Nuria Bouzas Antas
2. A paisaxe como construción social: o caso galego
Just as conductors guide currents in electric circuits, magnetic cores guideflux in mag-netic circuits. But there is an important difference. In electric circuits, the conductivity of copper is approximately 1020times higher than that of air, allowing leakage currents to be neglected at dc or at low frequencies such as 60 Hz. In magnetic circuits, however, the permeabilities of magnetic materials are, at best, only 104 times greater than that of air.
Because of this relatively low ratio, the core window in the structure of Figure 7.4a has
“leakage” flux lines, which do not reach their intended destination that may be another winding, for example in a transformer, or an air gap in an inductor. Note that the coil
(a) (b)
i
e
i
e φl
φm
FIGURE 7.4 (a) Magnetic and leakage fluxes; (b) equivalent representation of magnetic and leakage fluxes.
134 Power Electronics: A First Course
shown in Figure 7.4a is drawn schematically. In practice, the coil consists of multiple layers, and the core is designed to fit as snugly to the coil as possible, thus minimizing the unused “window” area.
The leakage effect makes accurate analysis of magnetic circuits more difficult that requires sophisticated numerical methods, such as finite element analysis. However, we can account for the effect of leakage fluxes by making certain approximations. We can divide the total fluxφ into two parts:
1. The magnetic flux φm, which is completely confined to the core and links allN turns, and,
2. The leakage flux, which is partially or entirely in air and is represented by an
“equivalent” leakage flux φ‘, which also links all N turns of the coil but does not follow the entire magnetic path, as shown in Figure 7.4b.
In Figure 7.4b,φ ¼ φmþ φ‘, whereφ is the equivalent flux that links all N turns.
Therefore, the total flux linkage of the coil is λ ¼ Nφ ¼ Nφ|ffl{zffl}m
λm
þ Nφ|{z}‘
λ‘
¼ λmþ λ‘ (7.18)
The total inductance (called the self-inductance) can be obtained by dividing both sides of Equation 7.18 by the currenti:
λ
|{z}i
Lself
¼ λm
|{z}i
Lm
þ λ‘
|{z}i
L‘
(7.19)
‘ Lself ¼ Lmþ L‘ (7.20)
whereLmis often called themagnetizing inductance due to φmin the magnetic core, and L‘ is called theleakage inductance due to the leakage flux φ‘. From Equations 7.19 and 7.20, the total flux linkage of the coil in Equation 7.18 can be written as,
λ ¼ ðLmþ L‘Þi (7.21)
Hence, from Faraday’s law in Equation 7.16, eðtÞ ¼ Lmdi
|ffl{zffl}dt
emðtÞ
þ L‘di
dt (7.22)
This results in the electrical circuit of Figure 7.5a. In Figure 7.5b, the voltage drop due to the leakage inductance can be shown separately so that the voltage induced in the coil is solely due to the magnetizing flux. The coil resistanceR can then be added in series to complete the representation of the coil.
Magnetic Circuit Concepts 135
7.4.1 Mutual Inductances
Most magnetic circuits, such as those encountered in inductors and transformers consist of multiple coils. In such circuits, the flux established by the current in one coil partially links the other coil or coils. This phenomenon can be described mathematically by means of mutual inductances, as examined in circuit theory courses. However, we will use simpler and more intuitive means to analyze mutually coupled coils, as in a Flyback converter discussed in Chapter 8 dealing with transformer-isolated dc-dc converters.
7.5 TRANSFORMERS
In power electronics, high-frequency transformers are essential to switch-mode dc power supplies. Such transformers often consist of two or more tightly coupled windings where almost all of the flux produced by one winding links the other windings. Including the leakage flux in detail makes the analysis very complicated and not very useful for our purposes here. Therefore, we will include only the magnetizing fluxφmthat links all the windings, ignoring the leakage flux whose consequences will be acknowledged separately.
To understand the operating principles of transformers, we will consider a three-winding transformer shown in Figure 7.6 such that this analysis can be extended to any number of windings.
FIGURE 7.5 (a) Circuit representation; (b) leakage inductance separated from the core.
FIGURE 7.6 Transformer with three windings.
