Large Loop (ka ) For larger values of ka, the radiation pattern has symmetrical ring-shaped main lobes above and below the x-y plane and sidelobes elevation angles away from the x-y plane. As the loop radius increases relative to the wavelength, the constant-current assumption in (3.38) breaks down due to phase variation of the timeharmonic current around the loop. For large loop antennas, a better current model or numerical method must be used to obtain accurate radiated fields. 3.4 Comparison of Dipole and Loop Antennas The loop and dipole antennas have roughly the same gain, but the radiation resistances are quite different. This has practical ramifications when choosing the antenna type for a given application. From the definition of radiation efficiency, it is easy to show that η rad R rad R rad + R ohmic (3.52) In order to determine the radiation efficiency, we need to determine the ohmic resistance of the antenna. A DC current is spread evenly over the cross section of a conductor, but at high frequencies the skin effect causes currents to be confined to the surface of the conductor. The current distribution decays exponentially according to J(x) e x/δ s, where x is the distance from the surface of the conductor. The skin depth is δ s πfµσ (3.53) where σ (S/m) is the conductivity. By integrating the exponential current distribution for a conductor with a circular cross section of circumference C, it can be shown that the resistance of a wire of length l is R l Cσδ s (3.54) The quantity /(σδ s ) has units of ohms, and can be identified as the conductor surface resistance R s σδ s πfµ σ (3.55)
ECEn 665: Antennas and Propagation for Wireless Communications 5 at the frequency f. This quantity is sometimes referred to as having units of Ohms per square, as this is the resistance from one edge to another of a square sheet of the material. For the loop antenna, the wire length is 2πa and the wire circumference is C 2πb, and the resistance is R ohmic a b R s (3.56) where b is the radius of the wire. The radiation efficiency of the loop antenna is therefore η rad For a short dipole antenna, the radiation efficiency is 6 ηπ(ka)4 6 ηπ(ka)4 + a b R s (Loop) (3.57) η rad 6π η(kl)2 6π η(kl)2 + l 2πb R s (Dipole) (3.58) The most significant difference between the dipole and loop is that the radiation efficiency of the loop is much smaller than the efficiency of the dipole for electrically small antenna dimensions (kl, ka ). For a loop antenna of radius a 4 mm made of one turn of copper wire with radius b.25 mm, at 3 MHz the radiation efficiency is η rad 3.5 7. A dipole with length 8 mm made of the same wire has radiation efficiency η rad 5.8 4, which is small but much larger than that of the loop antenna. Because of the low radiation efficiency, electrically small loop antennas are generally not used as transmitters. 3.4. Dipole and Loop Antenna Receivers In order to compare the loop and the dipole as receivers, we need to consider the signal to noise ratio realized by the antenna. From (2.4), the equivalent noise temperature delivered to a conjugate matched load at an antenna feed port is T n η rad T a + ( η rad )T p (3.59) where T p is the physical temperature of the antenna and T a is the effective brightness temperature of the scene around the antenna (i.e., the equivalent temperature of the noise at the antenna output port if the antenna were lossless). This temperature can be converted to power using P n k B BT n, where B is the system bandwidth. The signal power is This leads to a signal to noise ratio at the antenna output of inc λ2 P sig S 4π η radd (3.6) SNR Sinc λ2 4π Dη rad k B T n B k B B Sinc λ2 ( 4π D ) (3.6) T a + η rad η rad T p This expression holds for any antenna type, including dipoles and loops. The second term in the denominator is the noise produced by losses in the antenna referred to an equivalent external isotropic brightness temperature. If the radiation efficiency is small, ( η rad )/η rad is large, and loss has a bigger impact on the total noise power. The design choice between the loop antenna and the dipole antenna for a given application depends on the operating frequency. The ramifications of low radiation efficiency depends on the noisiness of the
ECEn 665: Antennas and Propagation for Wireless Communications 52 external environment, which in turn is frequency dependent. At low radio frequencies, galactic radiation is relatively intense, and the effective noise temperature of the sky is many tens of decibels larger than the physical temperature. At MHz, for example, the sky noise equivalent brightness temperature can range from 3 to 2 db higher than a reference temperature of 29 K! At MHz, the ratio is closer to 4 db. With such large values of T a in (3.6), the loss contribution is negligible even for the very low radiation efficiency of the loop antenna, and the performance of the loop and dipole are essentially identical. In practice, at low frequencies the transmitted signal level must be extremely high in order to be larger than the ambient noise. The antenna loss noise is insignificant in relation to the ambient noise for both the loop and the dipole, and either antenna can be chosen. At higher frequencies, the situation is different. The effective brightness temperature of the environment around the antenna is comparable to the physical temperature of the environment, in which case the low radiation efficiency of the loop causes a significant SNR penalty, and the performance of the dipole antenna is generally much better than that of the loop. A multi-turn loop antenna can be closer in performance to the dipole, but generally is larger and less convenient than a dipole or dipole-like antenna. 