Distance Protection Impedance Calculator — Mho & Quadrilateral Zones

A distance relay decides by comparing the measured impedance Z = V / I against a characteristic drawn in the R–X plane. This calculator plots Mho, offset Mho and quadrilateral characteristics, places a measured impedance point on the same plane, and shows which zone it falls in and how much margin remains to the boundary.

1. What this calculator does — and what it does not

It draws the distance characteristic in the R–X plane from the settings you enter, computes the measured impedance from a secondary voltage, current and impedance angle, and tests that point against up to three enabled zones. For each zone it also returns the distance from the origin to the boundary along the measured angle, so you can see how close the test point is to a zone edge.

It is a geometry reference, not a relay model. It does not model load encroachment, power-swing blocking, VT fuse failure, CT saturation, mutual coupling, series compensation, relay-specific tolerances or the trip logic between zones. Where the relay’s setting sheet or instruction manual differs from these shapes, that document governs.

2. Distance protection impedance calculator

3. How to read the result

Work in secondary ohms unless you deliberately convert

The relay sees the secondary quantities at its terminals. A measured 20 V and 5 A gives Z = 4 Ω secondary. To compare with a primary-impedance setting sheet, multiply by the VT ratio and divide by the CT ratio: Z_primary = Z_secondary × (VT ratio / CT ratio). Mixing the two planes is the most common source of a “wrong zone” result that is not a relay fault.

The angle convention matters more than the magnitude

The impedance angle φ is measured from the +R axis. A positive angle means inductive reactance (X > 0), which is the normal forward-fault direction for an overhead line. A point in the second or third quadrant is behind the relay or has a reversed current direction. If the injected current is defined as flowing into the line, the angle you enter should match the relay’s own convention; if it is defined as flowing out of the relay, add or subtract 180° before trusting the zone result.

A Mho circle is fixed by one reach and one angle

For a reach Z and MTA φ, the circle centre is at (Z/2) ∠ φ and its radius is Z/2. The circle always passes through the origin. That is why a Mho element is inherently directional: a fault behind the relay sits outside the circle.

An offset Mho adds a reverse region

With a reverse offset Z_o, the circle passes through −Z_o and +Z_f along the MTA line. Its centre becomes ((Z_f − Z_o)/2) ∠ φ and its radius (Z_f + Z_o)/2. The offset is usually a small percentage of the forward reach and is used for reverse-looking or blocking elements.

A quadrilateral is defined by edges, not by a reach

The polygon is set by a forward resistive reach R_f, a forward reactance reach X_f and a reverse reactance reach X_r. A positive tilt θ drops the top line as R increases, which reduces overreach on resistive faults and helps with arc resistance. Keep X_f − R_f tanθ > X_r, otherwise the top and bottom edges cross and the shape is no longer a usable zone.

Margin is measured along the test angle, not straight down

For a measured point at radius |Z| and angle φ, the calculator finds where the ray from the origin at that angle crosses the zone boundary. The margin is the difference between that boundary distance and |Z|. A negative margin means the point is outside that zone by that amount. This is more useful than a perpendicular distance because it answers the practical question: how much more impedance can this fault have before it leaves the zone?

4. Formula and geometry

Quantity	Expression	Note
Measured impedance	Z = V / I; R = |Z| cosφ; X = |Z| sinφ	Secondary volts and amperes.
Mho circle	centre (Z/2) ∠ φMTA; radius Z/2	Passes through the origin.
Offset Mho	centre ((Zf − Zo)/2) ∠ φMTA; radius (Zf + Zo)/2	Reverse offset Zo along the MTA line.
Quadrilateral vertices	(0, Xr), (Rf, Xr), (Rf, Xf − Rf tanθ), (0, Xf)	Tilt θ = 0 gives a rectangle.
Circle test	|Z − Zc| ≤ Rc	Inside when the distance to the centre is within the radius.
Polygon test	ray-casting point-in-polygon	Inside when an odd number of edges are crossed.

All angles are in degrees in the input fields and converted to radians internally. The calculator assumes a balanced, sinusoidal steady state and an impedance plane with equal scales on both axes, so circles appear as circles rather than ellipses.

5. Worked example — Zone 1 Mho

Zone 1 is set to 4.5 Ω secondary at an MTA of 78°. A secondary injection applies 20 V and 5 A with the current lagging the voltage by 78°. The measured impedance is 4.0 Ω at 78°, which lies inside the Zone 1 circle. The boundary along the same angle is at 4.5 Ω, so the point has 0.5 Ω of margin, or 12.5% of the measured magnitude. If the same injection is repeated at 30 V and 5 A, the measured 6.0 Ω point falls outside Zone 1 and inside Zone 2; the margin to the Zone 1 boundary is then −1.5 Ω.

Switch the characteristic to quadrilateral with R_f = 4 Ω, X_f = 5 Ω and X_r = −1 Ω. The same 4.0 Ω at 78° point is inside the polygon, and the boundary along the test angle is about 5.11 Ω, giving roughly 1.11 Ω of margin. The larger margin is expected: a quadrilateral with these reaches encloses more of the R–X plane than a 4.5 Ω Mho circle.

6. Frequently asked questions

What is the difference between a Mho and a quadrilateral distance characteristic?

A Mho characteristic is a circle through the origin defined by one reach and one maximum torque angle. It is inherently directional and simple to set. A quadrilateral is a polygon defined by separate resistive and reactive reaches, so it can be shaped to cover fault resistance without extending the reactive reach. Quadrilaterals are common on modern numerical relays; Mho circles remain common on electromechanical and older static relays.

Should I enter primary or secondary impedance?

Enter secondary impedance unless your setting sheet is explicitly in primary ohms. The relay measures secondary voltage and current, so its settings are normally expressed in secondary ohms. Convert with Z_primary = Z_secondary × (VT ratio / CT ratio).

Why does my test point show as outside the zone when the relay trips?

Check the angle convention, the CT/VT ratios and whether the relay applies a different characteristic than the one you selected. A point can also be inside the measured-impedance plane but blocked by load encroachment, power-swing blocking or a VT supervision element. The calculator shows geometry only; the relay’s trip logic is separate.

What does a negative margin mean?

The measured point lies outside that zone. The magnitude of the negative margin is the distance from the point back to the boundary along the measured angle. It is not the same as the perpendicular distance to the nearest edge.

Can I use this for generator or transformer distance protection?

The geometry is the same, but the settings and the angle conventions differ. For transformer distance protection, pay particular attention to the winding connection, the CT ratio matching and the relay’s compensation for the transformer impedance. Use the relay’s setting sheet for acceptance.

7. Where to go next

For the injection method and the classic mistakes, see secondary injection testing. For overcurrent elements and their curves, see overcurrent relay testing. The VAE-430 and VAE-660 relay test systems provide the voltage and current channels these tests need; for a configuration matched to your protection schemes, request a quote.

Related tools: IDMT relay curve calculator · relay test lead voltage drop calculator · transformer turns ratio calculator · all test calculators.