2.5.3 Antenna Temperature

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1 ECEn 665: Antennas and Propagation for Wireless Communications Antenna Temperature We now turn to thermal noise received by an antenna. An antenna in a warm environment receives not only a signal of interest, but also blackbody radiation, which produces a noise signal at the antenna terminals. An antenna can be modelled as a warm resistor with an equivalent temperature that produces noise with the same standard deviation and time average power as the antenna. The goal of this analysis is to compute the equivalent temperature or antenna temperature as a function of the properties of the antenna and the physical temperature of the environment around the antenna. As a shorthand notation, let us denote the spherical angle pair θ, ϕ by Ω. If the temperature distribution of the scene around an antenna is T (Ω), where Ω denotes a particular spherical angle of arrival, then from the Rayleigh-Jeans law the thermal radiation flux density arriving at the antenna is F (Ω) k B λ T (Ω) (W/m /Hz) (.134) This quantity is a power spectral density with respect to frequency, and a per-unit area density with respect to the spatial distribution of the thermal radiation. This expression assumes that objects around the antenna are perfect blackbodies. If not, then the effective temperature is lower than the physical temperature by a factor called the emissivity, which is analogous to the factor L 1 in (.13). Good conductors have a very low emissivity, and so give off much less thermal radiation than other objects at a similar temperature. This means that metallic surfaces, such as a parabolic reflector antenna, produce much less thermal radiation than, say, warm ground behind the reflector. A typical temperature distribution at 1 GHz is 5 K towards the zenith direction from the sky, increasing gradually to the ambient outdoor temperature of around 8 K at angles closer to the horizon, and 8 K from any direction pointing to ground, non-metallic structures, or vegetation. The sky noise temperature is a combination of atmospheric noise, radiation from galactic hydrogen, and the cosmic background radiation. At frequencies below and above 1 GHz, the sky temperature is higher. At very low frequencies (< 1 MHz), the sky noise temperature is many orders of magnitude larger than the ambient temperature. The thermal noise power density delivered to a matched load at the antenna feed port is p (W/Hz) 1 F (Ω)A eff (Ω) dω (.135) where the 1/ factor is due to the fact that thermal radiation is randomly polarized and a single antenna port receives half the available incident power. The temperature of an equivalent warm resistor that delivers this same power density to the load is T a 1 p k B 1 k B 1 λ 1 4π kb λ T (Ω)A eff(ω) dω T (Ω) λ G(Ω) dω 4π T (Ω)G(Ω) dω (.136) This expression is a weighted integral of the antenna gain pattern. If the brightness temperature distribution is a constant T, then the integral of the gain over a sphere is equal to the radiation efficiency, and T a η rad T. As an example, we will consider spillover noise from the warm ground around a reflector feed antenna. From the point of view of the feed, the thermal noise distribution is very small over the angular extent of the reflector, since the brightness temperature of the metallic surface is small, and the sky noise reflected Warnick & Jensen January, 15

