A transformer nameplate carries a short code such as Dyn11 or YNyn0, and that code decides three things at once: the phase displacement between the two sides, the relationship between the line voltage ratio and the actual winding turns ratio, and whether zero-sequence current can flow through the unit at all. Get the first wrong and differential protection misoperates. Get the second wrong and your short-circuit study is out by a factor of √3. Get the third wrong and the earth-fault current you calculated does not exist.
This calculator takes the two winding connections, the clock number, and the nameplate ratings, then reports the phase shift, both ratios, the impedance converted onto your system base and split into R and X, and the zero-sequence behaviour. It draws the winding arrangement, the clock phasor diagram, and the zero-sequence equivalent circuit, all of which redraw as you change the connection.
Transformer Connections & Vector Groups
Phase shift, turns ratio, per-unit impedance and the zero-sequence equivalent circuit
The clock notation, and the convention used here
IEC 60076-1 writes the vector group as an uppercase letter for the HV winding, a lowercase letter for the LV winding, and a clock number. Y or y is star, D or d is delta, Z or z is zigzag, and an added N or n means that winding’s neutral is brought out and earthed. The clock number is read on a clock face: put the HV line-to-neutral phasor of the reference phase at 12 o’clock, then read off the hour at which the corresponding LV phasor sits.
The convention adopted in this tool, stated plainly because the standards describe it differently: the clock face is read clockwise, phasors rotate anticlockwise, so an LV phasor sitting at hour \(n\) lags the HV phasor by
\[ \theta_{LV}=30^{\circ}\times n \]
On this convention Dyn11 means the LV lags the HV by 330 degrees, which is the same angle as leading by 30 degrees. That is the usual reading of Dyn11 in IEC practice. IEEE C57.12.00 approaches the same physics from the other end: it defines the standard three-phase delta-wye connection so that the HV quantities lead the LV quantities by 30 degrees, which corresponds to the IEC clock number 1 rather than 11. Both statements are internally consistent; what matters is that you know which one your data follows. Whenever you copy a vector group from a nameplate into a relay or a study case, check whether the target software wants an IEC clock number or an IEEE lead angle, and check the reference phase.
The parity rule is not arbitrary. A star and a delta winding on the same core produce line-to-neutral phasors 30 degrees apart, so any pairing of one star winding with one delta or zigzag winding can only land on odd hours. Two windings of the same kind can only land on even hours. The calculator offers only the hours that are physically reachable for the pair you select.
Line ratio is not the turns ratio
The nameplate quotes line-to-line voltages. The windings themselves see phase voltages, and the relationship between the two depends on the connection. A star winding sees \(V_{LL}/\sqrt{3}\); a delta winding sees the full \(V_{LL}\). When both sides are the same kind the two factors cancel and the winding turns ratio equals the line voltage ratio:
\[ \frac{N_{HV}}{N_{LV}}=\frac{V_{HV}}{V_{LV}} \qquad \text{(Yy and Dd)} \]
When the two sides differ, a factor of √3 appears and it does not cancel. For a delta HV winding feeding a star LV winding:
\[ \frac{N_{HV}}{N_{LV}}=\sqrt{3}\,\frac{V_{HV}}{V_{LV}} \qquad \text{(Dy)}, \qquad \frac{N_{HV}}{N_{LV}}=\frac{1}{\sqrt{3}}\cdot\frac{V_{HV}}{V_{LV}} \qquad \text{(Yd)} \]
This is the classic source of error. A 132/33 kV transformer has a line voltage ratio of 4 : 1 whatever its connection, but if it is Dyn11 the HV winding carries 132 kV across it while each LV winding carries 33/√3 = 19.05 kV, so the turns ratio is 4√3 = 6.928 : 1. Tap changers make it worse, because a tap expressed as a percentage on the HV side moves the HV winding voltage and therefore both ratios together. The calculator applies the tap to the HV side and reports both numbers side by side so the difference is visible rather than assumed.
Zigzag windings are a third case. Each phase is split into two half-windings placed on different limbs, so the line-to-neutral voltage is √3 times a half-winding voltage rather than 2 times it. For the same line voltage a zigzag winding needs about 2/√3, roughly 15%, more copper than a star winding, which is the price paid for its zero-sequence behaviour.
