Observación no estructurada

The power of electric current flowing through electric line conductors is trans-ported by an electromagnetic field surrounding these conductors. Physically it is explained that although electrons inside the conductors move along the electric field with velocity barely ca. 0.2 mm/s, the impulse and electromagnetic field move with velocity near 300,000 km/s. With the same speed is transmitted electric energy.

Mathematically, the whole flux of power flowing along conductors can be calculated by the integration of the power flux density, that is, the Poynting vector, through an infinite plane perpendicular to the conductor axes.

Let us consider the power flow in the case when two current-carrying conductors pass through a metal wall (Figure 3.1), which may be a cover of a power transformer, or a metal wall with a thickness d > λ. From the total power flux flowing in sur-roundings of the conductors and meeting the barrier in the form of impenetrable transformer cover, one can distinguish the following components of Poynting vector in particular points of space (Figure 3.1):

• The main vector S_g corresponding to the power density flowing into the transformer. It is directed along the conductors. This vector can be split into

i i

FIGURE 3.1 The field components, H and E, and the Poynting vectors S, at conductors passing through a steel cover of transformer tank (Turowski [2.31]): E₁, H₁, S₁—the field inci-dent on steel surface, E₂, H₂, S₂—the reflected field, E₃, H₃, S₃—refracted (penetrating) to the steel body of cover; Scu—into copper bushings; Sg—density of the general (main) power flux (W/m²) incoming and outgoing from the inside of transformer.

two components: the component of power which is the incident on the cover surface, S₁, and the component of reflected power, S₂.

• The vector of power penetrating the solid steel of cover, S₃. It represents the power losses in the cover.

• The vector of power penetrating into the current-carrying conductors, S_Cu. It is directed perpendicularly to the surface of the conductors and its active component equals to the per-unit losses in the conductors.

The above considerations follow from the following simple equations

E i

l H i

r S E H R i

p = R , ₀ = , = ₀ = rl²

2p ^ρ 2p

which, after considering Poynting’s theorem (3.1), gives the total losses in conductors:

∆P S i

rl rl Ri

A

cu =

 ∫

⋅d^A= ^2Rp p^{2 2} = ²

In an analogical way, the reactive component of the S_Cu vector determines the internal reactance of conductors.

An investigation of the electromagnetic field and power losses in the cover alone, the component S_Cu can be skipped, assuming that the bushings are made of a very good conductor.

• The refracted vector S₃ of the power-penetrating metal of the cover through all its surfaces; it is directed perpendicularly to the cover surfaces. This vector will be used in the calculation of losses in covers and other construc-tional metal parts of electric machines and apparatus.

The main flux of power S_main≡ Sg (Figure 3.1), carrying the power flowing into or out of the transformer, squeezes almost totally through the isolation (porcelain) gap around the bushing conductor if the thickness of the cover is sufficiently big (d > λ) and has no other holes filled with dielectric in which an electromagnetic field would exist. The power flux flowing into the transformer does not depend on the thickness of the mentioned isolation gap, because the thinner is the gap the bigger will be value of the vector E, and therefore vector S. In the case of a metallic connection, when the isolation gap does not exist, a short-circuit occurs and then no power flux or current can penetrate through the cover.

The total power flow into a transformer exclusively through holes filled by dielec-tric can be easily checked quantitatively. In a single-phase transformer, both on the primary and the secondary side, only two bushings are present. The electric field intensity in the central plane of the hole in the steel sheet amounts to

E u

= rln (R/r₁)

where u is half of the line voltage, that is, 2u = uline; R the radius of the hole for bush-ing; r₁ the radius of the conductor in bushing. Usually, the distance between bushings is so large in comparison with the radius of hole that one can assume that in the hole exists the magnetic field intensity H = i/(2πr).

Since both fields are perpendicular to each other, the modulus of the vector of power density in the hole per surface unit has the value

S = E H = ui ⋅ R r r 2

1 p ln /( ) 3

(3.14)

and an element of the considered surface is dA = 2πr ⋅ dr. The instant power flow-ing into the transformer tank by two holes, accordflow-ing to Poyntflow-ing’s theorem, there-fore, equals

p S dA ui r r

r ui u i

A r

R

= ²

∫∫

= ^{2 2} 1

∫

² 2⋅ = ² =

p 1 p

ln(R/r)

d

p

which proves that the entire instantaneous power passes only by the isolation of the bushing holes (Figure 3.2).

