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Vector Diagram of Transformer: An Essential Tool for Fault Analysis

Key learnings:

Vector Diagram Definition: A transformer’s vector diagram visually depicts the phase relationships between its primary and secondary voltages and currents using phasors.
Drawing Vector Diagrams: Constructing a transformer’s vector diagram requires knowledge of its rated voltages and currents, winding configuration, vector group, and load impedance.
Fault Analysis: Vector diagrams are essential for detecting and troubleshooting transformer issues like open circuits, short circuits, and ground faults.
Three-Phase Vector Diagram: For three-phase systems, vector diagrams illustrate the phase shifts dictated by the transformer’s vector group, providing insight into power flow.
Importance in Fault Analysis: Vector diagrams are key tools for analyzing transformer performance, efficiency, and fault conditions, and they assist in selecting and coordinating protective devices.

Transformers utilize electromagnetic induction to transfer electrical energy between circuits. Commonly employed in power systems, they adjust voltage levels, isolate circuits, and balance loads. Transformers are categorized by construction, winding configuration, and vector group.

A transformer’s vector diagram visually represents the phase relationships between its primary and secondary voltages and currents, providing key insights into its performance and behavior under various operating conditions and fault scenarios.

This article explains transformer vector diagrams, including their construction, application in fault analysis, and the impact of different transformer connections and vector groups on power system protection and coordination.

Table of Contents

What is a Vector Diagram?

A vector diagram is a graphical representation used to depict one or more vectors. A vector itself is a quantity possessing both magnitude and direction. In the realm of electrical engineering, alternating quantities like voltage and current are frequently represented as vectors due to their inherent characteristic of varying in both magnitude and direction over time. This dynamic nature makes vectors a suitable tool for their analysis.

Within a vector diagram, these alternating quantities are visually represented by arrows. The length of each arrow corresponds to the root mean square (RMS) value of the respective alternating quantity. This length provides a direct visual indication of the magnitude of the voltage or current. Furthermore, the angular position of the arrow, measured with respect to a chosen reference axis or another relevant quantity, represents the phase angle of the quantity. This angle is crucial as it indicates the temporal relationship between different alternating quantities. Finally, the arrowhead signifies the direction in which the quantity is acting. This directional information is vital for understanding the flow of current and voltage within a circuit.

A convention is used to establish the polarity of these vectors. When an electrical quantity acts away from the source and towards the load, the vector representing that quantity is considered positive. Conversely, when the quantity acts towards the source, originating from the load, the vector is considered negative. This convention provides a consistent framework for interpreting the direction and relationship of electrical quantities within a vector diagram.

What is a Vector Diagram of a Transformer?

A transformer’s vector diagram graphically represents the phasor relationships between its primary and secondary voltages and currents, illustrating both the phase shift introduced by the transformer and the winding polarity.

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:

Transformer analysis involves several key tasks:

Parameter Determination: Calculating the equivalent circuit parameters, including impedance, resistance, reactance, and losses.
Performance Evaluation: Analyzing performance and efficiency under various load conditions, ranging from no-load to full-load, overload, and short-circuit.
Fault Diagnosis: Identifying and diagnosing faults within the transformer or its associated circuitry, such as open circuits, short circuits, ground faults, or inter-turn faults.
Protection Design: Selecting and coordinating appropriate protective devices like fuses, circuit breakers, relays, or differential protection schemes.
Installation Verification: Confirming correct connection and polarity during transformer installation and commissioning.

How to Draw a Vector Diagram of a Transformer?

Constructing a transformer vector diagram requires specific information: the rated voltage and current of both primary and secondary windings, the winding configuration (e.g., star or delta), the vector group (indicating phase shift and winding polarity), and the load impedance and power factor.

The following steps outline the drawing process:

Establish a Reference: Select a reference axis, typically the horizontal axis.
Primary Voltage (V1): Draw the primary voltage vector (V1) along the reference axis, pointing right (positive direction).
Secondary Voltage (V2): Draw the secondary voltage vector (V2), its length proportional to its RMS value and its angle determined by the transformer’s vector group.
Primary Current (I1): Draw the primary current vector (I1), its length proportional to its RMS value and its angle determined by the power factor. I1’s direction opposes V1 if the primary winding supplies power.
Secondary Current (I2): Draw the secondary current vector (I2), its length proportional to its RMS value and its angle determined by the power factor. I2’s direction opposes V2 if the secondary winding receives power.
Load Impedance (ZL): Draw the load impedance vector (ZL), its length proportional to its value and its angle determined by the power factor. ZL’s direction opposes I2 for passive loads.
Additional Vectors: Include any other relevant vectors, such as impedance, resistance, reactance, or losses, as needed for the analysis.

