In electric energy metering, incorrect wiring of the current transformer can lead to inaccuracies in both active and reactive power measurements. While the impact on active energy is more commonly discussed, errors in the transformer's wiring also significantly affect the measurement of reactive energy, which can result in misleading readings for the user. Below are several examples illustrating how improper wiring affects reactive power measurement.
1. **Full Polarity Reversal of Three-Phase Current Transformers**
When the polarity of all three-phase current transformers is reversed, the reactive power expressions for each phase become:
$ Q_a = I_a U_{bc} \cos(90^\circ + \phi_a) $
$ Q_b = I_b U_{ca} \cos(90^\circ + \phi_b) $
$ Q_c = I_c U_{ab} \cos(90^\circ + \phi_c) $
The total reactive power is:
$ Q = Q_a + Q_b + Q_c = - (I_a U_{bc} \sin\phi_a + I_b U_{ca} \sin\phi_b + I_c U_{ab} \sin\phi_c) $
In a balanced system where $ I_a = I_b = I_c $ and $ U_a = U_b = U_c $, this simplifies to:
$ Q = -3 I_a U_{bc} \sin\phi_a $
Meanwhile, the actual reactive power is $ Q' = 3 I_a U_{bc} \sin\phi_a $. This means the reactive energy meter will show a reverse reading, essentially equal in magnitude to the correct value but in the opposite direction.
2. **Incorrect Connection of Two-Phase Voltage Components**
If the voltage components for phases A and C are connected incorrectly, the reactive power expressions become:
$ Q_a = I_a U_{ba} \cos(150^\circ - \phi_a) $
$ Q_b = I_b U_{ac} \cos(90^\circ + \phi_b) $
$ Q_c = I_c U_{cb} \cos(30^\circ - \phi_c) $
In a balanced system, the sum of these values equals zero, meaning the reactive power meter does not register any change, effectively stopping the meter from turning.
3. **Incorrect Connection of Two-Phase Current Components**
When the current components for phases A and B are incorrectly connected, the reactive power expressions are:
$ Q_a = I_b U_{bc} \cos(30^\circ + \phi_a) $
$ Q_b = I_a U_{ca} \cos(150^\circ + \phi_b) $
$ Q_c = I_c U_{ab} \cos(90^\circ - \phi_c) $
Again, under balanced conditions, the total reactive power sums to zero, causing the meter to remain stationary.
4. **Simultaneous Incorrect Connection of Two-Phase Current and Voltage Components**
If both current and voltage components for two phases are incorrectly connected, the reactive power expressions are:
$ Q_a = I_c U_{ba} \cos(90^\circ + \phi_a) $
$ Q_b = I_b U_{ac} \cos(90^\circ + \phi_b) $
$ Q_c = I_a U_{cb} \cos(90^\circ + \phi_c) $
In a balanced system, the total becomes $ Q = -3 I_c U_{ba} \cos\phi_a $, which results in the reactive power meter reversing at the same speed as it would normally operate.
5. **Phase Sequence Shift (A→B, B→C, C→A)**
When the phase sequence is shifted, the reactive power expressions for each phase become:
$ Q_a = I_a U_{ca} \cos(150^\circ + \phi_a) $
$ Q_b = I_b U_{ab} \cos(150^\circ + \phi_b) $
$ Q_c = I_c U_{bc} \cos(150^\circ + \phi_c) $
Under balanced conditions, the total becomes $ Q = 3 I_a U_{ca} \cos(150^\circ + \phi_a) $, while the actual reactive power is $ Q' = 3 I_a U_{ca} \sin\phi_a $. Since $ \cos(150^\circ + \phi_a) $ falls in the second or third quadrant, the meter reverses its reading.
6. **Another Phase Sequence Shift (A→C, B→A, C→B)**
In this case, the reactive power expressions are:
$ Q_a = I_a U_{ab} \cos(30^\circ + \phi_a) $
$ Q_b = I_b U_{bc} \cos(30^\circ + \phi_b) $
$ Q_c = I_c U_{ca} \cos(30^\circ + \phi_c) $
The total reactive power is $ Q = 3 I_a U_{ab} \cos(30^\circ + \phi_a) $, while the actual reactive power is $ Q' = 3 I_a U_{ab} \sin\phi_a $. Depending on the angle $ \phi_a $, the meter may under-read, over-read, or even stop.
7. **Summary**
These examples demonstrate that incorrect wiring of current and voltage transformers can have significant effects on reactive power measurements. Although this analysis covers only a few scenarios, understanding the vector diagram approach allows for accurate evaluation of any miswiring situation. By applying these principles, technicians can correctly calculate the reactive equivalent for users and ensure accurate meter readings, avoiding potential billing discrepancies.
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