3 phases and 5 limbs (fun hidden secrets)

Here's what you'll find in this article:

• Why would someone build a 5 limb transformer core instead of a 3 limb?
• What are the advantages?
• What are the downsides?
• And so? What's so cool about it? (hope you don't skip straight here)

Why would someone build a 5 limb transformer core instead of a 3 limb?

I saw two reasons that could lead to this drastic change in the design of the transformer's magnetic circuit, they are:

1. In large power transformers, the height for transportation is a determining limitation for the design and some manufacturers use the return limbs to reduce the height of the equipment. See in the Figure 1 that the inclusion of the return limbs allows to reduce the cross section (and consequently the dimensions of the sheets) of the upper and lower yokes by half. Considering a core with a diameter of 1m, this same 1m can be reduced in the final height of the transformer. This is usually enough reason to justify the change (perfectionists will tell you that yokes need to use more than 50% of the cross section of the main limb, and in this case they are right. The reason will be explained later).
2. Transformers with star connection in all windings, that is, YNyn, which may be exposed to unbalanced voltages or even due to single-phase loads, may have heating in the clamps/frames and in the tank (if it is a transformer immersed in oil) or emanate intense magnetic fields (if it is a dry transformer). In these situations, the magnetic flux does not "add to zero" and ends up needing to circulate outside the limbs with windings. The return limb(s) allow the circulation of this magnetic flux, which is the zero sequence flux, without it permeating structural steel materials. It is quite common in the distribution network of some countries, although it can be applied with only one return limb intead of two as indicated.

What are the advantages?

• Reduction of the height of the equipment
• Reduction of stray losses: part of the magnetic flux dispersed from the coils will circulate through the return limbs without causing losses in the steel frames or tank
• Low reluctance path for zero sequence flux (important in YNyn): in addition to avoiding heating in the tank, it also reduces the intense magnetic fields propagated by the transformer during inrush events, for example.

What are the downsides?

• Increased dimension towards the core limbs, i.e. in the length of the transformer
• Increase in the mass of silicon steel required (even with the reduction in yokes, there is some overall increase, although not that dramatic) and structural steel, and also an increase in the volume of oil
• Low reluctance path for zero-sequence flux (yes, this is not always an advantage): when subjected to some continuous component in the neutral, such as during a GIC (Geomagnetic Induced Current) event, this characteristic ends up allowing the core to saturate more quickly, which sharply increases the amount of reactive power drained from the power system and can bring stability problems to the network.

And so? What's so cool about it?

Non-sinusoidal flux in return limbs

Considering an unloaded transformer, the flux in the core is the time integral of the voltage in its primary winding, therefore, it typically has a sinusoidal waveform, since the integral of the sine function also has the aspect of the sine function, but with a phase shift and amplitude change. This happens in all segments of the core of a three-phase three-limb transformer (the most conventional one), as all segments are "in series" with the limbs that has coils. Thus, taking some arbitrary angular instant, the flux will always go up one limb and down another, while the third will have no flux at all, or something like that.

The first 'fun' feature of the three-phase five-limb transformer is the non-sinusoidal flux, as the topography of the magnetic circuit does not impose any flux directly on the return lilmbs, but the flux split between the two possible paths will occur: the first is the path of the other phases and the second option is the return limbs, mainly the one that is closest. The Figure 2 displays an illustrative diagram.

When at low flux densities, the preferred path is the return limb, but as the flux density increases and the permeability of the steel drops, the participation of the return limb becomes smaller when compared to the coiled limbs that are "pulling" the flux, regardless of the constitutive characteristic of the core. This phenomenon ends up forming a plateau in the waveform of the flux that circulates through the return limb. The Figure 3 shows the flux waveforms (note that the flux in the return limb is multiplied by 2, to refer to the comparative flux density with the main limb). It is also important to mention that the appearance of the plateau depends a little on the model used to characterize hysteresis and saturation. So if we wind a coil around this limb and measure the voltage, the voltage will not be sinusoidal either.

Flux distribution in yokes and return limbs

If the magnetic flux is not sinusoidal in the return limbs, it can be deduced that the same is true for the central yokes. The deflected magnetic flux that forms the plateau in the return limbs appears as an excess flux in the central yokes (see Figure 4). For this reason, it is common to use more than 50% of the cross section of the main limbs in these segments, avoiding high levels of flux density, reducing losses and avoiding hot spots. Note also that at any instant the sum of the black and green curves results in the red curve, multiplied by -1.

Different building schemes, including a non-intuitive one

While the constructive scheme of the three-phase three-limb core is practically unanimous, for the three-phase five-limb core one can find several different schemes, each with its pros and cons. It is even possible to build the core using only trapeze-type plates, which eliminates all scrap, making the net weight equal to the gross weight. However, in this constructive style, continuous vertical gaps are formed that separate the core into four independent frames, as shown in Figure 5.

Thinking again at an arbitrary instant, for example, when the flux is at its maximum upwards in phase 1 and at its maximum downwards in phase 3, there is an impasse. At the same time that the return limb do not have enough cross-section to support the entire flux of the adjacent coiled limb, the flux also cannot circulate over the central phase due to the vertical gaps.

Despite this, this constructive scheme works and the solution is surprising. In the central phase where the net flux needs to be zero (due to an imposition of winding voltage) there is magnetic flux in both directions, in half of the section the magnetic flux follows "upwards" and in the other half it presents the same value, however following "downwards", so that the net flux is zero. Despite the advantages related to the reduction of scrap, this type of construction "in frames" presents higher values of specific losses, in terms of W/kg when compared to the conventional construction scheme without the vertical gaps. This happens because, on average, the flux density shows higher values in the core as a whole. Observing the animation below carefully, it is possible to visualize the phenomenon.

Inrush in Yy dry transformers

When a transformer is subjected to an inrush event (see articles 1, 2 and 3), the core enters the saturation region and the excess flux overflows the transformer, reaching high levels in the vicinity of the equipment (unless it is confined in a delta winding or in the tank). In oil-insulated transformers, because they have a tank of magnetic material of considerable thickness, most of this flux ends up being confined and only a small portion can effectively circulate outside the transformer. However, in dry Yy transformers, it is quite common for nearby sensitive equipment to be affected quickly (I have seen some digital clocks reset, it would certainly affect the hands of a compass). Return limbs help to mitigate these effects, although not completely eliminate them.

To clear up any misunderstandings before they arise

This article deals only with core-type transformers. If you want to know a little better the difference, you can consult this post by Doug, or Kulkarni's book, for example (when a core-type transformer has limbs without windings, the so-called return limbs, it is commonly called shell-type transformer, but I don't think this is consistent, as this change in the magnetic circuit does not integrate all the constructive differences of a true shell-type transformer).

References

This time there isn't... what appears in this article, in general, is the result of personal technical experience in real applications. The animations were created using automated routines in the FEMM 4.2 program and the flux curves were calculated using a program I wrote during my master's degree to solve coupled electric and magnetic circuits.

Important note about the flux GIF: Unfortunately, in FEMM 4.2, the windings cannot be excited with voltage, this implies some inaccuracies in the flux distribution (since I had to estimate and impose the excitation current).