# A Brief Introduction to Antennas & Transmission Lines Prof. John Vesecky Outline of Presentation Maxwell’s Equations & EM Waves EM Spectrum Antenna Characterization.

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A Brief Introduction to Antennas & Transmission Lines Prof. John Vesecky Outline of Presentation Maxwell’s Equations & EM Waves EM Spectrum Antenna Characterization Dipoles and Monopoles End Fires (Yagis & Log-Periodics) Apertures (Parabolic Reflectors) Patches & Arrays Transmission Lines Friis’ Equation

I. Outline for Wire, Aperture and Patch Antennas EM Spectrum Antenna Characterization Dipoles and Monopoles End Fires (Yagis & Log-Periodics) Apertures (Parabolic Reflectors) Patches & Arrays

EM waves in free space v 2 = 1/(  o µ o ) so v = 3 x 10 8 m/s –  o = 8.855 x 10 -12 Farads/m –µ o = 1.2566 x 10 -6 Henrys/m EM waves in free space propagate freely without attenuation What is a plane wave? –Example is a wave propagating along the x-direction –Fields are constant in y and z directions, but vary with time and space along the x-direction –Most propagating radio (EM) waves can be thought of a plane waves on the scale of the receiving antenna

E & H fields and Poynting Vector for Power Flow Power flow in the EM field –P = E x H (P is Poynting vector) In free space E and H are perpendicular P is perpendicular to both E and H Plane wave radiated by an antenna –P = E x H -> E o H o Sin 2 (  t-kx) –P = [E o 2 /  ] Sin 2 (  t-kx) –P avg = (1/2) [E o 2 /  ] in W/m 2 –  = impedance of free space = 377 

Electromagnetic Spectrum After Kraus & Marhefka, 2003

Frequencies & Wavelengths After Kraus & Marhefka, 2003

RF Bands, Names & Users After Kraus & Marhefka, 2003

Radiation from a Short Antenna Element or Hertzian Dipole Using the Electrodynamic Retarded Potential A (Vector) we can derive (see Ramo et al., 1965 or Skilling, 1948, Ulaby, 2007 or any EM theory book) E and H fields associated with a small element of current of length l (<< ) that has the current varying as i = I Sin (  t) This could be a wire or charge moving in space, e.g. in the plasma of the ionosphere or a star or nebula E and H fields at r could be in the r,  or  directions

Radiation from a Short Antenna Element or Hertzian Dipole A(r) = (µ o /4  ) (e -jkr /r)∫ V [J (e -jkr’ /r’)] dv’ H = (1/µo) curl A H  = (I o l k 2 /4  ) e -jkr [j/(kr) + 1/(kr) 2 ] sin  H r = 0 and H  = 0 E = (1/j  o ) curl H E r = (2I o l k 2 /4  )  o e -jkr [1/(kr) 2 - j/(kr) 3 ] E  = (I o l k 2 /4  )  o e -jkr [j/kr + 1/(kr) 2 - j/(kr) 3 ]  o = Sqrt(µ o /  o ) = 377 Ω

Radiation from a Short Antenna Element Terms that fall off as 1/r 3 or 1/r 2 are small at any significant distance from an antenna Remaining “radiation” terms fall off only as 1/r and thus transmit energy for long distances also E and H fields are in phase When one is in the “near field” the 1/r 3 or 1/r 2 the other terms are important

Antenna Field Zones The dividing line “Rule of Thumb” is R = 2L 2 / The near field or Fresnel zone is r < R The far field or Fraunhofer zone is r > R

Polarization of EM Waves AR = Axial Ratio

Simple Dipole Antenna

Antenna Characterization Directivity Power Pattern Antenna Gain Effective Area Antenna Efficiency

Antenna Directivity An omnidirectional antenna radiates power into all directions (4  steradians) equally Typically an antenna wants to beam radiation in a particular direction Directivity D = 4  / ,  is the antenna beam solid angle What  would be for one octant (x,y,z all > 0) ?

P n ( ,  ) = S(  )/S(  ) max S(  ) = Poynting vector magnitude = [E  2 + E  2 ]/   = 376.7  free space) After Kraus (2003) Normalized Antenna Power Pattern

Antenna Gain Gain is like directivity, but includes losses as well G(  ) ≈  /(  ) is nondimensional° --  accounts for losses dB = 10 log(x/x ref ) -- always refers to power Gain for Typical Antenna with significant directivity G(  ) ≈ 2500/(  °  °), taking into account beam shape and typical losses

Estimating Effective Antenna Area & Gain Definition: G = (4  A e )/ 2 A e =  A, where A is the physical area and  is the antenna efficiency To get the average power available at the antenna terminals we use P av,Ant = P av,Poynting (Average Poynting Flux) A e A crude estimate of G can be obtained by letting –  ≈ ( /d), where d is the antenna dimension along the direction of the angle  -- big antenna means small  –and G(  ) ≈  /(  )

Radiation Resistance & Antenna Efficiency Radiation resistance (R rad ) is a fictitious resistance, such that the average power flow out of the antenna is P av = (1/2) 2 R rad Using the equations for our short (Hertzian) dipole we find that R rad = 80  2 (l/ ) 2 ohms Antenna Efficiency  = R rad /(R rad + R loss ) where R loss = ohmic losses as heat Gain =  x Directivity --- G =  D

Antenna Family

Short Dipole Antenna Analysis Consider a finite, but short antenna with l <<  situated in free space Current is charging the uniformly distributed capacitance of the antenna wire & so has a maximum at the middle and tapers toward zero at the ends Each element dl radiates per our radiation equations (previous slide), namely In the far field E  = (  I dl sin  /(2 r )) cos {  [t-(r/c)]} The  direction is in the same plane as the element dl and the radial line from antenna center to observer and perpendicular to r

