Transformer Differential Protection (87T): Slope & Harmonics

Match the CT currents

For each winding, rated line current is $I_n=S_n/(\sqrt{3}U_{LL})$. The CT ratio is $C=I_{CT,p}/I_{CT,s}$ and the rated-load secondary tap is $T=I_n/C$. A measured secondary RMS phasor divided by its own tap is on the common transformer per-unit base. ABB RET 54_/Diff6T Application and Setting Guide, 1MRS 755916 (2005), attached PDF page 8, gives the rated-current and ratio-matching method and the 25 MVA example.

Compensate balanced phase displacement

All current arrows point into the zone. For balanced ABC positive sequence, the LV nameplate voltage lag is $30^\circ c$ for clock number $c$. Rotate the LV inward-current phasor by $+30^\circ c$. Ideal through currents then oppose and cancel. This is a balanced positive-sequence equivalent of vector-group matching, not a full three-phase relay compensation matrix. The preset through current has an LV angle $180^\circ-30^\circ c$; the preset internal contribution has angle $-30^\circ c$. Star-connected CT secondaries are assumed.

Calculate differential and average restraint

With compensated matched phasors $J_H,J_L$, $D=|J_H+J_L|$ and $R=(|J_H|+|J_L|)/2$. Ehsan et al., Energies 14 (2021) 8447, DOI 10.3390/en14248447, attached PDF page 9, describes this average-restraint convention. Other relays use different restraint definitions.

Apply this explicit continuous characteristic

Slopes entered in percent are divided by 100. The two rising segments meet continuously at $R_k$; the minimum pickup is a floor. The input condition keeps the knee above that floor. Percentage pickup requires strictly $D>L$. This selectable example curve is not a reproduction of any manufacturer's entire relay characteristic.

Apply harmonic blocking and optional high set

Supplied $H_2=100D_2/D$ and $H_5=100D_5/D$ are ratios of compensated differential RMS components. An enabled harmonic block asserts at or above its threshold while $D>0$. Either block inhibits the low-set percentage element. This implements inrush restraint as blocking, not additive harmonic restraint. If enabled, the unrestrained element trips for $D>I_{high}$ regardless of either block or slope. The result is a static logical decision, not a trip time. ABB attached PDF pages 11–13 discusses the distinction and limitations; Ehsan et al. page 16 reports harmonic injection tests.

• Two-winding, balanced ABC positive-sequence equivalent with ideal ratio scaling and star-connected CT secondaries; no delta-connected CT circuits.
• Nameplate MVA and rated voltages define both winding bases. No measured tap-position compensation, automatic winding identification or polarity detection.
• The four operating presets are author-defined teaching examples: through fault 6 / 5.4 pu; internal fault 4 + 2 pu; inrush 3 / 0 pu with 20% second harmonic; overexcitation 1.2 / 0 pu with 40% fifth harmonic. Other preset harmonic ratios are 2% second and 1% fifth.
• The comparison table uses the through and internal examples with the selected compensation and protection settings, and 2% / 1% harmonics.
• Manual inputs represent prefiltered fundamental phasors and already computed differential harmonic ratios. Ratios are not calculated from winding harmonic percentages or from waveforms.
• Equality at the percentage or high-set threshold does not trip; equality at a harmonic threshold blocks. No hysteresis, harmonic sharing or time delay is modeled.

Use for balanced current matching, characteristic studies and interpretation of balanced three-phase injection tests. Clock rotation alone cannot represent unbalanced or earth-fault compensation: zero-sequence removal, negative-sequence transformation and the actual relay matrices are outside this model. It is not suitable for three-winding transformers, differential zones with multiple breakers or autotransformer neutral/tertiary studies.

CT under-reading in the through-fault preset is a static magnitude error, not a CT saturation simulation or a guarantee of security. CT excitation, burden, remanence, fault X/R, waveform filtering and adaptive restraint can change actual performance. No CT class or saturation adequacy is checked.

Fifth-harmonic blocking neither measures V/Hz nor protects the transformer from overexcitation damage; separate ANSI 24 protection and manufacturer withstand data are needed. Harmonics can also occur during faults, and inrush can have low second-harmonic content. The high-set bypass and all harmonic settings require the relay's own application rules.

IEEE C37.91-2021 was consulted for terminology and scope. Check against the standard in force and the relay manual in force. No standard tables or figures are reproduced; this calculator does not certify compliance.