ELECTRICAL TECHNOLOGY EET 103/4

Transcription

1 ELECTRICAL TECHNOLOGY EET 103/4 Define and analyze the rincile of transformer, its arameters and structure. Describe and analyze Ideal transformer, equivalent circuit, and hasor diagram Calculate and justify efficiency, losses, erformance 1

2 TRANSFORMERS (CHAPTER 22) 2

3 FUNCTION OF A TRANSFORMER The main function of an electrical ower transformer is to transfer electrical energy from one side (rimary) to the other side (secondary). The secondary current and voltage may or may not be at the same level as that of the rimary current and voltage. The energy is transferred by means of magnetic couling. The magnetic flux roduced by the current in rimary winding links the secondary winding. Since the flux varies with time, this flux linkage results in an induced voltage in the secondary winding. If the secondary winding is terminated with a load, the induced voltage will drive a secondary current through the load. 3

4 22.1 Introduction This chater covers: Mutual inductance that exists between coils of the same or different dimensions. Basic to the oeration of a transformer Three of the basic oerations of a transformer. ste u/down imedance matching isolation The dot convention. 4

5 22.2 Mutual Inductance Transformers are constructed of two coils laced so that the charging flux develoed by one will link the other. The coil to which the source is alied is called the rimary coil. The coil to which the load is alied is called the secondary coil. 5

6 22.2 Mutual Inductance 6

7 22.2 Mutual Inductance 7

8 22.2 Mutual Inductance 8

9 22.2 Mutual Inductance Primary transformer formula for voltage using Faraday s Law: Voltage induced across the rimary and selfinductance is directly related to the number of turns in the coil and the rate of change of magnetic flux linking the rimary coil: 9

10 22.2 Mutual Inductance The magnitude of e S, the voltage inducted across the secondary is determined by; where N S is the number of turns in the secondary winding and m is the ortion of the rimary flux P that links the secondary winding 10

11 22.2 Mutual Inductance If all of the flux linking the rimary links the secondary: The coefficient of couling (k) between two coils is determined by; 11

12 22.2 Mutual Inductance Since the maximum level of m is P, the coefficient of couling between two coils can never be greater than 1. Coils with low coefficients of couling are termed loosely couled. 12

13 22.2 Mutual Inductance Mutual inductance between two coils is roortional to the instantaneous change in flux linking one coil due to an instantaneous change in the current through the other coil. In terms of inductance of each coil and the coefficient of couling, the mutual inductance is: 13

14 22.3 The Iron-core Transformer 14

15 22.3 The Iron-core Transformer The iron core will serve to increase the coefficient of couling between the coils by increasing the mutual flux m.. The magnetic flux lines will always take the ath of least reluctance, which in this case is the iron core. When the current I through the rimary circuit of the iron-core transformer is a maximum, the flux m linking both coils is also a maximum. The flux is directly roortional to the current through the winding. 15

16 The Iron-core Transformer t I i sin 2 If; then; t m m sin By Faraday s law; t dt d N dt d N dt d N e m m sin Or; 90 sin cos t N t N e m m

17 22.3 The Iron-core Transformer The effective value of e is; E N m fn m Since the flux linking the secondary flux equals that of the rimary; E s N s m fn s m 17

18 22.3 The Iron-core Transformer Dividing; E E s N N s N N s a Transformation ratio For and ideal transformer; E Vg and Es VL Hence; V V g L N N s 18

19 22.3 The Iron-core Transformer The ratio of the magnitudes of the induced voltages is the same as the ratio of the corresonding turns: The ratio N /N s, usually reresented by the lowercase letter a, is referred to as the transformation ratio. If a < 1, the transformer is a ste-u transformer. If a > 1, the transformer is a ste-down transformer. 19

20 22.3 The Iron-core Transformer Examle22.2 (a) Find the maximum flux m. (b) Find the secondary turns N s. 20

21 22.3 The Iron-core Transformer Examle22.2 Solution (a) E fn m m E 4.44 fn mwb 21

22 22.3 The Iron-core Transformer Examle22.2 Solution (cont d) (b) E E s N N s N s N E E s turns 22

23 22.3 The Iron-core Transformer The rimary and secondary current of a transformer are related by the inverse ratio of the turns: 23

24 22.4 Reflected Imedance and Power The imedance of the rimary circuit of an ideal transformer is related to the imedance load by the transformation ratio If the load is caacitive or inductive, the reflected imedance will also be caacitive or inductive. For an ideal iron-core transformer; 24

