Two-Source Swing Loci Plots

Plots the system impedance line, the equal- and unequal-EMF swing loci, the δ angle construction, and mho zone characteristics on the R–X diagram of a line relay, in the style of Blackburn Fig. 14.3. δ is the angle by which VR lags VS.

VS VR 3 4 G H ZS ZL ZR

Impedance BaseiScaling for the plot axes only. Enter every impedance in per unit on one common base. With Zbase = 1 the axes are in pu. Otherwise the axes (and the zone table's electrical-center value) are shown in pu × Zbase, e.g. primary ohms with Zbase = kV²/MVA, or secondary ohms by also multiplying by CTR/PTR.

Zbase — impedances entered in pu; plot axes in pu × Zbase.

Sending Source ZS (behind bus G)iPositive-sequence Thévenin impedance of everything behind bus G on the VS side (generation plus the rest of the system as seen from G), in pu.

RS:    XS:

Protected Line ZL (G – H)iPositive-sequence impedance of the protected line between buses G and H, in pu. Zone reaches and the Zone 3 offset are percentages of |ZL|, and the mho characteristic angle defaults to the angle of ZL.

RL:    XL:

Receiving Source ZR (behind bus H)iPositive-sequence Thévenin impedance of everything behind bus H on the VR side, in pu.

RR:    XR:

Swing Loci

|VS|/|VR| ratiosiEach value n = |VS|/|VR| adds an unequal-EMF swing locus, which is a circle. n > 1 (red) bends around the VR end; n < 1 (green) bends around the VS end. Only the ratio matters, not the absolute voltages: VR is the 1∠0° reference and VS = n∠δ. The equal-EMF locus (n = 1, a straight line) is always drawn.
Up to 4, comma-separated. The equal-EMF locus is always drawn.
δ angles to mark (deg)iAngles marked with a dot and label on the equal-EMF locus, and with small dots on each unequal-EMF locus. δ is the angle by which VR lags VS. Useful property: at any point on any locus, the lines drawn to the VS and VR points meet at an angle equal to δ.
Outer lines (VR → δ → VS, with 360° − δ)iDraws the Blackburn angle construction for the chosen pair: straight lines from VR to the δ point on the equal-EMF locus and on to VS, plus the same for the mirror point 360° − δ. The four lines form the diamond in Blackburn Fig. 14.3. Any missing δ point is marked automatically.:

Relay LocationiBus G: relay 3 at G, forward direction toward H. Power flowing S → R appears as load in the first quadrant.
Bus H: relay 4 at H, forward direction toward G. The whole diagram is redrawn from H: the origin moves to H and the picture is flipped end for end, so S → R load appears in the second quadrant.

Mho Zones (reach in % of ZL; 0 = off)iStatic mho circles. For a forward zone the circle passes through the relay location with a diameter of reach × |ZL| along the MTA; e.g. 85% for Zone 1, 120% for Zone 2. A reach of 0 turns the zone off. Zone 3 can be set forward or reverse and can have an offset. Memory polarization, load encroachment and quadrilateral shapes are not modelled.

Zone 1:   Zone 2:
Zone 3:   Offset (%)iOffset in percent of |ZL|, the same units as the zone reaches. It is measured from the relay along the zone's characteristic angle (MTA, default = line angle), on the opposite side from the reach: behind the relay for a forward Zone 3, in front of it for a reverse Zone 3. The circle's diameter runs from the offset point to the reach point, i.e. (reach + offset) × |ZL|.

Example: |ZL| = 0.2154 pu, 10% offset → 0.0215 pu past the relay. 0 = ordinary mho through the relay location.:
MTA: deg (0 = line angle)iMaximum torque angle (characteristic angle) used for all three mho zones, in degrees. 0 uses the angle of the line impedance ZL, which is the usual setting.
iThe PRC-026-2 Attachment B unstable power swing boundary. The lens is traced at δ = 120° (right) and 240° (left) as the source voltage ratio runs from 0.7 to 1.43. Full Shape adds the loss-of-synchronism circles for ratios 0.7 and 1.43. Under PRC-026-2, a relay characteristic lying completely inside this boundary is not expected to trip for a stable power swing; check the standard for the full criteria.
iRemoves angle dots and labels, the VS/VR/G/H markers, the ZS/ZL/ZR labels, the zone labels and the electrical-center ×, leaving only the lines, axes and legend.

☰  Line colors & thickness

ScaleiZoom for the plot. 1 fits VS, VR, the marked angle points and the zones. Values above 1 zoom out (useful for seeing the full PRC-026-2 shape or large unequal-EMF circles); values below 1 zoom in. Does not affect the zone table.
Swing loci on R-X diagram
Slip frequency Δf Hz
What this table shows and how to use it

Each row is one swing locus (the equal-EMF locus plus each |VS|/|VR| ratio you entered); each column is an enabled mho zone. As δ increases from 0° to 360°, the table gives the angle at which the locus enters the zone, the angle at which it leaves, the angle span Δδ spent inside, and the time that span takes at the slip frequency above (time = Δδ ÷ (360° × Δf); 1 Hz of slip = 360°/s). “not entered” means that locus never reaches the zone. The table is computed in your browser and updates as you type; Run Analysis is not needed.

Typical uses:

Choosing the slip frequency. The best source is transient stability studies of the worst credible contingencies. Without them, IEEE PSRC Working Group J5, Application of Out-of-Step Protection Schemes for Generators (2020), summarizes typical values: local generator-vs-system oscillations about 1–2 Hz; a common general maximum of about 2.5 Hz; oscillations between areas seldom above 1 Hz and as low as 0.2–0.3 Hz; and a widely quoted range of 4–7 Hz for system swing rates, consistent with a fully loaded generator reaching about 3.6 Hz at its first pole slip. For OOS timer settings the report recommends using the maximum expected slip with added margin, and its examples use 4–5 Hz. The default of 5 Hz follows that guidance. For Zone 2 / Zone 3 security during stable swings, a slow slip (the lower end of the ranges) gives the longest, worst-case dwell times.

Limitations. The entry and exit angles are exact for each fixed voltage ratio. The times assume a constant slip frequency, which a real swing does not have: the rotor accelerates and decelerates, and a stable swing reverses before reaching 180°. Treat the times as order-of-magnitude values, or as a bound if you enter the fastest credible slip. For a time-accurate trajectory, use the swing simulator page.