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Overcurrent Protection Theory

IDMT curve equations, time multiplier mathematics, coordination principles, and standards references.

In Detego

See the Overcurrent Protection Guide for how to configure and use this tab in the app. This page covers the mathematical theory.

IDMT Operating Principle

The Inverse Definite Minimum Time (IDMT) characteristic defines the relay operating time as a function of the current multiple $I_r = I_{\text{fault}} / I_s$ . The key property is that operating time decreases as current increases -- an "inverse" relationship. This allows relays at different locations to coordinate by time alone: a relay closer to the fault sees higher current, operates faster, and clears the fault before the upstream relay times out.

Current Multiple (Ir)

The current multiple (also called the plug setting multiple or PSM) is the ratio of the measured fault current to the pickup current:

Current multiple

I_r = \frac{I_{\text{fault}}}{I_s}

Where

$I_{\text{fault}}$ Measured RMS fault current (primary amps)
$I_s$ Pickup (plug) setting current (primary amps)

The relay only starts timing when Ir > 1 (current exceeds pickup). Below pickup, the operating time is infinite -- the relay does not operate.

Unified IDMT Formula

Both IEC and IEEE curves can be expressed with a single unified formula. IEC curves have $B = 0$ , while IEEE curves have a non-zero additive constant $B$ that produces subtly different coordination behaviour at high multiples:

Unified IDMT operating time

t = \text{TMS} \times \left( \frac{A}{I_r^p - 1} + B \right)

Where

$t$ Operating time in seconds
$\text{TMS}$ Time Multiplier Setting -- scales the entire curve vertically
$A$ Curve constant (determines steepness)
$p$ Curve exponent (determines inverseness)
$B$ Additive constant (0 for IEC, non-zero for IEEE)
$I_r$ Current multiple = I_fault / I_s (must be > 1)

For Definite Time (DT) curves: t = TMS regardless of current (flat time). Set TMS = 0 for instantaneous operation.

The denominator $I_r^p - 1$ creates an asymptote at $I_r = 1$ -- operating time approaches infinity as current approaches the pickup value. For Standard Inverse (p = 0.02), the curve is nearly flat at high multiples; for Extremely Inverse (p = 2), the curve drops very steeply, making it suitable for fuse coordination.

Time Multiplier Setting (TMS)

The TMS is a linear scaling factor that shifts the entire curve up or down without changing its shape. Doubling TMS doubles all operating times; halving TMS halves them. This allows fine-tuning of coordination margins between relays:

TMS = 1.0 -- Standard (reference) curve time
TMS = 0.5 -- Half the operating time at every current level
TMS = 0.1 -- Very fast operation (common for high-set stages)

DT Stages

For Definite Time (DT) curves, TMS is the trip time in seconds. Set TMS = 0 for instantaneous operation. A DT stage with TMS = 0.3 trips in exactly 300 ms regardless of how much current exceeds pickup.

IEC 60255 Curves

The IEC 60255-151 standard defines four inverse-time curves plus a definite-time option. The IEC formula has no additive constant ( $B = 0$ ):

t = \text{TMS} \times \frac{A}{I_r^p - 1}

Where

$A, p$ Curve constants (see table below)

Curve	A	p	t @ Ir=2	t @ Ir=10	Typical Application
Standard Inverse (SI)	0.14	0.02	10.02 s	2.97 s	General purpose -- most common worldwide. Good for networks where fault current does not vary significantly with location.
Very Inverse (VI)	13.5	1	13.50 s	1.50 s	Networks where fault current varies significantly with distance to the fault. Better discrimination than SI for variable fault levels.
Extremely Inverse (EI)	80	2	26.67 s	0.81 s	Fuse coordination in distribution feeders. The steep curve closely matches fuse melting characteristics.
Long-Time Inverse (LTI)	120	1	120.0 s	13.33 s	Thermal overload protection. Very slow operation tolerates long overloads without nuisance tripping.
Definite Time (DT)	--	--	TMS	TMS	Flat operating time: $t = \text{TMS}$ regardless of current magnitude. Used for high-set and instantaneous stages.