S H^→ E^→

S

S

→

d E ≈ 0 E ≈ 0

i i

E ≈ 0

FIGURE 3.2 A schematic picture of the power flow through the cover of a transformer tank, the Poynting vector (S) lines, with ignoring the eddy current losses. (Adapted from Turowski J.: Electromagnetic field and losses in the transformer housing. (in Polish). “Elektryka”

Science Papers, Technical University of Lodz., No. 3, 1957, pp. 73–63.)

In Figure 3.2, the continuous lines represent the power flux lines (lines of the Poynting vector S) and the dashed lines represent the lines of the force of an elec-tric field E. Lines of vector H lie in planes parallel to the surface of the tank cover.

Figure 3.2 does not present quantitative interdependences. Such plots are created on the basis of perpendicularity of the lines of vector S to the planes of vectors E and H.

In case when the thickness d of a cover or screen is smaller than the wavelength (d < λ) in metal (Table 2.1), a certain fraction of the field power penetrates into the tank directly through the cover, which in this case becomes transparent to the elec-tromagnetic field (Section 4.3).

3.4 POWER FLUX IN A CONCENTRIC CABLE AND SCREENED BAR Reasoning as above and assuming that in formula (3.14) u means the voltage between conductors of the cable, we can see that the total power ui transmitted by the cable is transported by the electromagnetic field moving in parallel to the cable axis in dielectric enclosed between internal and external conductors of cable. This power flows, however, not as a uniform flux but is more concentrated near the surface of the internal conductor and decays according to Equation 3.14 inversely proportionally to square of the distance r from the cable axis (Figure 3.3).

A similar picture of the power flux distribution occurs in screened bars used in power generators and transformer systems of power stations. In such a case, each of three bars of a three phase system is enclosed in a cylindrical screen. The screens are either grounded or connected together on both ends, directly or through reactors. In such systems, the magnetic field of bar gets out indeed outside of the screen, but the electric field exists practically only in the insulation space between bar and screen.

Therefore, the power, similarly as in a concentric cable, is transported in this case also only through the enclosed space.

i

2r 2R

S~ 1r₂

u 2 S

i i

r

FIGURE 3.3 Distribution of the Poynting vector (power density) in cross-section of a con-centric cable.

3.4.1 FactorSoF utilizationoF conStructional Space

One of the most important factors of technology advancement is reduction of space (limiting outlines) occupied by electromagnetic equipment. Especially it is seen in switching stations and devices with SF₆, occupying several times smaller space than conventional constructions.

The main reserves are in a nonuniform distribution of the Poynting vector (power density), measure of which can be the factor of utilization of constructional space

ηS = Smin/S_max (3.15)

The factor (3.15) can assume values within the range 0 ≤ ηS ≤ 1. For instance, for a concentric cable, hs =r R₁²/ ², which means that a small change of one of the diam-eters causes a significant change of ηS. For the interwinding gap of a transformer, however, ηS ≈ 1.

Another factor could be the ratio ηP = Pvar/P_unif of the total powers P = ∫∫AS dA at a nonuniform P_var distribution to the flux P_unif at a uniform distribution of the Poynting vector (power density).

Another factor here is the highest possible value S_max, which in turn is limited by the electric field strength of space. As per Equation 3.14, for cables with a constant utilization of conductors (u = const, i = const, r = const)

h^S = −p



 exp 

max

ui

r S12 (3.16)

Assuming, for example, for a concentric cable: R r/₁² = =e 2 718. , we get ηS = 1/e² = 0.136. The maximum value of the Poynting vector for such a cable, with parame-ters 35 kV, 400 A, cross-section 185 mm², r1 = 7.5 mm, R = er1 = 2.7 . 7.5 = 20.2 mm, would be

S_max = ⋅ .

