Antennas Transmitter I Transmitting Antenna Transmission Line Electromagnetic Wave I Receiver I Receiving Antenna Transmission Line Electromagnetic Wave I Antennas are transducers that transfer electromagnetic energy between a transmission line and free space. Amanogawa, 2006 Digital Maestro Series 1
Here are a few examples of common antennas: Linear dipole fed by a two-wire line Linear monopole fed by a single wire over a ground plane Coaxial ground plane antenna Ground plane Linear elements connected to outer conductor of the coaxial cable simulate the ground plane Loop antenna Uda-Yagi dipole array Loop dipole Multiple loop antenna wound around a ferrite core Parabolic (dish) antenna Log periodic array Passive elements Amanogawa, 2006 Digital Maestro Series 2
From a circuit point of view, a transmitting antenna behaves like an equivalent impedance that dissipates the power transmitted Transmitter Transmission Line I Transmitting Antenna Electromagnetic Wave I Transmitter Z g I Pt ()= 1 2 ReqI 2 V g Transmission Line Z eq = R eq + jx eq I The transmitter is equivalent to a generator. Amanogawa, 2006 Digital Maestro Series 3
A receiving antenna behaves like a generator with an internal impedance corresponding to the antenna equivalent impedance. Receiver I Receiving Antenna Transmission Line Electromagnetic Wave I Pt () = 1 2 R I in 2 I Z eq Z R Transmission Line Z in V eq I The receiver represents the load impedance that dissipates the time average power generated by the receiving antenna. Amanogawa, 2006 Digital Maestro Series 4
Antennas are in general reciprocal devices, which can be used both as transmitting and as receiving elements. This is how the antennas on cellular phones and walkie talkies operate. The basic principle of operation of an antenna is easily understood starting from a two wire transmission line, terminated by an open circuit. I Z g V g Z R Open circuit I Note: This is the return current on the second wire, not the reflected current already included in the standing wave pattern. Amanogawa, 2006 Digital Maestro Series 5
Z g I Antennas Imagine to bend the end of the transmission line, forming a dipole antenna. Because of the change in geometry, there is now an abrupt change in the characteristic impedance at the transition point, where the current is still continuous. The dipole leaks electromagnetic energy into the surrounding space, therefore it reflects less power than the original open circuit the standing wave pattern on the transmission line is modified I 0 V g Z 0 I I 0 Amanogawa, 2006 Digital Maestro Series 6
In the space surrounding the dipole we have an electric field. At zero frequency (d.c. bias), fixed electrostatic field lines connect the metal elements of the antenna, with circular symmetry. E E Amanogawa, 2006 Digital Maestro Series 7
At higher frequency, the current oscillates in the wires and the field emanating from the dipole changes periodically. The field lines propagate away from the dipole and form closed loops. Amanogawa, 2006 Digital Maestro Series 8
The electromagnetic field emitted by an antenna obeys Maxwell s equations E= jω µ H H= J+ jω ε E Under the assumption of uniform isotropic medium we have the wave equation: 2 E= jω µ H = jωµ J+ ω µεe H= J+ jωε E 2 = J+ ω µεh Note that in the regions with electrical charges ρ E = E 2 E = 2 E ( ρε) Amanogawa, 2006 Digital Maestro Series 9
In general, these wave equations are difficult to solve, because of the presence of the terms with current and charge. It is easier to use the magnetic vector potential and the electric scalar potential. The definition of the magnetic vector potential is B = A Note that since the divergence of the curl of a vector is equal to zero we always satisfy the zero divergence condition We have also B= A = 0 ( ) E= jω µ H= jω A E+ A = 0 ( jω ) Amanogawa, 2006 Digital Maestro Series 10
