Transformer Voltage Regulation: Explained with Formulas

When we talk about voltage regulation we are looking at how good a transformer is at keeping its secondary voltage steady even when the load changes.

Sometimes the output voltage we get from the secondary side is not exactly what we were expecting and that is where voltage regulation comes into play.

Now when you power up the primary winding of a transformer, then it generates a secondary voltage and current. The amount of this secondary voltage and current is determined by something called the turns ratio TR.

For example if you have a single-phase transformer, that has a step-down turns ratio of 2:1, and you apply 220 volts to the high voltage primary winding, then you might think that you would get an output terminal voltage of 110 VAC on the secondary winding.

This assumption comes from thinking about it as if everything is perfect and ideal.

But in real life things do not always work out that way. Since transformers are essentially wound magnetic circuits, they experience some losses. These losses come in two main forms, I²R copper losses and magnetic core losses. Because of these losses, the ideal secondary voltage we expected can drop by a few percent. So instead of getting that perfect 110 VAC, we might actually see something like 105 VAC coming out, and honestly that is pretty normal.

Now there is one more important concept related to transformers and electrical machines that also influences this secondary voltage when the transformer is working at full power.

This concept is known as “regulation.”

Understanding Voltage Regulation in Single-Phase Transformers

The voltage regulation of single-phase transformers is how much the secondary terminal voltage changes as a percentage or in per unit value compared to its original no-load voltage. This happens when the load on the secondary side varies.

In simpler terms the voltage regulation helps us understand how much the secondary terminal voltage fluctuates inside the transformer.

These fluctuations occur because of changes in the connected load of the transformer. If these variations lead to significant losses then it can seriously impact the performance and efficiency of the transformer. This is especially true if the secondary voltage ends up being too low.

Now let us consider what happens when there is no load connected to the secondary winding of the transformer.

This means that the output terminals are open-circuited and there is no closed-loop condition. As a result there is no output load current flowing which we refer to as I_L being equal to zero.

In this situation the transformer behaves like a single winding with high self-inductance.

It is important to note that the no-load secondary voltage we observe is determined by two main factors, the fixed primary voltage that is applied and the turns ratio of the transformer itself.

When you hook up a basic load to the secondary winding of a transformer it gets this current flowing through the internal winding of the transformer regardless of what the power factor happens to be.

This current causes some voltage to drop because of the resistance and reactance that are present inside the transformer itself. As a result this drop in voltage affects what you see at the output terminals.

Now when we talk about how the voltage regulation of a transformer changes we are really measuring the difference in the secondary terminal voltage.

We look at this difference by comparing two situations, one where there is no load at all which we refer to as I_L = zero or an open circuit and then we look at it again when it is fully loaded which means I_L = I_MAX or maximum current. And all of this is done while making sure that the primary voltage stays exactly the same throughout the process, as expressed in the following formula:

Formula for Voltage Regulation in Transformers as Fractional Change

Regulation = Change in Actual Voltage / No-Load Output Voltage

∴ Regulation = V_(no-load) – V_(full-load) / V_(no-load)

Now it is important for us to note that this voltage regulation when we express it as a fraction or as a change in units of the no-load terminal voltage can actually be defined in one of two different ways.

We have what is known as voltage regulation-down which we can call Reg_down and then we have voltage regulation-up which we can refer to as Reg_up.

So this means is that when we connect a load to the secondary output terminal the terminal voltage tends to go down.

Then on the flip side when we remove that load the secondary terminal voltage goes up. Thus the regulation of the transformer will really depend on which voltage value we decide to use as our reference point whether it is the load value or the non-load value.

We can also take this transformer voltage regulation and express it as a percentage change between what happens under no-load conditions and what happens under full-load conditions and we can do that in the following way:

Formula for Transformer Voltage Regulation as a Percentage Change

%Reg_(down) = ((V_(no-load) – V_(full-load))/ V_(no-load)) * 100%

%Reg_(up) = ((V_(no-load) – V_(full-load))/ V_(full-load)) * 100%

Let’s think about a single-phase transformer for a second. Imagine this transformer has a no-load terminal voltage of 100 volts when nothing is connected to it.

