Vector Diagram of Transformer: An Essential Tool for Fault Analysis

A transformer is a device that transfers electrical energy from one circuit to another by electromagnetic induction. Transformers are widely used in power systems to step up or step down voltages, isolate circuits, and balance loads. Transformers can be classified into different types based on their construction, winding configuration, and vector group.

A vector diagram of a transformer graphically shows the phasor relationships between the primary and secondary voltages and currents. It is crucial for understanding how a transformer performs and behaves under different conditions and during faults.

In this article, we will explain what a vector diagram of a transformer is, how to draw it, and how to use it for fault analysis. We will also discuss the different types of transformer connections and vector groups and their implications for power system protection and coordination.

What is a Vector Diagram?

A vector diagram is a diagram on which one or more vectors can be represented. A vector is a quantity that has both magnitude and direction. In electrical engineering, alternating quantities such as voltages and currents are often represented by vectors, because they vary in magnitude and direction with time.

On a vector diagram, alternating quantities are represented by arrows. The length of the arrow represents the rms value of the alternating quantity. The angular position represents the phase angle of the quantity with respect to a reference axis or another quantity. The arrowhead represents the direction in which the quantity is acting.

When an electrical quantity acts away from the source towards the load, the vector representing the quantity is considered positive. When it acts towards the source from the load, it is considered negative.

What is a Vector Diagram of a Transformer?

A vector diagram of a transformer is a vector diagram that shows the phasor relationships between the primary and secondary voltages and currents of a transformer. It also shows the phase shift and polarity of the transformer windings.

You can draw a vector diagram for any transformer type, whether it’s single-phase or three-phase, star or delta connected, with various winding configurations and vector groups.

A vector diagram of a transformer can help us to:

• Determine the equivalent circuit parameters of a transformer, such as impedance, resistance, reactance, and losses.
• Analyze the performance and efficiency of a transformer under various loading conditions, such as no-load, full-load, over-load, or short-circuit.
• Identify and diagnose faults in a transformer or its associated circuits, such as open-circuit, short-circuit, earth-fault, or inter-turn fault.
• Select and coordinate protective devices for a transformer, such as fuses, circuit breakers, relays, or differential protection schemes.
• 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 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 vector diagram of a transformer are:

1. Choose a reference axis for the diagram. Usually, the horizontal axis is chosen as the reference axis.
2. Draw the primary voltage vector along the reference axis with its positive direction pointing to the right. Label it as V1.
3. Draw the secondary voltage vector with its length proportional to its rms value and its angle according to its vector group. Label it as V2.
4. Draw the primary current vector with its length proportional to its rms value and its angle according to its power factor. Label it as I1. The direction of I1 should be opposite to V1 if the primary winding is supplying power to the secondary winding.
5. Draw the secondary current vector with its length proportional to its rms value and its angle according to its power factor. Label it as I2. The direction of I2 should be opposite to V2 if the secondary winding is receiving power from the primary winding.
6. Draw the load impedance vector with its length proportional to its value and its angle according to its power factor. Label it as ZL. The direction of ZL should be opposite to I2 if the load is passive (resistive or inductive).
7. Draw any other vectors that are relevant for analysis, such as impedance, resistance, reactance, losses, etc.

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: Star-star
• Vector group: Yy0
• Load impedance: 6 ohms resistive

The vector diagram of this transformer is shown below:

The vector diagram shows the following phasor relationships:

• The primary and secondary voltages are in phase, as indicated by the vector group Yy0. This means that there is no phase shift between the windings.
• The primary and secondary currents are in phase, shown by a power factor of 1 (resistive load).
• The primary and secondary voltages are inversely proportional to their currents, as indicated by the transformer ratio of 2:1.
• 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:

• The apparent power of the transformer: S = V1I1 = V2I2 = 2400 VA
• The active power of the transformer: P = VIcosφ = S = 2400 W
• The reactive power of the transformer: Q = VIsinφ = 0 VAR
• The power factor of the transformer: cosφ = P/S = 1
• The impedance of the transformer: Z = V/I = 24 ohms on the primary side and 6 ohms on the secondary side
• The resistance of the transformer: R = Zcosφ = Z = 24 ohms on the primary side and 6 ohms on the secondary side
• The reactance of the transformer: X = Zsinφ = 0 ohms on both sides
• The losses of the transformer: Ploss = I2R = 2400 W on both sides

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

• Open-circuit fault: If one of the windings is open-circuited, the corresponding voltage vector will disappear, and the current vector will become zero. The other winding will still have its normal voltage and current vectors, but they will not be transferred to the open-circuited winding. This will result in a loss of power and efficiency.
• Short-circuit fault: If one of the windings is short-circuited, the corresponding voltage vector will become zero, and the current vector will become very large. The other winding will still have its normal voltage vector, but its current vector will also increase due to the increased load. This will result in a high current and low voltage condition, which can damage the transformer and its associated circuits.
• Earth-fault: If one of the windings is earthed, the corresponding voltage vector will be reduced and shifted by 90 degrees. The current vector will also be reduced and shifted by 90 degrees. The other winding will still have its normal voltage and current vectors, but they will not be balanced with the earthed winding. This will result in a distorted waveform and a leakage current to earth, which can affect the insulation and safety of the transformer.

