Efficiency of Transformer – All Day Efficiency & Maximum Efficiency

Formulas of Transformer’s Efficiency

Efficiency = η = Output / Input

Efficiency = η= Output / (Output + Losses) … (As Input = Output + Losses)

Efficiency = η= Output / (Output + Copper Losses + Iron Losses)

Efficiency = η = (Input – Losses) / Input … (As Output = Input – Losses)

Efficiency = η = 1 – (Losses / Input)

As we know that the rating of transform is expressed in kVA not in kW. But the efficiency doesn’t depend on VA i.e. it would be expressed in power – watts ( or kW) not in kVA. Although, the losses are directly proportional to VA (Volt-Amperes), thus, efficiency depends on power factor on any kind of VA load. Keep in mind the efficiency would be maximum on unity (1) power factor.

• Related Post: Power Transformers Maintenance, Diagnostic & Monitoring

Good to Know

We can also find the transformer’s efficiency by determining:

• Core losses by open circuit test or No Load test, and
• Copper losses by short circuit test.

Condition for Maximum Efficiency of Transformer

We know that,

Copper Loss = W_C = I₁² x R₁   or I₂² x R₂

Iron Loss= W_I = Hysteresis Loss + Eddy Current Loss = W_I = W_H + W_E

Suppose to the primary side of transformer…

Primary Input = P₁ = V₁ x I₁ Cosθ₁

Efficiency = η = Output / Input

Efficiency = η = (Input – Losses) / input … (As Output = Input – Losses)

Efficiency = η = (Input – Copper losses – Iron Losses)/Input

Efficiency = η = (P₁ – W_C – W_I) / P₁

Efficiency = η = (V₁ x I₁ Cosθ₁ – I₁²  x R₁ – W_I) / V₁ x I₁ Cosθ₁

Taking LCM

Efficiency = η = 1- (I₁² x R₁ / V₁I₁ Cosθ₁) – (W_I / V₁ x I₁ Cosθ₁)

Or

Efficiency = η = 1- (I₁ x R₁ / V₁ Cosθ₁) – (W_I / V₁ x I₁ Cosθ₁)

Differentiating both sides w.r.t I₁.

Dη / dI₁ = 0 – ( R₁ / V₁ Cosθ₁) + (W_I /V₁ x I₁² Cosθ₁)

Dη / dI₁= – ( R₁ / V₁ Cosθ₁) + (W_I / V₁ x I₁² Cosθ₁)

For maximum efficiency, the value of (Dη / dI₁) should be minimum i.e.

Dη / dI₁ = 0

Thus, the above equation can be rewritten as:

R₁ / (V₁ Cosθ₁) = (W_I /V₁ x I₁² Cosθ₁)

Or

W_I = I₁² x R₁or       I₂² x R₂

Iron Loss = Copper Loss

The value of Output current (I₂) on which maximum efficiency can be gained:

I₂ = √ (W_I / R₂)

The value of output current (I₂) is the factor which makes it possible to equal the value of copper loss and iron loss (i.e. copper loss = iron loss).

• Also read: Transformer Phasing: The Dot Notation and Dot Convention

Good to Know

• Efficiency is usually less than 1 and it is often expressed as a percent (%).
• Ideal Transformer is 100% efficient i.e. the efficiency of ideal transformer is 1.
• Practical transformers efficiencies are generally quite high in compression to other electrical machines and electronics devices (i.e. motors, generators etc.) on the order of 90 t0 98%.

All Day Efficiency of Transformer

As we know that the commercial or typical efficiency of a transformer is the ratio of its output and input in watts.

Efficiency = Output (in Watts) / Input (in Watts)

But there are a number of transformers whose performance can’t be monitored according to the general efficiency formula given above.

Those distribution transformers which supply electrical energy to the lighting and other general circuits, their primary energizes for 24 hours, but the secondary windings do not energize all the time at once. In other words, secondary windings only energize at the night time when they supply electrical energy to lighting circuits. I.e. secondary windings supply eclectic power for very small loads or no load for maximum time in 24 hours. It means that core loss occurs for 24 hours a day regularly (during the operation) but copper loss occurs only when the transformer is on load.

• Also read: Difference between Power Transformers and Distribution Transformers?

Therefore, the designers realize the necessity to design a transformer with as low core losses as possible. As copper loss depends on load, therefore, they should be neglected. In these types of transformers, we can only track their performance by determining the all day efficiency.

All day efficiency is also known as “Operational Efficiency”. On the basis of usable energy, we estimate the all day efficiency for a specific time (during the 24 hours = one day) and it can be calculated using the following formula.

All Day Efficiency = Output (in kWh) / Input (in kWh)

To know the exact value of all day efficiency, we must know about the load cycle i.e. how much load is connected to the transformer and for how much time (in 24 hours).

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