Why SI Is So Common

The SI curve has the weakest inverse characteristic (p = 0.02 means the curve is nearly flat). This makes it forgiving of CT errors, fault level uncertainty, and infeed effects. For most radial distribution networks, SI provides adequate discrimination with the simplest coordination study.

IEEE C37.112 Curves

The IEEE C37.112 standard defines three inverse-time curves using the unified formula with a non-zero additive constant $B$ . The IEEE curves produce slightly different coordination behaviour at high current multiples compared to IEC curves.

t = \text{TMS} \times \left( \frac{A}{I_r^p - 1} + B \right)

Where

$B$ Additive constant -- produces a non-zero minimum operating time even at very high Ir

Curve	A	B	p	IEC Equivalent
Moderately Inverse	0.0515	0.114	0.02	Similar to IEC SI, but with a finite minimum time at high currents
Very Inverse	19.61	0.491	2	Similar to IEC EI, with steeper drop and different asymptotic behaviour
Extremely Inverse	28.2	0.1217	2	Very steep inverse -- well-suited for fuse coordination on US distribution

IEC vs IEEE

The key difference is the additive constant B. IEC curves approach zero operating time as current increases indefinitely, while IEEE curves approach a finite minimum time of

\text{TMS} \times B

. In practice, this difference is small because physical relays have inherent minimum operating times (~20-50 ms) regardless of the mathematical curve.

Curve Comparison

The diagram below shows all seven IDMT curves at TMS = 1.0 on a log-log scale. The X-axis is the current multiple $I_r$ and the Y-axis is the operating time in seconds. Key observations:

All curves share the same asymptote at $I_r = 1$ (vertical dashed line at pickup)
SI and IEEE MI are nearly flat -- operating time varies little with current
EI, IEEE VI, and IEEE EI are very steep -- operating time drops dramatically with increasing current
LTI is the slowest curve at all current levels -- used for thermal protection

IDMT curve family at TMS = 1.0. IEC curves (solid) follow t = TMS × A / (I_r^p − 1). IEEE curves (dashed) add a constant B: t = TMS × (A / (I_r^p − 1) + B). All curves approach infinity as I_r → 1 (the pickup asymptote).

CT Ratio Scaling

CT ratio scaling

I_{\text{primary}} = I_{\text{measured}} \times \frac{\text{CT Primary}}{\text{CT Secondary}}

Where

$I_{\text{measured}}$ RMS current computed from the COMTRADE sample data
$I_{\text{primary}}$ Primary system current used for pickup comparison

Example: 1000/1 CT with measured 4.5 A secondary -> 4500 A primary.

Residual Current Computation

The residual current (also called neutral current or zero-sequence current) is the vector sum of all three phase currents. In a balanced system, the residual current is zero. During an earth fault, the faulted phase carries additional current that returns through earth, creating a non-zero residual.

Residual current

3I_0[n] = I_A[n] + I_B[n] + I_C[n]

Where

$3I_0[n]$ Residual current at sample n (instantaneous)
$I_A, I_B, I_C$ Phase current samples at sample n

The residual current is computed sample-by-sample, then RMS-processed like a phase current. This ensures the earth fault stages see the same measurement as a real IDMT relay connected to a residual CT.

Operating Point Evaluation

Detego evaluates protection operating points by measuring the RMS fault current at a specific time and computing the IDMT operating time for each configured stage.

Measurement Time Selection

The operating point is evaluated at the measurement time, which is determined in this order of priority:

Cursor position -- If you click on the RMS Trend chart or set the cursor elsewhere in the viewer, operating points are computed at the cursor time.
Post-inception (auto) -- If no cursor is set, Detego uses the auto-detected fault inception time plus 2 fundamental cycles. The 2-cycle delay allows the RMS computation to settle into the fault-level value.
Recording midpoint (fallback) -- If no fault is detected, the midpoint of the recording is used.

