# Calculating Voltage Regulation in Transformers

## Transformers experience losses that reduce voltage output at the secondary winding core. Here, you will learn about transformer voltage regulation and how to compute it.

During the design process, power system engineers have to compute the output voltage from the transformer terminals. The voltage must be regulated to meet design requirements. Voltage regulation measures how perfectly a transformer can constantly maintain a secondary voltage under a particular loading condition.

Anytime a transformer’s primary winding is energized, it induces secondary current and voltage where the transformer’s turn ratio determines the amount of current and voltage induced. For example, for a single-phase ideal transformer with a stepdown turn ratio of 2:1 and an input of 230 volts at the primary winding, the secondary winding output terminal is expected to output 115 volts. In the real world, this cannot be achieved as the transformer windings are expected to be affected by energy losses such as magnetic core and copper losses which will reduce the output voltage at the secondary windings by a small percentage. Apart from these normal losses, there is another parameter that affects the performance of transformers–regulation.

### Regulating Transformer Voltage

For a single-phase transformer, voltage regulation can be defined as the secondary terminal voltage percentage change as compared to its initial voltage while under zero loading and varying load conditions. Regulation determines the secondary terminal voltage variations that occur inside a given transformer due to variation in the loads connected to the transformer, and affects its efficiency and performance if the losses are high, hence reducing the secondary voltage.

Transformer voltage regulation can be calculated as a functional change using the formula below:

\[Regulation=\frac{Output\,Voltage\,Change}{No-Load\,Output\,Voltage}\]

\[Regulation=\frac{V_{no-load}-V_{full-load}}{V_{no-load}}\]

From the equation above, the transformer’s voltage regulation can be defined in two ways:

- Voltage regulation-down,Reg
_{down}: this occurs when a load is connected to the transformer secondary terminal and the voltage output at the terminal goes down. - Voltage regulation-up,Reg
_{up}: this happens when the load at the transformer’s secondary terminal is removed and the voltage at the terminal goes up.

Here the transformer’s voltage regulation is determined by the reference voltage used, which is no-load or load.

From the equation above, it is possible for us to compute the transformer’s voltage regulation in percentile using the formulas below:

\[\%Reg_{down}=\frac{V_{no-load}-V_{full-load}}{V_{no-load}}\times100\%\]

\[\%Reg_{up}=\frac{V_{no-load}-V_{full-load}}{V_{full-load}}\times100\%\]

#### Problem 1

A certain transformer has a no-load open-circuit voltage of 120 V and the voltage drops to 110 when a load is applied. Calculate the percentage voltage regulation of the transformer.

#### Solution

\[\%Reg_{down}=\frac{V_{no-load}-V_{full-load}}{V_{no-load}}\times100\%\]

\[\%Reg_{down}=\frac{120-110}{120}\times100\%=8.33\%\]

#### Problem 2

A single-phase transformer that has a 5% voltage regulation has 115.5 volts at the secondary terminal when fully loaded. Compute the transformer’s no-load terminal if the load is removed.

#### Solution

\[\%Reg_{up}=\frac{V_{no-load}-V_{full-load}}{V_{full-load}}\times100\%\]

\[5\%=\frac{V_{no-load}-115.5}{115.5}\times100\% \]

\[V_{no-load}=115.5+115.5\times\frac{5}{100}=121.275V\]

Note from the computation that a change in the loads connected to the transformer induces a change in the terminal voltage of the transformer between full load and the no-load voltages and this makes the voltage regulation of the transformer an external function to the given transformer. Therefore, to make the transformer more stable, the voltage regulation should be very small. To achieve this, the connected loads should be more resistive.

Let’s examine why transformer voltages at the secondary terminal change with the load current change.

### Transformers While On-load

**Figure 1. **Transformer on load. Image used courtesy of Simon Mugo

**Figure 1.**Transformer on load. Image used courtesy of Simon Mugo

A transformer supplying power to the load through the secondary winding experiences losses through magnetic iron losses that occur on laminated core and copper losses that happen due to windings resistance. The opposing forces inject resistance and reactance into the windings of the transformer, creating an impedance channel through which current output at the secondary, I_{s} terminal must flow through.

From the diagram above, the secondary winding is made up of both reactance and resistance. The reactance and the resistance cause a voltage drop in the transformer winding which depends on the load current supplied and the effective impedance of the secondary windings.

