Impairments#

This guide covers realistic impairment models that affect phased array performance in practice: mutual coupling, phase quantization, element failures, and scan blindness.

Mutual Coupling#

Electromagnetic coupling between adjacent elements affects both the element patterns and the input impedance of each element.

Theoretical Model#

Approximates coupling based on element spacing using a simple dipole model:

import phased_array as pa
import numpy as np

geom = pa.create_rectangular_array(8, 8, dx=0.5, dy=0.5)

# Create coupling matrix
coupling_matrix = pa.mutual_coupling_matrix_theoretical(
    geom.x, geom.y,
    coupling_coefficient=0.3,  # Coupling at d=lambda/2
    coupling_exponent=2.0      # Decay rate
)

print(f"Matrix shape: {coupling_matrix.shape}")  # (64, 64)
print(f"Self-coupling: {coupling_matrix[0, 0]:.2f}")  # 1.0
print(f"Neighbor coupling: {np.abs(coupling_matrix[0, 1]):.2f}")

Measured Coupling#

For more accurate modeling, import measured S-parameters:

# S-parameter data from measurement or EM simulation
s_params = np.load('measured_s_params.npy')  # (N, N) complex matrix

coupling_matrix = pa.mutual_coupling_matrix_measured(s_params)

Applying Coupling Effects#

k = pa.wavelength_to_k(1.0)
weights_ideal = pa.steering_vector(k, geom.x, geom.y, theta0_deg=30, phi0_deg=0)

# Apply coupling (modifies effective weights)
weights_coupled = pa.apply_mutual_coupling(weights_ideal, coupling_matrix)

# Compare patterns
theta, phi, pattern_ideal = pa.compute_full_pattern(
    geom.x, geom.y, weights_ideal, k
)
theta, phi, pattern_coupled = pa.compute_full_pattern(
    geom.x, geom.y, weights_coupled, k
)

Coupling causes:

  • Beam pointing errors

  • Increased sidelobes

  • Main beam distortion

  • Input impedance variations

Active Element Pattern#

The element pattern when embedded in an array differs from an isolated element:

# Compute active element pattern including coupling
theta = np.linspace(0, np.pi/2, 91)
phi = np.zeros_like(theta)

aep = pa.active_element_pattern(
    theta, phi, geom, coupling_matrix,
    element_idx=32  # Center element
)

Phase Quantization#

Digital phase shifters have finite resolution (typically 3-8 bits). Quantization causes beam pointing errors and increased sidelobes.

Basic Quantization#

# Ideal weights
weights = pa.steering_vector(k, geom.x, geom.y, theta0_deg=20, phi0_deg=0)

# Quantize to 4-bit (16 levels, 22.5 deg steps)
weights_q = pa.quantize_phase(weights, n_bits=4)

# Check quantization levels
phases_deg = np.rad2deg(np.angle(weights_q))
unique_phases = np.unique(np.round(phases_deg * 16/360) * 360/16)
print(f"Number of unique phases: {len(unique_phases)}")

Quantization Error Analysis#

# RMS phase error
for bits in [3, 4, 5, 6]:
    rms_error = pa.quantization_rms_error(bits)
    sll_increase = pa.quantization_sidelobe_increase(bits)
    print(f"{bits}-bit: RMS error = {rms_error:.1f} deg, SLL increase ~ {sll_increase:.1f} dB")

Typical results:

Bits

Levels

RMS Error

Effect on Pattern

3

8

13 deg

Significant beam errors, high quantization lobes

4

16

6.5 deg

Moderate errors, visible quantization lobes

5

32

3.3 deg

Small errors, acceptable for most applications

6

64

1.6 deg

Minimal impact, near-ideal performance

Full Analysis#

results = pa.analyze_quantization_effect(
    weights, geom, k,
    n_bits=4,
    theta_range=(0, np.pi/2),
    n_points=361
)

# Plot comparison
import matplotlib.pyplot as plt
plt.plot(results['theta_deg'], results['pattern_ideal_dB'], label='Ideal')
plt.plot(results['theta_deg'], results['pattern_quantized_dB'], label='4-bit')
plt.xlabel('Theta (deg)')
plt.ylabel('Pattern (dB)')
plt.legend()
plt.grid(True)

Element Failures#

Large arrays can tolerate element failures with graceful degradation. The library models random failures with different failure modes.