Example: Single-Phase Transformer Vector Diagram
Consider a single-phase transformer with these specifications:

Primary Voltage: 240 V
Secondary Voltage: 120 V
Primary Current: 10 A
Secondary Current: 20 A
Winding Configuration: Star-star
Vector Group: Yy0
Load Impedance: 6 ohms (resistive)

The vector diagram reveals several key relationships:

Ground Fault: A grounded winding leads to reduced and 90° shifted voltage and current vectors. The other winding retains normal vectors, but imbalance occurs, resulting in waveform distortion and leakage current, potentially compromising insulation and safety.

Constructing a three-phase transformer vector diagram requires the following information: the rated primary and secondary voltages and currents, the winding configuration (star or delta), the vector group (indicating phase shift and polarity), and the load impedance and power factor.
Here’s the step-by-step process:

Reference Axis: Choose a reference axis, typically horizontal.
Primary Voltages (V1R, V1Y, V1B): Draw the primary voltage vectors (V1R, V1Y, V1B) along the reference axis, representing the red, yellow, and blue phases (or 1U, 1V, 1W for positive sequence), ensuring correct phase sequence (R-Y-B or 1U-1V-1W).
Secondary Voltages (V2R, V2Y, V2B): Draw the secondary voltage vectors (V2R, V2Y, V2B), their lengths proportional to their RMS values. Their angles are determined by the vector group. The vector group notation (e.g., Dyn11) indicates the phase relationship. For example, Dyn11 means V2R is at the 11 o’clock position when V1R is at 12 o’clock, implying a 30° phase shift.
Primary Currents (I1R, I1Y, I1B): Draw the primary current vectors (I1R, I1Y, I1B), their lengths proportional to their RMS values, and their angles based on the power factor. These currents oppose the primary voltages if the primary winding is supplying power.
Secondary Currents (I2R, I2Y, I2B): Draw the secondary current vectors (I2R, I2Y, I2B), their lengths proportional to their RMS values, and their angles based on the power factor. These currents oppose the secondary voltages if the secondary winding is receiving power.
Load Impedances (ZLR, ZLY, ZLB): Draw the load impedance vectors (ZLR, ZLY, ZLB), their lengths proportional to their values, and their angles based on the power factor. They oppose the secondary currents for passive loads.
Additional Vectors: Add any other relevant vectors (impedance, resistance, reactance, losses, etc.) as needed for the analysis.

Example: Vector Diagram of a Three-Phase Transformer

Consider a three-phase transformer with the following specifications:

Primary Voltage: 11 kV
Secondary Voltage: 400 V
Primary Current: 50 A
Secondary Current: 1500 A
Winding Configuration: Delta-star
Vector Group: Dyn11
Load Impedance: 0.16 ohms (resistive)
The vector diagram for this transformer (shown below) illustrates the following phasor relationships:
Phase Displacement: The Dyn11 vector group indicates a 30° phase shift between the primary and secondary voltages.
Current Phasing: The primary and secondary currents are in phase due to the resistive load (power factor = 1).
Voltage/Current Proportionality: The transformer’s 27.5:1 turns ratio is reflected in the inverse proportionality between voltages and currents.
Ohm’s Law Verification: Ohm’s Law (load impedance = secondary voltage / secondary current) is confirmed by the diagram.
The vector diagram facilitates the calculation of various quantities:
Apparent Power (S): S = V1I1 = V2I2 = 605 kVA
Active Power (P): P = VIcosφ = S = 605 kW
Reactive Power (Q): Q = VIsinφ = 0 kVAR
Power Factor (cosφ): cosφ = P/S = 1
Impedance (Z): Z = V/I = 220 ohms (primary), 0.27 ohms (secondary)
Resistance (R): R = Zcosφ = Z = 220 ohms (primary), 0.27 ohms (secondary)
Reactance (X): X = Zsinφ = 0 ohms (both sides)
Losses (Ploss): Ploss = I²R = 202.5 kW (both sides)

Conclusion

In summary, a transformer’s vector diagram is a valuable tool for understanding and analyzing the phasor relationships between its primary and secondary voltages and currents. It aids in determining equivalent circuit parameters, performance, efficiency, and fault conditions, as well as in selecting and coordinating protective devices.
Applicable to various transformer types (single-phase or three-phase, star or delta connected, and with diverse winding configurations and vector groups), the vector diagram visually represents the phase shift and polarity of the windings, which are crucial factors in power transfer and fault analysis.
Constructing a vector diagram requires knowledge of the transformer’s rated voltage and current, winding configuration and connection, vector group, and load impedance and power factor. The process involves:

Selecting a reference axis.
Drawing the primary voltage vector.
Drawing the secondary voltage vector (based on the vector group).
Drawing the primary current vector (based on the power factor).
Drawing the secondary current vector (based on the power factor).
Drawing the load impedance vector (based on the power factor).
Including any other relevant vectors for the analysis.

Vector| Definition, physics & Facts