A single-phase, 22 kVA, 2200/220 V, 50 Hz, distribution transformer is to be connected as an auto-transformer to get an output voltage of 2420 V. Its maximum kVA rating as an auto-transformer is

Understanding Auto-Transformer Conversion

The question asks for the maximum kVA rating when a single-phase, 22 kVA, 2200/220 V two-winding transformer is reconfigured as an auto-transformer to achieve an output voltage of 2420 V.

Auto-Transformer Configuration

To obtain a higher output voltage ($V_{out}=2420$ V) from the given transformer ($V_{HV}=2200$ V, $V_{LV}=220$ V), the auto-transformer connection must involve adding the voltages of the two windings. The output voltage $2420$ V is the sum of the high voltage (HV) and low voltage (LV) winding voltages: $V_{out}=V_{HV}+V_{LV}=2200+220=2420$ V.

There are two ways to connect a two-winding transformer as an auto-transformer:

1. Using the HV winding as the common winding and the LV winding as the series winding. The input voltage is $V_{HV}=2200$ V, and the output voltage is $V_{HV}+V_{LV}=2420$ V.
2. Using the LV winding as the common winding and the HV winding as the series winding. The input voltage is $V_{LV}=220$ V, and the output voltage is $V_{LV}+V_{HV}=2420$ V.

Calculating the Maximum kVA Rating

The kVA rating of an auto-transformer derived from a two-winding transformer can be calculated using the relationship between the original rating and the voltage levels involved. The formula depends on which winding acts as the common winding.

The general formula relating the auto-transformer kVA ($S_{auto}$) to the two-winding transformer kVA ($S_{2w}$) is:

$$S_{auto}=S_{2w}\times \frac{V_{out}}{V_{terminal}}$$

Where $V_{terminal}$ is the voltage rating of the winding connected to the input terminals (the common winding).

To achieve the maximum possible kVA rating, we should choose the configuration where the winding with the lower voltage acts as the common winding, as this results in a higher voltage multiplication factor.

Let's analyze the two configurations:

1. Configuration 1: HV winding (2200 V) as common - Input Voltage ($V_{in}$ or $V_{terminal}$) = $V_{HV}=2200$ V - Output Voltage ($V_{out}$) = $2420$ V - $S_{auto}=S_{2w}\times \frac{V_{out}}{V_{HV}}=22\text{ kVA}\times \frac{2420\text{ V}}{2200\text{ V}}=22\times 1.1=24.2\text{ kVA}2.\ast \ast Configuration2:LVwinding(220V)ascommon\ast \ast -InputVoltage($ V_{in}$or$ V_{terminal}$)=$ V_{LV} = 220$V-OutputVoltage($ V_{out}$)=$ 2420$V-$ S_{auto} = S_{2w} \times \frac{V_{out}}{V_{LV}} = 22 \text{ kVA} \times \frac{2420 \text{ V}}{220 \text{ V}} = 22 \times 11 = 242 \text{ kVA}

Comparing the two configurations, the maximum kVA rating is obtained when the LV winding (220 V) is used as the common winding, resulting in a rating of 242 kVA.

This calculation aligns with the formula derived from power transfer perspectives: $S_{auto}\le S_{2w}(1+V_{HV}/V_{LV})$ for the case where the LV winding serves as the common element in a step-up auto-transformer configuration yielding $V_{out}=V_{LV}+V_{HV}$.

Calculation:

$$S_{auto}\le 22\text{ kVA}\times \left(1+\frac{2200\text{ V}}{220\text{ V}}\right)$$

$$S_{auto}\le 22\text{ kVA}\times (1+10)$$

$$S_{auto}\le 22\text{ kVA}\times 11$$

$$S_{auto}\le 242\text{ kVA}$$

Conclusion

The maximum kVA rating of the auto-transformer, when connected to achieve an output voltage of 2420 V, is 242 kVA.