Turns Ratio and Voltage Transformation Ratio

As discussed in the previous chapter, the EMF of equation of a transformer is given by,

$$mathrm{mathit{E}:=:4.44:mathit{fphi _{m}:N}}$$

For primary winding,

$$mathrm{mathit{E_{mathrm{1}}}:=:4.44:mathit{fphi _{m}:N_{mathrm{1}}}:cdot cdot cdot (1)}$$

For secondary winding,

$$mathrm{mathit{E_{mathrm{2}}}:=:4.44:mathit{fphi _{m}:N_{mathrm{2}}}:cdot cdot cdot (2)}$$

Turns Ratio of Transformer

From equations (1) and (2), we have,

$$mathrm{frac{mathit{E_{mathrm{1}}}}{mathit{E_{mathrm{2}}}}:=:frac{mathit{N_{mathrm{1}}}}{mathit{N_{mathrm{2}}}}:=mathrm{a}::cdot cdot cdot (3)}$$

The constant "a" is known as the turns ratio of the transformer. It may be defined as under,

The ratio of number of turns in the primary winding the number of turns in the secondary winding of a transformer is known as turns ratio.

Voltage Transformation Ratio of Transformer

The ratio of output voltage to the input voltage of transformer is known as voltage transformer ratio, i.e.,

$$mathrm{mathrm{Transformation: Ratio}:=:frac{Output :Voltage}{Input :Voltage}}$$

Thus, if V₁ is the input voltage and V₂ is the output voltage of a transformer, then its transformation ratio is given by,

$$mathrm{mathrm{Transformation: Ratio}:=:frac{mathit{V_{mathrm{2}}}}{mathit{V_{mathrm{1}}}}:cdot cdot cdot (4)}$$

For an ideal transformer, V₁ = E₁ and V₂ = E₂, then

$$mathrm{mathrm{Transformation: Ratio}:=:frac{mathit{V_{mathrm{2}}}}{mathit{V_{mathrm{1}}}}:=:frac{mathit{E_{mathrm{2}}}}{mathit{E_{mathrm{1}}}}:=::frac{mathit{N_{mathrm{2}}}}{mathit{N_{mathrm{1}}}}:=:frac{1}{a}cdot cdot cdot (5)}$$

However, in a practical transformer, there is a small difference between V₁ and E₁, and V₂ and E₂, due to winding resistances. Although, this difference is very small so for analysis purposes, we take V₁ = E₁ and V₂ = E₂.

Numerical Example (1)

A transformer with 1000 primary turns and 400 secondary turns is suppped from a 220 V AC supply. Calculate the secondary voltage and the volts per turn.

Solution

Given data,

$$mathrm{mathit{N_{mathrm{1}}}:=:1000:mathrm{and}:mathit{N_{mathrm{2}}}:=:400}$$

$$mathrm{mathit{V_{mathrm{1}}}:=:220:V}$$

The turns ratio of transformer is,

$$mathrm{frac{mathit{V_{mathrm{1}}}}{mathit{V_{mathrm{2}}}}:=:frac{mathit{N_{mathrm{1}}}}{mathit{N_{mathrm{2}}}}}$$

$$mathrm{Rightarrow mathit{V_{mathrm{2}}}:=:mathit{V_{mathrm{1}}} imes frac{mathit{N_{mathrm{2}}}}{mathit{N_{mathrm{1}}}}:=:220 imes frac{400}{1000}}$$

$$mathrm{ hereforemathit{V_{mathrm{2}}}:=:88:mathrm{Volts}}$$

The volts per turn is given by,

$$mathrm{mathrm{For: primary: winding}:=:frac{mathit{V_{mathrm{1}}}}{mathit{N_{mathrm{1}}}}:=:frac{200}{1000}:=:0.22:mathrm{Volts}}$$

$$mathrm{mathrm{For: Secondary: winding}:=:frac{mathit{V_{mathrm{2}}}}{mathit{N_{mathrm{2}}}}:=:frac{88}{400}:=:0.22:mathrm{Volts}}$$

Hence, from this example, it is clear that the volts per turn for a transformer remain the same on both primary and secondary windings.

Numerical Example (2)

A transformer with an output voltage of 2200 V is suppped at 220 V. If the secondary winding has 2000 turns, then calculate the number of turns in primary winding.

Solution

Given data,

$$mathrm{mathit{V_{mathrm{1}}}:=:200:mathit{V}:mathrm{and}:mathit{V_{mathrm{2}}}:=:2200:mathit{V}}$$

$$mathrm{mathit{N_{mathrm{2}}}:=:2000:mathrm{turns}}$$

The turns ratio of transformer is,

$$mathrm{frac{mathit{V_{mathrm{1}}}}{mathit{V_{mathrm{2}}}}:=:frac{mathit{N_{mathrm{1}}}}{mathit{N_{mathrm{2}}}}}$$

$$mathrm{Rightarrow {mathit{N_{mathrm{1}}}}:=:mathit{N_{mathrm{2}}}: imes :frac{mathit{V_{mathrm{1}}}}{mathit{V_{mathrm{2}}}}:=:mathrm{2000}: imes :frac{220}{2200}:=:mathrm{200:turns}}$$