We define the scalar potential φ first noticing that ( ± φ ) = 0 and then choosing (with sign convention as in electrostatics) E+ ωa = φ E= jω A φ ( j ) ( ) Note that the magnetic vector potential is not uniquely defined, since for any arbitrary scalar field ψ B= A= A+ ψ ( ) In order to uniquely define the magnetic vector potential, the standard approach is to use the Lorenz gauge A+ jω µεφ = 0 Amanogawa, 2006 Digital Maestro Series 11
From Maxwell s equations 1 H= B= J+ jωεe µ B= µ J+ jωµεe = + ( A) µ J jωµε( jωa φ) From vector calculus 2 ( ) = ( ) 2 2 A = A= J A ω µε φ ( ) ( A) µ + ω µε j Lorenz Gauge A = jω µ εφ = jω µ ε φ ( A) Amanogawa, 2006 Digital Maestro Series 12
Finally, the wave equation for the magnetic vector potential is 2 2 2 2 A + ω µε A = A + β A = µ J For the electric field we have ρ D= ρ E= ( jωa φ) = ε 2 2 ρ φ + jω A = φ + jω( jωµεφ) = ε The wave equation for the electric scalar potential is 2 2 2 2 ρ φ + ω µε φ = φ + β φ = ε Amanogawa, 2006 Digital Maestro Series 13
The wave equations are inhomogenoeous Helmholtz equations, which apply to regions where currents and charges are not zero. We use the following system of coordinates for an antenna body z J( r ') ρ( r ') r r' Observation point r ' dv' r x Radiating antenna body y Amanogawa, 2006 Digital Maestro Series 14
The generals solutions for the wave equations are A ( r) = µ π J r ( ') e 4 V r r' jβ r r' dv' φ ( r) = 1 ρ ( r' ) e 4 πε V r r' jβ r r' dv' The integrals are extended to all points over the antenna body where the sources (current density, charge) are not zero. The effect of each volume element of the antenna is to radiate a radial wave e jβ r r' r r' Amanogawa, 2006 Digital Maestro Series 15
x J(0) ρ(0) z I = z Infinitesimal Antenna S dv' constant phasor r ' = 0 Observation point r = r r' y Infinitesimal antenna body z << λ Dielectric medium (ε, µ) Amanogawa, 2006 Digital Maestro Series 16
The current flowing in the infinitesimal antenna is assumed to be constant and oriented along the z axis I = S J ( r' ) = S J( 0 ) V' = V'J r' I z ( ) = iz S z The solution of the wave equation for the magnetic vector potential simply becomes the evaluation of the integrand at the origin A 1 H A jβ r = µ I ze µ = iz 4π r 1 E= H jωε Amanogawa, 2006 Digital Maestro Series 17
There is still a major mathematical step left. The curl operations must be expressed in terms of spherical coordinates z Polar angle θ r i ϕ i r i θ x ϕ Azimuthal angle y i ϕ Amanogawa, 2006 Digital Maestro Series 18
In spherical coordinates A = r 2 i ri rsinθ i r 1 sinθ r θ ϕ A ra rsinθ A r 1 ( ) = sinθ Aϕ ( Aθ ) i r sinθ θ ϕ 1 1 ( A ) ( ) + r raϕ iθ r sinθ ϕ r 1 + ( raθ ) ( Ar ) iϕ r r θ θ θ ϕ ϕ r Amanogawa, 2006 Digital Maestro Series 19
We had Antennas jβr µ I ze A= iz with iz = ircosθ iθ sinθ 4π r jβr jµβ I ze 1 A= iϕ 1+ sinθ 4π r jβr For the fields we have j r 1 β jβ I ze 1 H= A= iϕ 1+ sinθ µ 4π r jβr Amanogawa, 2006 Digital Maestro Series 20
E 1 µ jβ I ze = H= jωε ε 4πr 1 1 2cosθ + jβ r 2 ( jβ r) 1 1 + sinθ 1 + + i j r 2 θ β ( jβ r) i r jβr The general field expressions can be simplified for observation point at large distance from the infinitesimal antenna 1 >> 1 1 r 2 π r j r j r 2 β 1 β >> = λ >> ( β ) Amanogawa, 2006 Digital Maestro Series 21
At large distance we have the expressions for the Far Field H E i θ i ϕ jβ I µ jβ I ε ze 4π r jβr ze 4π r jβr sinθ sinθ 2π r >> λ At sufficient distance from the antenna, the radiated fields are perpendicular to each other and to the direction of propagation. The magnetic field and electric field are in phase and E µ = H = η H ε These are also properties of uniform plane waves. Amanogawa, 2006 Digital Maestro Series 22