2 ECEn 665: Antennas and Propagation for Wireless Communications 37 from the surface to the feed at microwave frequencies is also small. The brightness temperature of the warm ground behind the reflector is typically much larger than the apparent brightness temperature of the reflector surface. Thermal radiation from the ground arrives from an angle from the reflector rim to the horizon. Since the backlobes of a feed are typically small, to simplify the integrals we can assume that the brightness temperature distribution T (Ω) is equal to T ground over the full sphere except for the solid angle of the reflector. If we denote the complement of the solid angle of the reflector as Ω spill, then the received noise temperature from the ground is approximately Ω T spill T sp G(Ω) dω ground (.137) G(Ω) dω For a reflector feed, the spillover efficiency η spill is defined to be the ratio of the power incident on the reflector to the total power radiated by a feed as a transmitter, which means that the factor on the right is equal to 1 η spill. This leads to T spill T ground (1 η spill ) (.138) for the antenna temperature due to spillover noise. If the ohmic loss of an antenna is significant, then the noise received from the scene around the antenna is reduced by the radiation efficiency, but the antenna loss represents a warm resistance and contributes an additional noise component. The noise temperature at the antenna feed port is the sum of the ideal received noise temperature T a reduced by the antenna loss factor and the additional thermal noise contributed by the antenna structure itself given by (.13), which leads to T a T a L + (L 1)T p L (.139) where T p is the physical temperature of the antenna. Since L 1/η rad, this becomes for the output noise temperature of a lossy antenna. T a η rad T a + (1 η rad )T p (.14).6 Receiver Sensitivity For a receiver system, the key performance metric is the SNR at the receiver output. In a communication system, millions of dollars may be spent just to improve the SNR by a few db. Using (.118), the receiver output SNR can be expressed as where SNR λ S inc 4πk B B G T sys (.141) T sys η rad T a + (1 η rad )T p + T rec (.14) T rec is the equivalent noise temperature due to amplifiers and other components in the receiver attached to the antenna. T a represents external noise sources such as sky noise or spillover noise. The last factor in (.141) is intrinsic to the receiver, since it is independent of the incident power density, and so is a convenient performance metric for the receiver. This is the receiver sensitivity, figure of merit, or simply the receiver G over T, G 4πk BB T sys λ SNR (1/K) (.143) Sinc Warnick & Jensen January, 15

3 ECEn 665: Antennas and Propagation for Wireless Communications 38 When converged to db, the units of this measure of sensitivity are referred to as db/k. Another common metric for sensitivity is the effective area relative to the system temperature, A eff k BB T sys S inc SNR (m /K) (.144) Both forms of the receiver figure of merit are related by a scale factor to the receiver output SNR. Warnick & Jensen January, 15

4 Chapter 3 Linear and Loop Antennas The simplest linear antenna is a dipole consisting of two straight conductors fed by a balanced transmission line at a feed gap in the center of the dipole. A monopole is a single conductor over a ground plane, fed by an unbalanced transmission line. The electromagnetic dual of the dipole antenna is the wire loop antenna, which radiates fields with similar polarization but with the electric and magnetic fields exchanged. We will consider the dipole antenna first. 3.1 Dipole Current Models In order to analyze the radiation characteristics of a dipole, we need to come up with a model for the current on the dipole arms. A very accurate approximation to the current distribution can be obtained using a numerical method such as the method of moments. In order to gain mathematical insight, however, it is desirable to have a closed form approximation to the current distribution. The closed form approximation may be significantly different from the true current distribution, but simple approximations in analytical form are very valuable in engineering analysis and design Hertzian Dipole Model (Constant Current) The Hertzian dipole or constant current model was developed in Section.3. This model is reasonable only for very short dipole antennas. In reality, the signal introduced at the feed point creates a standing wave on the dipole arms with zero current at the ends. This standing wave is approximately sinusoidal. Better approximations to this standing wave include a triangular current distribution when the antenna is short enough that the current follows a half cycle or less of a sinusoid, or more generally a sinusoidal distribution. The true current on a dipole antenna is not precisely sinusoidal, so a numerical algorithm can be used to find an even more accurate current distribution Triangular Current Model For a short dipole, the partial sinusoid can be approximated as triangular, so that J(r) ẑδ(x)δ(y)i (1 z /l), z l/ (3.1) If the dipole is short, the phase term in the integrand of (.67) can be approximated as unity as with the Hertzian dipole model, and the radiated fields are exactly half that of the Hertzian dipole. The radiation resistance drops by one fourth, but the radiation pattern is the same and the directivity remains 3/. 39