Per-Stage Evaluation

For each phase (A, B, C) and each configured stage, Detego computes:

Stage evaluation

I_r = \frac{I_{\text{RMS,phase}}}{I_s}, \quad t_{\text{op}} = \begin{cases} \text{IDMT}(I_r) & \text{if } I_r > 1 \\ \infty & \text{if } I_r \leq 1 \end{cases}

Where

$I_{\text{RMS,phase}}$ RMS current at measurement time for this phase (primary amps)
$I_s$ Pickup current for this stage
$I_r$ Current multiple
$t_{\text{op}}$ Operating time for this stage

After evaluating all stages, Detego identifies the fastest stage for each phase -- the stage with the minimum operating time. This is the stage that would trip first in a real relay.

Earth Fault Evaluation

Earth fault stages are evaluated identically, but against the residual current $3I_0$ RMS instead of phase currents.

Multi-Stage Coordination

When multiple stages are configured, they form a composite protection characteristic -- the effective operating time at any current level is the minimum of all individual stage times. Detego renders this composite on the TCC chart as a composite curve.

Stage Domination

A stage is dominated at a particular current level if another stage produces a faster operating time. The composite curve logic:

IDMT clipping -- An inverse-time curve is clipped (not drawn) at current levels where a faster DT or IDMT stage takes over
Vertical connectors -- At the transition point (the pickup current of the faster stage), a vertical dotted line connects the slower stage's time to the faster stage's time
DT right boundaries -- A DT stage is drawn as a horizontal line from its pickup current to the point where an even faster stage takes over

Multi-stage TCC coordination: S1 (IEC Standard Inverse, 200 A, TMS 0.3) handles moderate faults; S2 (Definite Time instantaneous, 2000 A) provides fast clearance for high-current faults. The composite curve transitions from S1 to S2 via a vertical connector at the S2 pickup current. Operating points show per-phase fault currents plotted on the fastest-tripping stage.

Grading Margin

When coordinating multiple relays in series, the upstream relay must be slower than the downstream relay by a sufficient grading margin at every current level. The required margin accounts for:

CT errors -- Typically +/-5% at rated burden
Relay timing tolerance -- Typically +/-5% of operating time (IEC 60255 Class 5)
Circuit breaker clearance time -- 50-80 ms for modern breakers
Safety margin -- Additional 50-100 ms

Minimum grading margin

\Delta t = 2 \times \epsilon_{\text{relay}} \times t + t_{\text{CB}} + t_{\text{overshoot}} + t_{\text{safety}}

Where

$\epsilon_{\text{relay}}$ Relay timing tolerance (e.g. 0.05 for 5%)
$t$ Operating time of the downstream relay at the fault current
$t_{\text{CB}}$ Circuit breaker clearance time (typically 50-80 ms)
$t_{\text{overshoot}}$ Relay overshoot time after fault clearance (typically 50 ms)
$t_{\text{safety}}$ Additional safety margin (typically 50-100 ms)

A typical fixed grading margin of 300-400 ms is often used as a simplified approach when detailed relay and breaker data is not available.

Worked Example

Consider a relay with the following settings:

S1: IEC Standard Inverse, Is = 200 A, TMS = 0.3
S2: Definite Time, Is = 2000 A, TMS = 0 (instantaneous)
CT ratio: 1000/1

A COMTRADE recording shows a phase A fault current of 4500 A primary.

S1 Evaluation

S1 operating time

I_r = \frac{4500}{200} = 22.5, \quad t = 0.3 \times \frac{0.14}{22.5^{0.02} - 1} = 0.3 \times \frac{0.14}{0.0637} = 0.659 \text{ s}

S2 Evaluation

Since 4500 A > 2000 A (S2 pickup), S2 picks up with: $t = \text{TMS} = 0$ (instantaneous).

Result

S2 is faster (0 s vs 0.659 s), so the relay trips on S2 instantaneously. On the TCC chart, the Phase A operating point marker appears on the S2 horizontal line at 4500 A, labelled "INST".

