Phased Array Fundamentals¶

Theory of phased array antenna operation.

Overview¶

A phased array is an antenna system that uses multiple radiating elements with controlled phase and amplitude to electronically steer the beam without mechanical movement.

Array Factor¶

The array factor describes the radiation pattern contribution from element positioning:

\[ AF(\theta, \phi) = \sum_{n=1}^{N} a_n e^{j(k \mathbf{r}_n \cdot \hat{\mathbf{r}} + \alpha_n)} \]

Where: - $a_n$ = amplitude weight of element $n$ - $k = 2\pi/\lambda$ = wavenumber - $\mathbf{r}_n$ = position of element $n$ - $\hat{\mathbf{r}}$ = unit vector in observation direction - $\alpha_n$ = phase of element $n$

Linear Array¶

For a uniform linear array along the z-axis with spacing $d$:

\[ AF(\theta) = \sum_{n=0}^{N-1} e^{jn(kd\cos\theta + \beta)} \]

Where $\beta$ is the progressive phase shift.

Closed form:

\[ AF(\theta) = \frac{\sin(N\psi/2)}{\sin(\psi/2)}, \quad \psi = kd\cos\theta + \beta \]

Rectangular Planar Array¶

For an $N_x \times N_y$ array:

\[ AF(\theta, \phi) = AF_x(\theta, \phi) \cdot AF_y(\theta, \phi) \]

\[ AF_x = \frac{\sin(N_x \psi_x / 2)}{\sin(\psi_x / 2)}, \quad \psi_x = kd_x \sin\theta\cos\phi + \beta_x \]

\[ AF_y = \frac{\sin(N_y \psi_y / 2)}{\sin(\psi_y / 2)}, \quad \psi_y = kd_y \sin\theta\sin\phi + \beta_y \]

Antenna Gain¶

Directivity¶

The directivity is the ratio of radiation intensity in a given direction to the average:

\[ D = \frac{4\pi U(\theta, \phi)}{P_{rad}} \]

Peak directivity for a uniform array (boresight):

\[ D_0 \approx \frac{4\pi A_{eff}}{\lambda^2} = \frac{4\pi}{\lambda^2} \cdot N_x d_x \cdot N_y d_y \cdot \lambda^2 = 4\pi N_x d_x N_y d_y \]

Gain¶

Gain includes efficiency:

\[ G = \eta_a \cdot D \]

Where aperture efficiency $\eta_a$ typically ranges from 0.5 to 0.8.

For uniform amplitude and half-wavelength spacing ($d = 0.5\lambda$):

\[ G \approx \eta_a \cdot \pi \cdot N_x \cdot N_y \]

In dB:

\[ G_{dB} = 10\log_{10}(\eta_a \cdot \pi \cdot N) \]

For $\eta_a = 0.65$ and $N = 64$ elements:

\[ G_{dB} = 10\log_{10}(0.65 \cdot \pi \cdot 64) \approx 22.1 \text{ dB} \]

Beamwidth¶

Half-Power Beamwidth (HPBW)¶

For a uniform linear array:

\[ \theta_{3dB} \approx \frac{0.886 \lambda}{N d \cos\theta_0} \]

At boresight ($\theta_0 = 0$) with $d = 0.5\lambda$:

\[ \theta_{3dB} \approx \frac{1.77}{N} \text{ radians} = \frac{101°}{N} \]

For a rectangular array:

\[ \theta_{3dB,x} \approx \frac{101°}{N_x}, \quad \theta_{3dB,y} \approx \frac{101°}{N_y} \]

Examples¶

Array	Elements	HPBW
8×8	64	12.6° × 12.6°
16×16	256	6.3° × 6.3°
32×32	1024	3.2° × 3.2°

Beam Steering¶

Electronic beam steering is achieved by applying a progressive phase shift:

\[ \beta_x = -kd_x \sin\theta_0 \cos\phi_0 $$ $$ \beta_y = -kd_y \sin\theta_0 \sin\phi_0 \]

Where $(\theta_0, \phi_0)$ is the desired beam direction.

Scan Loss¶

When scanning off boresight, gain is reduced:

\[ G(\theta) \approx G_0 \cos^p(\theta) \]

Where $p$ depends on element pattern (typically $p \approx 1.2-1.5$).

Approximate scan loss:

Scan Angle	Approximate Loss
0°	0 dB
30°	1-2 dB
45°	2-3 dB
60°	3-5 dB

Grating Lobes¶

Grating lobes appear when element spacing exceeds $\lambda$. To avoid grating lobes when scanning to angle $\theta_0$:

\[ d < \frac{\lambda}{1 + |\sin\theta_0|} \]

For $\theta_0 = 60°$:

\[ d < \frac{\lambda}{1 + \sin(60°)} = \frac{\lambda}{1.866} \approx 0.54\lambda \]

This is why half-wavelength spacing ($d = 0.5\lambda$) is common.

Sidelobe Level¶

For a uniform amplitude array, the first sidelobe is approximately -13.2 dB below the main lobe.

Tapering¶

Amplitude tapering reduces sidelobes at the cost of beamwidth and gain:

Taper	First SLL	Beamwidth Factor	Efficiency
Uniform	-13.2 dB	1.0	1.0
Hamming	-42 dB	1.36	0.73
Taylor -25 dB	-25 dB	1.1	0.95
Taylor -35 dB	-35 dB	1.2	0.87

Power and EIRP¶

Total Radiated Power¶

\[ P_{rad} = N \cdot P_{elem} \cdot \eta_{feed} \]

Where: - $N$ = number of elements - $P_{elem}$ = power per element - $\eta_{feed}$ = feed network efficiency

EIRP¶

Effective Isotropic Radiated Power:

\[ EIRP = P_{rad} \cdot G = N \cdot P_{elem} \cdot \eta_{feed} \cdot G \]