Circuit Breaker Interruption · Training Guide
A companion to the TRV guide. Where the first guide explained what transient recovery voltage means in practice, this page explains the calculation concepts behind it — current interruption at current zero, the current-injection method, inductive interruption and stray capacitance, transformer-limited and short-line faults, line surge impedance, and the first pole to clear — so the engineer can understand why a TRV has its shape, and why some switching duties are more severe than others.
Section 1
The first TRV guide explained the practical meaning of transient recovery voltage — RRRV, reignition, restrike, TRV envelopes, EMT studies and mitigation. This companion page explains the calculation concepts behind TRV.
The objective is not to replace EMT simulation. It is to help the engineer understand why the TRV has a particular shape, and why some switching duties are more severe than others.
Section 2
TRV is normally calculated as a prospective network stress. The breaker is represented as an ideal switch opening at current zero, and the detailed arc physics inside the breaker is not normally modelled. Instead, the network response after current interruption is calculated, and the voltage across the open contacts is compared with the breaker’s tested TRV capability.
The breaker succeeds if its dielectric recovery is faster and stronger than the recovery voltage imposed by the network.
Section 3
When a circuit breaker receives a trip command, the contacts start to separate mechanically — but the current does not stop immediately. An arc remains between the contacts, and current continues through the ionised path until a current zero is reached.
Section 4
The voltage across the breaker contacts after interruption can be considered as the sum of a steady-state and a transient component:
The transient component appears immediately after current interruption and is governed by the energy exchange between inductances, capacitances and travelling waves. The later steady-state component is the power-frequency recovery voltage set by the post-fault network condition. The engineer must check the early transient waveform against the breaker TRV envelope.
Section 5
The current-injection method is a useful way to calculate TRV. Before opening, the breaker carries current \(i(t)\); at current zero it opens. Opening the breaker can be represented by injecting an equal and opposite current into the post-opening network, which cancels the current that had been flowing through the breaker. The network response to this injected current produces the voltage across the open breaker contacts.
The post-opening network can be treated as a linear network, so the voltage across the open contacts can be calculated using superposition. If the two breaker terminals are side 1 and side 2, the recovery voltage is their difference:
Each side may have a different transient response, so the total TRV is not always created by one side only — it is the difference between both terminal voltages.
Section 6
Power-system fault circuits are usually predominantly inductive, and in an inductive circuit the current lags the voltage. So when the current reaches zero, the source voltage may be close to its maximum value — creating a severe recovery voltage immediately after interruption. In a purely inductive ideal circuit the voltage across the breaker would rise instantaneously after current zero, implying an infinite initial RRRV. In practice this does not happen, because real equipment and buswork have stray capacitance.
Section 7
Near current zero, the sinusoidal current can be approximated by a ramp:
This is useful because TRV is mainly concerned with the first short interval after current zero, where the ramp approximation is accurate and practical. The steeper the current ramp, the higher the initial voltage rise can be — so the fault-current magnitude and frequency directly influence the RRRV.
Section 8
Every substation has stray capacitance — from transformer windings to tank, bushings, circuit-breaker grading capacitors, buswork, current transformers, voltage transformers, and connected cables or lines. It is important because it prevents the voltage from changing instantaneously. Without capacitance, a simplified inductive circuit predicts an unrealistically sharp voltage rise; with capacitance included, the TRV becomes oscillatory or damped. A simple LC response can be expressed conceptually as:
This is the classical “one-minus-cosine” TRV waveform.
Small capacitance \(\Rightarrow\) high natural frequency \(\Rightarrow\) high RRRV.
Section 9
A transformer-limited fault is an important TRV duty. Here the transformer leakage reactance may dominate the fault-current limitation, while the stray capacitance at the breaker terminal may be small. This combination can produce a relatively limited fault current together with a high natural frequency, a steep TRV and a high RRRV. So a transformer-limited fault may need special attention even when its current is not the maximum possible short-circuit current.
A lower-current fault can still be severe if the associated capacitance is small and the voltage rises very quickly — the rate of rise, not just the peak, decides the duty.
Section 10
The voltage across the open breaker is the difference of the two terminal voltages:
Each side may have its own inductance, capacitance and natural frequency, so the total TRV can contain contributions from both. For example, the source side may produce a damped or oscillatory recovery voltage, the line side may produce a travelling-wave response, and the total TRV is the difference between the two. This is why EMT simulation is often required for complex substations.