But when we connect a resistive load, the voltage drops to 95 volts. This change allows us to figure out the voltage regulation of the transformer which comes out to be 0.05 or 5 percent.

We find this by subtracting the loaded voltage from the no-load voltage (100 minus 95) and then dividing that by the no-load voltage (100), and finally multiplying by 100 percent.

In this situation we can describe the transformers voltage regulation in two ways. We can either say it’s a unit change of 0.05 or a percentage change of 5 percent based on the original no-load voltage.

Solving a Transformer Voltage Regulation Problem #1

We have a single-phase step-down transformer that can handle 600 VA of power. That means it can manage up to 600 volt-amperes! The turns ratio for this transformer is 12:2 meaning for every 12 turns on the primary side, there are 2 turns on the secondary side.

This transformer is hooked up to a constant supply voltage of 220 volts RMS.

When we connect this transformer to a load of 1.2 ohms on the secondary side, we want to see how well it performs under that load. So we want to calculate the percentage regulation of the transformer.

Data in Hand:
Transformer Rating = 600 VA
Turns Ratio (N1 : N2) = 12 : 2 = 6 : 1
Primary Voltage (V1) = 220 Vrms
Load Impedance (ZL) = 1.2 Ω

Step 1: Secondary Voltage at No-Load
V2_no_load = V1 / Turns Ratio
= 220 / 6
V2_no_load = 36.67 Vrms

Step 2: Secondary Current Under Load
I2 = S / V2_no_load
I2 = 600 / 36.67
I2 = 16.36 A

Step 3: Voltage Drop Across Load Impedance
V_drop = I2 * ZL
V_drop = 16.36 * 1.2
V_drop = 19.63 V

Step 4: Secondary Voltage Under Load
V2_full_load = V2_no_load - V_drop
= 36.67 - 19.63
V2_full_load = 17.04 Vrms

Step 5: Percentage Voltage Regulation
Regulation (%) = ((V2_no_load - V2_full_load) / V2_no_load) * 100
Regulation (%) = ((36.67 - 17.04) / 36.67) * 100
Regulation (%) = 53.54%

Solving a Transformer Voltage Regulation Problem #2

We have a single-phase transformer with a voltage regulation of 5% and shows that when it’s under full load, then voltage at the secondary terminal is 110.4 volts. To find out what the voltage is when there’s no load connected, let us calculate what happens when the load is completely unplugged.

Data we Have:
Voltage Regulation = 5%
Full Load Voltage (V_full_load) = 110.4 V

Step 1: Rearranging the Voltage Regulation Formula
%Reg(up) = ((V_no_load - V_full_load) / V_full_load) * 100

Rearrange to find V_no_load:
V_no_load = V_full_load * (1 + (Voltage Regulation / 100))

Step 2: Substituting the values
V_no_load = 110.4 * (1 + (5 / 100))
V_no_load = 110.4 * 1.05
V_no_load = 115.92 V

So the Final Answer:
No-Load Voltage (V_no_load) = 115.92 V

When we connect different loads to a transformer then it affects the voltage at the transformer’s terminals.

There are two important types of voltage to know, the “no-load” voltage which is what we measure when nothing is connected and the “full-load” voltage which is what we get when a device is connected.

This shows that the voltage changes based on what we connect, making transformer voltage regulation depend on external factors.

If we think about it, a lower percentage of voltage regulation means that the voltage at the secondary terminal stays more stable, even if we add more load current.

When the load is purely resistive then voltage drop is smaller. In a perfect scenario, an ideal transformer would have zero voltage regulation, meaning the full-load voltage would be exactly the same as the no-load voltage, with no losses at all.

With this in mind we can say that a transformer’s voltage regulation is about finding the difference between its full-load voltage and no-load voltage compared to its maximum rated secondary current.

We can show this difference as a ratio or a percentage. But the big question is: why does the secondary voltage change or drop when we adjust the load current?

Voltage Transformers with a Load Connected (On-load)

So when we think about a transformer that is supplying a load through its secondary winding we need to keep in mind that there are some magnetic iron losses happening inside the laminated core.

