How to draw transformer phasor diagram

As an indispensable equipment in power system, transformer plays a key role in energy transmission and voltage conversion. It is very important for power engineers to understand the working principle of transformer, especially the phase relationship between its internal voltage and current. As an intuitive graphic expression, Phasor Diagram can help us clearly analyze the electrical characteristics of transformers under different load conditions. This paper will introduce the drawing method and significance of transformer phasor diagram in detail.

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1. What is a transformer phasor diagram?

Transformer phasor diagram is a vector diagram used to express the relationship between amplitude and phase between voltage and current on primary side and secondary side of transformer. The relative relationship between voltage and current can be seen intuitively by phasor diagram, which is convenient for analyzing the running state and performance of transformer.

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Second, the basic assumptions and premises

When drawing transformer phasor diagram, it is usually based on the following assumptions:

1. Ignore the iron loss: regard the transformer as an ideal transformer and ignore the active component in the excitation current.
2. Attribution to the same side: Usually, the secondary side parameters are attributed to the primary side or vice versa for comparison in the same diagram.
3. Linear load: It is assumed that the load is linear, that is, the current is proportional to the voltage and the phase difference is constant.
4. Consistent frequency: the primary and secondary voltage and current frequencies are the same.

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Third, the basic steps of drawing phasor diagram

# 1. Determine the reference phasor

Usually, the primary voltage $ U_1 $ is taken as the reference phasor, which is set to the horizontal direction to the right.

# 2. Said the excitation current $ I_0 $

In an ideal transformer, the excitation current $ I_0 $ lags behind the primary voltage $ U_1 $ by about 90 degrees, because the excitation branch is an inductive element.

# 3. Reduce the secondary side current to $ I_2' $

According to the transformer ratio $ a = frac{N_1}{N_2} $,the secondary side current $ I_2 $ can be reduced to the primary side:

$$
I_2' = frac{I_2}{a}
$$

The current direction is related to the load property. For example, under resistive load, $ I_2' $ is in phase with the reduced secondary voltage $ U_2' $; Inductive load is lagging behind, capacitive load is leading.

# 4. Synthesize primary current $ I_1 $

According to the balance relation of magnetomotive force:

$$
I_1 = I_0 + I_2'
$$

The amplitude and phase of the primary current $ I_1 $ can be obtained by superimposing $ I_0 $ and $ I_2' $ vectors.

# 5. Draw the voltage drop

Consider the voltage drop caused by the internal resistance $ R_1 $ and the leakage reactance $ X_{σ1} $ of the primary winding, and show it on the phasor diagram:

$$
U_1 = E_1 + I_1(R_1 + jX_{σ1})
$$

Among them, $ E_1 $ is the induced electromotive force, which usually lags behind $ U_1 $ by a certain angle.

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Four, phasor diagram characteristics under different loads

1. No-load operation: At this time, $ I_2 = 0 $,$ I_1 = I_0 $,and the current is very small and lags behind the voltage by nearly 90.
2. Resistive load: the secondary current $ I_2 $ is in phase with the voltage $ U_2 $,and the primary current lags behind slightly.
3. Inductive load: the secondary current lags behind the voltage, the primary current lags behind more obviously, and the power factor is lower.
4. Capacitive load: The secondary current leads the voltage, which may improve the power factor of the primary side and even lead the current.

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V. Significance and application of phasor diagram

-Analysis of transformer performance: voltage regulation rate, power factor and other performance indicators can be directly seen through phasor diagram.
-Design and selection: It is helpful to select the appropriate transformer capacity and type to meet different load requirements.
-Teaching and research: it is an important tool to explain the principle of transformer in electrical engineering teaching.

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VI. Conclusion

To sum up, transformer phasor diagram is not only a graphical tool, but also an important means to understand the working principle of transformer. Mastering its drawing method is helpful for power engineers to understand the operating characteristics of transformers more deeply and provide theoretical support for practical engineering design and operation. With the increasing complexity of power system, mastering the drawing and analysis of phasor diagram will become one of the necessary skills for electrical engineers.