Impedance and the change of base
A transformer’s percentage impedance is quoted on its own MVA rating and its own rated voltage. A system study needs it on the common study base, and the conversion is the standard change-of-base relation:
\[ Z_{pu,sys}=Z_{pu,rated}\cdot\frac{S_{sys}}{S_{rated}}\cdot\left(\frac{V_{rated}}{V_{sys}}\right)^{2} \]
The MVA term is linear and the voltage term is squared, which is why choosing the voltage base to match the transformer’s rated voltage on each side removes the second factor entirely. That is the whole reason per-unit is used across transformers: pick the voltage bases in the same ratio as the transformer, and the ideal ratio disappears from the network model, leaving only the series impedance.
The X/R ratio splits that impedance into its components, which is what a fault study needs for the DC offset and what a load flow needs for the loss calculation:
\[ R=\frac{Z}{\sqrt{1+(X/R)^{2}}}, \qquad X=\frac{Z\,(X/R)}{\sqrt{1+(X/R)^{2}}} \]
The ohmic value referred to the HV side follows from the base impedance at that voltage:
\[ Z_{HV}=\frac{Z\%}{100}\cdot\frac{V_{HV}^{2}}{S_{rated}} \]
Zero-sequence paths: the part that decides your earth-fault study
Positive and negative sequence pass through a transformer more or less unchanged apart from the phase shift. Zero sequence does not. Whether zero-sequence current can flow depends entirely on the winding connection and on whether the neutral is earthed, and the standard way to capture that is the three-terminal zero-sequence equivalent circuit that the tool draws.
Read it as a series zero-sequence impedance between two internal nodes, one per winding, with a switch at each terminal and a switch from each internal node down to the zero-potential reference bus. The rules are:
- Earthed star (YN): the terminal switch closes. Zero-sequence line current can enter the winding and return through the neutral earth connection.
- Isolated star (Y): the terminal switch opens. There is no return path, so no zero-sequence current flows in the lines at all.
- Delta (D): the terminal switch stays open because no zero-sequence current appears in the delta’s lines, but the shunt switch to the reference bus closes, because the equal zero-sequence voltages drive a circulating current around the closed delta. A delta therefore provides a path for the other winding while blocking transfer beyond itself.
- Zigzag (Z, ZN): the two half-windings of a phase sit on different limbs and their zero-sequence ampere-turns cancel inside the winding. An earthed zigzag therefore presents a low zero-sequence impedance to its own network but does not couple zero sequence through to the other winding, which is exactly what an earthing transformer is for.
Put those rules together and the standard cases fall out. YNyn0 with both neutrals earthed transfers zero-sequence current between the two networks. YNd1 gives an earth-fault source on the star side and blocks transfer to the delta side. Dyn11 gives an earth-fault source on the LV star side, which is why it is the near-universal distribution connection: the LV network gets a solid earth reference while the HV network is screened from LV residual current. Yy with neither neutral earthed blocks zero sequence completely, so the transformer contributes nothing to earth-fault current on either side.
Worked example: a 40 MVA 132/33 kV Dyn11
Load the example in the calculator. The vector group is Dyn11, so on the convention above the LV lags the HV by 330 degrees, or equivalently leads by 30 degrees, and the clock diagram shows the LV phasors sitting one hour anticlockwise from the HV set.
The line voltage ratio is 132/33 = 4.000 : 1. The winding turns ratio is 4√3 = 6.928 : 1, because the delta winding sees 132 kV while each star winding sees 19.05 kV. Rated currents are 175 A on the HV side and 700 A on the LV side.
The nameplate impedance of 12% is on 40 MVA. On a 100 MVA study base at 132 kV the change of base gives 12 × (100/40) × 1 = 30%, that is 0.30 pu. With X/R = 20 that splits into R = 0.0150 pu and X = 0.2996 pu. Referred to the HV side the impedance is 0.12 × 132²/40 = 52.3 ohm.
The zero-sequence circuit shows the HV terminal switch open with the delta shunt closed, and the LV terminal switch closed. Zero-sequence current can flow into the LV network and return through the LV neutral, circulating in the HV delta, but none of it appears as residual current on the 132 kV lines. Earth-fault protection on the 33 kV network can therefore be graded without reference to the 132 kV earth-fault scheme.
Zero-sequence classification bands
- Transfers zero sequence: both neutrals earthed and neither winding cancels the zero-sequence ampere-turns (YNyn, and variants of it). Residual current crosses between the two networks, so earth-fault grading has to be coordinated across the transformer and non-directional residual elements can over-reach.
