Transformer Magnetic Inrush Current: Closing Angle, Residual Flux, Core Saturation, Second Harmonic Restraint & Duration
Abstract
When a power transformer is energized, the transient magnetizing current — the inrush current — can exceed the rated full-load current by 6-12 times, lasting from several cycles to over one second. This current waveform is highly asymmetric (unipolar) and rich in even harmonics, particularly the second harmonic. If not properly discriminated by the differential protection relay, inrush current will cause a false trip on energization, delaying restoration and creating operational confusion. This article explains the physics of inrush current from first principles: the role of the closing angle on the voltage waveform, the contribution of residual core flux (remanence), the mechanism of core saturation that produces the current spike, the second harmonic content as the basis for the industry-standard restraint criterion, and the factors that determine inrush duration. Practical guidance is provided for relay setting engineers facing the challenge of low-second-harmonic transformers (amorphous core, domain-refined steel).
1. The Physics of Inrush Current
1.1 Steady-State Magnetization
Under normal steady-state operation, the applied voltage V(t) and the core flux Φ(t) are in quadrature:
V(t) = V_m × sin(ωt)
Φ(t) = (1/N) ∫ V(t) dt = -Φ_m × cos(ωt) + C
Where:
- V_m = peak applied voltage
- Φ_m = V_m / (ω × N) = peak steady-state flux
- C = integration constant = 0 in steady state (no DC flux component)
The peak flux Φ_m is designed to be approximately 80-90% of the core saturation flux (Φ_sat). For a transformer designed with B_max = 1.7 T operating voltage, the saturation flux density B_sat ≈ 2.0 T, giving:
Φ_m / Φ_sat ≈ 1.7 / 2.0 = 0.85
At 1.0 pu voltage, the core operates in the linear region of the B-H curve, well below saturation.
1.2 Energization Transient
Consider the transformer energized at an arbitrary point on the voltage wave: V(t) = V_m × sin(ωt + θ), where θ is the closing angle (θ = 0 corresponds to energizing at voltage zero-crossing).
The flux after energization is:
Φ(t) = -Φ_m × cos(ωt + θ) + Φ_m × cos(θ) × e^(-t/τ) + Φ_residual
Where:
- The first term (-Φ_m × cos(ωt + θ)) is the steady-state AC flux
- The second term (Φ_m × cos(θ) × e^(-t/τ)) is the DC transient flux, decaying with the time constant τ = L_m/R_winding
- The third term (Φ_residual) is any residual flux remaining in the core from prior de-energization
1.3 Worst-Case Scenario
The worst-case inrush occurs when:
- The transformer is energized at voltage zero-crossing (θ = π/2): Then cos(θ) = 0 → the DC transient term is Φ_m (maximum). The peak flux during the first half-cycle reaches 2Φ_m + Φ_residual.
- The core has maximum residual flux (Φ_residual = +Φ_m × BR/B_max): After de-energization at a voltage peak (current zero, no arcing), the flux remains at the remanence value. For grain-oriented silicon steel, BR ≈ 1.0-1.5 T (60-85% of B_max), so Φ_residual ≈ 0.6-0.85 × Φ_m.
Worst-case peak flux: Φ_peak = 2Φ_m + 0.85Φ_m = 2.85 × Φ_m
Since Φ_m corresponds to B_max = 1.7 T, and saturation begins at B_sat ≈ 2.0 T:
Φ_peak / Φ_m = 2.85 → instantaneous flux density ≈ 4.8 T
The core saturates massively at B ≈ 2.0 T. The incremental permeability drops to near μ₀, and the inrush current is limited only by the air-core reactance of the winding plus the system impedance. This saturation current can reach 6-12× rated current.
2. Factors Affecting Inrush Magnitude and Duration
2.1 Closing Angle
The inrush current as a function of closing angle is:
| Closing Angle (θ) | cos(θ) | DC Flux Component | Relative Inrush Peak |
|---|---|---|---|
| 0° | 1.0 | Φ_m (Maximum DC) | Maximum inrush (6-12× I_rated) |
| 30° | 0.866 | 0.866 × Φ_m | Moderate inrush (3-6×) |
| 60° | 0.5 | 0.5 × Φ_m | Low inrush (1-3×) |
| 90° (voltage peak) | 0 | 0 | Minimal (≈ rated magnetizing current, 0.5-2% I_rated) |
Controlled switching (point-on-wave closing): Modern circuit breakers with controlled switching can close each phase independently at the voltage peak (θ = 90° per phase), virtually eliminating inrush current. This requires:
- A circuit breaker with independent pole operation (IPO) and a closing time scatter <±1 ms
- Knowledge of the residual flux in each core limb (requires flux measurement or estimation from the de-energization waveform)
- A controller that commands closing at the optimal angle for each phase
2.2 Residual Flux (Remanence)
After the transformer is de-energized, the core flux does not decay to zero — it remains at the remanence value:
| Core Material | B_max (T) | B_R (T) | B_R / B_max | Inrush Amplification |
|---|---|---|---|---|
| Hot-rolled steel (historical) | 1.4-1.5 | 1.0-1.3 | 70-90% | Highest |
| Conventional CGO steel | 1.7-1.8 | 1.0-1.4 | 60-80% | High |
| Hi-B (domain-refined) steel | 1.7-1.9 | 0.9-1.3 | 50-70% | Moderate |
| Amorphous metal | 1.5-1.6 | 0.3-0.5 | 20-30% | Low |
The lower remanence of amorphous metal cores is an advantage for inrush — the peak flux is reduced (2Φ_m + 0.3Φ_m = 2.3Φ_m vs. 2.85Φ_m for CGO steel). However, the disadvantage is that the inrush current waveform for amorphous cores contains less second harmonic (see Section 3), making it harder for differential relays to discriminate from internal faults.