Practical Note

In practice, no relay trips in exactly 0 ms. Physical relays have inherent minimum operating times of 20-50 ms even for instantaneous stages. The "INST" label in Detego means the configured TMS is 0, not that the actual trip time is zero.

Curve Rendering Details

Detego uses an adaptive two-zone sampling strategy to generate smooth IDMT curves on the log-log TCC chart. This is necessary because the curve has an asymptote at $I_r = 1$ where the slope approaches infinity, and a nearly flat region at high multiples.

Two-Zone Sampling

Zone 1 (30% of points) -- Geometric spacing in the $(I_r - 1)$ domain, concentrating points very close to the pickup asymptote (Ir = 1.000001 to ~1.5).
Zone 2 (70% of points) -- Logarithmically spaced from Ir = 1.5 to the maximum current ratio (default 100).

Times exceeding the maximum display limit are capped (not discarded), which produces a visual vertical asymptote on the chart. 400 points per curve provides smooth rendering at all zoom levels.

Standards Reference

Standard	Relevance
IEC 60255-151	Defines the four IEC IDMT curves (SI, VI, EI, LTI) and the DT characteristic. Specifies timing tolerances (Class 5 = +/-5%).
IEEE C37.112	Defines the three IEEE IDMT curves (MI, VI, EI) with the additive constant B formula. Standard in North American practice.
IEC 60255-1	General requirements for measuring relays and protection equipment. Defines terms like pickup, dropout, TMS, and IDMT.
IEC 61850-7-4	Defines logical nodes for overcurrent protection (PTOC) in modern IEC 61850-based substation automation systems.

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Overcurrent Protection Theory

IDMT curve equations, time multiplier mathematics, coordination principles, and standards references.

In Detego

See the Overcurrent Protection Guide for how to configure and use this tab in the app. This page covers the mathematical theory.

IDMT Operating Principle

Current Multiple (Ir)

The current multiple (also called the plug setting multiple or PSM) is the ratio of the measured fault current to the pickup current:

Current multiple

I_r = \frac{I_{\text{fault}}}{I_s}

Where

$I_{\text{fault}}$ Measured RMS fault current (primary amps)
$I_s$ Pickup (plug) setting current (primary amps)

The relay only starts timing when Ir > 1 (current exceeds pickup). Below pickup, the operating time is infinite -- the relay does not operate.

Unified IDMT Formula

Unified IDMT operating time

t = \text{TMS} \times \left( \frac{A}{I_r^p - 1} + B \right)

Where

$t$ Operating time in seconds
$\text{TMS}$ Time Multiplier Setting -- scales the entire curve vertically
$A$ Curve constant (determines steepness)
$p$ Curve exponent (determines inverseness)
$B$ Additive constant (0 for IEC, non-zero for IEEE)
$I_r$ Current multiple = I_fault / I_s (must be > 1)

For Definite Time (DT) curves: t = TMS regardless of current (flat time). Set TMS = 0 for instantaneous operation.

Time Multiplier Setting (TMS)

TMS = 1.0 -- Standard (reference) curve time
TMS = 0.5 -- Half the operating time at every current level
TMS = 0.1 -- Very fast operation (common for high-set stages)

DT Stages

IEC 60255 Curves

The IEC 60255-151 standard defines four inverse-time curves plus a definite-time option. The IEC formula has no additive constant ( $B = 0$ ):

t = \text{TMS} \times \frac{A}{I_r^p - 1}

Where

$A, p$ Curve constants (see table below)

Curve	A	p	t @ Ir=2	t @ Ir=10	Typical Application
Standard Inverse (SI)	0.14	0.02	10.02 s	2.97 s	General purpose -- most common worldwide. Good for networks where fault current does not vary significantly with location.
Very Inverse (VI)	13.5	1	13.50 s	1.50 s	Networks where fault current varies significantly with distance to the fault. Better discrimination than SI for variable fault levels.
Extremely Inverse (EI)	80	2	26.67 s	0.81 s	Fuse coordination in distribution feeders. The steep curve closely matches fuse melting characteristics.
Long-Time Inverse (LTI)	120	1	120.0 s	13.33 s	Thermal overload protection. Very slow operation tolerates long overloads without nuisance tripping.
Definite Time (DT)	--	--	TMS	TMS	Flat operating time: $t = \text{TMS}$ regardless of current magnitude. Used for high-set and instantaneous stages.

