1 8 () ( ) 1 2 V, I, N 8.2 8.2 (primary) (secondary) 1 2 V 1 I 1 = V 2 I 2, V 2 V 1 = N 2 N 1. V 1 = ±jωl 1 I 1 ± jωmi 2, V 2 = ±jωmi 1 ± jωl 2 I 2, ± 8.1 () () 8.1 [1 3] 8.2 ( ) 8.1 (, ransformer, ) [1 3] i 1 i flux Φ 2 v 1 primary secondary v 2 ferromagneic core i 1 M i 2 v 1 L 1 L 2 v 2 8.2 Load
2 8 () (p s ) ( ) V 2 V 1 = I 1 I 2 = N 2 N 1. (8.1) V 1 V 2 I 1 I 2 N 1 N 2 500 kv 100 V *1 1 V 1 I 1 = V 2 I 2 (8.2) 8.3 () 8.3.1 8.3 ( ) v 1 = L 1 di 1 + M di 2, (8.3) v 2 = M di 1 + L 2 di 2. (8.4) i M 1 i 2 v 1 L 1 L 2 v 2 8.3 L 1 L 2 M di v 1 1 L 1 di 1 v 2 2 L 2 2 * 2 v 1 2 M di 2 v 2 1 M di 1 * 2 8.3.2 8.4 : L k di k (k = 1,2) (8.5) *1 *2
8.4. (DOT CONVENTION) 3 i i i 1 i 1 v 1 v 1 i 1 i 1 Φ Φ i 2 i 2 v 2 v 2 M di 1 L di 2 = + 2 v 2 v 2 M di 1 L di 2 = + 2 8.4 8.4 ( ) () di 2 v 2 = L 2 + M di 1 (8.6) 8.4 () () di 2 v 2 = L 2 M di 1 M (8.7) ( ) 8.4 (Do convenion) J. W. Nilsson and S. A. Riedel [4] When he reference direcion for a curren eners he doed erminal of a coil, he reference polariy of he volage ha i induces in he oher coil is posiive a is doed erminal. When he reference direcion for a curren leaves he doed erminal of a coil, he reference polariy of he volage ha i induces in he oher coil is negaive a is doed erminal. = = 8.5 = =
4 8 () i 1 M i 2 v 1 L 1 L 2 v 2 I 1 M I 2 di v 1 = 1 L 1 + di 1 v 2 = M + di 2 M di 2 L 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 8.7 8.5 ( ) 8.5 di/ ω i() I di = jωi (8.8) 8.6 ( ) *3 () ( 1 ) + *3 v v > 0 8.7 V 1 = jωl 1 I 1 + jωmi 2, (8.9) V 2 = jωmi 1 + jωl 2 I 2. (8.10)
8.7. K 5 8.6 How o + 8.6.1 L 1 L 2 ± V 1 = ±jωl 1 I 1 ± jωmi 2, (8.11) V 2 = ±jωmi 1 ±jωl 2 I 2. (8.12) V 1 I 1 (+jωl 1 I 1 ) ( jωl 1 I 1 ) V 2 I 2 (+jωl 2 I 2 ) ( jωl 2 I 2 ) 8.6.2 M ± V 1 = ±jωl 1 I1 ±jωmi 2, (8.13) V 2 = ±jωmi1 ± jωl 2 I 2. (8.14) () () V 1 ( (> 0) ) (+jωmi 2 ) ( jωmi 2 ) () () V 2 ( (> 0) ) (+jωmi 1 ) ( jωmi 1 ) 8.7 k k M k L 1 L 2 M = k L 1 L 2 ( k 1). (8.15) 1 8.8
6 8 () Loss i 1 i 2 v 1 v 2 L1 L 2 8.8 k M I 1 I 2 L 1 M L 2 M High loss in magneic flux Low loss in magneic flux V 1 M I 1 +I 2 V 2 8.10 k < 1 k 1 Loosely coupled Tighly coupled 8.9 () V 1 = jω(l 1 M)I 1 + jωm(i 1 + I 2 ), (8.18) V 2 = jω(l 2 M)I 2 + jωm(i 1 + I 2 ). (8.19) 8.10 k < 1 M k k < 1 k < 1 k 1 8.8 8.10 () 8.10 8.10 8.10 V 1 = jωl 1 I 1 + jωmi 2, (8.16) V 2 = jωmi 1 + jωl 2 I 2. (8.17) 8.9 8.11 ( ) Z 1 = V 1 /I 1 V 1 = jωl 1 I 1 + jωmi 2, (8.20) V 2 = jωmi 1 + jωl 2 I 2, (8.21) V 2 = Z 2 I 2. (8.22) Z 1 = V 1 I 1 = jωl 1 + ω2 M 2 jωl 2 + Z 2. (8.23) Z 2 = Ω Z 1 = jωl 1. (8.24)