136 Power Electronics: A First Course
Faraday’s Law: In this transformer, all windings are linked by the same flux φm. Therefore, from the Faraday’s law, the induced voltages at the dotted terminals with respect to their undotted terminals are as follows:
e1¼ N1dφm
dt (7.23)
e2¼ N2
dφm
dt (7.24)
e3¼ N3dφm
dt (7.25)
The above equations based on Faraday’s law result in the following relationship that shows that the volts-per-turn induced in each winding are the same due to the same rate of change of flux that links them,
dφm dt ¼ e1
N1¼e2
N2¼e3
N3
(7.26)
Equation 7.26 shows how desired voltage-ratios between various windings can be achieved by selecting the appropriate winding turns-ratios. The instantaneous fluxφmis obtained by expressing Equation 7.26 in its integral form below with proper integral limits,
φm¼ 1 N1
Z
e1dt ¼ 1 N2
Z
e2dt ¼ 1 N3
Z
e3dt (7.27)
Ampere’s Law: In accordance with the Ampere’s law given in Equation 7.9, the fluxφmat any instant of time is supported by the net magnetizing ampere-turns applied to the core in Figure 7.6,
N1i1þ N2i2þ N3i3¼ Rmφm (7.28) In Equation 7.28,Rmis the reluctance in the flux path of the core of Figure 7.6, and the currents are defined positive into the dotted terminals of each winding such as to produce flux lines in the same direction. The net ampere-turns consist of various winding currents that depend on the circuits connected to them.
Equations 7.27 and 7.28 are the key to understanding transformers: to one of the windings, the applied voltage, equal to the induced voltage in it if the winding resistance and the leakage flux are ignored, results in fluxφm which is supported by the net mag-netizing ampere-turns given by Equation 7.28, overcoming the core reluctance.
7.5.1 Transformer Equivalent Circuit
It is often useful to have an equivalent circuit of a transformer such as that shown in Figure 7.7b. Prior to developing this equivalent circuit, consider this to be an ideal transformer, with an infinite core permeability resulting inRm¼ 0. Therefore, the net
Magnetic Circuit Concepts 137
magnetizing ampere-turns in Equation 7.28 are zero, and such an ideal transformer is shown in Figure 7.7a.
A practical transformer such as that in Figure 7.6 doesn’t have infinite core per-meability and hence needs net ampere-turns to support the core flux. Although any of the windings could have been selected, let us select winding 1 to deliver the net magnetizing ampere-turns, with a magnetizing current im1 flowing through N1 turns. Therefore, in Equation 7.28,
N1im1¼ Rmφm (7.29)
From Equations 7.28 and 7.29, we can write the following, N1ði|fflfflfflfflffl{zfflfflfflfflffl}1 im1Þ
i10
þ N2i2þ N3i3¼ 0 (7.30)
wherei1 can be considered as consisting of the sum of two components,
i1¼ imþ i10 (7.31)
and, from Equation 7.30,
N1i10 þ N2i2þ N3i3¼ 0 (7.32) The net ampere-turns in Equation 7.32 equal zero, and hence this equation corre-sponds to the ideal-transformer portion of the equivalent circuit, as shown in Figure 7.7b.
In i1 of Equation 7.31 and Figure 7.7b, the magnetizing current im1 flows through the magnetizing inductanceLm1, as justified on the following page.
e1 e2
FIGURE 7.7 Equivalent circuits of transformers: (a) ideal, and (b) actual (leakage impedances not shown).
138 Power Electronics: A First Course
From Equation 7.29,
φm¼N1im1
Rm (7.33)
Substituting forφm from Equation 7.33 into Equation 7.23, e1¼ N12
Rm
dim1
dt (7.34)
where, using Equation 7.11, the quantity within the brackets in the equation above is the magnetizing inductance of winding 1,
Lm1¼N12
Rm (7.35)
The analysis above is based on neglecting the leakage flux, assuming that the flux produced by a winding links all the other windings. In a simplified analysis, the leakage flux of a winding can be assumed to result in a leakage inductance, which can be added, along with the winding resistance, in series with the induced voltageeðtÞ in the winding in the equivalent circuit representation. A systematic description of the principle on which transformers operate is presented in [1].
REFERENCE
1. N. Mohan, T.M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3rd Edition (New York: John Wiley & Sons, 2003).