However, there are significant differences with respect to a uniform plane wave: The surfaces of constant phase are spherical instead of planar, and the wave travels in the radial direction. The intensities of the fields are inversely proportional to the distance, therefore the field intensities decay while they are constant for a uniform plane wave. The field intensities are not constant on a given surface of constant phase. The intensity depends on the sine of the polar angle θ. The radiated power density is 1 { * } 1 µ Pt () = Re E H = ir H 2 2 ε 2 η β I z 2 = i r sin θ 2 4π r 2 ϕ Amanogawa, 2006 Digital Maestro Series 23
z Pt () H ϕ θ J ϕ r E θ The spherical wave resembles a plane wave locally in a small neighborhood of the point ( r, θ, ϕ ). Amanogawa, 2006 Digital Maestro Series 24
Radiation Patterns Electric Field and Magnetic Field z θ E θ or H ϕ y x x Fixed r Plane containing the antenna proportional to sinθ Plane perpendicular to the antenna omnidirectional or isotropic Amanogawa, 2006 Digital Maestro Series 25
Time average Power Flow (Poynting Vector) z θ Pt () y x x Fixed r Plane containing the antenna proportional to sin 2 θ Plane perpendicular to the antenna omnidirectional or isotropic Amanogawa, 2006 Digital Maestro Series 26
Total Radiated Power The time average power flow is not uniform on the spherical wave front. In order to obtain the total power radiated by the infinitesimal antenna, it is necessary to integrate over the sphere Ptot = d d r P t 2π π 2 ϕ θ sin θ ( ) 0 0 = 2π 2 2 β I z 2 π 3 4πη β I z 2π r dθ sin θ 2 4π r 0 3 4π = 43 η = = Note: the total radiated power is independent of distance. Although the power density decreases with distance, the integral of the power over concentric spherical wave fronts remains constant. Amanogawa, 2006 Digital Maestro Series 27
P = P tot1 tot2 P tot2 P tot1 Amanogawa, 2006 Digital Maestro Series 28
The total radiated power is also the power delivered by the transmission line to the real part of the equivalent impedance seen at the input of the antenna P tot 2 2 1 2 4πη 2π I z 1 2 2πη z = I Req= = I 2 3 λ 4π 2 3 λ Req The equivalent resistance of the antenna is usually called radiation resistance. In free space η = η = o µ ε o o 2 [ ] R 8 π [ Ω] = 120π Ω = 0 eq z λ 2 Amanogawa, 2006 Digital Maestro Series 29
The total radiated power is also used to define the average power density emitted by the antenna. The average power density corresponds to the radiation of a hypothetical omnidirectional (isotropic) antenna, which is used as a reference to understand the directive properties of any antenna. Power radiation pattern of an omnidirectional average antenna z Power radiation pattern of the actual antenna θ x P ave Ptθ (, ) Amanogawa, 2006 Digital Maestro Series 30
The time average power density is given by P ave Total Radiated Power = = Surface of wave front Ptot η ( ) 2 1 η β I z = = β I z = 2 12 2 4π r π 4π r 3 4π r 2 The directive gain of the infinitesimal antenna is defined as 2 2 Ptr (,, θ) η β I z 2 η β I z D( θ, ϕ) = = sin θ Pave 2 4π r 3 4π r 1 = 3 sin 2 2 θ Amanogawa, 2006 Digital Maestro Series 31
The maximum value of the directive gain is called directivity of the antenna. For the infinitesimal antenna, the maximum of the directive gain occurs when the polar angle θ is 90 3 2 π Directivi ty = max { D( θϕ, )} = sin = 1. 5 2 2 The directivity gives a measure of how the actual antenna performs in the direction of maximum radiation, with respect to the ideal isotropic antenna which emits the average power in all directions. z 90 Pave P x max Amanogawa, 2006 Digital Maestro Series 32