8.11. () 7 M I 1 I 2 V 1 L 1 L 2 V 2 I 1 V 1 Z 2 Z 1 = jωl 1 + ω 2 M 2 jωl 2 + Z 2 L M 1 L 2 L M 1 L 2 L = L + L 2M + 1 2+ L = L + L 2M 1 2 8.13 8.11 I M 1 I 2 R 1 R 2 V 1 L 1 L 2 V 2 Z 2 I 1 V 1 Z 1 = R 1 + jωl 1 + ω 2 M 2 jωl 2 + R 2 + Z 2 8.12 8.14 (k ) M [5] Z 2 = 0 Ω ) Z 1 = jω (L 1 M2. (8.25) L 2 M 2 = k 2 L 1 L 2 k = 1 Z 1 = 0 8.12 ω 2 M 2 Z 1 = R 1 + jωl 1 +. (8.26) jωl 2 + R 2 + Z 2 8.10 ( ) 8.13 M L + L M 8.9 8.14 [5] 8.15 [6] 8.11 () 8.16 [7] 8.17
8 8 () J 2 L 2 W 2 M W 1 J 1 L 1 8.18 8.15 [6] I 1 L 1 +M V 1 I 2 V 2 L 1 L 1 +L 2 +2M 8.19 W 1 = V 1, (8.27) W 2 = V 2 V 1 (8.28) J 1 = I 1 I 2 (8.29) J 2 = I 2. (8.30) 8.16 [7] 8.18 W 1 = jωl 1 J 1 + jωmj 2, (8.31) I 2 I 2 W 2 = jωmj 1 + jωl 2 J 2. (8.32) V 1 I 1 V 2 I 1 V 1 I 1 I 2 I 2 L 2 V 2 V 1 M V 2 L 1 V 1 = jωl 1 (I 1 + I 2 ) + jωmi 2, (8.33) V 2 V 1 = jωm(i 1 + I 2 ) + jωl 2 I 2. (8.34) 8.17 V 1 = jωl 1 I 1 + jω(l 1 + M)I 2, (8.35) V 2 = jω(l 1 + M)I 1 + jω(l 1 + L 2 + 2M)I 2 (8.36) 8.19 8.10 T 8.20 8.20 8.10
8.12. 9 I 2 I 2 M L 1 +M I 1 L 2 +M M V 2 V 2 L 1 +L 2 +2M V 1 L 1 +M I 1 V 1 L 1 L 1 L 1 L 2 =M 2 /L 1 L 1 :M 8.20 T M:L 2 (c) L 2 8.21 8.22 L 1 L 1 : M (c) L 2 M : L 2 8.12 V 2 = nv 1, (8.37) I 2 = I 1 n. (8.38) n n = N 2 /N 1 n L 1 L 2 M () 1 () R L C 8.21 8.12.1 V 1 I 1 = V 2 n ni 2 = V 2I 2 (8.39) Z 1 = V 1 I 1 = 1 n 2 V 2 I 2 = 1 n 2 Z 2 (8.40) 8.12.2 8.22 8.23
10 8 () M L 1 L 2 L 1 M 2 /L 2 = (1 k 2 )L 1 M L 1 =M 2 /L 2 =k 2 L 1 L 2 (c) M L 2 M 2 /L 1 = (1 k 2 )L 2 L 1 L 2 =M 2 /L 1 =k 2 L 2 8.23 L 1 M 2 /L 2 = (1 k 2 )L 1 M:L 2 (c) L 1 =M 2 /L 2 =k 2 L 1 8.24 L 1 M 2 /L 2 M:L 2 L 2 (c) M:L 2 (L 1 L 2 M 2 )L 2 /M 2 L 2 8.24 ( 8.23) (c)
8.12. 11 (1a) (1b) 8.25 8.25 +jωl 1 I 1 +jωl 2 I 2 +jωl 1 I 1 *4 jωl 2 I 2 +jωmi 2 +jωmi 2 V 1 = +jωl 1 I 1 +jωmi 2 (8.41) V 1 = +jωl 1 I 1 +jωmi 2 (8.43) +jωmi 1 jωmi 1 V 2 = +jωl 2 I 2 +jωmi 1 (8.42) V 2 = jωl 2 I 2 jωmi 1 (8.44) I 1 M I 2 V 1 L 1 L 2 V 2 I 1 M I 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 jωl 2 I 2 8.25 (1) *4 ()
