Vector Diagram of Transformer: An Essential Tool for Fault Analysis

A transformer transfers AC electrical energy between coupled windings by electromagnetic induction. In power systems, transformers change voltages, provide galvanic isolation when designed for it and connect systems with specified winding arrangements. Three-phase units are also identified by their vector group.

A transformer vector diagram, more precisely called a phasor diagram, shows the steady-state magnitude and phase relationships between primary and secondary voltages and currents. It shows how winding displacement and internal voltage drop affect a stated load condition. Fault analysis still needs a system model and measured or calculated fault quantities.

This article explains the phasor convention, a practical drawing sequence and the effect of three-phase winding connections. It also shows where a phasor diagram supports protection work and where fault studies require sequence networks, transformer impedance, earthing details and relay settings.

What is a Vector Diagram?

A phasor is a complex-number representation of a sinusoidal quantity at one fixed frequency. It stores magnitude and phase while suppressing the common time factor. This lets engineers compare alternating voltages and currents without drawing each waveform against time.

On a phasor diagram, an arrow’s length represents magnitude and its angle represents phase relative to a chosen reference. Transformer diagrams commonly use the rms value. Peak magnitude is another valid convention when it is stated and used consistently. The arrowhead indicates phase angle. Circuit reference arrows separately define current direction and power flow.

Voltage polarity and current reference direction must be defined before a diagram is drawn. A negative phasor means a 180-degree reversal from the selected reference, not simply motion from load to source. Dot markings and passive or generator sign conventions determine the signs used in transformer equations.

What is a Vector Diagram of a Transformer?

A transformer phasor diagram relates terminal voltages, induced emfs, winding currents, exciting current and internal voltage drops under stated operating conditions. For a three-phase transformer, it also shows the displacement caused by the connection and polarity of the windings.

A single-phase diagram uses the selected polarity and current references. A three-phase diagram must also distinguish phase quantities from line quantities and apply the stated star, delta or zigzag connection. IEC vector-group clock notation belongs to three-phase transformer connections.

A vector diagram of a transformer can help us to:

• Interpret an equivalent circuit containing winding impedance, resistance, leakage reactance and exciting-branch quantities. Open-circuit and short-circuit test data are needed to determine their values.
• Analyse voltage regulation and phase relationships at no load or under a specified load. Efficiency also needs measured or estimated core and winding losses.
• Compare expected and measured phasors during an open-circuit, short circuit, earth fault or winding fault. A diagram supports diagnosis but does not identify a fault by itself.
• Apply vector-group compensation in differential protection and interpret relay quantities. Selecting fuses, circuit breakers or relays also requires fault-current, inrush, thermal and time-current studies.
• Verify the correct connection and polarity of a transformer during installation or commissioning.

How to Draw a Vector Diagram of a Transformer?

To draw a vector diagram of a transformer, we need to know the following information:

• The rated voltage, turns ratio and current bases for the primary and secondary windings.
• The winding configuration and connection of the transformer, such as star or delta.
• The vector group of the transformer, which indicates the phase shift and polarity of the windings.
• The load impedance and power factor, plus whether current leads or lags voltage.

The steps to draw a vector diagram of a transformer are:

1. Choose one phasor, often the applied primary voltage V1, as the zero-degree reference and state the polarity and current directions.
2. Draw V1 on the reference axis. If internal drops matter, relate V1 to the primary induced emf E1 through the winding resistance and leakage reactance.
3. Refer E1 to the secondary by the turns ratio and selected polarity to obtain E2. Then obtain terminal voltage V2 after the secondary internal voltage drop. A single-phase transformer has no IEC clock-number vector group.
4. Draw secondary current I2 at the angle set by the load power factor relative to V2. It lags for an inductive load, leads for a capacitive load and is in phase for an ideal resistive load.
5. Refer I2 to the primary using the inverse turns ratio and the chosen dot convention. Add the exciting current phasor I0 to the referred load current to obtain primary current I1.
6. Draw the voltage-drop phasors across winding resistance and leakage reactance. Load impedance ZL is the complex ratio V2/I2, not a rotating time-domain phasor that points opposite to current.
7. Add induced emf, core flux, exciting current or referred quantities only when the selected equivalent-circuit model needs them.