2.3 Inrush Duration
The DC transient flux decays with the time constant:
τ = L_m / R_winding
Where L_m is the unsaturated magnetizing inductance (large, typically 100-1,000 H for a power transformer) and R_winding is the primary winding resistance (small, typically 0.1-2 Ω).
Typical τ = 1-10 seconds for large power transformers. The inrush current decays slowly because the DC component has a long time constant — the winding resistance in the primary is small compared to the large magnetizing inductance.
Inrush envelope over time (approximate):
| Time After Energization | Inrush Current (% of Peak) |
|---|---|
| 0 (first peak) | 100% |
| 0.1 s (5 cycles at 50 Hz) | 80-90% |
| 0.5 s (25 cycles) | 50-70% |
| 1.0 s (50 cycles) | 35-50% |
| 5.0 s | 10-20% |
| 10.0 s | 2-5% |
3. Second Harmonic Restraint — The Standard Discrimination Criterion
3.1 Harmonic Content of Inrush Current
The asymmetric, unipolar inrush current waveform is rich in even harmonics. The harmonic content (as a percentage of fundamental) for a typical power transformer with CGO steel core:
| Harmonic | Frequency (50 Hz system) | Typical Content (% of fundamental) |
|---|---|---|
| Fundamental | 50 Hz | 100% |
| 2nd | 100 Hz | 15-65% |
| 3rd | 150 Hz | 10-25% |
| 4th | 200 Hz | 5-15% |
| 5th | 250 Hz | 3-8% |
| DC (decaying) | 0 Hz | 40-80% |
The second harmonic is the most discriminating because: (1) it is the strongest even harmonic, (2) internal faults produce negligible second harmonic (transformer internal faults are typically symmetric or near-symmetric), and (3) CT saturation during an external fault also produces harmonics, but the third harmonic rather than the second harmonic tends to dominate for CT saturation.
3.2 Relay Setting
The second harmonic restraint setting is applied as:
I_2nd / I_fundamental > Setting → Block differential trip (classify as inrush)
Standard settings:
- 15%: Standard setting for transformers with conventional CGO steel cores. Provides reliable inrush discrimination in 95%+ of energizations.
- 20%: Conservative setting. Reduces the risk of false tripping on inrush but may delay tripping on internal faults if CT saturation adds second harmonic.
- 10%: Aggressive setting for low-loss core transformers (amorphous, Hi-B) that produce low second harmonic content. Risk: increased probability of maloperation on inrush.
Cross-blocking: In some relays, the detection of second harmonic in ANY phase blocks tripping for ALL phases (cross-blocking). This is conservative — inrush may not be present in all three phases at every energization, but a single inrush-detected phase blocks the entire relay. Alternative: phase-segregated second harmonic restraint blocks only the phase where inrush is detected, allowing the other phases to trip on an internal fault that coincides with energization.
3.3 The Low-Second-Harmonic Problem
Transformers with low-loss core materials (domain-refined Hi-B, amorphous metal) produce inrush currents with 7-12% second harmonic content. A standard 15% second harmonic restraint will NOT block tripping for these transformers during energization. This has caused documented false trips at newly commissioned amorphous-core distribution transformers and Hi-B transmission transformers.
Mitigations:
- Lower the second harmonic threshold to 10% (and accept the reduced security margin)
- Use waveform symmetry detection (dead-angle principle) — the inrush current waveform has intervals near current zero where the current is very low (the "dead angle"); internal faults do not have this feature
- Use flux-based restraint — the relay calculates the core flux from the terminal voltage and compares it to the saturation flux; inrush corresponds to flux exceeding saturation; internal fault corresponds to flux within normal range
- Combined 2nd + 5th harmonic restraint — add a fifth harmonic criterion (5th harmonic > 5-10% indicates overexcitation rather than inrush), but this addresses a different phenomenon
FAQ
Q: Why doesn't the inrush current immediately trip the transformer on overcurrent?
Three reasons: (1) the inrush current is typically 6-12× rated current, which is comparable to a through-fault current — but it is flowing into the transformer primary only, not through it (the secondary is open-circuited during energization), so the differential relay sees primary current with zero secondary current and classifies this as an internal fault unless the second harmonic restraint blocks tripping, (2) the magnitude of inrush current is highest at the first peak and decays over seconds — for the first 0.1-0.5 seconds, the current may exceed the pickup setting but is blocked by the harmonic restraint, and (3) the transformer overcurrent protection (backup 50/51) is typically set with a definite-time or inverse-time characteristic that does not trip for the inrush current duration (<1 second).
