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Retarded Potential, Radiation Field and Power for a Short Dipole Antenna

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Retarded Potential, Radiation Field and Power for a Short Dipole Antenna Module 1- Antenna: Retarded potential, radiation field and power for a short dipole antenna ELL 212 Instructor: Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract An antenna is a device that acts as interface between electromagnetic waves prop- agating in free space and electric current flowing in metal conductor. It is one of the most beautiful devices that we study in electrical engineering since it combines the concepts of flow of electricity in circuits and propagation of waves in free space. The governing physics behind antenna, e.g. how and why antenna radiates power, can be confusing to learn. It is only after a careful study of the Maxwell's equations that we can start understanding the physics of antenna. In this module we shall discuss the physics of radiation of an antenna in details. We will first learn Green's functions because that will help us in understanding the concept of retarded vector potential, without which we will not be able to derive the radiation field for time varying charge and current and show its "1/r" dependence. We will then derive the expressions for radiated field and power for time varying current flowing through a short dipole antenna. (Reference: a) Classical electrodynamics- J.D. Jackson b) Electromagnetics for Engineers- T. Ulaby) 1 We need the Maxwell's equations throughout the module. So let's list them here first (for vacuum): ρ r~ :E~ = (1) 0 r~ :B~ = 0 (2) @B~ r~ × E~ = − (3) @t 1 @E~ r~ × B~ = µ J~ + (4) 0 c2 @t Also scalar potential φ and vector potential A~ are defined as follows: @A~ E~ = −r~ φ − (5) @t B~ = r~ × A~ (6) Note that equation (1)-(4) are independent equations but equation (5) is dependent on equation (3) and equation (6) is dependent on equation (4). 1 Green's function and retarded potential Let L be an operator such that L (~r) = f(~r) (7) L is a function of position vector ~r) If f(~r) is known and (~r) needs to be evaluated, using Z f(~r) = δ(~r − r~0)f(r~0)d3r~0 (8) and LG(r; r0) = δ(~r − r~0) (9) we can show that Z (~r) = G(r; r0)f(r0)d3r~0 (10) is a solution of equation (7) as below: Z Z Z L (~r) = L G(r; r0)f(r0)d3r~0 = LG(r; r0)f(r0)d3r~0 = δ(~r − r~0)f(r0)d3r~0 = f(~r) (11) 2 Note that this Green's function method is a multi variable extension of the formalism in continuous time unit impulse response and convolutional integral representation of Linear Time Invariant (LTI) systems (Reference: Signals and Systems- Oppenheim). In that formalism, if the input signal is given by x(t) and output signal is y(t) then we can write: Z 1 Z 1 x(t) = x(τ)δ(t − τ)dτ; y(t) = x(τ)h(t − τ)dτ (12) −∞ −∞ where h(t) is the impulse response to δ(t). Now we can think of the LTI system as solution to a ordinary differential equation, with initial condition equal to 0. Without stating an initial condition, multiple outputs are possible for a given input, then the system is non- deterministic. By making the initial/boundary condition 0 we make the system linear. Let the differential operator corresponding to the differential equation be D and it is only a function of independent variable t. We can write Dy(t) = x(t) (13) Also Dh(t − τ) = δ(t − τ) (14) We can show Z 1 y(t) = x(τ)h(t − τ)dτ (15) −∞ is solution to equation (13) as follows: Z 1 Z 1 Dy(t) = D x(τ)h(t − τ)dτ = x(τ)Dh(t − τ)dτ −∞ −∞ Z 1 = x(τ)δ(t − τ)dτ = x(t) (16) −∞ What we showed for Green's function above is exactly same as this, just that there is more than one variable and the equation is a partially differential equation. 