 # Three-phase Transformers

Just as with general three-phase circuits, there are four possible configurations for a 3-phase transformer setup:

1. wye-wye
2. delta-delta
3. wye-delta
4. delta-wye

### Power in three-phase transformers:

With all of the four configurations above, the following power expressions can be used:

## Total apparent power:

$$S_T = \sqrt{3} V_L \; I_L$$

## Total real power:

$$P_T = S_T \; \cos(\theta)$$ $$\;\;\;\; = \sqrt{3} V_L \; I_L \; \cos(\theta)$$

## Total reactive power:

$$Q_T = S_T \; \sin(\theta)$$ $$\;\;\;\; = \sqrt{3} V_L \; I_L \; \sin(\theta)$$

### Wye-Wye connections Recall the turns ratio for an ideal transformer: $$\frac{V_2}{V_1} = \frac{N_2}{N_1} = n$$ For the Y-Y circuit above, we have: $$\frac{V_{Ls}}{V_{Lp}} = n$$

$$V_{Ls} = n V_{Lp}$$

Additionally, we know that the turns ratio for input and output current is: $$\frac{I_2}{I_1} = \frac{N_1}{N_2} = \frac{1}{n}$$ So, for the Y-Y circuit we have: $$\frac{I_{Ls}}{I_{Lp}} = \frac{1}{n}$$

$$I_{Ls} = \frac{I_{Lp}}{n}$$

### Delta-Delta connections In a similar manner we have the following expressions for a delta-delta transformer configuration:

$$V_{Ls} = nV_{Lp}$$ $$I_{Ls} = \frac{I_{Lp}}{n}$$

### Wye-Delta connections For the wye-delta transformer we have:

$$V_{Ls} = \frac{nV_{Lp}}{\sqrt{3}}$$ $$I_{Ls} = \frac{\sqrt{3}\;I_{Lp}}{n}$$

### Delta-Wye connections $$V_{Ls} = n\sqrt{3} V_{Lp}$$ $$I_{Ls} = \frac{I_{Lp}}{n\sqrt{3}}$$

Next we will look at an example problem involving a three-phase transformers.

Continue on to three-phase transformer example problem...