07 TT 02 Electromagnetic

energize - March 2011 - Page 36

TRANSMISSION AND DISTRIBUTION

Transformers in general consist of primary and secondary windings, across which a voltage and current transformation is induced at constant power. During normal operation the transformer will be supplied on the primary at rated voltage and current. Current flowing in the primary windings will induce a magnetic field that will link the windings of the secondary, inducing current flow. Current flow in the secondary will be inversely proportional to the turn ratio between the primary and secondary windings.

The generation of an electromagnetic field inside the transformer is therefore intrinsic to transformer operation. This electromagnetic field will however lead to the generation of forces inside the transformer windings stipulated by the laws of magnetism viz. Faradays law of induction and Lenz's law of electromagnetic force[1].

During normal operation, these forces are relatively low and for this condition the transformer design is based primarily on the dielectric and thermal considerations – loss reduction and insulation integrity. Under short-circuit fault conditions the current excitation increases significantly, possibly 8 – 10 times nominal current, resulting in extreme forces in the transformer windings. The short circuit withstand capability of a transformer, is mainly a function of its thermal and mechanical performance. However, due to the speed at which these faults occur and are cleared, the primar y concern is of a mechanical nature, and design considerations shift to control the electromagnetic forces and prevent mechanical failure. The basics

Electromagnetic forces in transformers under short circuit conditions

by Nadim Mahomed, Powertech Transformers

Short circuit events generate high current conditions in transformer windings. These currents in turn induce excessive forces in a transformer. Electromagnetic forces are important considerations in the design, manufacturing and operation of transformers. These forces can be subdivided into axial and radial forces each with unique considerations and mitigating measures.

of electromagnetic forces are presented here to guide toward an appreciation of the significance of short circuit events in transformer design. The discussion begins with a description of the short circuit current, the induced electromagnetic forces and finally methods of preventing transformer failure.

Short circuit current

The most commonly occurring short circuit event is a single-line to earth fault in which one phase is short circuited to ground. This type of fault may occur due to lightning strikes, debris, pollution effects, animals and vegetation. Other types of faults such as three-phase to ground, double earth fault are also considered in short circuit current calculations. In general the short circuit current is calculated using symmetr ical components for var ious situations taking into account:

Tapping arrangement

Fault position (e.g. low voltage or high voltage winding)

Shor t c i rcu i t power combinat ion (network and transformer)

Short circuit type (e.g. single phase to earth, three phase symmetrical, etc.)

T he sho r t c i r cu i t cu r r en t g i ven i n Eqn 1.consists of two components: a steady state component at power frequency, and an exponentially decreasing unidirectional component as shown in Fig. 1 [2].

(1)

where:

i(t) = instantaneous short circuit current

Ip = peak short circuit current

= angular frequency in rad/s

= voltage angle at which short circuit occurs

= impedance phase angle

Fig. 1 shows that the first current peak is the maximum peak, becoming progressively smaller as the unidirectional (green line in Fig. 1) component decays. For this reason the first peak is usually used to calculate maximum occurring forces.

To aid the understanding of short circuit current and its relationship to the power transformer, the simplest fault scenario, viz. three-phase symmetrical short circuit fault, is discussed. This allows the use of an equivalent single phase circuit model, as all phases remain balanced and nominal or rated values of the transformer current can be used. The steady state component of short circuit current for this simple case can be expressed as a multiple of nominal current by the overcurrent factor "r" shown in Eqn. 2. The steady state current under short circuit condition is then r times the nominal current.

(2)

where:

Zt = transformer impedance in percentage

Zs = system impedance in percentage

To account for the initial direct current (DC) offset and therefore the first peak, a further factor called the asymmetry factor "k" is used [2]. This results in a more accurate representation of the current to precisely calculate the peak electromagnetic force. Values of k are specified in the applicable standards such as the IEC 60076-5 [3]. The asymmetry factor is dependent on the ratio of the transformer reactance (X) and resistance (R). Values of k increase for increasing X/R ratios. Values of 1,8 for transformers up to 100 MVA and 1,9 for transformers in excess of 100 MVA are indicated when the X/R ratio is unavailable [3].

Electromagnetic forces

The current carrying conductors of the Fig. 1: Short circuit current [2].



transformer windings are situated in the magnetic leakage field. By the fundamental postulates of electromagnetics these conductors w i l l exper ience a force due to the interact ion between the electric (current) and magnetic fields. This electromagnetic force is calculated as the vector product of current density and magnetic field intensity as given in Eqn. 3 [1].