Then there are also copper losses which occur because of the resistivity in its windings. This situation applies to both the primary and secondary windings of the transformer.

These losses create both reactance and resistance within the transformer’s windings which together provide an impedance path that the secondary output current which we can call I_S has to flow through just like we can see in the diagram.

Now since the secondary winding is made up of both resistance and reactance it makes sense that there will be an internal voltage drop occurring in the transformer’s windings.

The amount of this drop depends on the effective impedance and the load current that is being supplied. This relationship is explained by Ohm’s Law which tells us that V = I * Z.

Then as we observe what happens when the secondary load current increases we can see that the voltage drop happening within the transformer’s windings must also increase.

And because we are keeping the primary supply voltage constant this means that the secondary output voltage has to fall as a result.

The impedance which we refer to as Z of the secondary winding is actually the phasor sum of both its resistance which we call R and the leakage reactance which we denote as X.

Each of these components produces a different voltage drop across them. So now we can define the secondary impedance along with the no-load and full-load voltages as follows:

Impedance, Z = √(R² + X²)

So we can define the no-load voltage of the secondary winding of a transformer as:

V_S(no-load)  =  E_S

Likewise we can define the full-load voltage of the transformer as:

V_S(full-load)  =  E_S – I_SR – I_SX

or V_S(full-load)  =  E_S – I_S(R + jX)

∴ V_S(full-load)  =  E_S – I_S*Z

So here is what we can clearly see. The transformer’s winding is made up of a reactance that is in series with a resistance and the load current flows through both of these elements.

Now since we are dealing with resistance the voltage and current are in sync or what we call in-phase. This means that the voltage drop across the resistor which we write as ISR has to be in sync with the secondary current IS.

But then if we look at a pure inductor that has inductive reactance which we call XL we find that the current actually lags behind by 90 degrees.

This means that the voltage drop across the reactance which we write as ISX actually leads the current by an angle ΦL because it is an inductive load.

Now when we talk about the impedance Z of the secondary winding it is really just the combination of the resistance and reactance added together in what we call a phasor sum. Each of these components has its own phase angle which we can describe as follows:

cosΦ_R = R/Z,

∴ R = ZcosΦ_R, and

sinΦ_X = X/Z

∴ X = ZsinΦ_X

ZcosΦ = Z((cosΦ_R * cosΦ_X ) + (sinΦ_R * sinΦ_X))

ZcosΦ = RcosΦ + XsinΦ

So with V = I*Z we get the voltage drop across the secondary impedance which can be therefore expressed as:

V_drop  =  I_S(RcosΦ + XcosΦ)

Also because V_S(full-load) = V_S(no-load) – V_drop, so we can exxpress the percentage regulation in the following way:

Formula for the Lagging Power Factor

V_S(full-load) = V_S(no-load) – V_drop

V_S(no-load) = V_S(full-load) + I_S(RcosΦ + XcosΦ)

%Reg = (I_S(RcosΦ + XcosΦ)/V_S(no-load))* 100%

So when we talk about a positive regulation expression that involves cos(Φ) and sin(Φ), we can see that the transformer’s secondary terminal voltage actually goes down or falls. This drop tells us that we are dealing with a lagging power factor which means we have an inductive load connected.

Now on the flip side if we have a negative regulation expression between cos(Φ) and sin(Φ, then transformer’s secondary terminal voltage goes up or rises.

This rise indicates a leading power factor which means we are dealing with a capacitive load. So basically the regulation expression for a transformer works the same way for both leading and lagging loads, it is just that the sign changes to show whether the voltage is rising or falling.

Therefore when we have a positive regulation condition it causes a voltage drop within the secondary winding while in contrast a negative regulation condition causes the voltage to increase in the winding.

Even though leading power factor loads are not as common as inductive loads like coils or solenoids, there are times when a transformer supplying a light load with low currents might experience a capacitive condition which can cause the terminal voltage to rise.

