Array Phase-Error Budget in Phased Arrays and RIS Metasurfaces

Ten degrees RMS random phase error costs only about 0.13 dB gain. Yet a 10° phase slope can move a λ/2 beam significantly.

That contrast exposes the real calibration problem.

Phase error is not defined only by magnitude. Its spatial correlation, frequency dependence, temperature dependence, and origin matter equally.

That is why one RMS phase number can hide serious array weaknesses.

Start With the Correct Random-Phase Model

For independent, zero-mean Gaussian phase errors:

εₙ ~ N(0, σφ²)

the coherent field expectation is reduced approximately by:

E[e^{jε}] = e^{−σφ²/2}

Therefore, coherent power approaches:

ηφ ≈ e^{−σφ²},

for a sufficiently large uniform array.

A detailed probabilistic treatment gives this same exponential phase-error dependence. [Ref. 1]

The corresponding gain penalty is:

Lφ = 10log₁₀(ηφ); with σφ expressed in radians.

The Familiar RMS Numbers

Using the large-array approximation:

• 10° RMS → −0.13 dB

• 20° RMS → −0.53 dB

• 30° RMS → −1.19 dB

• 45° RMS → −2.68 dB

At first glance, 10° RMS looks almost harmless.

That interpretation is dangerous. Peak gain is often the least sensitive array metric.

There Is Also a Finite-Array Correction

For N equal-amplitude elements, phase-only Gaussian errors give:

ηφ,N ≈ e^{−σφ²} + [1−e^{−σφ²}]/N

This follows from the expected power-pattern expression for excitation errors. [Ref. 1]

For large N, the second term becomes negligible.

For smaller arrays, the classical exponential slightly overstates coherent loss.

This nuance matters in compact Radar and SATCOM subarrays.

But Random Phase Error Is Only One Error Class

This is the central engineering point.

A real phase-error budget contains:

• random phase noise

• systematic phase offsets

• spatial phase gradients

• quantisation residuals

• frequency-dependent errors

• temperature-driven drift

• calibration uncertainty

These error classes do not produce identical radiation consequences. They should not automatically share one RMS bucket.

A Spatial Gradient Can Be Far More Dangerous

For a uniform linear array:

Δφ = 2π(d/λ)sinθ

Analog Devices uses this fundamental steering relationship. [Ref. 2], [Ref. 3]

With d = λ/2, the expression becomes:

Δφ = πsinθ

Now introduce a systematic 10° phase increment error per element, then:

sinθerror = 10° / 180°

giving approximately: θerror ≈ 3.2°.

That is a large pointing error. Yet 10° random RMS error loses only around 0.13 dB.

Same order of phase magnitude, but completely different system consequence.

This Is Why RMS Alone Can Mislead

Random uncorrelated phase errors mainly reduce coherent summation.

A linear phase gradient primarily moves the beam.

A periodic error can create parasitic lobes.

A local element error can disturb sidelobes asymmetrically.

A common phase offset may change almost nothing spatially.

Therefore, phase-error topology matters as much as phase-error magnitude.

Key Point 1: Sidelobes Reveal Errors Before Peak Gain Does

Random excitation errors redistribute energy across the radiation pattern.

IEEE analysis shows phase errors reduce ideal-pattern power. They also create an approximately angle-independent error contribution. [Ref. 1]

That additional energy raises the sidelobe floor. The main beam can still look surprisingly healthy.

A Radar Measurement Demonstrates the Difference

A calibrated X-band phased-array weather Radar achieved major sidelobe improvement. Amplitude and phase correction suppressed sidelobes by over 40 dB. It also improved suppression by over 10 dB versus uncorrected adaptive DBF. [Ref. 4]

That result matters more than another tenth-decibel boresight gain.

For Radar, spatial dynamic range often exposes calibration weakness first.

Peak Gain Can Therefore Pass While Radar Performance Fails

Suppose an array has σφ = 10° RMS.

Peak gain penalty is roughly: 0.13 dB.

That looks excellent. But a raised sidelobe floor can increase:

• clutter susceptibility

• jammer sensitivity

• angular ambiguity

• false detections

So a gain-only acceptance test can miss the real failure.

The Same Applies to Null Depth

Nulls require near-perfect destructive interference. A small residual phasor can dominate a deep theoretical null.

That makes null depth highly sensitive to:

• phase residual

• amplitude residual

• mutual coupling

• channel drift

Adaptive arrays therefore need tighter calibration than gain-only arrays.

A Useful Error-Floor Approximation

For random amplitude and phase errors, IEEE analysis gives:

SLLerror ∝ (σa² + σφ²)/(NηA)

under the stated small-error assumptions. [Ref. 1]

Here:

• σa = RMS fractional amplitude error

• σφ = RMS phase error in radians

• ηA = aperture taper efficiency

The key message is important: Amplitude and phase errors jointly create a sidelobe floor.