Example: Vector Diagram of a Single-Phase Transformer

Let us consider a single-phase transformer with the following specifications:

• Rated primary voltage: 240 V
• Rated secondary voltage: 120 V
• Rated primary current: 10 A
• Rated secondary current: 20 A
• Winding configuration: Two-winding single-phase transformer
• Vector group: Not applicable; corresponding voltage polarities are assumed in phase
• Load impedance: 6 ohms resistive

The vector diagram of this transformer is shown below:

The vector diagram shows the following phasor relationships:

• With the assumed polarity, the ideal primary and secondary voltage phasors are in phase. The in-phase relation comes from that polarity choice. IEC Yy0 notation applies to a three-phase winding connection.
• Each local current is in phase with its local terminal voltage for the resistive load. A 180-degree reversal can appear between referred winding currents depending on the chosen current directions and dot convention.
• For an ideal transformer, V1/V2 = N1/N2 = 2 and I1/I2 = N2/N1 = 1/2. The inverse relationship applies to the voltage and current ratios, not to two phasors merely because they are drawn together.
• The load impedance is equal to the secondary voltage divided by the secondary current, as indicated by Ohm’s law.

The vector diagram can be used to calculate the following quantities:

• Ideal apparent power at full load: S = V1I1 = V2I2 = 2400 VA
• Ideal load active power at unity power factor: P = VIcosφ = 2400 W
• Load reactive power for the stated resistive load: Q = VIsinφ = 0 var
• Load power factor: cosφ = P/S = 1
• Load impedance: ZL = V2/I2 = 6 ohms; referred to the primary of an ideal transformer, ZL’ = 24 ohms
• Load resistance: R = ZL = 6 ohms on the secondary and 24 ohms when referred to the primary
• Load reactance: X = 0 ohms for the stated purely resistive load
• Transformer loss: it cannot be calculated from the stated load resistance; I2R = 2400 W here is power delivered to the load, while transformer loss needs winding-resistance and core-loss data

The vector diagram can also be used to analyze the fault conditions of the transformer, such as:

• Open-circuit condition: If the secondary circuit opens while the primary remains energised, secondary current becomes zero but secondary terminal voltage can remain near its no-load value. The primary draws only exciting current under normal no-load conditions. An open primary instead de-energises the transformer.
• Short-circuit fault: Voltage at the fault collapses according to fault impedance, while current is limited by source impedance, transformer leakage impedance and the fault path. The phasors must come from that network model; neither the entire winding voltage nor every fault resistance is automatically zero.
• Earth fault: Voltage and current depend on fault location, winding connection, neutral earthing, zero-sequence paths and fault impedance. There is no universal 90-degree shift. Sequence networks and vector-group compensation are used to calculate and compare the quantities seen by earth-fault and differential protection.

How to Draw a Three-Phase Vector Diagram of a Transformer?

To draw a three-phase vector diagram of the transformer, we need to know the following information:

• The rated voltage and current of the primary and secondary windings of the transformer.
• The winding configuration and connection of the transformer, such as star or delta.
• The vector group of the transformer, which indicates the phase shift and polarity of the windings.
• The load impedance and power factor of the transformer.

