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Impedance Analysis

Fault impedance, resistance, reactance, and distance-to-fault estimation from voltage and current phasors.

In Detego

See the Compute Tab Guide for how to create this signal via the Compute Tab. This page covers the mathematical theory.

Overview

Impedance analysis computes the apparent impedance seen at the relay location by dividing the voltage phasor by the current phasor. The resulting complex impedance is decomposed into its resistive and reactive components, which are plotted on the $R\text{-}X$ impedance plane. Distance protection relays use this measurement to determine whether a fault lies within their protected zone and to estimate the physical location of the fault along a transmission line.

Detego computes four derived signals from a paired voltage and current channel: fault impedance magnitude $|Z|$ , resistance $R$ , reactance $X$ , and distance to fault.

R-X Impedance Plane

The R-X plane is the fundamental visualization used by distance protection relays. The horizontal axis represents resistance and the vertical axis represents reactance. The line impedance defines a characteristic angle (typically 60-85 degrees for overhead lines). Protection zones are drawn as circles (mho characteristic) or rectangles (quadrilateral characteristic) centered along the line impedance vector.

R-X impedance plane showing distance relay zones (mho characteristic). During a fault, the measured impedance moves from the load region to the fault impedance point. Zone 1 covers ~80% of the protected line; Zone 2 and Zone 3 provide backup.

Protection Zones

Zone 1 is set to cover approximately 80% of the protected line to avoid overreach due to measurement errors and CT/VT inaccuracies. Zone 2 extends to 120-150% of the line, covering the remainder plus a margin, but with a deliberate time delay (typically 300-500 ms). Zone 3 provides remote backup for the adjacent line section.

Computed Signals

1. Fault Impedance |Z|

The fault impedance magnitude is computed by performing complex division of the voltage phasor by the current phasor, then taking the absolute value. This gives the magnitude of the loop impedance seen at the relay location. If the current magnitude is zero (no fault current), the result is returned as zero to avoid division by zero.

Complex impedance

Z = \frac{V}{I} = \frac{V_{\text{re}} + jV_{\text{im}}}{I_{\text{re}} + jI_{\text{im}}}

Where

$V$ Voltage phasor (complex, from one-cycle DFT)
$I$ Current phasor (complex, from one-cycle DFT)

Impedance magnitude

|Z| = \sqrt{R^2 + X^2}

Where

$R$ Resistive component (real part of Z)
$X$ Reactive component (imaginary part of Z)

Returns zero when I = 0 to avoid division by zero. During load conditions, |Z| is large (hundreds of ohms). During a fault, |Z| drops to a few ohms.

Phasor Source

Both

V

and

I

are extracted using the same one-cycle DFT (Discrete Fourier Transform) used by the phasor computation engine. The DFT provides fundamental-frequency phasors with DC offset rejection, ensuring clean impedance measurements even during the transient period immediately after fault inception.

2. Resistance R

The resistance is the real part of the complex impedance. It represents the power-dissipating component of the fault loop. In a bolted fault (zero fault resistance), R approaches zero. High values of R indicate arc resistance at the fault point, tower footing resistance in ground faults, or load flow through the fault loop.

Resistance (real part of Z)

R = \text{Re}\!\left(\frac{V}{I}\right) = \frac{V_{\text{re}} \cdot I_{\text{re}} + V_{\text{im}} \cdot I_{\text{im}}}{|I|^2}

Where

$V_re, V_im$ Real and imaginary parts of the voltage phasor
$I_re, I_im$ Real and imaginary parts of the current phasor
$|I|²$ Squared magnitude of the current phasor = I_re² + I_im²

Arc resistance is a significant factor in practical fault analysis. For short arcs (e.g., insulator flashover), the Warrington formula gives typical values of 1-5 ohms. For long arcs on high-voltage lines, arc resistance can be 10-50 ohms. This resistance causes the measured impedance to shift to the right on the R-X plane, which can cause underreaching of mho relays and is one reason quadrilateral characteristics are preferred for ground distance elements.

3. Reactance X

The reactance is the imaginary part of the complex impedance. For overhead transmission lines, the reactance per unit length is relatively constant and well-characterized by the line geometry and conductor configuration. This makes reactance the primary indicator of electrical distance to the fault point.