Short Dipole Antenna Result The resultant field at the observer at r is the sum of the contributions from the elemental lengths dl –Each contribution is essentially the same except that the current I varies –Radiation contribution to the sum is strongest from the center and weakest at the ends This can be summarized as the rms field strength in volts per meter as E ,rms = [  I o l e sin  /(2 r )] -- V/m What do you think the effective length l e & current I o are? The radiated power is P av = (E ,rms ) 2 /(2 

Modifications for Half Wavelength Dipole For antennas comparable in size to –Current distribution is not linear –Phase difference between different parts of the antenna Current distribution on /2 dipole –Antenna acts like open circuit transmission line with uniformly distributed capacitance –Sinusoidal current distribution results

Fields from /2 Dipole To take account of the phase differences of the contributions from all the elements dl we need to integrate over the entire length of the antenna as shown by the figure (from Skilling, 1948) E  = ∫ ± /4 (  I o sin  e /2 r e ) cos kx cos  [t-(r e /c)] dx –Integral is from - /4 to /4, i.e. over the antenna length Result of integration E  = (  I o /2  r) cos  [t-(r/c)] {cos [(  /2) cos  ] / sin  } We know that E r = E  = 0 as for the Hertzian dipole

/2 and Dipole Antenna Pattern (E-field)

Monopole over a Conducting Plane -- /4 Vertical

/4 Vertical over Ground Plane & Real Earth Solid line is for perfectly conducting Earth Shaded pattern shows how the pattern is modified by a more realistic Earth with dielectric constant k = 13 and conductivity G = 0.005 S/m

Yagi - Uda Driven element induces currents in parasitic elements When a parasitic element is slightly longer than /2, the element acts inductively and thus as a reflector - - current phased to reinforce radiation in the maximum direction and cancel in the opposite direction The director element is slightly shorter than  /2, the element acts inductively and thus as a director -- current phased to reinforce radiation in the maximum direction and cancel in the opposite direction The elements are separated by ≈ 0.25 

3 Element Yagi Antenna Pattern

2.4 GHz Yagi with 15dBi Gain G ≈ 1.66 * N (not dB) N = number of elements G ≈ 1.66 *3 = 5 = 7 dB G ≈ 1.66 * 16 = 27 = 16 dB

A log periodic is an extension of the Yagi idea to a broad-band, perhaps 4 x in wavelength, antenna with a gain of ≈ 8 dB Log periodics are typically used in the HF to UHF bands Log-Periodic Antennas

Parabolic Reflectors A parabolic reflector operates much the same way a reflecting telescope does Reflections of rays from the feed point all contribute in phase to a plane wave leaving the antenna along the antenna bore sight (axis) Typically used at UHF and higher frequencies

Stanford’s Big Dish 150 ft diameter dish on alt-azimuth mount made from parts of naval gun turrets Gain ≈ 4   A/ 2 ≈ 2 x 10 5 ≈ 53 dB for S-band (l ≈15 cm)

Patch Antennas Radiation is from two “slots” on left and right edges of patch where slot is region between patch and ground plane Length d =  /  r 1/2 Thickness typically ≈ 0.01  The big advantage is conformal, i.e. flat, shape and low weight Disadvantages: Low gain, Narrow bandwidth (overcome by fancy shapes and other heroic efforts), Becomes hard to feed when complex, e.g. for wide band operation After Kraus & Marhefka, 2003

Patch Antenna Pattern

Array Antennas

Patch Antenna Array for Space Craft The antenna is composed of two planar arrays, one for L-band and one for C- band. Each array is composed of a uniform grid of dual- polarized microstrip antenna radiators, with each polarization port fed by a separate corporate feed network. The overall size of the SIR-C antenna is 12.0 x 3.7 meters Used for synthetic aperture radar

Very Large Array http://www.vla.nrao.edu/ Organization: National Radio Astronomy Observatory Location:Socorro NM Wavelength: radio 7 mm and larger Number & Diameter 27 x 25 m Angular resolution: 0.05 (7mm) to 700 arcsec

Impedance Matching SWR = (1 + |  |)/ (1 - |  |)

Friis’ Transmission Formula P r = P t {(A et A er )/( 2 r 2 )} S/N = Signal to noise ratio = P r /(kT sys B) where T sys = system noise temperature, typically 10’s to 1000’s of K depending on receiver characteristics k = 1.38 x 10 -23 J/K B = bandwidth in Hz

References 1 Balanis, C.A., Antenna Theory, Analysis and Design, 2nd ed., Wiley (1997) Cloude, S., An Introduction to Electromagnetic Wave Propagation & Antennas, Springer-Verlag, New York (1995) Elmore, W. C. and M. A. Heald, Physics of Waves, Dover, NY (1969) Fusco, V. F., Foundations of Antenna Theory & Techniques, Pearson Printice-Hall (2005) Ishimaru, A., Electromagnetic Wave Propagation, Radiation and Scattering, Prentice-Hall, Englewood Cliffs NJ (1991) Jones, D. S., Acoustic and Electromagnetic Waves, Oxford Science Publications, Oxford (1989)

References 2 Kraus, J. D., Antennas, 2nd ed., McGraw-Hill, New York (1988) Kraus, J. D. and R. J. Marhefka, Antennas, 3rd ed., McGraw-Hill, New York (2004) Kraus, J. D., Electromagnetics, 3rd ed., McGraw-Hill, New York (1983) Ramo, S., J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, Wiley NY (1965) Skilling, H. H., Fundamentals of Electric Waves, 2nd ed., Wiley, NY (1948) Ulaby, F., Fundamentals of Applied Electromagnetics, 5th Ed., Pearson Printice-Hall (2007)

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