(3)

where:

F = force in N

J = current density in A/m2

B = magnetic flux density in T

Short circuit current will influence both flux density B and current density J in Eqn. 3 implying that force is proportional to the square of the current. Fig. 2 shows the relationship between current and force and the nature of electromagnetic forces. The electromagnetic force pulsates at approximately twice the power frequency and is unidirectional. Considering that the overcurrent factor is typically between 8 and 10, the force generated can easily be in the region of a hundred times the nominal force magnitude and may reach several thousand kN [2].

The direction of the forces are stipulated by the vector product in Eqn. 3 indicating that the force will act perpendicular to the plane formed by the magnetic field intensity and current direction. The force direction can be determined using the left hand rule, as indicated in Fig. 3 which describes the field relationships for the cross-section of a single conductor.

Electromagnetic force tends to minimise the magnet ic energy dens i ty in the volume. In this way the forces tend to :

Reduce the radius of inner windings

Increase the radius of outer windings

Reduce the height of windings – windings are compressed toward the median horizontal

Viewing the transformer from a cross sectional view (Fig. 4), and taking into account the direction of the current as being constant, it is noted that the force is perpendicular to, and follows the bending of the flux lines. In the middle of the windings this results in a radial force pushing outward, and toward the ends of the windings this results in an axial force pushing onto the windings.

The explanation of the transformer cross section is shown in Fig. 5. The 3D figure of

the full 3-phase transformer on the left is a 360° rotation of the 2D diagrams on the right about the axis (dashed line). This is a single phase axi-symmetric simplification of the transformer geometry that may be analysed in two dimensions as is the case in Fig. 4.

Electromagnetic forces in transformer windings can be sub-divided into axial and radial force by means of associated axial and radial modes of failure. Axial forces occur in a direction parallel to the winding height. Radial forces occur perpendicular to the winding height. Axial and radial forces, although shar ing a common origin, can for the most part, be treated as mutually exclusive modes [2].

Radial forces

The flux at mid winding height is for the most part parallel to the winding height. According to the left hand rule, the resulting force therefore acts perpendicular to the winding height. For windings on the inside of the main flux field (situated between primary and secondary windings) the force acts inward, and for outside windings, outward as depicted in Fig. 5.

Forces acting on the inside windings result in a compressive stress whereas on the outer winding this force leads to a tensile

Fig. 2: Current and induced force waveforms.

Fig. 3: Directional relationship between fields.Fig. 4: Magnetic field and associated

force directions.



stress acting to elongate the winding turn – as shown in Fig. 6 [2]. This is as a result of the cylindrical profile of the windings where forces, perpendicular to the winding circumference, create tangential stresses.

Axial forces

Axial forces are generated parallel with the winding height. Due to the pattern of the magnetic leakage field, the windings experience opposing forces at the winding ends, leading to compressive forces. Axial force is generated when the transformer's magnetic field lines are radially orientated. Fig. 4. shows that the highest ‘bending' of the magnetic field occurs at the winding ends, consequently maximum axial force is generated here. The local force generation accumulates toward the middle of the windings resulting in a maximum occurring force at mid-winding height. This force is compressive towards the winding centre as the circulation direction of the magnetic field leads to opposing forces at each end of the winding. A schematic representation of the axial force distribution, in a typical transformer winding, is shown in Fig. 7.

In addit ion to the compressive force occurring in the winding the effect of recoil is also considered as transformer windings are essentially springs from a mechanical point of view. This spring action can be explained by noting the nature of the electromagnetic wave in Fig. 2. The force magnitude fol lows a s inuso ida l pat te rn and therefo re compressive tension stored in the winding is released as the force approaches zero. This force will be exerted against the core yokes and end insulation structures, and will have a magnitude of less than the compressive force generating it.

Winding displacement from the centre line result in excessive axial forces, and provision for this possibility is necessary – shown in Fig. 8. This is due to the difference in the ampere-turns (i.e. mmf) distribution along the winding heights resulting in more pronounced bending of the field lines at the point of missing ampere-turns or where the displacement occurs. The forces then

work to increase the displacement i.e. windings are pushed further apart, resulting in iteratively worse displacements and therefore higher forces [2].

Mitigating the effects of short circuit

In essence there are two approaches to reducing the destructive effects of short circuit forces. At the outset transformer Designers aim to decrease the occurring electromagnetic forces by sound design choices. Electromagnetic forces however will always be of significant levels, and measures to mitigate the effect of these induced forces must be taken. Measures against e lect romagnet ic forces are

primarily of a mechanical nature, involving st ructural countermeasures whereas reducing the origin of these forces is primarily a magnetic field problem and therefore electrical in nature.

