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# Determination of Voltage Regulation of a Three Winding Transformer

The procedure for determining the voltage regulation of a 3-winding transformer is given as follows −

## Step 1

Determine the kVA in each winding (primary, secondary and tertiary) for the given load. Determine x for each winding. Where, x is the ratio of actual kVA loading of the winding to the base kVA used in determining the network parameters. Hence,

$$\mathrm{For\:primary\:winding,\:𝑥_{1 }=\frac{Primary\:kVA\:loading}{Base\:kVA}}$$

$$\mathrm{For\:secondary\:winding,\:𝑥_{2 }=\frac{Secondary\:kVA\:loading}{Base\:kVA}}$$

$$\mathrm{For\:tertiary\:winding,\:𝑥_{3 }=\frac{Tertiary\:kVA\:loading}{Base\:kVA}}$$

## Step 2

Calculate the voltage regulation for each winding at its operating power factor.

Let cosφ1, cosφ2 and cosφ3 are the operating power factors of primary winding, secondary winding and tertiary winding respectively. If v_{r1}, v_{r2} and v_{r3} are the per unit voltage drops in the resistances of primary winding, secondary winding and tertiary winding respectively and v_{x1}, v_{x2} and v_{x3} are the per unit voltage drops in the leakage reactances of primary winding, secondary winding and tertiary winding respectively. Then,

The per voltage regulation for the primary winding is given by,

$$\mathrm{\nu_{1}=𝑥_{1}(\nu_{𝑟1}\:cos\:\varphi_{1}\: + \:\nu_{x1}\:sin\:\varphi_{1})}$$

The per voltage regulation for the secondary winding is given by,

$$\mathrm{\nu_{2}=𝑥_{2}(\nu_{𝑟2}\:cos\:\varphi_{2}\: + \:\nu_{x2}\:sin\:\varphi_{2})}$$

The per voltage regulation for the tertiary winding is given by,

$$\mathrm{\nu_{3}=𝑥_{3}(\nu_{𝑟3}\:cos\:\varphi_{3}\: + \:\nu_{x3}\:sin\:\varphi_{3})}$$

## Step 3

The voltage regulation of any pair of windings is obtained by the algebraic sum of the individual voltage regulations of the pair under consideration, if power flows from one to another. Otherwise, a negative sign is used.

Now, for a 3-winding transformer with the primary winding energized from an AC supply and the load is connected to the both secondary and tertiary windings, the voltage regulations are given by,

$$\mathrm{For\:primary\:to\:secondary,\:\nu_{12 }=\nu_{1}\:+\:\nu_{2}}$$

And,

$$\mathrm{For\:primary\:to\:tertiary,\:\nu_{13} = \nu_{1}\:+\:\nu_{3}}$$

Here, v_{12} and v_{13} are obtained by adding respective voltage regulations because the power flows form primary winding to secondary winding and primary winding to tertiary winding respectively. Since, power does not flow from secondary winding to the tertiary winding or from tertiary winding to the secondary winding, then the voltage regulation form secondary winding to tertiary winding is given by,

$$\mathrm{\nu_{23} = \nu_{2}\:−\:\nu_{3}}$$

And the voltage regulation from tertiary winding to the secondary winding is given by,

$$\mathrm{\nu_{32} = \nu_{3}\:−\:\nu_{2}}$$

Also, the voltage regulation from secondary winding to primary winding is

$$\mathrm{\nu_{21} = -(\nu_{1}\:+\:\nu_{2})}$$

The negative sign shows that the voltage regulation is determined form secondary to primary winding, whereas the power flows from primary winding to secondary winding.

Similarly, the voltage regulation from tertiary winding to the primary winding is

$$\mathrm{\nu_{31} = -(\nu_{1}\:+\:\nu_{3})}$$

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