A “phase-only” calibration strategy may therefore remain insufficient.

A Hardware Research Reinforces That Warning

A multichannel impairment model examined numerous beamforming error sources. It included phase-shifter errors, timing mismatches, and I/Q imbalance. [Ref. 5]

Its Monte Carlo study found sidelobes especially sensitive to gain errors. EVM was dominated more strongly by timing errors.

So one phase budget cannot describe every system metric.

That is the uncomfortable engineering truth.

Key Point 2: Timing Error Becomes Phase Error Extremely Fast

Phase error from timing mismatch is: Δφ = 2πfΔt.

This becomes unforgiving at mmWave.

Consider:

Δt = 1 ps

At:

10 GHz → 3.6°

28 GHz → 10.1°

30 GHz → 10.8°

77 GHz → 27.7°

One picosecond can therefore consume an entire phase budget.

Even 100 fs Becomes Relevant

At 30 GHz, 100 fs produces approximately: 1.08°.

At 77 GHz, 100 fs produces approximately: 2.77°.

This connects array calibration directly to:

• LO distribution

• clock synchronization

• package delay

• PCB path matching

• converter timing

Phase calibration is therefore partly a timing problem.

Sub-THz Hardware Makes This Even Harder

A 300 GHz transmitter required explicit path-mismatch calibration. Calibration improved measured EVM by approximately 3.3 dB. [Ref. 6]

The system operated between approximately 237 and 250 GHz. At those frequencies, tiny timing mismatches create enormous phase rotations.

This is where packaging becomes part of the calibration architecture.

PCB Material Variation Also Enters the Phase Budget

Transmission-line phase is:

φ = βl

where:

β ≈ 2πf√εeff / c

Therefore:

Δφ ≈ βΔl + lΔβ

and approximately:

Δβ/β ≈ ½·Δεeff/εeff

This connects phase error directly to:

• Dk tolerance

• processed dielectric thickness

• copper roughness

• routing-length mismatch

The phase-shifter IC is only one contributor.

A 6-Bit BFIC Cannot Repair Unlimited PCB Error

Suppose the BFIC phase LSB is 5.625°.

But PCB uncertainty contributes 8°.

Thermal drift contributes another 6°.

Then finer digital resolution alone cannot solve the system. The array is no longer phase-quantisation limited.

It is hardware-uncertainty limited.

Current Wideband Hardware Shows How Small IC Error Can Become

A 2–18 GHz phased-array receiver achieved RMS phase errors of 0.4° to 3.3° across its operating band. [Ref. 7]

A 18–40 GHz phase shifter reported simulated RMS error 1.1° to 2.0° across its band. [Ref. 8]

Those are strong component-level numbers. But system calibration must still include PCB and antenna effects.

Key Point 3: Temperature Makes Calibration Multidimensional

An array calibrated at 25°C is not permanently calibrated.

RF component characteristics change with temperature.

Phase paths also drift across the physical aperture.

Modern calibration research explicitly targets frequency and temperature dependence. [Ref. 9]

That means the correction table should look more like:

Cₙ = Cₙ(f, T, beam, power)

not:

Cₙ = constant

A Current 16-Channel Hardware Benchmark

Analog Devices demonstrated self-contained X-band calibration in 2026. The 16-channel platform reached roughly 23 dB transmit combining gain. Receive SNR improved by approximately 12 dB after coherent combination. [Ref. 9]

The workflow builds correction tables across frequency and temperature.

This is what practical calibration increasingly looks like.

A K-Band Array Uses RMS Error Explicitly

A 24-channel K-band array incorporated onboard orthogonal calibration. The system covers approximately 18–20 GHz. It steers approximately −40° to +40°. [Ref. 10]

Its calibration assessment uses RMS amplitude and phase errors. Those residual errors are then used for further correction.

This makes calibration measurable instead of qualitative.

But RSS Error Budgets Need One Critical Qualification

Engineers often write:

σtotal = √(σ₁² + σ₂² + σ₃² + …)

That is valid only when error terms are sufficiently independent. It is not valid for every array impairment.

This distinction is essential.

Independent Errors Can Usually Be RSS Combined

Examples may include:

• residual random quantisation error

• uncorrelated component spread

• independent measurement noise

• local random phase noise

Then, σrandom² = Σσi², is a useful engineering approximation.

Correlated Errors Must Remain Separate

Do not RSS these blindly:

• temperature gradients

• deterministic PCB length gradients

• LO distribution phase slope

• mutual-coupling bias

• calibration reference-plane bias

• frequency-dependent Dk error

These terms contain spatial structure. A covariance or deterministic vector model is more appropriate.