25 22.4 Reflected Imedance and Power Examle22.3 (a) Find I and V g. (b) Find Z. 25

26 22.4 Reflected Imedance and Power Examle22.3 Solution (a) I I s N N s I N N s I s ma 26

27 22.4 Reflected Imedance and Power Examle22.3 Solution (cont d) (b) Z a 2 Z L a N N s Z k 128 k 27

28 22.4 Reflected Imedance and Power Examle 22.4 For the residential suly, determine (assuming a totally resistive load) the following : a. The value of R to ensure a balanced load b. the magnitude of I 1 and I 2 c. The line voltage V L d. The turn ratio a=n /N s 28

29 22.4 Reflected Imedance and Power Examle Solution a. P T = (10) (60) W+ 200 W W = 600 W W W = 2800 W P in = P out V I =V s I s =2800 W (urely resistive load) (2400 V) I = 2800 W and I =1.17 A R=V hase /I =2400 V/1.17 A = ohms 29

30 22.4 Reflected Imedance and Power Examle 22.4 Solution (cont d) b. P 1 =600 W = VI 1 = (120V) I 1 And I 1 = 5 A P 2 = 2000 W = VI 2 = (240W) I 2 And I 2 = 8.33 A c. V L 3V 1.73(2400V ) 4152V d. a N N s V V s 2400V 240V 10 30

31 22.6 Equivalent Circuit (Iron- core Transformer) For the non ideal or ractical iron-core transformer, the equivalent circuit aears below. Part of the equivalent circuit includes an ideal transformer. 31

32 22.6 Equivalent Circuit (Iron- core Transformer) 32

33 22.6 Equivalent Circuit (Iron- core Transformer) All elements in the iron-core transformer other than the ideal transformer are the elements of the transformer that contribute to the nonideal characteristics of the device. Resistance R and R s are simly the geometric resistance of the rimary and secondary windings. Leakage flux L and L s is a small amount of flux that links each coil but does not ass through the core. 33

34 22.6 Equivalent Circuit (Iron- core Transformer) The resistance R C reresents the hysteresis and eddy current losses (core losses) within the core due to an ac flux through the core. The inductance L m (magnetizing inductance) is the inductance associated with the magnetization of the core, that is, the establishing of the flux m in the core. The caacitances C P and C S and the lumed caacitances of the rimary and secondary circuits, resectively, and C w reresent the equivalent lumed caacitance between the windings of the transformer. 34

35 22.6 Equivalent Circuit (Iron- core Transformer) For ower transformer; Winding caacitances have negligible effects because of the low oerating frequency. Under normal oerating condition, normally i >> i m. Hence, C, C s, C w, R C and L m may be omitted. The aroximate equivalent circuit becomes as follows; 35

36 22.6 Equivalent Circuit (Iron- core Transformer) Reduced equivalent circuit for the nonideal ironcore transformer. 36

37 22.6 Equivalent Circuit (Iron- core Transformer) Normally, L and L s are reresented by their equivalent reactances; X and X s Ls L X X s 37

38 22.6 Equivalent Circuit (Iron- core Transformer) X X s X s and R s may be reflected into the rimary circuit using the relationshi; R 2 ' a and X ' a X s 2 R s 38

39 22.6 Equivalent Circuit (Iron- core Transformer) The equivalent circuit now becomes; 39

40 22.6 Equivalent Circuit (Iron- core Transformer) Combining the series resistances and reactances, results in the following circuit; R e R R ' X e X X ' 40

41 22.6 Equivalent Circuit (Iron- core Transformer) The load resistance R L may also be reflected into the rimary circuit; R ' a 2 L R L 41

42 22.6 Equivalent Circuit (Iron- By voltage divider rule; core Transformer) av L g R R jx e R i i e V R a 2 i R L is the load resistance reflected into the rimary circuit of the transformer 42

43 22.6 Equivalent Circuit (Ironcore Transformer) Examle 22.7 (a) Determine R e and X e. (b) Determine the magnitude V L and V g. 43

44 Equivalent Circuit (Iron- core Transformer) Examle 22.7 Solution (a) s e R a R R s e X a X X

45 22.6 Equivalent Circuit (Ironcore Transformer) Examle 22.7 Solution (b) av L I a 2 R L 2400 V V a V g I R e V L a 2 R L jx e V g j V 100 j 2452 V