Introduction & Motivation
What is a Phased Array?
A phased array antenna is a group of multiple radiating elements (antennas) whose relative phases and amplitudes are controlled to form and steer a beam of radio waves in a desired direction without physically moving the antenna structure.
Key Advantage
Electronic beam steering allows for microsecond switching between directions, enabling rapid scanning, tracking of multiple targets, and adaptive interference rejection—impossible with mechanically steered antennas.
Principle of Operation
By adjusting the phase shift between adjacent elements, constructive interference is achieved in the desired direction while destructive interference suppresses radiation in other directions.
🎓 Learning Objectives
After studying these notes, you should be able to:
- Explain the fundamental principles of phased array operation
- Calculate the array factor for uniform linear arrays
- Determine beam steering angles using phase shift equations
- Analyze grating lobes and side lobe levels
- Compare passive (PESA) and active (AESA) electronically scanned arrays
- Understand beamforming techniques and their applications
Fundamental Concepts
Array Geometry & Coordinate Systems
Uniform Linear Array (ULA)
The simplest phased array configuration consists of N identical elements arranged along a straight line with equal spacing d between adjacent elements. For analysis, we use spherical coordinates (θ, φ) where:
- θ (theta): Elevation angle from the array axis (0° to 180°)
- φ (phi): Azimuth angle in the plane perpendicular to the array
- Broadside: Direction perpendicular to the array axis (θ = 90°)
- Endfire: Direction along the array axis (θ = 0° or 180°)
Phase Difference Between Adjacent Elements
where k = 2π/λ (wavenumber), d = element spacing, θ = observation angle, β = progressive phase shift
🔑 Key Concept: Pattern Multiplication
- The total radiation pattern of an array = Element Pattern × Array Factor
- Element Pattern (EP): Radiation characteristics of a single element
- Array Factor (AF): Depends only on array geometry and excitation
- For isotropic elements, Total Pattern = Array Factor
Phase Steering Principle
Broadside Radiation
When all elements are fed in phase (β = 0), maximum radiation occurs perpendicular to the array axis. This is called broadside radiation.
Beam Steering
To steer the beam to angle θ₀, introduce a progressive phase shift between elements:
Beam Steering Equation
The Array Factor
Mathematical Derivation
The Array Factor (AF) represents the far-field radiation pattern of an array of isotropic point sources. For a uniform linear array of N elements with equal amplitude excitation:
Array Factor (General Form)
Closed-Form Solution (Uniform Array)
Normalized: AFn = (1/N) × sin[Nψ/2] / sin[ψ/2]
Interactive Array Factor Visualization
Array Factor Pattern Simulator
Polar plot showing normalized array factor pattern. Adjust parameters to observe beam steering, side lobes, and grating lobes.
📊 Array Characteristics
- Main Lobe Width: Decreases as N increases (≈ 2λ/Nd radians)
- Side Lobe Level: First side lobe ≈ -13.2 dB for uniform illumination
- Directivity: Increases with number of elements (~2N for large arrays)
- Grating Lobes: Appear when d > λ, causing multiple maxima
Grating Lobes
Condition for Grating Lobes
Grating lobes are unwanted secondary maxima that appear when the element spacing is too large relative to wavelength. To avoid grating lobes in visible space for broadside arrays:
For broadside (θ₀ = 90°): dmax = λ
For endfire (θ₀ = 0°): dmax = λ/2
Beamforming Techniques
Conventional Beamforming
Also known as delay-and-sum beamforming. Each element's signal is delayed (phase-shifted) to compensate for path length differences, then summed coherently.
Adaptive Beamforming
Weights are dynamically adjusted to maximize signal-to-interference ratio. Algorithms include MMSE, LMS, and RLS for real-time optimization.
Null Steering
Places nulls (directions of zero response) in the radiation pattern toward interfering sources while maintaining gain in the desired direction.
Weighting Methods (Tapering)
| Window Function | First Side Lobe | Main Lobe Width | Applications |
|---|---|---|---|
| Uniform | -13.2 dB | Narrowest | Maximum directivity |
| Binomial | No side lobes | Widest | Interference-free |
| Dolph-Chebyshev | Equal level (-20 to -60 dB) | Variable | Radar, controlled SLL |
| Taylor | Decreasing away from main | Moderate | General purpose |
Phase Shifter Technologies
Electronic beam steering requires precise phase control at each element. Common implementations include:
- Digital Phase Shifters: PIN diodes or FET switches with discrete phase steps (e.g., 5-bit = 32 states, 11.25° resolution)
- Analog Phase Shifters: Ferrite or varactor diode based, continuous phase adjustment
- True Time Delay (TTD): Uses transmission line lengths, frequency-independent (wideband)
- Digital Beamforming: Phase shifting performed in baseband/DSP (most flexible)
Types of Phased Arrays
PESA vs AESA
| Feature | PESA (Passive) | AESA (Active) |
|---|---|---|
| Architecture | Central transmitter/receiver | Distributed T/R modules |
| Power Source | Single high-power tube | Multiple solid-state amplifiers |
| Reliability | Single point of failure | Graceful degradation |
| Beam Agility | Limited | Simultaneous multiple beams |
| Cost | Lower | Higher (but decreasing) |
| Applications | Early radar, satellite | Modern radar, 5G, SatCom |
Array Configurations
T/R (Transmit/Receive) Modules
Active electronically scanned arrays (AESAs) use T/R modules at each element. A typical T/R module contains:
- High Power Amplifier (HPA) for transmit
- Low Noise Amplifier (LNA) for receive
- Phase shifter (analog or digital)
- Attenuator for amplitude control
- Circulator or switch for duplexing
Applications
Radar Systems
Air traffic control, weather monitoring, military surveillance, and missile guidance. Enables tracking of multiple targets simultaneously.
5G Communications
Massive MIMO arrays with 64-256 elements provide beamforming gain and spatial multiplexing for increased capacity.
Satellite Communications
Ground terminals and satellite payloads use phased arrays for electronic steering toward moving satellites without mechanical tracking.
Automotive Radar
77 GHz phased arrays in vehicles for adaptive cruise control, collision avoidance, and autonomous driving.
Medical Imaging
Ultrasound and MRI systems use beamforming to focus energy and improve image resolution.
Radio Astronomy
Arrays like the VLA and ALMA combine signals from multiple dishes to achieve high angular resolution.
🎓 Summary of Key Equations
- Array Factor: AF = sin(Nψ/2) / sin(ψ/2), where ψ = kd cos θ + β
- Beam Steering: β = -kd cos θ₀
- Beamwidth: BW ≈ 0.886 λ/(Nd) radians (broadside, uniform)
- Directivity: D ≈ 2Nd/λ (for broadside linear array)
- Grating Lobe Avoidance: d < λ/(1 + |cos θ₀|)
Study Checklist
Conceptual Understanding
- ☐ Understand constructive/destructive interference
- ☐ Explain pattern multiplication principle
- ☐ Describe electronic vs mechanical steering
- ☐ Compare PESA and AESA architectures
Mathematical Skills
- ☐ Derive the array factor equation
- ☐ Calculate beam steering phase shifts
- ☐ Determine null positions and side lobe levels
- ☐ Analyze grating lobe conditions
Practical Analysis
- ☐ Plot radiation patterns for various N and d
- ☐ Design arrays for specific beamwidth requirements
- ☐ Select appropriate window functions
- ☐ Calculate array gain and directivity