3.5 Electrically Small Antennas and Bandwidth Limitations For most applications, all else being equal, the smaller the antenna that can be used, the better. We have seen from our treatment of dipole and loop antennas that radiation resistance is low when the antenna is much smaller than the operating wavelength, which leads to poor radiation efficiency. But poor radiation efficiency is not the only difficulty associated with the design of small antennas for low frequency operation. Even lossless antennas have limits when the dimensions are small relative to the electromagnetic wavelength. At low frequencies, the loop antenna has a very small radiation resistance and a finite inductance, so the antenna is essentially a reactive load to its feeding transmission line. Similarly, a short dipole is primarily capacitative. In order for a generator to supply substantial power to either antenna, the generator must be conjugate matched to the antenna. For a dipole, this means attaching an inductive source to a capacitative load. If the loss resistances in the system are small, this results in a high quality factor (Q) LC resonant circuit and a narrow usable bandwidth. Since a high Q circuit corresponds to a low bandwidth, the antenna becomes difficult to use for transmission of a modulated signal. If the signal frequency changes slightly, the generator and antenna are no longer conjugate matched, and the radiated signal power drops rapidly as the operating frequency moves away from the designed resonant frequency. Another way to interpret this is that the near fields around the antenna store a substantial amount of power in relation to the radiated fields, and it is difficult for the generator to change those near fields once they have built up to a steady state. Similar considerations that limit the performance of small antennas hold on the receive side. In a fundamental body of work on antenna theory, Chu, Harrington, Wheeler, and others have developed bounds on the Q of an antenna when matched to a generator or load [3, 4]. These bounds have been given in various mathematical forms, but one commonly used result is Q + 2(kr)2 (kr) 3 [ + (kr) 2 ] (kr) 3 (3.62) where r is the radius of the smallest sphere entirely enclosing the antenna. This bound sets a fundamental limit on how small an antenna can be made at a given frequency. The fractional bandwidth of the antenna when matched to a source or load is B/f /Q, where f is the center frequency. If Q is large, the system has only a small usable bandwidth. For r 8 mm at 3 MHz, the bound implies that Q 8 6, which corresponds to a bandwidth limit of only 4 Hz. Furthermore, this is a theoretical limit on antenna Q, so any real antenna will have an even smaller usable bandwidth.
ECEn 665: Antennas and Propagation for Wireless Communications 53 The bound given above is for lossless antennas. By adding ohmic loss to the antenna, bandwidth can be increased, at the expense of poorer radiation efficiency. From the classical results of small antenna theory, a bound on the bandwidth efficiency product of a small antenna can be derived [5], B η rad [ f 2 kr + ] (kr) 3 (3.63) In this expression, B is the impedance matching bandwidth defined with respect to a voltage standing wave ratio (VSWR) of 2. This result illustrates several fundamental tradeoffs of antenna design. As antenna ohmic loss becomes large, radiation efficiency drops and wide bandwidth performance becomes easier to obtain. The antennas used for terrestrial wireless communication systems, where environmental noise and interference are relatively large, often have fairly low efficiency (7-8%), and broad bandwidth small antennas are not too difficult to realize. On the other hand, when highly efficient antennas are required, such as for radio astronomy, remote sensing, and satellite communications, it is more challenging to obtain broad bandwidth with a low-loss, electrically small antenna. Fortunately, those kinds of applications also typically require large antenna aperture areas and antenna size requirements are not as stringent. As the bounding sphere radius r increases, the bound in (3.63) has a larger value, and wider bandwidths become easier to achieve. There are many techniques to improve the performance of small antennas and approach these fundamental limits on antenna bandwidth. One is to load the antenna with dielectric. This reduces the effective wavelength of the fields around the antenna, making the structure appear electrically larger. The bound (3.62) still applies, so such an antenna can only move closer to the fundamental limit. In practice, the dielectric material also introduces loss, which has the effect of increasing the bandwidth and decreasing radiation efficiency, as implied by (3.63). Antennas such as the loaded dipole, sleeved dipole, hemispherical dipole, biconical dipole, or spirals that use the volume inside the sphere of radius r more efficiency than the dipole have been developed which move closer to the theoretical performance limi. These antennas are nonplanar and can be difficult or expensive to fabricate, but offer wider bandwidths and lower Q than the dipole. If the antenna is not required to be electrically small, the small antenna bounds given above do not apply, and ultrawideband antennas can be designed. UWB antennas include the Vivaldi or traveling wave slot antenna, self-complementary spiral antenna, and many others. For communication applications, 5-% relative bandwidths are typical, but bandwidths with ratios of maximum to minimum operating frequency as large as 3: to : or more have been demonstrated. UWB antennas typically have very small geometrical features that resonate at the high end of the band combined with large features that resonate at the low end of the band.