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)

Active Impedance and VSWR

In a phased array, mutual coupling causes each element to see a different impedance depending on the scan angle and the excitations of neighboring elements. This “active impedance” can vary significantly from the isolated element impedance.

Active Reflection Coefficient

The active reflection coefficient accounts for coupling from all other elements when the array is excited with given weights:

import phased_array as pa
import numpy as np

geom = pa.create_rectangular_array(8, 8, dx=0.5, dy=0.5)
k = pa.wavelength_to_k(1.0)

# Create coupling matrix
C = pa.mutual_coupling_matrix_theoretical(geom, k, coupling_coeff=0.2)

# Compute steering weights
weights = pa.steering_vector(k, geom.x, geom.y, theta0_deg=30, phi0_deg=0)

# Active reflection coefficient for center element
gamma = pa.active_reflection_coefficient(C, weights, element_idx=32)
print(f"Active reflection coeff: {np.abs(gamma):.3f} at {np.rad2deg(np.angle(gamma)):.1f} deg")

Active Impedance

# Active impedance seen at element port
Z_active = pa.active_impedance(C, weights, element_idx=32, Z0=50.0)
print(f"Active impedance: {Z_active.real:.1f} + j{Z_active.imag:.1f} ohms")

# Compare with nominal 50 ohms
# Active impedance varies with scan angle and position in array

# Get active impedance for all elements at a scan angle
Z_all = pa.active_scan_impedance_matrix(
    geom, C, k,
    theta_deg=30, phi_deg=0,
    Z0=50.0
)
print(f"Impedance range: {Z_all.real.min():.1f} to {Z_all.real.max():.1f} ohms (real)")

VSWR vs Scan Angle

VSWR is a critical metric that indicates how well elements remain matched as the array scans. High VSWR indicates potential scan blindness or poor matching conditions.

# Compute VSWR for all elements vs scan angle
theta_deg, vswr_all, vswr_max = pa.vswr_vs_scan(
    geom, C, k,
    theta_range=(0, 60),
    n_angles=31,
    phi_deg=0.0
)

# Find maximum VSWR at each scan angle
for i, theta in enumerate(theta_deg[::5]):  # Every 5th angle
    print(f"Scan {theta:.0f} deg: Max VSWR = {vswr_max[i*5]:.2f}:1")

# Plot VSWR vs scan angle
import matplotlib.pyplot as plt
plt.plot(theta_deg, vswr_max)
plt.xlabel('Scan Angle (deg)')
plt.ylabel('Maximum VSWR')
plt.title('VSWR vs Scan Angle')
plt.grid(True)
plt.axhline(y=2.0, color='r', linestyle='--', label='2:1 VSWR spec')
plt.legend()

VSWR guidelines:

VSWR	Reflection Loss	Interpretation
1.0:1	0.0 dB	Perfect match
1.5:1	0.2 dB	Excellent
2.0:1	0.5 dB	Good (typical spec)
3.0:1	1.2 dB	Marginal
>5:1	>2.5 dB	Potential scan blindness

Mismatch Loss

# Compute mismatch loss from reflection coefficient
gamma = 0.333  # Corresponds to 2:1 VSWR
loss = pa.mismatch_loss(gamma)
print(f"Mismatch loss at 2:1 VSWR: {loss:.2f} dB")  # ~-0.5 dB

# Array of reflection coefficients
gamma_array = np.array([0.0, 0.1, 0.2, 0.333, 0.5])
loss_array = pa.mismatch_loss(gamma_array)
for g, l in zip(gamma_array, loss_array):
    vswr = (1 + g) / (1 - g)
    print(f"Gamma={g:.2f}, VSWR={vswr:.1f}:1, Loss={l:.2f} dB")

Edge Effects

Elements near the array edges typically have different active impedance than interior elements due to the asymmetric coupling environment:

# Compare edge and center element active impedance
Z_center = pa.active_impedance(C, weights, element_idx=27, Z0=50.0)  # Center
Z_corner = pa.active_impedance(C, weights, element_idx=0, Z0=50.0)   # Corner
Z_edge = pa.active_impedance(C, weights, element_idx=3, Z0=50.0)     # Edge

print(f"Center element Z: {Z_center.real:.1f} + j{Z_center.imag:.1f}")
print(f"Edge element Z: {Z_edge.real:.1f} + j{Z_edge.imag:.1f}")
print(f"Corner element Z: {Z_corner.real:.1f} + j{Z_corner.imag:.1f}")

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.

6. Check VSWR across the full scan range to identify potential blind spots.

7. Account for edge effects when specifying element matching requirements.