Reactance (imaginary part of Z)

X = \text{Im}\!\left(\frac{V}{I}\right) = \frac{V_{\text{im}} \cdot I_{\text{re}} - V_{\text{re}} \cdot I_{\text{im}}}{|I|^2}

Where

$X > 0$ Inductive reactance (fault in the forward direction)
$X < 0$ Capacitive reactance (possible reverse fault or load)

Reactance and Distance

On overhead lines, the positive-sequence reactance is typically 0.3-0.5 ohms per kilometer. Because reactance is largely unaffected by fault resistance (arc resistance adds only to R, not X), the reactance measurement provides a more reliable estimate of distance to fault than the impedance magnitude. This is the principle behind reactance-type distance relay elements used for ground fault protection.

4. Distance to Fault

The distance to fault is estimated by dividing the measured impedance magnitude by the known line impedance per kilometer. This requires a parameterized value for the line's positive-sequence impedance, which must be entered by the user when creating the computed signal.

Distance estimation

d = \frac{|Z|}{Z_{\text{line/km}}}

Where

$d$ Estimated distance to fault (km)
$|Z|$ Measured fault impedance magnitude (ohms)
$Z_line/km$ Line impedance per kilometer (ohms/km), user-specified

This is a simplified single-ended impedance-based estimate. Accuracy depends on the fault type, fault resistance, load flow, and source impedance ratio.

For a phase-to-phase fault, the positive-sequence impedance is used directly. For a single-line-to-ground fault, the calculation should use the appropriate loop impedance that accounts for the zero-sequence impedance of the line. The zero-sequence compensation factor $k_0 = (Z_0 - Z_1) / (3 Z_1)$ adjusts for the difference between zero-sequence and positive-sequence impedances.

Accuracy Limitations

Single-ended impedance-based fault location is subject to errors from fault resistance, load flow (infeed/outfeed), mutual coupling on parallel lines, and CT/VT errors. Typical accuracy is 1-5% of line length. For more accurate results, two-ended fault location methods using synchronized data from both line terminals are preferred.

Engineering Context

Distance Relay Characteristics

Distance relays operate by comparing the measured impedance against predefined zones on the R-X plane. The two most common characteristics are:

Mho characteristic -- a circle passing through the origin with its diameter along the line impedance vector. Inherently directional (only operates for forward faults). Used widely for phase distance elements on transmission lines.
Quadrilateral characteristic -- an independently adjustable rectangle in the R-X plane with separate reach settings for resistance and reactance. Provides better coverage for resistive faults and is preferred for ground distance elements.

Protection Zone Philosophy

Zone	Reach	Time	Purpose
Zone 1	80-85% of line	Instantaneous	Primary high-speed protection. Set below 100% to prevent overreach.
Zone 2	120-150% of line	300-500 ms	Covers the remaining 15-20% of the line plus remote bus. Time-delayed to coordinate with Zone 1 of the next line.
Zone 3	200-300% of line	600-1200 ms	Remote backup. Covers adjacent line sections. Must be checked against maximum load impedance to avoid tripping on heavy load.

Fault Loop Selection

The choice of voltage and current channels determines the fault loop being measured. Standard loop configurations for different fault types are:

Phase-to-phase (A-B, B-C, C-A): $Z = (V_A - V_B) / (I_A - I_B)$ -- uses delta voltage and delta current.
Phase-to-ground (A-G, B-G, C-G): $Z = V_A / (I_A + k_0 \cdot I_0)$ -- uses phase voltage and compensated current with the zero-sequence compensation factor.
Three-phase: any phase-phase loop gives the correct positive-sequence impedance.

In Detego

When creating an impedance computed signal, you select one voltage channel and one current channel. For phase-to-phase loops, use the Basic Operations compute category first to create a delta voltage (e.g., Va - Vb) and delta current (e.g., Ia - Ib), then use those as inputs to the impedance calculation.

PreviousPower Quantities NextHarmonic Analysis

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Impedance Analysis

Fault impedance, resistance, reactance, and distance-to-fault estimation from voltage and current phasors.

