MC-A5 · Efficiency & voltage regulation

Efficiency and Voltage Regulation

A transformer's two losses differ in nature. Core loss is a fixed loss that does not depend on the load, and copper loss is a variable loss that grows with the square of the load, so vary the load and find for yourself the condition under which efficiency peaks.

At what load is efficiency greatest?

On the graph, the horizontal line is the core loss (1 kW), which does not depend on the load, and the parabola is the copper loss (4 kW at rated load), which grows with the square of the load. Push the load factor with the slider and the points on the two loss curves move together; at the load where the two curves meet stands a gold dashed line (ηmax). Why efficiency is highest at that load is the question of this lesson.

Load fraction mm = 0.90

The slider changes the load factor continuously. Where the two losses meet is the peak.

Efficiency and the two losses

η = 95.5% · Pcu=3.24 / Pfe=1.00 kW

The load is far from the maximum-efficiency point. Copper loss and core loss differ widely, and one of them drags the efficiency down.

Misaligned

Fixed loss and variable loss

Core loss arises in the shunt exciting branch. Once the voltage and frequency are fixed the flux is fixed, and the hysteresis and eddy-current losses that the flux sets occur even at 0 load, so this is a fixed loss that does not depend on the load. Copper loss arises in the winding resistance of the series branch, in proportion to the square of the load current, and at load factor m it is m² times the rated copper loss. Core loss is reduced by stacking thin silicon-steel laminations to cut the eddy-current paths or by using amorphous cores, and copper loss by using thicker conductors.

ObservePcu(m) = m² Pcu

Copper loss is a variable loss that grows with the square of the load factor.

The condition for peak efficiency

Efficiency is the output divided by the sum of the output and the losses. At load factor m, η = mScosφ / (mScosφ + Pfe + m²Pcu). Differentiating with respect to m and solving for where the derivative is 0 gives Pfe = m²Pcu, that is, the condition that fixed and variable losses are equal, and rearranging gives the maximum-efficiency load factor m = √(Pfe / Pcu). With the values in the picture, that point is half of the rated load.

Choosemax η: Pfe = ?

Maximum efficiency occurs where variable loss = fixed loss.

Fill inm = √(?)

Solving for the load factor gives the square root of the ratio of core loss to copper loss.

Voltage regulation comes from the series drop

When the load current passes through the series impedance, the voltage on the load side drops below its no-load value. The fractional change is the voltage regulation, approximated as ε ≈ p cosφ + q sinφ, where p is the percent resistance drop and q the percent reactance drop. At a lagging power factor (inductive load) the voltage falls and the regulation is positive; at a leading power factor (capacitive load) the voltage can actually rise and the regulation can turn negative.

On your ownε = p cosφ + ?

Voltage regulation is made up of the resistance drop and the reactance drop.

Back to the first screen

Efficiency peaked at the load where the two loss curves meet. Core loss is flat and does not depend on the load, while copper loss grows with the square of the load and equals the core loss at some load; at exactly that point the loss relative to output is smallest, so efficiency is highest. In terms of load factor, m = √(Pfe / Pcu), the load at which variable and fixed losses are equal. Distribution transformers run at light load most of the day, so they are built with small core loss to put this peak below rated load and raise the all-day efficiency, measured by the energy over a whole day. Meanwhile, the voltage drop across the same series impedance shows up as the voltage regulation ε ≈ p cosφ + q sinφ.

The efficiency: η = output / (output + core loss + copper loss). Copper loss goes as the square of load (m²Pcu); core loss is fixed. Peak efficiency occurs at m²Pcu = Pfe, i.e. variable loss = fixed loss, at load fraction m = √(Pfe / Pcu). The voltage regulation from the series drop is ε ≈ p cosφ + q sinφ, positive for a lagging factor and possibly negative for a leading one.

Closing out group A

That completes magnetic circuits and transformers. We read flux with the magnetic Ohm’s law (A1), saw the energy pooled in it (A2), swapped voltage and current by the turns ratio (A3), captured the non-idealities in an equivalent circuit (A4), and computed performance from the losses (A5). The next group B is the DC machine, which makes rotation on the same magnetic circuit. It begins with how the commutator turns AC into DC.