Why SI Is So Common

IEEE C37.112 Curves

t = \text{TMS} \times \left( \frac{A}{I_r^p - 1} + B \right)

Where

$B$ Additive constant -- produces a non-zero minimum operating time even at very high Ir

Curve	A	B	p	IEC Equivalent
Moderately Inverse	0.0515	0.114	0.02	Similar to IEC SI, but with a finite minimum time at high currents
Very Inverse	19.61	0.491	2	Similar to IEC EI, with steeper drop and different asymptotic behaviour
Extremely Inverse	28.2	0.1217	2	Very steep inverse -- well-suited for fuse coordination on US distribution

IEC vs IEEE

The key difference is the additive constant B. IEC curves approach zero operating time as current increases indefinitely, while IEEE curves approach a finite minimum time of

\text{TMS} \times B

. In practice, this difference is small because physical relays have inherent minimum operating times (~20-50 ms) regardless of the mathematical curve.

Curve Comparison

The diagram below shows all seven IDMT curves at TMS = 1.0 on a log-log scale. The X-axis is the current multiple $I_r$ and the Y-axis is the operating time in seconds. Key observations:

All curves share the same asymptote at $I_r = 1$ (vertical dashed line at pickup)
SI and IEEE MI are nearly flat -- operating time varies little with current
EI, IEEE VI, and IEEE EI are very steep -- operating time drops dramatically with increasing current
LTI is the slowest curve at all current levels -- used for thermal protection

CT Ratio Scaling

CT ratio scaling

I_{\text{primary}} = I_{\text{measured}} \times \frac{\text{CT Primary}}{\text{CT Secondary}}

Where

$I_{\text{measured}}$ RMS current computed from the COMTRADE sample data
$I_{\text{primary}}$ Primary system current used for pickup comparison

Example: 1000/1 CT with measured 4.5 A secondary -> 4500 A primary.

Residual Current Computation

Residual current

3I_0[n] = I_A[n] + I_B[n] + I_C[n]

Where

$3I_0[n]$ Residual current at sample n (instantaneous)
$I_A, I_B, I_C$ Phase current samples at sample n

Operating Point Evaluation

Detego evaluates protection operating points by measuring the RMS fault current at a specific time and computing the IDMT operating time for each configured stage.

Measurement Time Selection

The operating point is evaluated at the measurement time, which is determined in this order of priority:

Cursor position -- If you click on the RMS Trend chart or set the cursor elsewhere in the viewer, operating points are computed at the cursor time.
Post-inception (auto) -- If no cursor is set, Detego uses the auto-detected fault inception time plus 2 fundamental cycles. The 2-cycle delay allows the RMS computation to settle into the fault-level value.
Recording midpoint (fallback) -- If no fault is detected, the midpoint of the recording is used.

Per-Stage Evaluation

For each phase (A, B, C) and each configured stage, Detego computes:

Stage evaluation

I_r = \frac{I_{\text{RMS,phase}}}{I_s}, \quad t_{\text{op}} = \begin{cases} \text{IDMT}(I_r) & \text{if } I_r > 1 \\ \infty & \text{if } I_r \leq 1 \end{cases}

Where

$I_{\text{RMS,phase}}$ RMS current at measurement time for this phase (primary amps)
$I_s$ Pickup current for this stage
$I_r$ Current multiple
$t_{\text{op}}$ Operating time for this stage

After evaluating all stages, Detego identifies the fastest stage for each phase -- the stage with the minimum operating time. This is the stage that would trip first in a real relay.