Section 11
For high-frequency TRV behaviour, a transmission line should not always be represented by a simple nominal-frequency impedance. Before reflections return from the remote end, a long line behaves like its surge impedance \(Z\), so during the early TRV interval the line can often be represented by \(Z\) alone. This matters especially for short-line faults.
Before the reflected waves return \(\Rightarrow\) the surge-impedance model is appropriate. After reflections return, the full travelling-wave behaviour must be considered.
Section 12
In a three-phase circuit breaker, the three poles do not interrupt at exactly the same electrical condition, and the first pole to clear can experience the most severe recovery voltage. For a three-phase ungrounded fault, the fault point can shift in potential after the first pole opens, which increases the voltage across the first open pole — the first-pole recovery voltage can approach \(1.5\) times the phase voltage:
For grounded three-phase faults the recovery voltage is generally lower, because the fault point is fixed closer to ground potential. The practical message: an ungrounded three-phase fault gives a higher first-pole recovery voltage.
Section 13
Three-phase faults are often used for standard breaker duties, but a single-phase-to-ground fault can also be critical — particularly where the zero-sequence impedance is low, the single-phase fault current is high, the fault is close to the substation, or the connected line produces severe travelling-wave behaviour.
The engineer should not assume that the three-phase fault is always the worst TRV case.
Section 14
A short-line fault occurs when an overhead-line fault is located a short distance from the circuit breaker. It is one of the most severe TRV duties, because the line-side travelling wave can create a very steep voltage rise. The short-line-fault TRV has two components — a source-side TRV, checked against the standard terminal-fault envelope, and a line-side TRV, checked using the short-line-fault RRRV requirement. The line-side rise is strongly controlled by the surge impedance of the faulted line, \(Z_{\ell}\):
Section 15
The simplified equations are useful for understanding the physical behaviour, but practical TRV studies may involve several lines of different lengths, transformers and reactors, cables, busbar capacitance, grading capacitors, surge arresters, different fault types, different opening sequences, and damping or frequency-dependent parameters. For these cases EMT simulation is preferred — it calculates the voltage on each side of the breaker, the total TRV, the RRRV, the peak TRV, the effect of line reflections, the effect of capacitance and damping, and the comparison with breaker envelopes.
Section 16
TRV is not only a peak-voltage problem — it is a network response problem after current interruption. Current interruption occurs at current zero; the breaker opening can be represented by current injection into the post-opening network; inductance creates the recovery voltage while capacitance controls the initial rate of rise; small capacitance can produce high RRRV; transformer-limited faults can be severe even when the current is not maximum; both sides of the breaker can contribute to the TRV; transmission lines behave as surge impedances during the early TRV interval; the first pole to clear can see the highest recovery voltage; and short-line faults can produce very high RRRV.
TRV severity depends on current, voltage, capacitance, inductance, line surge impedance, fault type and breaker pole sequence. A breaker is acceptable only when the calculated TRV waveform remains below its rated capability envelope for the relevant duty.
This page is the calculation companion to the Transient Recovery Voltage training guide, which covers the practical meaning of TRV, RRRV, reignition, restrike, the TRV envelopes and mitigation.
Section 17
Section 18
| Symbol | Meaning |
|---|---|
| TRV / RRRV | Transient recovery voltage / rate of rise of recovery voltage |
| \(\Delta v\) | Voltage across the open breaker contacts |
| \(\Delta v_s,\ \Delta v_t\) | Steady-state and transient (TRV) components of the recovery voltage |
| \(v_1,\ v_2\) | Transient voltages on the two breaker terminals |
| \(i(t),\ I_0\) | Instantaneous current; crest of the interrupted current |
| \(k,\ \omega\) | Current slope at zero (\(k=\omega I_0\)); angular power-frequency |
| \(v_c,\ E_0,\ \omega_0\) | LC response voltage; driving voltage; natural frequency \(1/\sqrt{LC}\) |
| \(L,\ C\) | Circuit inductance; stray capacitance |
| \(Z,\ Z_{\ell}\) | Surge impedance of a line; surge impedance of the faulted (short-line) line |
| \(V_{\text{phase}}\) | Phase (line-to-ground) voltage |
Two-Part Technical Series
A two-part guide to transient recovery voltage — Part One explains what TRV is and how to read it (the interruption race at current zero, reignition and restrike, waveforms, the breaker envelope and mitigation); Part Two develops the calculation concepts (current injection, inductive interruption and stray capacitance, transformer-limited and short-line faults, line surge impedance and the first pole to clear).
The calculation concepts behind the shape — current injection, inductive interruption and stray capacitance, transformer-limited and short-line faults, line surge impedance and the first pole to clear.