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 a reference axis for the diagram. Usually, the horizontal axis is chosen as the reference axis.
2. Draw the primary voltage vectors along the reference axis with their positive directions pointing to the right. Label them as V1R, V1Y, and V1B for the red, yellow, and blue phases, respectively. The phase sequence should be R-Y-B or 1U-IV-1W for a positive sequence system.
3. Draw the secondary voltage vectors with their lengths proportional to their rms values and their angles according to their vector group. Label them as V2R, V2Y, and V2B for the red, yellow, and blue phases, respectively. The vector group indicates the clock hour figure that the secondary voltage hand is at when the primary voltage hand is at 12 o’clock. For example, if the vector group is Dyn11, it means that V2R is at 11 o’clock when V1R is at 12 o’clock. This implies a 30-degree phase shift between the primary and secondary voltages.
4. Draw the primary current vectors with their lengths proportional to their rms values and their angles according to their power factor. Label them as I1R, I1Y, and I1B for the red, yellow, and blue phases, respectively. The direction of I1 should be opposite to V1 if the primary winding is supplying power to the secondary winding.
5. Draw the secondary current vectors with their lengths proportional to their rms values and their angles according to their power factor. Label them as I2R, I2Y, and I2B for the red, yellow, and blue phases respectively. The direction of I2 should be opposite to V2 if the secondary winding is receiving power from the primary winding.
6. Draw the load impedance vectors with their lengths proportional to their values and their angles according to their power factor. Label them as ZLR, ZLY, and ZLB for the red, yellow, and blue phases, respectively. The direction of ZL should be opposite to I2 if the load is passive (resistive or inductive).
7. Draw any other vectors that are relevant for analysis, such as impedance, resistance, reactance, losses, etc.

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: 50 A
• Rated secondary current: 1500 A
• Winding configuration: Delta-star
• Vector group: Dyn11
• Load impedance: 0.16 ohms resistive

The vector diagram of this transformer is shown below:

The vector diagram shows the following phasor relationships:

• The primary and secondary voltages are displaced by 30 degrees, as indicated by the vector group Dyn11. This means that there is a 30-degree phase shift between the windings.
• The primary and secondary currents are in phase, as indicated by the power factor of 1 (resistive load).
• The primary and secondary voltages are inversely proportional to their currents, as indicated by the transformer ratio of 27.5:1.
• 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:

• The apparent power of the transformer: S = V1I1 = V2I2 = 605 kVA
• The active power of the transformer: P = VIcosφ = S = 605 kW
• The reactive power of the transformer: Q = VIsinφ = 0 kVAR
• The power factor of the transformer: cosφ = P/S = 1
• The impedance of the transformer: Z = V/I = 220 ohms on the primary side and 0.27 ohms on the secondary side.
• The resistance of the transformer: R = Zcosφ = Z = 220 ohms on the primary side and 0.27 ohms on the secondary side
• The reactance of the transformer: X = Zsinφ = 0 ohms on both sides
• The losses of the transformer: Ploss = I2R = 202.5 kW on both sides

Conclusion

A vector diagram of a transformer is a useful tool for understanding and analyzing the phasor relationships between the primary and secondary voltages and currents of a transformer. It can help us to determine the equivalent circuit parameters, performance, efficiency, and fault conditions of a transformer. It can also help us to select and coordinate the protective devices for a transformer.

A vector diagram of the transformer can be drawn for any type of transformer, such as single-phase or three-phase, star or delta connected, or with different winding configurations and vector groups. The vector group of a transformer indicates the phase shift and polarity of the windings, which affects the power transfer and fault analysis.

To draw a vector diagram of the transformer, we need to know the rated voltage and current, winding configuration and connection, vector group, and load impedance and power factor of the transformer. We can then use the following steps:

1. Choose a reference axis for the diagram.
2. Draw the primary voltage vector along the reference axis.
3. Draw the secondary voltage vector according to its vector group.
4. Draw the primary current vector according to its power factor.
5. Draw the secondary current vector according to its power factor.
6. Draw the load impedance vector according to its power factor.
7. Draw any other vectors that are relevant for analysis.

We hope this article has helped you to learn more about vector diagrams of transformers and how to draw them. If you have any questions or feedback, please feel free to leave a comment below. Thank you for reading!