The steps to draw a three-phase vector diagram of the transformer are:

1. Choose one primary phase-to-neutral or line-to-line voltage as the zero-degree reference and state the positive phase sequence.
2. Draw the three primary voltage phasors 120 degrees apart. Label them V1R, V1Y and V1B, or use terminal labels 1U, 1V and 1W. Only the selected reference phasor lies on the horizontal axis.
3. Apply the winding connection, polarity and clock numeral to draw the secondary voltages at the correct scale and displacement. In Dyn11, D denotes an HV delta, yn denotes an LV star with neutral and 11 denotes a 330-degree displacement from the HV reference in the IEC clock convention. For positive sequence this is commonly described as the LV voltage leading the corresponding HV voltage by 30 degrees.
4. Draw each secondary phase current at its load power-factor angle relative to the corresponding secondary phase voltage. Derive secondary line currents from the star or delta connection.
5. Refer the winding currents through the turns ratio and polarity, then derive the primary line currents from the primary connection. The line-current displacement is affected by the winding connection and cannot be obtained from load power factor alone.
6. Calculate each complex phase impedance from ZL = Vphase/Iphase. Show its angle on a complex-impedance plane if useful, but do not treat impedance as a time-varying phasor with a physical direction.
7. Add internal voltage drops, exciting current or positive-, negative- and zero-sequence quantities when the analysis requires them.

Example: Vector Diagram of a Three-Phase Transformer

Let us consider a three-phase transformer with the following specifications:

• Rated primary voltage: 11 kV
• Rated secondary voltage: 400 V
• Rated primary current: 52.5 A
• Rated secondary current: 1443 A
• Winding configuration: Delta-star
• Vector group: Dyn11
• Load impedance: 0.16 ohms per secondary star phase, resistive

The vector diagram of this transformer is shown below:

The vector diagram shows the following phasor relationships:

• Dyn11 specifies a 30-degree displacement between corresponding HV and LV voltage references under the IEC clock convention, with the stated connection and phase sequence.
• Each secondary phase current is in phase with its secondary phase voltage for the resistive load. Primary and secondary line currents are not simply in phase when both are referred to one common reference because delta-star connection and polarity must be included.
• The line-to-line voltage ratio is 11,000/400 = 27.5. At equal apparent power, the line-current magnitudes have approximately the inverse ratio.
• For the star-connected secondary, phase voltage is 400/√3 V and phase current equals line current, so Ohm’s law gives about 0.16 ohms per phase.

The vector diagram can be used to calculate the following quantities:

• Three-phase apparent power: S = √3VlineIline, approximately 1.00 MVA on either side
• Ideal active load power at unity power factor: P = Scosφ, approximately 1.00 MW
• Load reactive power for the stated resistive load: Q = Ssinφ = 0 kvar
• Load power factor: cosφ = P/S = 1
• Secondary phase load impedance: ZL = (400/√3)/1443, approximately 0.16 ohms
• Secondary phase load resistance: R = ZLcosφ, approximately 0.16 ohms
• Secondary phase load reactance: X = ZLsinφ = 0 ohms
• Transformer loss: it cannot be derived from the load impedance; core-loss and winding-resistance or efficiency-test data are required

Conclusion

A transformer phasor diagram shows steady-state voltage and current magnitudes and angles under a stated reference convention. Combined with an equivalent circuit, it explains turns ratio, load power factor, internal voltage drop and voltage regulation. Test data determine model parameters and losses, while network and sequence models determine fault quantities.

Single-phase diagrams use dot polarity and stated current references. Three-phase diagrams add line-to-phase relationships, phase sequence and the IEC vector-group displacement. Those details matter for parallel operation, metering, differential protection and zero-sequence treatment.

Start with a voltage reference and the winding polarity. Then apply the turns ratio, connection, load power factor and internal impedance model in a consistent RMS phasor convention:

1. Choose one voltage phasor as the zero-degree reference and state the phase sequence.
2. Draw all primary voltage phasors using the selected line or phase convention.
3. Draw secondary voltages from turns ratio, polarity, connection and vector group where applicable.
4. Draw load current from the secondary voltage and load power factor.
5. Refer winding current through the transformer and derive line current for each connection.
6. Draw voltage drops across the transformer’s resistance and leakage reactance.
7. Add exciting current, induced emf or sequence quantities only when the analysis needs them.

A phasor diagram is an analysis aid, not a substitute for transformer tests or a protection study. Its value comes from explicit references, correct line-to-phase conversion and consistent treatment of winding polarity, current direction and vector-group displacement.