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Primary and secondary impedance for the 21

Calculate the line impedance from its ohms per kilometer, the ground compensation factor k0 in both of its forms, and the reaches of the zones you define, in primary and secondary ohms.

Blackburn and Dominsecondary Z = primary Z · CT ratio / VT ratioDistance relay (21) manualsk0 as magnitude and angle, RE/RL and XE/XL

Example cases
Primary and secondary impedance for the 21
E1Line
km
Ω/km
Ω/km
Ω/km
Ω/km
E2Instrument transformers
A

Enter each voltage with its base: the tool converts both to phase-to-phase before dividing.

E3Zone reaches, % of Z1
%
%
%
%

The preloaded percentages are an example: use the ones from your protection coordination study. Leave a zone empty to skip it.

E4Direct conversion
Ω

Use a decimal point. The calculation runs in your browser and updates as you type.

Line Z1, secondary

— Ω

Line Z1, primary

— Ω

k0 factor

—

k0 by components

—

Zone 1, secondary

— Ω

Direct conversion

— Ω

R-X plane, secondary ohms
  • Line Z1
  • Zone 1
  • Zones 2 to 4
Each zone is drawn as a reference mho circle with its diameter on the line angle, only to show the reach. Both axes use the same scale.

Step-by-step calculation

What it means for the settings

    What it is for

    Studies give the reaches in primary ohms, but many distance relays are set in secondary ohms and ask for ground compensation in their own format. This tool does that conversion without unit errors: use it to enter or review the settings of a 21, prepare the points of a secondary injection test, or check that a relay already set matches the line and the instrument transformers installed.

    How it is calculated

    Line impedance

    Z1 = (R1 + jX1) · L · Z0 = (R0 + jX0) · L

    With |Z| = √(R² + X²) and angle φ = arctan(X / R).

    Blackburn and DominProtective Relaying: Principles and Applications

    Primary to secondary

    Zsec = Zpri · CTR / VTR

    With the CT ratio CTR = Ipri / Isec and the VT ratio VTR = Vpri / Vsec, both voltages on the same base. If one is given phase-to-neutral, the tool multiplies it by √3 before dividing: mixing 115 kV phase-to-phase with 66.4 V phase-to-neutral without correction would give a √3 error. The angle does not change.

    Ground compensation

    k0 = (Z0 − Z1) / (3 · Z1) · RE/RL = (R0 − R1) / (3 · R1) · XE/XL = (X0 − X1) / (3 · X1)

    The first is a complex number given as magnitude and angle (the form used, for example, by SEL and ABB); the second, by components, is the one Siemens uses. In both cases the value is the same in primary and secondary. R0 can be lower than R1, and then RE/RL is negative.

    Reach of each zone

    Zzone = (percentage / 100) · |Z1|

    On the line angle, then converted to secondary with the same ratio. The zone 1 indicator is green below 90 % of the line, yellow between 90 % and 100 % (little margin for CT, VT and line data errors) and red from 100 %, because it would reach the remote bus. This is a Tiza convention; the percentage for each case comes from the study.

    Limits of the calculation

    • This is a conversion, not a setting: the percentages come from the protection coordination study, based on the adjacent line, the transformers at the remote substation, infeed and fault resistance.
    • It assumes a homogeneous line, without zero-sequence mutual coupling to parallel circuits and without series compensation; in those cases the effective k0 and the reaches change.
    • The mho in the chart is illustrative: if your relay uses a quadrilateral characteristic, the resistive reach is a separate setting this tool does not calculate.
    • Check whether your relay expects the reach in per-phase ohms (the usual case) and in which unit it takes the line angle.

    Frequently asked questions

    How are primary ohms converted to secondary ohms for a distance relay?

    Multiply the primary impedance by the CT ratio and divide it by the VT ratio: Zsec = Zpri · CTR / VTR. For example, with a 600/5 CT (CTR = 120) and a 115 kV/115 V VT (VTR = 1000), 10 primary ohms equal 1.2 secondary ohms.

    What is the k0 ground compensation factor?

    It is the ratio k0 = (Z0 − Z1) / (3·Z1) between the zero- and positive-sequence impedances of the line. The relay uses it so that the phase-to-ground loops measure the same positive-sequence impedance as the phase-to-phase loops. Some manufacturers ask for it as magnitude and angle, others as two numbers, RE/RL and XE/XL.

    Does k0 change between primary and secondary?

    No. k0 is a ratio of two impedances of the same line, so the CT and VT ratios cancel out. What does change with the ratios are the zone reaches in ohms.

    We set your distance protection

    We do the protection coordination study of the line, define the reach of each zone and the k0, and load the settings into the relay remotely, in the manufacturer's format.

    See remote configurationSee the coordination study

    Related guides

    Distance protection 21 (PDIS): how it worksWhat distance protection 21 is, how it measures impedance, how its zones are set and why it trips or blocks in the field.Line Protection Main 1 and Main 2: How It Is BuiltMain 1 and main 2 line protection: what is duplicated, how the two systems coordinate with backup and breaker failure 50BF, and why the scheme fails.Teleprotection Schemes (85): Types, Channel and SelectionLine teleprotection schemes: DUTT, PUTT, POTT, DCB, DCUB, DTT and phase comparison. What each one sends, which channel it needs and when to use it.

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