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Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation
Chapter 2Antennas and Propagation Slide 2
Outline
Vector Potentials, Wave Equation
Far-field Radiation
Duality/Reciprocity
Transmission Lines
Chapter 2Antennas and Propagation Slide 3
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.
Chapter 2Antennas and Propagation Slide 4
General Antenna Analysis
Problem Statement
Given:
Arbitrary volume V
Filled with sources
= electric currents (A/m2)
= magnetic currents (A/m2)
Problem:
Compute fields E and H generated by currents
Solution:
Maxwell’s equations gives exact solution
Fields:
Chapter 2Antennas and Propagation Slide 5
Maxwell’s Equations
Equations (Differential Form)
Gauss’ Law (electric field)
Gauss’ Law (magnetic field)
Faraday’s Law
Ampere’s Law
Constitutive Relationships
Chapter 2Antennas and Propagation Slide 6
Quantities
E electric field (V/m)
H magnetic field (A/m)
B magnetic flux density (tesla, T)
D electric displacement (C/m2)
J electric current density (A/m2)
M magnetic current density (V/m2)
ρ electric charge density (C/m3)
ε permittivity (F/m)
µ permeability (H/m)
η Characteristic impedance (Ohm)
ε = ε0 = 8.8542 x 10-12 F/m µ = µ0 = 4π x 10-7 H/m (free space)
ε = εr ε0 µ = µr µ0
Chapter 2Antennas and Propagation Slide 7
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).
Chapter 2Antennas and Propagation Slide 8
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
Chapter 2Antennas and Propagation Slide 9
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
Chapter 2Antennas and Propagation Slide 10
Vector Potential
Basic Antenna Analysis Problem
Given currents J̅ and M̅ , compute fields E̅ and H̅
Option 1: Direct
Option 2: Use potential
Static Problems
Nice to use electric scalar potential (V =voltage) instead of E̅
Why? Can solve scalar equations instead of vector equations
Dynamic Problems
Use the vector potential A̅ instead of B̅Can solve vector equations
instead of complicated dyadic (matrix) equations
Definition of A̅ chosen to make analysis as simple as possibleExploit physical and vectorial properties
Chapter 2Antennas and Propagation Slide 11
Vector Potential (2)
Properties we exploit in defining vector potential A
Gauss’ Law: ∇· B̅ = 0
Any vector: ∇· (∇× A̅ ) = 0
Therefore,B̅A =μ H ̅A = ∇× A̅ (No loss of generality)Subscript A means “arising from A̅” or from electric current J ̅Substitute into Faraday’s law (with M̅ = 0)∇× E̅A = -jωμ H̅A = -jω∇× A̅
∇× ( E̅A + jωA̅ ) = 0
Scalar identity: ∇× (∇φe) = 0 (φe any scalar function)
Finally: E̅A + jωA̅ = -∇φe
Note: when ω=0 , have normal definition of scalar potential
Chapter 2Antennas and Propagation Slide 12
Vector Potential (3)
Summarizing
We have chosen definition of A̅ so vector properties ensure certain physical conditions are automatically satisfied
Simplifies analysis!
Chapter 2Antennas and Propagation Slide 13
Vector Potential (4)
Substituting E̅A + jωA̅ = -∇φe into Ampere’s Law
Note: We specified ∇× A̅, but A̅ is not yet uniquely definedSo, choose ∇· A̅ = -jωµε φe (Lorenz Condition)
Chapter 2Antennas and Propagation Slide 14
Vector Potential (5)
Finally
Relationship for A̅ that is just the non-homogeneous wave equation
Individually satisfied for each component (e.g. Ax, Ay, Az)
If we need fields,
Chapter 2Antennas and Propagation Slide 15
Vector Potential (6)
Vector potential F̅ for magnetic current M̅
Almost identical procedure, but roles of E and H switch
See notes for derivation
Also, to get E
Chapter 2Antennas and Propagation Slide 16
Vector Potential Summary
Summarizing...