- Zero-sequence source on LV: the LV winding is an earthed star or earthed zigzag and the HV winding is a delta or zigzag (Dyn, Dzn, YNd viewed from the other side). The LV network gets a defined earth-fault level; the HV side sees no residual current. This is the standard distribution arrangement.
- Zero-sequence source on HV: the mirror case (YNd, ZNd). The delta side is screened, and the delta also acts as a stabilising winding for the zero-sequence flux.
- Source on both sides, no transfer: both windings offer an internal zero-sequence path and both neutrals are earthed, so each network is earthed independently.
- Blocks zero sequence: no terminal has a usable zero-sequence path (Yy, YNy, Dd). The transformer contributes nothing to earth-fault current. You need a separate earthing transformer or neutral earthing arrangement, or displacement protection instead of residual overcurrent.
Working with per-unit and change of base?
The free Per-Unit System Cheat Sheet sets out base current, base impedance, and the change-of-base relation used above, on one printable page.
Frequently asked questions
Does Dyn11 mean the LV leads or lags?
It depends on the convention, which is why this tool states its own. Reading the clock face clockwise with the HV phasor at 12 o’clock, the LV phasor at hour 11 lags the HV by 330 degrees, which is identical to leading by 30 degrees. IEC practice normally describes Dyn11 as the LV leading the HV by 30 degrees. IEEE C57.12.00 defines the standard delta-wye connection with the HV leading the LV by 30 degrees, which is clock number 1. Neither is wrong; they are different reference choices. Always confirm which one your relay or study software expects.
Why can the clock number only be odd or only be even?
Because the geometry allows nothing else. Two star windings, or two delta windings, produce phasor sets that can only be displaced by multiples of 60 degrees, giving even hours. A star winding paired with a delta or zigzag winding introduces the intrinsic 30 degree displacement between line and phase quantities, so only odd hours are reachable. Any other angle would require a phase-shifting transformer with a separate regulating winding.
Why does a delta winding block zero-sequence current?
The three zero-sequence voltages are equal and in phase, so around a closed delta they add to three times the phase value and drive a circulating current inside the delta loop. That circulating current balances the zero-sequence ampere-turns of the other winding, which is why the delta provides a path for the earthed-star side. But the circulating current never leaves the delta, so no zero-sequence current appears in the lines connected to it. The delta both enables and blocks, depending on which side you look from.
Why does the phase shift matter for differential protection?
A differential relay compares currents entering and leaving the protected zone. Across a Dy transformer those currents differ by 30 degrees and by the connection’s current distribution, so an uncompensated comparison shows a large standing differential current under healthy load. Traditionally the CT connections were arranged to compensate; modern numerical relays apply a matrix correction selected from the vector group setting. Entering the wrong clock number is a well-known cause of spurious tripping on energisation.
Is the zero-sequence impedance the same as the positive-sequence impedance?
Not in general. For a transformer with a delta winding the zero-sequence impedance is usually close to the positive-sequence value, often quoted around 0.85 to 1.0 times it, and that is the approximation this tool makes. For star-star units without a delta the zero-sequence impedance depends strongly on the core construction: a three-limb core forces the zero-sequence flux through the tank and gives a moderate impedance, while a five-limb or shell-type core gives a very high one. Use the manufacturer’s tested zero-sequence figure when it matters.
Related reading
- All EE Power School calculators: the full set of interactive power system tools.
- Symmetrical Components Calculator: decompose unbalanced phasors into positive, negative, and zero sequence, the framework this page’s zero-sequence circuit belongs to.
- Two-Bus Load Flow Calculator: see what the transformer impedance you converted here does to the voltage profile.
- SCR Calculator: screen grid strength at a point of interconnection using the same per-unit conventions.
References
- IEC 60076-1, Power transformers: General (vector group and connection symbol notation).
- IEEE C57.12.00, Standard for General Requirements for Liquid-Immersed Distribution, Power, and Regulating Transformers (angular displacement convention).
- J. D. Glover, M. S. Sarma and T. J. Overbye, Power System Analysis and Design, Cengage (transformer sequence networks).
- P. M. Anderson, Analysis of Faulted Power Systems, IEEE Press, 1995 (zero-sequence equivalent circuits for transformer connections).