As stated earlier, radial and axial short circuit forces and their associated modes of failure may be considered mutually exclusive for the most part. Preventative measures will involve manipulating the magnetic field, and making choices in the design that will lead to a more acceptable field distribution inside the transformer. One way to reduce the magnitude of the magnetic field would be to decrease the short circuit current by increasing the

Fig. 5: Transformer 2D diagrammatic representation explained. Fig. 6: Conductor force directions and associated stresses.



impedance values of the system and/or transformer thereby decreasing the overcurrent factor. However this is not always practicable since these parameters are usually dependent on a host of other requirements such as network topology and customer specifications.

Manipulating the field-winding relation can be achieved by changing the geometries of the transformer active part i.e. core and windings. The goal of any field manipulation would be, primarily, to reduce axial forces by straightening the field l ines in the region of the winding ends. Less bending of the field lines equals less axial forces. Adjusting winding height, winding duct width (space between windings), winding inner diameter, distances from winding to core are some of the considerations that can have a significant impact on the field pattern with limited effect to other design considerations.

In addition to field considerations the transformer must be built to mechanically withstand all forces generated under a short circuit event. The structural rigidity of the transformer active part is paramount

as high short circuit forces are inevitable. The short circuit withstand capability of a transformer is dependent on the occurring stress vs. mechanical withstand ability of the various structural parts.

Axial allowable stress limits are divided into two categories namely limits relating to the compressive forces in the windings themselves and those related to axial thrust forces toward the clamping structure. Axial thrust forces are a consequence of winding displacement (Fig. 8) and the spring action of the windings (after compressive forces). Compressive force is a concern in the windings themselves whereas thrust forces affect the end structures (between the yoke and windings). End thrust forces can lead to failure of the end insulation structures, which include those structures common to all windings. Since all windings in a multiple winding transformer may experience simultaneous end thrust forces (all currents in a winding block being in phase), the cumulative effect must be taken into account. Compressive forces on the other hand can lead to failure of the windings themselves. Various modes of failure are taken into account viz. conductors bending between spacers, conductor tilting, conductor telescoping, and spacers disintegrating [2]. Also worth noting is the pulsating nature of the forces, which can lead to deterioration of the insulation structure and demonstrate the importance of quick fault clearing times. The risk of failure from any of these forces is directly related to the processing and assembly of the windings. Reducing the risk of failure due to axial stresses, is concerned with minimising winding displacements, cor rect winding clamping, reduced moisture content due to proper processing and the use of quality materials.

Rad ia l fo rces lead to tens i l e and compressive stresses which display distinct modes of failure. Tensile stresses are due to forces acting to increase the diameter of the windings. These forces result in

stretching of the conductors and may lead to rupture, if the conductor yield strength is exceeded. Compressive stresses lead to buckling of the conductors, in which the conductors are forced inward and could potentially bulge outwards at the elastic limit of the conductor material. Mitigating radial forces primarily involves choosing the correct hardness of the conductor whi le taking into account cost and manufacturability. In addition, a sound winding design methodology such as self-supporting windings, correct drying and processing, and the use of appropriate conductors minimises the risk of radial failure.

Both axial and radial forces often lead to secondary failure which is dielectric in nature. Failure of the insulation structures due to the aforementioned failure modes is common, leading to arc formation, and dielectric breakdown.

Conclusion

Shor t c i rcu i t events that occur in a network induce high mechanical forces in transformer windings. The electromagnetic force is induced in the transformer windings and is due to the interaction of the magnetic field of the transformer and the current density in the transformer windings. As such the force is exponentially proport ional to the cur rent resul t ing in excessive force generation for any increase in current. The radial and axial force may be treated independently and could result in radial and axial modes of failure if the withstand capability is exceeded. The magni tude of these forces may be reduced by manipulating the magnetic field by changes in the transformer geometry. However provision for these forces must be made, consisting mainly of the mechanical withstand capabilities and positioning of the various transformer materials. Initially a transformer design is driven by thermal and dielectric properties; however short circuit provisions often supersede these design choices leading to s ignif icant cost increase. Short circuit withstand is a complicated interaction of electrical- and mechanical engineering and its significance in any electromagnetic machine should not be underestimated, least of all in power transformers.

References[1] D K C h e n : F i e l d a n d W a v e

Electromagnetics,Addison-Wesley Publishing Company Inc., USA, second edition, November 1992.

[2] G Bertagnolli: Short-circuit Duty of Power Transformers, ABB Ltd., Zurich, third edition, June 2006.

[3] IEC standard 60076-5: Power Transformers – Part 5: “Ability to withstand short circuit”, 2006.

Contact Nico Gunter, Powertech Transformers, Tel 012 318-9911, [email protected] Fig. 8: Winding displacement

and induced forces.

Fig. 7: Cumulative and local force distribution along winding height.

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07 TT 02 Electromagnetic