Ohm’s law directs that

\[V=IZ\]

And this explains that an increase in the secondary winding current leads to an increase in the voltage drop in the transformer winding.

The impedance of the secondary winding is calculated by taking the sum of the phasor of both the leaked reactance, X, and resistance, R where a different voltage is a producer across each parameter.

Hence the secondary impedance is determined by:

\[Impedance,Z=\sqrt{R^{2}+X^{2}}\]

The secondary voltage with no load can be determined by:

\[V_{S(no-load)}=E_{S}\]

At full load, the secondary winding voltage can be computed as

\[V_{S(full-load)}=E_{S}-I_{S}R-I_{S}X\]

Or

\[V_{S(full-load)}=E_{S}-I_{S}(R+jX)\]

Therefore,

\[V_{S(full-load)}=E_{S}-I_{S}Z\]

From the equations above, it is notably identified that the winding of a transformer is made up of the resistance which is in series with the reactance and the load current remains common to both the parameters. For resistance, the current and voltage are in phase and the voltage drop across this resistor which is determined by I_{S}R should be in phase with the transformer secondary current determined by I_{S}.

For the inductor, the current in the transformer lags by 90 degrees with the voltage across the inductor being determined by I_{S}X and a phasor-angle of φ_{L}.

The phasor angle can be calculated by:

\[cosφ_{R}=\frac{R}{Z}\]

Rearranging

\[R=Zcosφ_{R}\]

And

\[sinφ_{X}=\frac{X}{Z}\]

Therefore,

\[X=Zsinφ_{X}\]

\[Zcosφ=Z(cosφ_{R}\times cosφ_{X}+sinφ_{R}+sinφ_{X})\]

On simplification:

\[Zcosφ=Rcosφ+Xsinφ\]

Since V=IZ, the voltage drop across the transformer secondary impedance can be solved as

\[V_{drop}=L_{S}(Rcosφ+Xsinφ)\]

For the lagging power factor, the expression is as follows

\[V_{S(full-load)}=V_{S(no-load)}-V_{drop}\]

\[V_{S(no-load)}=V_{S(full-load)}+I_{S}(Rcosφ+Xsinφ)\]

\[\%Reg.=\frac{I_{S}(Rcosφ+Xsinφ)}{V_{s(no-load)}}\times100\%\]

The transformer’s secondary output voltage decreases with a positive voltage regulation between sin and cos resulting in a lagging power factor. A lagging power factor originates from an inductive load.

The transformer’s secondary output voltage increases with a negative voltage regulation between sin and cos resulting in a leading power factor. A leading power factor originates from a capacitive load.

It defies logic that the regulation expression for both lagging and leading loads remains the same and only the sign changes to indicate a fall or rise.

Therefore, the expression for regulation of leading power factor is:

\[\%Reg.=\frac{I_{S}(Rcosφ-Xsinφ)}{V_{s(no-load)}}\times100\%\]

#### Problem 3

A single-phase transformer rated at 10KVA provides a no-load 110V secondary voltage. It has a secondary winding resistance of 0.015 Ohms and a reactance of 0.04 Ohms. Compute the transform voltage regulation given that it has a lagging power factor of 0.85.

#### Solution

\[cosφ=0.85\]

\[φ=cos^{-1}(0.85)=31.8°\]

Therefore,

\[sinφ=sin\,sin\,31.8=0.527\]

The secondary current is calculated as

\[I_{S}=\frac{VA}{V}=\frac{10000}{110}=90.9\]

% voltage regulation can be computed as

\[\%Reg.=\frac{I_{S}(Rcosφ-Xsinφ)}{V_{s(no-load)}}\times100\%\]

\[\%Reg=\frac{90.9[(0.015\times0.85)+(0.04\times0.527)]}{100}\times100\%=2.8\%\]

### Key Takeaways of Transformer Voltage Regulation

- Loading the transformer secondary winding affects the output voltage.
- The effects of loading on the output voltage can be expressed as a percentile or a ratio.
- When there is no loading, the secondary current is absent, and therefore, the output voltage is at the maximum value.
- When loaded, the secondary currents are present and they induce losses within the transformer windings.
- Variations in the current of the load encourage variations in losses which ends up affecting regulation.
- Leading power factor increases secondary terminal voltage while the lagging power factor reduces it.

*Featured image used courtesy of Adobe Stock*

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