Introduction to Active Phased Arrays

1.0 Introduction to Phased Arrays

1.1 Phased Array History and Perspective
1.2 Fundamentals of Wave Propagation: The Wave Equation
1.2.1 Boundary Condition Cases
1.2.1.1 Standing Waves
1.2.1.2 Surface Waves and Scan Loss
1.2.1.3 Scan-Dependent Surface Impedance
1.3 Array Antennas
1.4 Aperture State Fundamentals
1.4.1 General Aperture State Relationships
1.4.2 Radiation Integrals for Circular Apertures
1.5 Array Far field Fundamentals
1.6 Frequency-Time Domains
1.6.1 Frequency-Time Domains: Fast Fourier Transform

 

2.1 Array Far Field Radiation

2.2 Array Far Field Fundamental Observations
2.3 General Array Theory
2.4 Two-Element Arrays
2.5 Linear Arrays
2.5.1 Linear Arrays in Sine Space
2.5.2 Linear Array Aperture Projection
2.6 Planar Arrays
2.6.1 Planar Arrays with No Real Space Grating Lobes
2.6.2 Planar Arrays with Real Space Grating Lobes
2.7 Conformal Arrays
2.7.1 Radius of Curvature Embedded Element Geometry
2.7.2 Conformal Array Phase Alignment
2.7.3 Eclipsed Elements in Conformal Arrays

 

3.1 Lattice Theory

3.2 Floquet’s Theorem
3.2.1 Phased array Surface Wave Condition
3.2.2 Phased Array Scan Volume
3.3 Lattice Theory
3.3.1 Rectangular Lattice
3.3.2 Equilateral Triangular Lattice
3.3.3 Isosceles Triangular Lattice
3.4 Reordered Lattice Theory
3.4.1 Ring Lattice
3.4.2 Spiral Lattice
3.5 Finite Array and Surface Wave Effects

 

4.1 Array Fundamentals: Supporting Theories I

4.2 Radiating Aperture Fundamentals, Three Domains
4.3 Array Architecture
4.3.1 Case 1: Hybrid Beamformed, Single Polarization
4.3.2 Case 2: Analog Beamformed, Dual Simultaneous Polarization
4.4 Practical Limits
4.4.1 Theorem of Reciprocity
4.4.2 Conservation of Energy
4.4.3 Superposition
4.4.4 Duality Theorem
4.5 Near- and Far-Fields
4.5.1 The Far Field Criterion
4.5.2 Array Reactive and Near Fields
4.6 Rotational Transforms
4.6.1 Coordinate Frames
4.6.2 Sine Space
4.6.3 Rotated Coordinate Frames
4.6.4 Inverted Rotated Coordinate Frames

 

5.1 Array Fundamentals: Supporting Theories II

5.2 Radiated Gain
5.3 Polarization Domain
5.3.1 Polarization Transforms
5.3.2 Stokes Parameters
5.3.3 Polarization Isolation
5.3.4 Cross-Polarization
5.3.5 Scan Dependent Polarization Properties
5.3.6 Polarization Compensation
5.4 Phased Array Noise Temperature
5.4.1 Antenna Noise Sources
5.4.2 Noise Wave Theory

 

6.1 Phased Array Radiating Elements

6.2 Single Element Dipole Over Ground Plane Radiators
6.2.1 Dipole Boundary Conditions
6.2.2 Dipole Radiation
6.3 Single Element Waveguide Radiators
6.3.1 Rectangular Waveguide
6.3.2 Circular Waveguide
6.3.3 Circular Waveguide Radiator
6.4 Single Element Patch Radiators
6.4.1 Square Patch Boundary Conditions
6.4.2 Square Patch Design Methods
6.4.3 Square Patch Radiation
6.4.4 Circular Patch Boundary Conditions
6.4.5 Circular Patch Radiation

 

7.1 Active Radiating Elements

7.2 Mutual Coupling and Embedded Elements in Arrays
7.2.1 Active Impedance and Reflection Coefficient
7.2.2 Wide Angle Impedance Matching
7.2.3 Real Space Grating Lobes
7.2.4 Surface Impedance Effects
7.2.5 Embedded Element Gain
7.3 Active Radiating Element Cases
7.4 Active Dipole Over Ground Plane Radiators
7.4.1 Linear Dipole Array
7.4.2 Vee Dipole Array
7.4.3 PUMA Array
7.5 Active Patch Radiators
7.5.1 Balanced Patch Radiator Array
7.5.2 Unbalanced Patch Radiator Array
7.5.3 Balanced Stacked Patch Radiator in a Rectangular Lattice Array

 

8.1 Far Field Synthesis I

8.2 Fourier Transform Method
8.3 Schelkunoff’s Form
8.4 Canonic Forms
8.5 Truncated Complex Gaussian Forms
8.5.1 Truncated Gaussian Magnitude Aperture Taper
8.5.2 Truncated Gaussian Phase Aperture Taper
8.6 Modified sin(x)/x Distribution

 

9.1 Far field Array Synthesis, Part II

9.2 Woodward-Lawson Synthesis
9.2.1 Non-Orthogonal Woodward-Lawson Method
9.2.2 Difference Patterns by the Woodward-Lawson Method
9.2.3 WLS Method with Controlled Sidelobes
9.3 Dolph-Chebyshev Synthesis
9.4 Taylor Line Source Synthesis
9.5 Planar 2-Dimensional Array Distributions
9.6 Circular Aperture Distributions
9.6.1 Taylor Circular Array Sources
9.7 Iterative Synthesis Methods
9.8 Maximal Likelihood Estimation

 

10.1 Stochastic Aperture Errors in Phased Arrays

10.1.1 Stochastic (Random) Errors in Arrays
10.1.2 Average (rms) Far-Field Characteristics
10.1.3 Beam Pointing Error
10.1.4 Peak and rms Sidelobes
10.1.5 Dispersion and its impact on Instantaneous Bandwidth
10.1.6 Polarization Isolation
10.2 Stochastic Error Budgets
10.3 Periodic (Correlated) Array Errors
10.3.1 Element-Level Phase Quantization
10.3.2 Subarray Spatial Effects
10.3.2.1 Subarray Gap Effects
10.3.2.2 Subarray Membrane Effects
10.3.3 Subarray Frequency Domain Effects
10.3.4 Aperture Blockage

 

11.1 Array Networks

11.1.1 Applied Array Networks
11.1.2 Networked Array Architectures (active, passive, reactive)
11.2 Networked Array Architectures
11.2.1 Linear Array Simultaneous Beams
11.2.2 Two-Dimensional Array Simultaneous Beam Set
11.2.3 Subarrayed Simultaneous Beam Systems
11.2.4 Spatially Randomized Subarrays
11.2.5 Overlapped Subarrays
11.2.6 Rotman lens arrays
11.3 MIMO Arrays

 

12.1 Array Calibration

12.2 Array Calibration Error Sources
12.3 Bench Calibration Methods
12.4 Far Field Range (NFR) Array Calibration
12.5 Near Field Range (FFR) Array Calibration
12.6 Satellite-Based Array (SATCAL) Calibration
12.7 Quasar-Based Array Calibration