In Detego

See the Compute Tab Guide for how to create this signal via the Compute Tab. This page covers the mathematical theory.

Overview

Detego computes four derived signals from a paired voltage and current channel: fault impedance magnitude $|Z|$ , resistance $R$ , reactance $X$ , and distance to fault.

R-X Impedance Plane

Protection Zones

Computed Signals

1. Fault Impedance |Z|

Complex impedance

Z = \frac{V}{I} = \frac{V_{\text{re}} + jV_{\text{im}}}{I_{\text{re}} + jI_{\text{im}}}

Where

$V$ Voltage phasor (complex, from one-cycle DFT)
$I$ Current phasor (complex, from one-cycle DFT)

Impedance magnitude

|Z| = \sqrt{R^2 + X^2}

Where

$R$ Resistive component (real part of Z)
$X$ Reactive component (imaginary part of Z)

Returns zero when I = 0 to avoid division by zero. During load conditions, |Z| is large (hundreds of ohms). During a fault, |Z| drops to a few ohms.

Phasor Source

Both

V

and

I

2. Resistance R

Resistance (real part of Z)

R = \text{Re}\!\left(\frac{V}{I}\right) = \frac{V_{\text{re}} \cdot I_{\text{re}} + V_{\text{im}} \cdot I_{\text{im}}}{|I|^2}

Where

$V_re, V_im$ Real and imaginary parts of the voltage phasor
$I_re, I_im$ Real and imaginary parts of the current phasor
$|I|²$ Squared magnitude of the current phasor = I_re² + I_im²

3. Reactance X

Reactance (imaginary part of Z)

X = \text{Im}\!\left(\frac{V}{I}\right) = \frac{V_{\text{im}} \cdot I_{\text{re}} - V_{\text{re}} \cdot I_{\text{im}}}{|I|^2}

Where

$X > 0$ Inductive reactance (fault in the forward direction)
$X < 0$ Capacitive reactance (possible reverse fault or load)

Reactance and Distance

4. Distance to Fault

Distance estimation

d = \frac{|Z|}{Z_{\text{line/km}}}

Where

$d$ Estimated distance to fault (km)
$|Z|$ Measured fault impedance magnitude (ohms)
$Z_line/km$ Line impedance per kilometer (ohms/km), user-specified

This is a simplified single-ended impedance-based estimate. Accuracy depends on the fault type, fault resistance, load flow, and source impedance ratio.

Accuracy Limitations

Engineering Context

Distance Relay Characteristics

Distance relays operate by comparing the measured impedance against predefined zones on the R-X plane. The two most common characteristics are:

Mho characteristic -- a circle passing through the origin with its diameter along the line impedance vector. Inherently directional (only operates for forward faults). Used widely for phase distance elements on transmission lines.
Quadrilateral characteristic -- an independently adjustable rectangle in the R-X plane with separate reach settings for resistance and reactance. Provides better coverage for resistive faults and is preferred for ground distance elements.

Protection Zone Philosophy

Zone	Reach	Time	Purpose
Zone 1	80-85% of line	Instantaneous	Primary high-speed protection. Set below 100% to prevent overreach.
Zone 2	120-150% of line	300-500 ms	Covers the remaining 15-20% of the line plus remote bus. Time-delayed to coordinate with Zone 1 of the next line.
Zone 3	200-300% of line	600-1200 ms	Remote backup. Covers adjacent line sections. Must be checked against maximum load impedance to avoid tripping on heavy load.

Fault Loop Selection

The choice of voltage and current channels determines the fault loop being measured. Standard loop configurations for different fault types are:

Phase-to-phase (A-B, B-C, C-A): $Z = (V_A - V_B) / (I_A - I_B)$ -- uses delta voltage and delta current.
Phase-to-ground (A-G, B-G, C-G): $Z = V_A / (I_A + k_0 \cdot I_0)$ -- uses phase voltage and compensated current with the zero-sequence compensation factor.
Three-phase: any phase-phase loop gives the correct positive-sequence impedance.

In Detego

PreviousPower Quantities NextHarmonic Analysis