Q: How can point-on-wave (controlled) switching eliminate inrush current?
Controlled switching closes the circuit breaker contacts at the voltage peak for each phase independently. At the voltage peak, the steady-state flux is zero — so the DC transient flux term is zero. There is no overshoot, no saturation, and the inrush current is identical to the normal magnetizing current (<0.5% of rated). The practical challenge: the residual flux in the core after de-energization is unknown without a flux measurement. If the residual flux is +0.7Φₘ and the steady-state flux at closing is zero, the initial flux is +0.7Φₘ — well within the core's linear range. But if the closing is mistimed and the flux adds to the residual, inrush can still occur. The most sophisticated controlled switching systems measure the residual flux from the de-energization voltage integral and calculate the optimal closing angle for each phase.
Q: Does inrush current cause any damage to the transformer?
Inrush current does not directly damage the transformer — the duration is too short (<10 seconds) to cause significant thermal heating (I²t is modest). However, repetitive inrush (multiple energization attempts during commissioning, auto-reclose sequences) can cause cumulative mechanical stress on the windings — the electromagnetic forces during inrush are proportional to I², so a 10× inrush applies 100× the rated-current force (but much less than a through-fault because the inrush is limited to the magnetizing branch and does not correspond to the full leakage-flux forces of a short-circuit). Inrush can also cause: (1) nuisance fuse blowing on distribution transformers if the fuse I²t rating is too tight, (2) voltage dip on the system — a large transformer drawing 5× rated inrush current from a weak grid can cause a 5-10% voltage sag for 0.5-1.0 seconds, and (3) sympathetic inrush — when one transformer is energized in parallel with an already-energized transformer, the inrush current of the incoming transformer flows through the already-energized transformer as a DC component, causing it to also saturate.
Q: What is sympathetic inrush?
Sympathetic inrush occurs when an energized transformer is connected in parallel with a transformer being energized. The inrush current of the incoming transformer — which contains a large DC component — flows through the system impedance and the winding of the already-energized transformer. The DC voltage drop across the already-energized transformer's winding resistance drives its core toward saturation. The result: both transformers experience inrush simultaneously, with the already-energized transformer's inrush typically being 1-3× rated and lasting longer than normal because the DC source (the incoming transformer's inrush time constant) decays slowly. Sympathetic inrush can cause false differential tripping of the already-energized transformer if its relay is not equipped with second harmonic restraint, or if the second harmonic content is below the threshold. Mitigation: second harmonic restraint on both transformers' differential relays, and sequential energization (energize one transformer, wait for inrush to decay, then energize the second).
Q: Why is the inrush current so much higher for the first half-cycle compared to subsequent cycles?
The first half-cycle after energization at voltage zero-crossing produces the maximum flux excursion (DC transient = Φ_m + steady-state AC oscillating = 2Φ_m peak). For the second half-cycle, the DC transient has decayed slightly (by e^(-0.01/τ)), reducing the peak flux to approximately 1.9Φ_m. The current is extremely nonlinear with flux — once the core saturates, a small reduction in peak flux produces a large reduction in peak current because the incremental permeability above saturation is very low (near μ₀), and the current is driven by the "excess flux" above saturation. This is why the inrush current decays from 100% to ~80% between the first and second peaks.
Q: Should I use cross-blocking or phase-segregated second harmonic restraint?
Cross-blocking (inrush detected in any phase → block trip on all phases) is the standard for most applications. It maximizes security (avoids false tripping) at the cost of a slight delay in tripping on an internal fault if one phase has a genuine fault and another phase has inrush second harmonic — but this scenario is rare because three-phase transformer energization produces inrush in two or three phases simultaneously, not just one. Phase-segregated blocking is used for: (1) single-phase transformer banks where each phase is independently energized, (2) applications where a simultaneous internal fault and energization is a credible scenario (e.g., energizing onto a fault after maintenance), and (3) where the grid code requires fastest possible fault clearance. The choice is a trade-off between security (cross-blocking) and speed (phase-segregated).
References / Standards
| Reference | Title |
|---|---|
| IEC 60076-1:2011 | Power transformers — Part 1: General |
| IEC 60255-187-1:2021 | Measuring relays and protection equipment — Part 187-1: Functional requirements for differential protection |
| IEEE C37.91-2021 | IEEE Guide for Protecting Power Transformers |
| IEEE C37.102-2023 | IEEE Guide for AC Generator Protection |
| CIGRE TB 419 | Protection of Power Transformers |
*Authored by Du Fu, Production Engineer at ZY POWER. Inrush current discrimination is the most fundamental challenge in transformer differential protection — every relay setting engineer must understand the physics and the limitations of second harmonic restraint, especially for modern low-loss core transformers.*
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