1.1 Electrostatics In the case of electrostatics, from equation (1) and (5), ρ(~r) r2φ(~r) = − (17) (Vector potential A~ and field B~ are not time varying). Using the method of Green's function above, we evaluate scalar potential φ(~r) for a given charge distribution ρ(~r). Now. 1 r2(− ) = δ(~r − r~0) (18) 4π(j~r − r~0j 3 (proof in Appendix A) Using equation (10), (17) and (18) Z 1 ρ(r~0) Z ρ(r~0) φ(~r) = (− )(− )d3r~0 = d3r~0 (19) 0 0 4π(j~r − r~ j 0 4π0(j~r − r~ j 1.2 Magnetostatics The vector potential A~(~r) can be evaluated using Green's function for the case of magne- tostatics, similar to the case of electrostatics above. Using equation (4) and (6), r~ × r~ × A~ = µ0J~ (20) (Field E~ does not vary with time) 2 ) r~ (r~ :A~) − r A~ = µ0J~ (21) Different values of potentials can lead to the same fields. Moving between these different potentials is called gauge transformation. Choosing vector potential A~ such that r~ :A~ = 0 (22) , which is called Coulomb gauge, we get 2 r A~ = −µ0J~ (23) 2 2 2 ) r Ax = −µ0Jx; r Ay = −µ0Jy; r Az = −µ0Jz (24) Using the same method of Green's function as in the case of electrostatics, we get Z ~0 Z ~0 Z ~0 µ0Jx(r ) 3 0 µ0Jy(r ) 3 0 µ0Jz(r ) 3 0 Ax(~r) = d r~ ; Ay(~r) = d r~ ; Az(~r) = d r~ (25) 4π(j~r − r~0j) 4π(j~r − r~0j) 4π(j~r − r~0j) Or, Z µ J~(r~0) A~(~r) = 0 d3r~0 (26) 4π(j~r − r~0j) 1.3 Electrodynamics For time varying charge ρ(~r; t) and and current J~(~r; t), @A~ @ ρ(~r; t) r~ :E~ (= r~ :(−r~ φ − ) = −∇2φ − (r~ :A~) = (27) @t @t 0 4 1 @E~ 1 @ @A~ r~ × B~ = µJ~ + ) r~ × r~ × A~ = µJ~ + (−r~ φ − ) c2 @t c2 @t @t 1 @ 1 @2 ) r~ (r~ :A~) − r2A~ = µJ~ − (r~ φ) − A~ c2 @t c2 @t2 1 @φ 1 @2 ) r~ (r~ :A~ + ) − µJ~ = r2A~ − A~ (28) c2 @t c2 @t2 Vector potential A~ can be chosen such that 1 @φ r~ :A~ + = 0 (29) c2 @t This choice of gauge is known as the Lorentz gauge. Using equation (29) in equation (27) we get 2 2 1 @ ρ(~r; t) r φ(~r; t) − 2 2 φ(~r; t) = − (30) c @t 0 and using equation (29) in equation (28) we get 1 @2 r2A~(~r; t) − A~(~r; t) = −µ J~(~r; t) (31) c2 @t2 0 It is to be noted that equation (28) can be broken down into three scalar equations identical to equation (25) with scalars Ax,Ay and Az. All these equations fit into the following form: 1 @2 (r2 − ) (~r; t) = f(~r; t) ) (~r; t) = f(~r; t) (32) c2 @t2 which is different from the equation for scalar potential in electrostatics (12) and equation for vector potential in magnetostatics (18) that were of the form: r2 (~r; t) = f(~r; t) (33) 1 @2 The c2 @t2 factor modifies the Green's function we obtained in the cases of electrostatics and magnetostatics and results in a time delay factor in the solution of the equation. That's why the potential functions and A~ obtained as solution of equations (31) and (32) are called retarded scalar and vector potentials respectively. We show how equation (33) is solved to obtain the retarded potential term in details in the next subsection. Note that 2 1 @2 r − c2 @t2 operator is known as the d'Alembert operator () in literature. 5 1.4 Retarded potential Following the method of Green's function described above, equation (33) can be solved as below: Z Z f(~r; t) = f(r~0; t0)δ(~r − r~0; t − t0)d3r~0dt0 (34) t0 r~0 If 1 @2 (r2 − )G(~r; t; r~0; t0) = δ(~r − r~0; t − t0) (35) c2 @t2 then Z Z 0 0 3 0 (~r; t) = G(~r; t; r~0; t )f(r~0; t )d r~0dt (36) t0 r~0 is the solution to equation (33). So basically we have to solve for G(~r; t; r~0; t0) in equation (35). Those solutions are called retarded and advanced Green's functions as we see next. Using the Fourier transform formalism as below: Z 1 1 Z 1 F (!) = f(τ)e−i!τ dτ; f(τ) = F (!)ei!τ d! (37) −∞ 2π −∞ Let r = j~r − r~0j,τ = t − t0 Then equation (35) becomes 1 @2 (r2 − )G(r; τ) = δ(~r − r~0)δ(τ) c2 @τ 2 1 @2 Z 1 1 1 Z 1 2 i!τ ~0 i!τ ) (r − 2 2 ) G(r; !)e d! = δ(~r − r ) e d! c @τ −∞ 2π 2π −∞ ! ) (r2 + ( )2)G(r; !) = δ(~r − r~0) (38) c (using standard Fourier transform result for delta function and exponential function:Appendix ! 2 B) which is same as equation (18) solved in Appendix A but the ( c ) factor. ! (r2 + ( )2)G(r; !) = δ(~r − r~0) c 1 d2 ! ) (rG(r; !)) + ( )2G(r; !) = δ(~r − r~0) r dr2 c d2 ! ) (rG(r; !)) + ( )2(rG(r; !)) = rδ(~r − r~0) (39) dr2 c When r 6= 0 the solution to equation (39) is: i( ! )r −i( ! )r i( ! )r −i( ! )r e c e c rG(r; !) = Ae c + Be c ) G(r; !) = A + B (40) r r 6 When ! = 0 (static case) equation (38) becomes equation (18) which we have solved in Appendix A.
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