⋅ =

35 400

2 ₂ 4000 ²

p 0.75 kW/cm

It is not difficult to estimate how the transmission power of the cable could be increased if the factor ηS was made higher. It is possible by means of multilayer structures. These conclusions should of course take into account material and pro-cessing limitations.

3.5 POWER FLUX IN A CAPACITOR, COIL, AND TRANSFORMER A capacitor and a cylindrical coil can be considered as the simplest constructional elements. They can, at the same time, serve as models of more complex systems, for instance—transformers. In a parallel-plate capacitor (Figure 3.4) connected to an alternating voltage u, in the part in which the field is uniform, we have

E =1_zu and H =1 a

i

Θ2pr (3.17)

Ignoring the edge deformations of the field (the so-called fringing fields) and tak-ing into account the sense of vectors (3.17) we can see that the Poynttak-ing vector in any considered moment is directed toward the capacitor center axis, and equals

S = E×H=−1r u i⋅ ra 2p

After multiplying this value by the lateral surface 2πra, we can see that the entire power flux entering to capacitor field by its lateral surface equals to u ⋅ i delivered to capacitor, as expected. The power flow into or out of the capacitor occurs along the equipotential lines, which in Figure 3.4 is shown by the arrows, at a moment of increasing voltage. At decreasing voltage, the sense of the instantaneous vectors S changes to the opposite one. The power flux in a capacitor connected to an alternat-ing voltage oscillates with frequency 2f.

In a resistance-less cylindrical coil (Figure 3.5), supplied by an alternating volt-age, the magnetic field H (dashed lines) inside the coil in a considered moment is directed toward the top, and outside—toward the bottom. The electric field E is tangential to the rings created by turns and directed according to the electromotive force (EMF) induced by the flux of the coil. As a result, the vector S (continuous lines) goes out from the coil surface to the ambient surroundings in both directions.

Similarly as before, it is easy to show (Turowski [1.15]) that the entire flux of an electromagnetic field delivered to the coil leaves (or enters) by its lateral surfaces—

external and internal. The direction of power flow changes with frequency 2f. During

aS

S S

E S

H i

i S Sr

H E

FIGURE 3.4 Distribution of power flux in a parallel-plate capacitor; S—Poynting vec-tor = power flux density (VA/m²).

one quarter period of the network frequency the magnetic energy is accumulated in the coil field, and then, during the next quarter period, it is given back to the network.

This is the so-called reactive power.

The active power flux (power losses) P₁= Sp (in W/m²) has always the same sense—toward the receiver. In Figure 3.5, the lines (continuous) are shown of the Poynting vector S at the moment when the reactive power is delivered to the coil from the network side. In a similar way, a distribution of power in a coil with iron core or in a transformer at no load will occur.

The described distribution of electromagnetic power concerns a smooth coil as a whole. It could be, of course, subdivided into particular twists around different discrete elements, such as turns of coil, leads, interconnections, and so on, as it was done in the work of Leites [3.2]. However, it is not necessary in practice.

In the short-circuit condition of a power transformer almost all the magnetic flux is displaced into the interwinding gap. There, as a leakage field, it induces in

S

i i

H U

H

H S S

S

S E

E

E

S E

H E

E S

H H

S S

S H E

H l₁

b

2R

S –

+

FIGURE 3.5 Distribution of power flux in a cylindrical coil.

both windings electric fields of inverse senses. Therefore, approximately in the center of the gap there exists a cylindrical surface on which E = 0, and hence S = 0. Thus, the gap is a barrier through which at short-circuit condition no power flux can pass from the primary to the secondary winding, except the power loss in secondary winding.

In a loaded power transformer the senses of E in both windings are the same.

Thanks to it the power flux S transfers from the primary winding to the secondary winding and to the magnetic core without obstacles. In the author’s book [1.15] the flow and distribution of power flux in a loaded transformer and in conductors in motion are discussed in more detail.

As we can see from the above figures, such a graphic illustration of the power flow S(x, y) is quite simple for any electromechanical system. It is only necessary to determine lines of fields H and E. It can be sometimes useful in practice, like for instance that case in Figure 3.2. In the next section, let us consider such power flow

In document TITEREHABLANDO SE DESENREDA LA ATENCIÓN (página 63-68)