Formula for the Leading Power Factor

%Reg = (I_S(RcosΦ + XsinΦ)/V_S(no-load))* 100%

So when we have a positive regulation condition it actually causes a decrease or drop in the voltage within the secondary winding of the transformer. On the other hand when we are dealing with a negative regulation condition it leads to an increase or rise in the voltage in that same winding.

Now it is important to note that leading power factor loads are not as common as those inductive loads that we often see like coils solenoids or chokes.

However there are times when a transformer that is supplying a light load with low currents, might find itself in a capacitive condition. This can happen, and when it does it makes the terminal voltage to rise.

Solving a Transformer Voltage Regulation Problem #3

Let’s say we have a 12KVA single-phase transformer which is rated to supply a no-load secondary voltage of 220 volts. In case the equivalent secondary winding resistance happens to be 0.022Ω and its total reactance is around 0.08Ω, then let us calculate what will be its voltage regulation when it is connected to a load at 0.85 power factor lagging.

Transformer Voltage Regulation Calculation

To calculate the voltage regulation of a transformer under load we can use the following formula:

Voltage Regulation (VR) = ((V_no-load - V_full-load) / V_full-load) × 100

For a transformer with known equivalent resistance (Re) and reactance (Xe), and operating at a power factor (pf), we estimate the voltage drop (V_drop) using:

V_drop = I_full-load × (Re × pf + Xe × sinθ)

Where:
- I_full-load = Full-load current
- Re = Equivalent resistance (in ohms)
- Xe = Equivalent reactance (in ohms)
- pf = Power factor of the load
- θ = cos⁻¹(pf)

Step 1: Calculate the full-load current
The full-load current of the transformer is given by:

I_full-load = S / V = 12000 / 220 ≈ 54.55 A

Step 2: Determine sinθ
For a power factor of 0.85 lagging:

sinθ = √(1 - pf²) = √(1 - 0.85²) ≈ 0.527

Step 3: Calculate the voltage drop
Substitute the given values:
- Re = 0.022 Ω
- Xe = 0.08 Ω
- I_full-load = 54.55 A
- pf = 0.85

V_drop = 54.55 × (0.022 × 0.85 + 0.08 × 0.527)

V_drop = 54.55 × (0.0187 + 0.0422) ≈ 54.55 × 0.0609 ≈ 3.32 V

Step 4: Calculate voltage regulation
Voltage regulation is given by:

VR = (V_drop / V_full-load) × 100

Substitute V_drop = 3.32 V and V_full-load = 220 V:

VR = (3.32 / 220) × 100 ≈ 1.51%

So the Final Answer is:
The voltage regulation of the transformer at a 0.85 power factor lagging is approximately 1.51%.

Conclusions

In this tutorial about Transformer Voltage Regulation we learned that when a transformer’s secondary winding is under load, its output voltage can change.

This change in voltage can be shown as a ratio or more often as a percentage. When theres no load connected, then there’s no current flowing in the secondary, so the voltage is at its highest point.

But when the transformer is fully loaded then current starts to flow, causing core losses and copper losses in the winding.

Core loss is a constant loss that happens because of the transformer’s magnetic circuit, which is influenced by the voltage from the primary winding. On the other hand, copper loss varies depending on how much current is being drawn by the load connected to the secondary winding.

Changes in losses resulting from variations in load current might affect the system’s ability to control voltage. For regulated power supply circuits, a transformer with improved voltage regulation implies that the secondary terminal voltage remains more constant even when the load fluctuates.

We also indicated that the secondary terminal voltage tends to decrease when there is a trailing power factor, such as with inductive loads. Large secondary currents may flow from a transformer with a very low lagging power factor, which leads to poor voltage control due to larger voltage dips in the winding.

However the output terminal voltage likely to rise when there is a leading power factor (such as with capacitive loads). This indicates that the winding’s voltage decreases under positive regulation and increases under negative regulation.

Although zero-voltage regulation is impossible (which is the only perfect transformers can achieve this) lowest regulation and, thus, optimum efficiency typically happen whenever the losses of copper and winding core losses are about identical.

References:

Transformer voltage regulation

Voltage regulation