Theory¶

Background theory and equations for phased array systems.

Contents¶

Topic	Description
Phased Array Fundamentals	Array factor, gain, beamwidth
Link Budget Equations	Communications link analysis
Radar Equation	Radar range and detection
Pareto Optimization	Multi-objective trade-offs

Quick Reference¶

Antenna Gain¶

For a uniform rectangular array with \(N = n_x \times n_y\) elements:

\[ G \approx \eta_a \cdot 4\pi \cdot n_x d_x \cdot n_y d_y \]

Where \(\eta_a\) is aperture efficiency (~0.6-0.7) and \(d_x, d_y\) are spacings in wavelengths.

Free Space Path Loss¶

\[ L_{FSPL} = 20 \log_{10}\left(\frac{4\pi d f}{c}\right) = 32.45 + 20\log_{10}(f_{MHz}) + 20\log_{10}(d_{km}) \]

Link Budget¶

\[ \begin{aligned} EIRP &= P_{tx} + G_{tx} - L_{tx} \\ P_{rx} &= EIRP - L_{path} + G_{rx} \\ SNR &= P_{rx} - N \\ Margin &= SNR - SNR_{required} \end{aligned} \]

Radar Range Equation¶

\[ SNR = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4 k T_s B L} \]

Pareto Optimality¶

Design \(A\) dominates design \(B\) if:

\(f_i(A) \leq f_i(B)\) for all objectives (minimization)
\(f_j(A) < f_j(B)\) for at least one objective

Key Constants¶

Constant	Symbol	Value
Speed of light	\(c\)	299,792,458 m/s
Boltzmann constant	\(k\)	1.38×10⁻²³ J/K
Reference temperature	\(T_0\)	290 K