The Better Mathematical Model Is a Covariance Matrix

Write the phase-error vector:

ε = [ε₁ ε₂ … εN]ᵀ

Then characterize:

Rε = E[εεᴴ]

Diagonal terms contain individual variances.

Off-diagonal terms capture spatial correlation.

That information is invisible inside one scalar RMS number. Yet it can determine beam pointing and sidelobe structure.

A Simple Phase Budget Example

Assume independent RMS contributions:

• quantisation residual: 6.5°

• BFIC residual: 4°

• PCB/material error: 3°

• random thermal residual: 5°

• measurement uncertainty: 2°

RSS combination gives σrandom ≈ 9.8°.

The corresponding large-array gain penalty is approximately: 0.13 dB.

Again, gain looks excellent. But that conclusion assumes every error is random.

Now Add a 10° Progressive Spatial Bias

Do not add that phase slope by RSS. It is deterministic.

For λ/2 spacing near broadside:

10° per-element slope → approximately 3.2° beam shift

The random error still controls coherence loss. The gradient controls pointing.

This is why phase-error budgeting requires classification first.

RIS Metasurfaces Make the Problem Even More Complicated

An active phased-array channel usually controls amplitude and phase electronically.

A RIS cell often controls a complex reflection coefficient:

Γₙ = ρₙe^{jφₙ}

The amplitude ρₙ may depend on phase state. So a phase error can also become an amplitude error.

The Phase-Only RIS Model Can Be Physically Wrong

Recent RIS research explicitly includes phase-dependent amplitude response. It finds ideal constant-amplitude assumptions unrealistic. [Ref. 11]

A 2026 stacked-metasurface model uses Lorentzian meta-atom dispersion. It explicitly models amplitude-phase tradeoffs in physical elements. [Ref. 12]

Ignoring those effects overestimated sum rate by up to 7.5 bps/Hz.

That is a substantial system-level error.

RIS Calibration Therefore Needs Complex-State Calibration

Do not calibrate only φstate, but also characterize ρstate across:

• frequency

• temperature

• incidence angle

• polarization

• manufacturing spread

The correct RIS state is a complex quantity, not a phase number.

2026 RIS Hardware Already Requires This

A 2026 EuCAP study demonstrated diagnostics and calibration for X-band RIS. It used a fabricated 1-bit transmission-type RIS. [Ref. 13]

The technique performs both amplitude diagnosis and phase calibration. Phase calibration is extracted using space-time modulation measurements.

That is a significant practical direction.

Hardware Degradation Makes RIS Calibration Persistent

An IEEE study explicitly considers RIS component degradation. It notes degraded phase shifters reduce beamforming performance. [Ref. 14]

The proposed calibration estimates both amplitude and phase states. Beamforming then uses the measured discrete-state alphabet.

That is more realistic than assuming ideal manufactured states forever.

Large RIS Apertures Can Amplify Small Hardware Mistakes

More RIS elements increase coherent aperture potential. But more elements also create more impaired states.

IEEE work shows hardware impairment can become increasingly significant. The effect grows as the number of reflected elements increases. [Ref. 15]

So scaling aperture does not remove calibration requirements.

It increases their strategic importance.

For Radar, Phase Error Is an Operational Variable

Manufacturing calibration captures static channel offsets. It does not capture every operating condition. Temperature changes device phase during operation.

IEEE Radar work explicitly identifies online calibration as necessary. [Ref. 16]

This is particularly important for long-duration AESA operation.

For LEO Payloads, Calibration Becomes a Reliability Function

Spaceborne phased arrays face strong temperature excursions. They also face radiation-induced channel degradation.

A 2026 study addresses in-orbit phased-array failure identification directly. Its simulated method achieved up to 93.06% fault-detection accuracy. [Ref. 17]

That moves calibration toward autonomous health monitoring.

For LEO systems, this distinction matters.

For Massive MIMO, Phase Error Becomes a Capacity Problem

Massive MIMO relies on spatial separation between users. Phase errors disturb channel orthogonality.

They can increase:

• inter-user interference

• EVM

• beam leakage

• multi-beam isolation errors

A high peak array gain therefore does not guarantee good multi-user performance.

Open RAN Adds Another Integration Boundary

The antenna and radio architecture may involve multiple vendors. Yet array calibration remains physically coupled.

BFIC state, antenna pattern, PCB phase, and temperature interact. An interface specification cannot repeal electromagnetic coherence.

Calibration ownership must therefore be defined architecturally.

The Strategic Engineering Solution

Top-tier teams should build a structured phase-error budget, not one RMS requirement.

1. Separate Random and Deterministic Errors

Create two categories.

