Antennas and Propagation : Basic Electromagnetic Analysis
Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2
Antenna Theory Problems Analysis Problem Given an antenna structure or source current distribution, how does the antenna radiate? Focus of this course (more mature and developed) Synthesis problem Given the desired operational characteristics (like the radiation pattern), find the antenna structure or source current that will generate this. Challenging! Much less developed. Topic for research. Antennas and Propagation Slide 3
General Antenna Analysis Problem Statement Given: Arbitrary volume V Filled with sources = electric currents (A/m 2 ) = magnetic currents (A/m 2 ) Problem: Compute fields E and H generated by currents Solution: Maxwell s equations gives exact solution Fields: Antennas and Propagation Slide 4
Maxwell s Equations Equations (Differential Form) Gauss Law (electric field) Gauss Law (magnetic field) Faraday s Law Ampere s Law Constitutive Relationships Antennas and Propagation Slide 5
Quantities E electric field (V/m) H magnetic field (A/m) B magnetic flux density (tesla, T) D electric displacement (C/m 2 ) J electric current density (A/m 2 ) M magnetic current density (V/m 2 ) ρ electric charge density (C/m 3 ) ε permittivity (F/m) µ permeability (H/m) η Characteristic impedance (Ohm) ε = ε 0 = 8.8542 x 10-12 F/m µ = µ0 = 4π x 10-7 H/m (free space) ε = ε r ε 0 µ = µ r µ 0 Antennas and Propagation Slide 6
Time Harmonic Fields Assuming Time-harmonic fields, or exp(jωt) variation Linear, isotropic media Maxwell s equations become where ω=2πf is the circular frequency (rad/s). Antennas and Propagation Slide 7
Boundary Conditions Tangential E and H are Continuous when no surface currents (J, M) (e.g. dielectric media) Differ according to the surface currents (conductive surface) What if region 2 is perfect conductor? PEC = perfect electrical conductor PMC = perfect magnetic conductor Antennas and Propagation Slide 8
The Wave Equation Maxwell famous for relating E and H fields by adding the D/ t term in Ampere s law Allows wave propagation to be predicted Given source free-environment, take curl of Faraday s law: Substituting into Ampere s law Wave equation Antennas and Propagation Slide 9
Vector Potential Basic Antenna Analysis Problem Given currents J and M, compute fields E and H Option 1: Direct Option 2: Use potential Static Problems Nice to use electric scalar potential (V =voltage) instead of E Why? Can solve scalar equations instead of vector equations Dynamic Problems Use the vector potential A instead of B Can solve vector equations instead of complicated dyadic (matrix) equations Definition of A chosen to make analysis as simple as possible Exploit physical and vectorial properties Antennas and Propagation Slide 10
Vector Potential (2) Properties we exploit in defining vector potential A Gauss Law: B = 0 Any vector: ( A ) = 0 Therefore, B A =μ H A = A (No loss of generality) Subscript A means arising from A or from electric current J Substitute into Faraday s law (with M = 0) E A = -jωμ H A = -jω A ( E A + jωa ) = 0 Scalar identity: ( φ e ) = 0 (φ e any scalar function) Finally: E A + jωa = - φ e Note: when ω=0, have normal definition of scalar potential Antennas and Propagation Slide 11
Vector Potential (3) Summarizing We have chosen definition of A so vector properties ensure certain physical conditions are automatically satisfied Simplifies analysis! Antennas and Propagation Slide 12
Vector Potential (4) Substituting E A + jωa = - φ e into Ampere s Law Note: We specified A, but A is not yet uniquely defined So, choose A = -jωµε φ e (Lorenz Condition) Antennas and Propagation Slide 13
Vector Potential (5) Finally Relationship for A that is just the non-homogeneous wave equation Individually satisfied for each component (e.g. A x, A y, A z ) If we need fields, Antennas and Propagation Slide 14
Vector Potential (6) Vector potential F for magnetic current M Almost identical procedure, but roles of E and H switch See notes for derivation Also, to get E Antennas and Propagation Slide 15
Vector Potential Summary Summarizing... Transforms complicated wave equation involving E and H Simpler scalar wave equations for comps of A and F Approach Given J and M solve for A and F When all finished, then find E and H Helps us manage the complexity of EM antenna problems Antennas and Propagation Slide 16
Finding A for Known J Governing equation Need to solve Direct solution wrt. some boundary conditions? Numerically... Alternative Approach Transform into integral equation Green s function technique Advantage: Automatically incorporates boundary condition Single equation Antennas and Propagation Slide 17
Green s Function Technique Roadmap 1. Find fields radiated by a point source at origin 2. Translate solution to arbitrary source location 3. Solution for arbitrary current distribution given by superposition of point source solutions Similar to signal analysis for LTI systems! Impulse response of system h(t) Response for arbitrary input is y(t) = h(t) x(t) Antennas and Propagation Slide 18
1. Fields due to Point Source at Origin Assume current distribution z We need to solve J y x Only have z-directed component on LHS, therefore also on RHS Now, solve for point source (unit impulse) at origin: call solution Problem: Antennas and Propagation Slide 19
Case 1: Away from origin Assume r 0 Simplify by invoking symmetry of problem RHS is only a function of r (point source), so g( r ) only should depend on distance from origin r = r Antennas and Propagation Slide 20
Case 1: Away from origin (2) Do a trick on the d/dr term So, need to solve Antennas and Propagation Slide 21
Case 1: Away from origin (4) Only depends on single var r, so have famous ODE Solutions Simplification: c 2 = 0 Antennas and Propagation Slide 22
Case 2: Include contribution at origin Have solution But how to find c 1? Need to include effect of source at origin Consider original problem Integrate both sides over a sphere Let sphere shrink to simplify solution Antennas and Propagation Slide 23
Case 2: Include contribution at origin (2) Note: Simplify first term with Divergence Theorem Integrating divergence of something over volume = Integration of outward normal component on surface Antennas and Propagation Slide 24
Case 2: Include contribution at origin (3) Becomes Just need to insert previous result and solve for c 1 Antennas and Propagation Slide 25
Case 2: Include contribution at origin (4) Antennas and Propagation Slide 26
2. Translate Solution Have solution for point at origin z r r - r For arbitrary source point r x r y Antennas and Propagation Slide 27
3. Arbitrary Current Distribution Have solution for point source What about arbitrary source J (r )? Big Idea: Can solve this equation by substituting the point source solution into the proper integral equation. Antennas and Propagation Slide 28
Arbitrary Current Distribution (2) Consider integral equation Use the vector identity: Antennas and Propagation Slide 29
Arbitrary Current Distribution (3) But, what is I equal to? Divergence Theorem S n Note: Surf. integral 0 as volume becomes large Called the radiation condition Intuitively: If we are far from all sources, A z and g will shrink to 0. V Antennas and Propagation Slide 30
Arbitrary Current Distribution (4) = 0 Recall that we have Substitute into volume equation Antennas and Propagation Slide 31
Arbitrary Current Distribution (5) Note: We have switched r and r (symmetry) Antennas and Propagation Slide 32
Arbitrary Current Distribution (6) Interpretation? Like a convolution! Since we have the same equation for each Cart. component Similarly for magnetic currents Antennas and Propagation Slide 33
Summary of Green s Function Analysis Goal Compute fields radiated by arbitrary current source (antenna). Solution Use vector potentials to make problem simpler Simple inhomogeneous wave equation for A Solve the equation for a point current source (= Green s function) Fields for arbitrary current given by integral equation (Superposition or convolution) Antennas and Propagation Slide 34