The infinitesimal antenna is a suitable model to study the behavior of the elementary radiating element called Hertzian dipole. Consider two small charge reservoirs, separated by a distance z, which exchange mobile charge in the form of an oscillatory curent I() t t + + + z + + + Amanogawa, 2006 Digital Maestro Series 33
The Hertzian dipole can be used as an elementary model for many natural charge oscillation phenomena. The radiated fields can be described by using the results of the infinitesimal antenna. Assuming a sinusoidally varying charge flow between the reservoirs, the oscillating current is d d I t = q t = q t o = jω q dt dt () () ocos( ω ) I current flowing out of reservoir charge on reference reservoir phasor o Radiation pattern I o q o Amanogawa, 2006 Digital Maestro Series 34
A short wire antenna has a triangular current distribution, since the current itself has to reach a null at the end the wires. The current can be made approximately uniform by adding capacitor plates. I max z I o I o The small capacitor plate antenna is equivalent to a Hertzian dipole and the radiated fields can also be described by using the results of the infinitesimal antenna. The short wire antenna can be described by the same results, if one uses an average current value giving the same integral of the current Io = I max 2 Amanogawa, 2006 Digital Maestro Series 35
Example A Hertzian dipole is 1.0 mm long and it operates at the frequency of 1.0 GHz, with feeding current I o = 1.0 Ampéres. Find the total radiated power. 8 9 λ = c f 3 10 10 = 0.3 m = 300 mm z = 1 mm λ Hertzian dipole P tot 2 4πη 2π Io z 1 2π = = 120 π ( ) ( 1 10 ) 3 λ 4π 12π 0.3 = 4.39 mw 2 3 2 For a short dipole with triangular current distribution and maximum current I max = 1.0 Ampére I = I 2 P = 4.39/4 1.09 mw o max tot η o λ I o z Amanogawa, 2006 Digital Maestro Series 36
Time dependent fields - Consider the far field approximation ( ) { jω t} jβ I z sinθ j( ω t βr) H t = Re H e iϕ Re e 4π r β I zsinθ ( 2 i Re jcos( t r) j sin( t r) ) ϕ ω β + ω β 4π r β I z sinθ iϕ s in( ω t βr) 4π r E t ( ) { jω t = Re E e } i θ β I z sinθ η sin( ω t 4π r βr) Amanogawa, 2006 Digital Maestro Series 37
Linear Antennas Antennas Consider a dipole with wires of length comparable to the wavelength. z' z z θ ' θ L 2 r' r i θ ' i θ L 1 Amanogawa, 2006 Digital Maestro Series 38
Because of its length, the current flowing in the antenna wire is a function of the coordinate z. To evaluate the far field at an observation point, we divide the antenna into segments which can be considered as elementary infinitesimal antennas. The electric field radiated by each element, in the far field approximation, is E' = i θ µ jβ I ze ε 4 π r' jβr' sin θ ' In far field conditions we can use these additional approximations θ θ ' r' r z'cosθ Amanogawa, 2006 Digital Maestro Series 39
The lines r and r are nearly parallel under these assumptions. z' z z θ ' θ L 2 θ z'cosθ r' r This length is neglected if r' r z'cosθ Amanogawa, 2006 Digital Maestro Series 40
The electric field contributions due to each infinitesimal segment becomes E' = i θ you can neglect here you cannot neglect here jβr jβz'cosθ µ jβ I ze e sinθ ε 4π r 4 πz'cosθ The total fields are obtained by integration of all the contributions jβr µ jβ e L2 jβzcosθ E= iθ sinθ I( z) e dz ε 4π r L1 jβr jβ e L2 jβzcosθ H= iϕ sinθ I( z) e dz 4π r L1 Amanogawa, 2006 Digital Maestro Series 41