Transforms complicated wave equation involving E̅ and H̅
Simpler scalar wave equations for comps of A̅ and F̅
Approach
Given J̅ and M̅ solve for A̅ and F̅When all finished, then find E̅ and H̅
Helps us manage the complexity of EM antenna problems
Chapter 2Antennas and Propagation Slide 17
Finding A ̅̅ ̅̅ for Known J ̅̅ ̅̅
Governing equationNeed to solve
Direct solution wrt. some boundary conditions? Numerically...
Alternative ApproachTransform into integral equationGreen’s function techniqueAdvantage:
Automatically incorporates boundary conditionSingle equation
Chapter 2Antennas and Propagation Slide 18
Green’s Function Technique
Roadmap1. Find fields radiated by a point source at origin
2. Translate solution to arbitrary source location
3. Solution for arbitrary current distribution givenby 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)
Chapter 2Antennas and Propagation Slide 19
1. Fields due to Point Source at Originat Originat Originat Origin
Assume current distribution
We need to solve
Only have z-directed component on LHS, therefore also on RHS
Now, solve for point source (unit impulse) at origin:call solution
Problem:
J ̅
z
x
y
Chapter 2Antennas and Propagation Slide 20
Case 1: Away from origin
Assume r̅ ≠ 0
Simplify by invoking symmetry of problem RHS is only a function of r (point source), so g( r̅) only should
depend on distance from origin r = |r̅|
Chapter 2Antennas and Propagation Slide 21
Case 1: Away from origin (2)
Do a trick on the d/dr term
So, need to solve
Chapter 2Antennas and Propagation Slide 22
Case 1: Away from origin (4)
Only depends on single var r, so have famous ODE
Solutions
Simplification: c2 = 0
Chapter 2Antennas and Propagation Slide 23
Case 2: Include contribution at origin
Have solution
But how to find c1?Need to include effect of source at origin
Consider original problem
Integrate both sides over a sphereLet sphere shrink to simplify solution
Chapter 2Antennas and Propagation Slide 24
Case 2: Include contribution at origin (2)
Note:
Simplify first term with Divergence TheoremIntegrating divergence of something over volume
= Integration of outward normal component on surface
∆∆∆∆
Chapter 2Antennas and Propagation Slide 25
Case 2: Include contribution at origin (3)
Becomes
Just need to insert previous resultand solve for c1
∆∆∆∆
Chapter 2Antennas and Propagation Slide 26
Case 2: Include contribution at origin (4)
∆∆∆∆
Chapter 2Antennas and Propagation Slide 27
2. Translate Solution
Have solution for point at origin
For arbitrary source point r̅ ′
z
x
yr ̅ ′
r ̅r ̅ - r ̅’
Chapter 2Antennas and Propagation Slide 28
3. Arbitrary Current Distribution
Have solution for point source
What about arbitrary source J̅ (r̅ ′)?
Big Idea: Can solve this equation by substituting the point source
solution into the proper integral equation.
Chapter 2Antennas and Propagation Slide 29
Arbitrary Current Distribution (2)
Consider integral equation
Use the vector identity:
Chapter 2Antennas and Propagation Slide 30
Arbitrary Current Distribution (3)
But, what is I equal to?
Divergence Theorem
Note: Surf. integral → 0 as volume becomes large
Called the “radiation condition”
Intuitively: If we are far from all sources, Az and g will shrink to 0.
V
Sn̂
Chapter 2Antennas and Propagation Slide 31
Arbitrary Current Distribution (4)
Recall that we have
Substitute into volume equation
= 0
Chapter 2Antennas and Propagation Slide 32
Arbitrary Current Distribution (5)
Note: We have switched r and r’ (symmetry)
Chapter 2Antennas and Propagation Slide 33
Arbitrary Current Distribution (6)
Interpretation? Like a convolution!
Since we have the same equation for each Cart. component
Similarly for magnetic currents
Chapter 2Antennas and Propagation Slide 34
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)