Simulating Failures#

# 5% random element failures
weights_failed, failure_mask = pa.simulate_element_failures(
    weights,
    failure_rate=0.05,
    mode='off',  # Failed elements produce no output
    seed=42
)

n_failed = np.sum(failure_mask)
print(f"Failed elements: {n_failed} / {len(weights)} ({100*n_failed/len(weights):.1f}%)")

Failure Modes#

  • ‘off’: Failed elements have zero output (most common)

  • ‘stuck’: Failed elements stuck at random phase, nominal amplitude

  • ‘full’: Failed elements at full power, random phase (worst case)

# Compare failure modes
for mode in ['off', 'stuck', 'full']:
    weights_f, mask = pa.simulate_element_failures(
        weights, 0.1, mode=mode, seed=42
    )
    theta, phi, pattern = pa.compute_full_pattern(
        geom.x, geom.y, weights_f, k
    )
    # 'full' mode causes highest sidelobe increase

Graceful Degradation Analysis#

Analyze how pattern degrades with increasing failure rate:

results = pa.analyze_graceful_degradation(
    geom, k, weights,
    failure_rates=[0.0, 0.02, 0.05, 0.10, 0.20],
    n_trials=10,  # Average over multiple random failures
    seed=42
)

# Results include:
# - Mean sidelobe level vs failure rate
# - Directivity loss
# - Beamwidth change

For a 256-element array, typical degradation:

  • 5% failures: ~0.5 dB gain loss, 2-3 dB SLL increase

  • 10% failures: ~1 dB gain loss, 4-5 dB SLL increase

  • 20% failures: ~2 dB gain loss, pattern significantly degraded

Scan Blindness#

At certain scan angles, surface waves can be excited, causing the array reflection coefficient to approach unity (scan blindness).

Surface Wave Angle#

Estimate the scan angle where blindness occurs:

blind_angle = pa.surface_wave_scan_angle(
    dx=0.5,  # Element spacing in wavelengths
    substrate_er=2.2,  # Substrate dielectric constant
    thickness=0.05  # Substrate thickness in wavelengths
)
print(f"Expected blindness near {blind_angle:.1f} degrees")

Scan Blindness Model#

theta = np.linspace(0, 90, 181)

# Get scan loss including blindness
loss = pa.scan_blindness_model(
    theta_deg=theta,
    dx=0.55,
    substrate_er=3.0,
    thickness=0.03,
    bandwidth=0.1  # Blindness bandwidth in sin(theta)
)

# Apply to pattern
scan_loss = pa.compute_scan_loss(theta, geom, k)

Applying to Patterns#

# Compute pattern at various scan angles
for scan_angle in [0, 30, 45, 60]:
    weights = pa.steering_vector(k, geom.x, geom.y, scan_angle, 0)

    # Apply scan blindness model
    weights_blind = pa.apply_scan_blindness(
        weights, geom, k,
        scan_angle_deg=scan_angle,
        substrate_er=2.5,
        blind_angle_deg=65
    )

    # Compare patterns with and without blindness

Combined Impairments#

In practice, multiple impairments occur simultaneously:

# Start with ideal steering
weights = pa.steering_vector(k, geom.x, geom.y, theta0_deg=30, phi0_deg=0)
weights *= pa.taylor_taper_2d(16, 16, sidelobe_dB=-30)

# Apply impairments in order
# 1. Mutual coupling
coupling = pa.mutual_coupling_matrix_theoretical(geom.x, geom.y)
weights = pa.apply_mutual_coupling(weights, coupling)

# 2. Phase quantization
weights = pa.quantize_phase(weights, n_bits=5)

# 3. Element failures
weights, _ = pa.simulate_element_failures(weights, failure_rate=0.03, mode='off')

# Compute final pattern
theta, phi, pattern_dB = pa.compute_full_pattern(geom.x, geom.y, weights, k)

Best Practices#

  1. Budget for coupling in beam pointing accuracy requirements.

  2. Use at least 5-bit phase shifters for most applications.

  3. Design for 5-10% failure tolerance in critical arrays.

  4. Avoid element spacings that place scan blindness in operational scan range.

  5. Combine impairment models for realistic performance prediction.