Short Dipole Antennas Consider a short symmetric dipole comprising two wires, each of length L << λ. Assume a triangular distribution of the phasor current on the wires ( ) ( ) Imax 1 z L z 0 I( z) = Imax 1 + z L z < 0 The integral in the field expressions becomes L L since jβzcosθ I( z) e dz I( z) dz= 1 2π max βz = β L= L 1 λ j zcos e β θ 1 L L for a 2L I 2 short max dipole Amanogawa, 2006 Digital Maestro Series 42
The final expression for far fields of the short dipole are similar to the expressions for the Hertzian dipole where the average of the triangular current distribution is used jβr z µ jβ e I E= iθ sinθ 2L ε 4π r 2 = i θ µ jβ ImaxL e ε 4π r jβr sinθ max average current H = i ϕ jβ ImaxL e 4π r jβr sinθ Amanogawa, 2006 Digital Maestro Series 43
Half wavelength dipole Consider a symmetric linear antenna with total length λ/2 and assume a current phasor distribution on the wires which is approximately sinusoidal I( z) = I cos( β z) max The integral in the field expressions is λ 4 λ 4 ( β ) jβzcosθ max max cos = 2 I z e dz 2I π cosθ cos βsin θ 2 Amanogawa, 2006 Digital Maestro Series 44
We obtain the far field expressions E= H= i θ i ϕ jβr µ je Imax π cosθ cos ε 2π r sinθ 2 jβr je Imax π cosθ cos 2π r sinθ 2 and the time average Poynting vector 2 Imax 2 2 2 µ 2 π cosθ Pt () = ir cos ε 8π r sin θ 2 Amanogawa, 2006 Digital Maestro Series 45
The total radiated power is obtained after integration of the time average Poynting vector ( u) 1 2 µ 1 2π 1 cos Ptot = Imax du 2 ε 4π 0 u 1 2 µ = Imax 0.193978 2 ε Req 2.4376 The integral above cannot be solved analytically, but the value is found numerically or from published tables. The equivalent resistance of the half wave dipole antenna in air is then µ R eq ( λ 2) = 0.193978 73.07 Ω ε Amanogawa, 2006 Digital Maestro Series 46
The direction of maximum radiation strength is obtained again for polar angle θ =90 ande we obtain the directivity D 2 max 2 2 µ I Ptr (,,90) ε 8π r = = 2 Ptot 4π r 1 2 µ I 2 2 max 2.4376 8π r ε 1.641 The directivity of the half wavelength dipole is marginally better than the directivity for a Hertzian dipole (D = 1.5). The real improvement is in the much larger radiation resistance, which is now comparable to the characteristic impedance of typical transmission line. Amanogawa, 2006 Digital Maestro Series 47
From the linear antenna applet Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 48
For short dipoles of length 0.0005 λ to 0.05 λ Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 49
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 50
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 51
For general symmetric linear antennas with two wires of length L, it is convenient to express the current distribution on the wires as I ( z) = I sin β ( L z) max { } The integral in the field expressions is now L L ( ) jβzcosθ Imax sin β L z e dz= 2I = max 2 βsin θ ( βl θ) ( βl) { cos cos cos } Amanogawa, 2006 Digital Maestro Series 52
The field expressions become jβr µ jβ e L2 jβzcosθ E= iθ sinθ I( z) e dz ε 4π r L1 jβr µ jimax e = iθ L L ε 2π r sinθ jβr ( β θ) ( β ) { cos cos cos } jβ e L2 jβzcosθ H= iϕ sinθ I( z) e dz 4π r L1 jβr jimax e = iϕ L L 2π r sinθ ( β θ) ( β ) { cos cos cos } Amanogawa, 2006 Digital Maestro Series 53
Examples of long wire antennas Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 54
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 55
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 56
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 57
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 58
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 59
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 60
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 61
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 62
Radiation Pattern for E and H Power Radiation Pattern Amanogawa, 2006 Digital Maestro Series 63