5 ECEn 665: Antennas and Propagation for Wireless Communications Sinusoidal Current Model The triangle current model includes the appropriate current boundary condition at the dipole ends, since the current must go to zero at the ends, but the form of the current still deviates from the actual current on a dipole. To obtain a better approximation, the dipole arms can be approximated as single wire transmission lines with a standing wave pattern on each of the arms. By combining a forward wave (e jkz for the z > arm) and a reverse wave (e jkz for the z > arm), we obtain the sinusoidal current distribution J(r) ẑδ(x)δ(y)i m sin[k(l/ z )], z l/ (3.) This expression can be derived by assuming that the current on the dipole arms is of the form Ae jkz +Be jkz and enforcing I(z) at the ends of the arms. The vector current moment is N(r) e jk r J(r ) dr l/ l/ l/ e jkzz I m sin[k(l/ z )] dz e jkzz I m sin[k(l/ z )] dz + e jkzz I m sin[k(l/ + z )] dz l/ where k kˆr and k z k cos θ. Using the identity e ax sin(bx + c) dx eax a [a sin(bx + c) b cos(bx + c)] (3.3) + b we obtain N z I m ejkzz k k z { jkz sin[k(l/ z )] + k cos[k(l/ z )] } l/ + ejk zz { jkz k kz sin[k(l/ + z )] k cos[k(l/ + z )] } l/ e jk zl/ k k ρ 1 kρ [jk z sin(kl/) + k cos(kl/)] + 1 kρ [jk z sin(kl/) k cos(kl/)] + e jk zl/ k kρ k sin θ [cos(k zl/) cos(kl/)] where k ρ k k z. The far electric field is E jωµ e jkr 4πr (ˆθN θ + ˆϕN ϕ ) jωµ e jkr 4πr ˆθ( sin θ)i m k sin θ [cos(k zl/) cos(kl/)] ji m η e jkr πr ˆθ cos( 1 kl cos θ) cos( 1 kl) } sin {{ θ } F (θ) Warnick & Jensen January, 15

6 ECEn 665: Antennas and Propagation for Wireless Communications 41 The power density is The total radiated power is S r E θ η 1 ηi m η πr F (θ) η 8π r I m F (θ) where P rad π π η 4π I m η π η 8π r I m F (θ)r sin θ dθ dϕ [cos( 1 kl cos θ) cos( 1 kl)] sin θ 4π I m { γ + ln(kl) Ci(kl) + 1 sin(kl)[si(kl) Si(kl)] + 1 cos(kl)[γ + ln(kl/) + Ci(kl) Ci(kl)]} η 4π I m f(kl) (3.4) Ci(kl) Si(kl) and γ.577 is Euler s constant. This leads to a radiation resistance of x x dθ cos(t) dt (Cosine integral) (3.5) t sin(t) dt (Sine integral) (3.6) t R rad P rad I() (3.7) η I m f(kl) π I m sin(kl/) ηf(kl) π sin (kl/) One problem with this result is that if the antenna length is an integer multiple of the wavelength, the denominator vanishes and (3.8) becomes infinite. This is because the sinusoidal current model has a zero at the antenna feed point. The actual current on the dipole does not vanish, so the true radiation resistance is finite. A different way to approximate the radiation resistance is to replace I() in (3.7) with I m. This is more accurate when the antenna length is an integer multiple of the wavelength, but less accurate when l is an odd number of half-wavelengths. (3.8) Warnick & Jensen January, 15

7 ECEn 665: Antennas and Propagation for Wireless Communications 4 The directivity of the dipole antenna with the sinusoidal current model is Half-wave Dipole S r D P rad /(4πr ) η I 8π r m F (θ) η 4π I m f(kl)/(4πr ) F (θ) f(kl) [ cos( 1 kl cos θ) cos( 1 kl) ] (3.9) f(kl) sin θ When the length of a dipole antenna is approximately a half wavelength at the desired operating frequency, the reactance of the antenna is close to zero. At this frequency, the antenna is said to be resonant. For this reason, the values of the antenna parameters of the dipole at l λ/ are particularly important. For the half-wave dipole, kl π, and the key antenna parameters are S r (r, θ) η I m cos ( π cos θ) 8π r sin θ (3.1) D(θ) cos ( π cos θ) f(π) sin cos ( π cos θ) θ 1.18 sin θ (3.11) D max (3.1) R rad 1.18η 73 Ω π (3.13) The directivity of the half-wave dipole with the sinusoidal current model is very close to the directivity obtained with the Hertzian dipole model (1.5). The radiation resistance is more sensitive to the details of the current model, and is quite different. The Hertzian dipole model gives about Ω and the triangular current model predicts 5 Ω. The sinusoidal current model leads to a result that is between those values. Warnick & Jensen January, 15

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