Earth Fault Evaluation

Earth fault stages are evaluated identically, but against the residual current $3I_0$ RMS instead of phase currents.

Multi-Stage Coordination

Stage Domination

A stage is dominated at a particular current level if another stage produces a faster operating time. The composite curve logic:

IDMT clipping -- An inverse-time curve is clipped (not drawn) at current levels where a faster DT or IDMT stage takes over
Vertical connectors -- At the transition point (the pickup current of the faster stage), a vertical dotted line connects the slower stage's time to the faster stage's time
DT right boundaries -- A DT stage is drawn as a horizontal line from its pickup current to the point where an even faster stage takes over

Grading Margin

When coordinating multiple relays in series, the upstream relay must be slower than the downstream relay by a sufficient grading margin at every current level. The required margin accounts for:

CT errors -- Typically +/-5% at rated burden
Relay timing tolerance -- Typically +/-5% of operating time (IEC 60255 Class 5)
Circuit breaker clearance time -- 50-80 ms for modern breakers
Safety margin -- Additional 50-100 ms

Minimum grading margin

\Delta t = 2 \times \epsilon_{\text{relay}} \times t + t_{\text{CB}} + t_{\text{overshoot}} + t_{\text{safety}}

Where

$\epsilon_{\text{relay}}$ Relay timing tolerance (e.g. 0.05 for 5%)
$t$ Operating time of the downstream relay at the fault current
$t_{\text{CB}}$ Circuit breaker clearance time (typically 50-80 ms)
$t_{\text{overshoot}}$ Relay overshoot time after fault clearance (typically 50 ms)
$t_{\text{safety}}$ Additional safety margin (typically 50-100 ms)

A typical fixed grading margin of 300-400 ms is often used as a simplified approach when detailed relay and breaker data is not available.

Worked Example

Consider a relay with the following settings:

S1: IEC Standard Inverse, Is = 200 A, TMS = 0.3
S2: Definite Time, Is = 2000 A, TMS = 0 (instantaneous)
CT ratio: 1000/1

A COMTRADE recording shows a phase A fault current of 4500 A primary.

S1 Evaluation

S1 operating time

I_r = \frac{4500}{200} = 22.5, \quad t = 0.3 \times \frac{0.14}{22.5^{0.02} - 1} = 0.3 \times \frac{0.14}{0.0637} = 0.659 \text{ s}

S2 Evaluation

Since 4500 A > 2000 A (S2 pickup), S2 picks up with: $t = \text{TMS} = 0$ (instantaneous).

Result

S2 is faster (0 s vs 0.659 s), so the relay trips on S2 instantaneously. On the TCC chart, the Phase A operating point marker appears on the S2 horizontal line at 4500 A, labelled "INST".

Practical Note

Curve Rendering Details

Two-Zone Sampling

Zone 1 (30% of points) -- Geometric spacing in the $(I_r - 1)$ domain, concentrating points very close to the pickup asymptote (Ir = 1.000001 to ~1.5).
Zone 2 (70% of points) -- Logarithmically spaced from Ir = 1.5 to the maximum current ratio (default 100).

Times exceeding the maximum display limit are capped (not discarded), which produces a visual vertical asymptote on the chart. 400 points per curve provides smooth rendering at all zoom levels.

Standards Reference

Standard	Relevance
IEC 60255-151	Defines the four IEC IDMT curves (SI, VI, EI, LTI) and the DT characteristic. Specifies timing tolerances (Class 5 = +/-5%).
IEEE C37.112	Defines the three IEEE IDMT curves (MI, VI, EI) with the additive constant B formula. Standard in North American practice.
IEC 60255-1	General requirements for measuring relays and protection equipment. Defines terms like pickup, dropout, TMS, and IDMT.
IEC 61850-7-4	Defines logical nodes for overcurrent protection (PTOC) in modern IEC 61850-based substation automation systems.

PreviousDistance Protection NextDirectional Elements