Random:

• phase noise

• measurement noise

• residual quantisation

• uncorrelated channel spread

Deterministic:

• PCB path gradient

• temperature gradient

• beam-state bias

• coupling-induced phase

• systematic calibration offsets

Do not combine them blindly.

2. Define a Metric for Every Error Class

For random error: σφ,RMS

For spatial gradient: dφ/dn

For temperature drift: dφ/dT

For frequency dispersion: dφ/df

For measurement uncertainty: uφ

One scalar cannot describe all of these.

3. Convert Timing Error Into Phase Explicitly

Always calculate: Δφ = 360°·f·Δt.

At mm-Wave, picoseconds matter.

Do this before specifying clock and LO distribution.

4. Budget PCB Material Uncertainty

Use: φ = βl, with extracted or measured εeff.

Include:

• Dk tolerance

• processed thickness

• copper roughness

• routing tolerance

HFSS or CST should use the same fabrication stack-up.

5. Use Measured BFIC State Data

Do not assume ideal phase commands.

Build φ(state,f,T) and 📐 A(state,f,T) from actual hardware.

Then use those states inside array simulation.

6. Model Spatial Correlation

Monte Carlo analysis should include covariance.

Random independent errors are easy.

Thermal gradients and feed-network errors are not.

Preserve their physical spatial structure.

7. Simulate More Than Peak Gain

Monitor:

• realized gain

• pointing error

• sidelobe level

• null depth

• axial ratio

• cross-polarization

• EIRP

• G/T

• EVM

The first failing metric defines the real phase requirement.

8. Calibrate Across Frequency

One correction at f₀ does not guarantee wideband accuracy.

Transmission-line phase is frequency dependent.

BFIC states are also dispersive.

Wideband arrays need frequency-indexed correction tables.

9. Calibrate Across Temperature

Build correction surfaces, not correction constants.

ADI's 2026 calibration work explicitly includes temperature variation. [Ref. 9]

For thermally dense arrays, this is unavoidable.

10. For RIS, Calibrate Complex Reflection States

Store: Γstate(f,T,θinc,pol), not merely φstate.

This captures phase-dependent amplitude. It also exposes defective metasurface cells.

11. Validate With OTA Measurement

Internal calibration verifies electronic paths.

OTA measurement verifies the complete electromagnetic system.

Compare:

• calibrated channel phase

• measured radiation pattern

• predicted pattern

The difference reveals unmodelled antenna-side errors.

12. Keep Measurement Uncertainty Inside the Budget

A calibration system has its own error.

Cable drift contributes phase uncertainty.

Fixture movement contributes error.

Chamber geometry contributes uncertainty.

A phase correction cannot be more trustworthy than its measurement reference.

A Better Engineering Specification

Do not write: “Array phase error < 10° RMS.”

Instead specify:

Random residual phase: <5° RMS

Progressive phase bias: <2° per channel

Temperature drift: <0.1°/°C

Frequency residual: <3° across operating bandwidth

Pointing error: <0.5°

Peak sidelobe: <−20 dBc

Null depth: >30 dB

These values are only illustrative.

The structure is the important part.

The Procurement Implication

When evaluating a phased-array supplier, request:

• RMS phase error

• RMS amplitude error

• phase versus frequency

• phase versus temperature

• channel-to-channel covariance

• calibration repeatability

• calibration frequency

• post-calibration sidelobes

• pointing accuracy

• EIRP versus beam

Do not accept “calibrated” as a sufficient specification.

For RIS Procurement, Ask Even More

Request:

• phase state accuracy

• reflection amplitude per state

• amplitude-phase coupling

• incidence-angle dependence

• polarization dependence

• defective-cell tolerance

• calibration method

• recalibration interval

A RIS metasurface is not an ideal phase-only matrix.

Modern hardware modelling makes that increasingly clear.

The Summary

For independent Gaussian phase errors: ηφ ≈ e^{−σφ²}.

Therefore:

10° RMS → −0.13 dB

20° RMS → −0.53 dB

30° RMS → −1.19 dB

45° RMS → −2.68 dB

Those numbers are useful, but they are not a complete phase-error budget.

A 10° per-element phase slope can shift a λ/2 beam roughly 3.2°while causing a very different gain response.

Radar measurements show calibration can improve sidelobes dramatically. [Ref. 4]

A 2026 phased-array hardware now calibrates across frequency and temperature. [Ref. 9]

RIS research increasingly calibrates both amplitude and phase. [Ref. 13]

So the engineering question is not:

“What is the RMS phase error?”

It is:

“What type of phase error remains, where, and under which conditions?”

Because in advanced beamforming hardware:

peak gain hides errors that sidelobes, nulls, and pointing expose first.