12 8 () (2a) (2b) 8.26 8.26 +jωl 1 I 1 +jωl 2 I 2 +jωl 1 I 1 *5 jωl 2 I 2 jωmi 2 jωmi 2 V 1 = +jωl 1 I 1 jωmi 2 (8.45) V 1 = +jωl 1 I 1 jωmi 2 (8.47) jωmi 1 +jωmi 1 V 2 = +jωl 2 I 2 jωmi 1 (8.46) V 2 = jωl 2 I 2 +jωmi 1 (8.48) I 1 M I 2 V 1 L 1 L 2 V 2 I 1 M I 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 jωl 2 I 2 8.26 (2) *5 ()
8.12. 13 (3a) (3b) 8.27 8.27 +jωl 1 I 1 *6 jωl 2 I 2 +jωl 1 I 1 +jωl 2 I 2 jωmi 2 jωmi 2 V 1 = +jωl 1 I 1 jωmi 2 (8.49) V 1 = +jωl 1 I 1 jωmi 2 (8.51) +jωmi 1 V 2 = jωl 2 I 2 +jωmi 1 (8.50) jωmi 1 V 2 = +jωl 2 I 2 jωmi 1 (8.52) I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 jωl 2 I 2 I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 8.27 (3) *6 ()
14 8 () (4a) 8.28 8.28 ( 8.28 ) +jωl 1 I 1 +jωl 2 I 2 +jωmi 2 V 1 = +jωl 1 I 1 +jωmi 2 (8.53) +jωmi 1 V 2 = +jωl 2 I 2 +jωmi 1 (8.54) L 1, L 2, M V 2 = nv 1 (8.55) V 1 = jωl 1 I 1 + jωmi 2, (8.55) V 2 = jωmi 1 + jωl 2 I 2 (8.56) I 1 = V 1 jωmi 2 jωl 1 (8.57) (8.56) V 2 = jωl 2 I 2 + M L 1 V 1 jωm2 L 1 I 2 (8.58) (M = L 1 L 2 ) L1 L 2 V 2 = jωl 2 I 2 + V 1 jωl 1L 2 I 2 L 1 L 1 L 2 = V 1 = nv 1 (8.59) L 1 n = L 2 /L 1 *7 I 2 = I 1 /n I 1 = V 1 jωmi 2 jωl 1 (8.60) I 1 = V 1 jωmi 2 jωl 1 = V 1 jωl 1 M L 1 I 2 (8.61) I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 I 1 M I 2 (M = L 1 L 2 ) n = L 2 /L 1 I 1 = V 1 L 2 I 2 jωl 1 L 1 = V 1 jωl 1 ni 2 (8.62) V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jωmi 2 V 2 = jωmi 1 + jωl 2 I 2 L 1 I 1 = ni 2 (8.63) 8.28 (4) *7 [8]
8.12. 15 n 0 n = L2 /L 1 L 1 L 2 L 2 M = L 1 L 2 M
16 8 () e = L di. (8.64) i L e ( ) (8.64) 8.29 v = L di (8.65) 8.30 i di + e (c) di e i e (d) i e i i+di Posiive direcion of he volage should be like his, if he volage of his elemen is regarded as elecromoive force. (c) B i i B B+dB B+dB e i+di i+di di > 0 db > 0 di e = L Elecromoive force o supress increase of i 8.29 Which is correc? (c) (d) i+di di e = L i+di di e = L If we wrie like his, e does no suppress increase of i. This does no correspond o he self-inducion heory, in which e should suppress he increase of i. Since e should suppress he increase of i, e should be expressed like his. 8.30
17 i di e = L The volage is reaed as elecromoive force. i di v = L The volage is reaed as volage drop. 8.31 8.31 e = L di (8.66) 8.31 v = L di (8.67) ( ) : ± M di k (k = 1,2) (8.68) + [1] di 1 > 0 [2] +
18 8 () [1] di 2 > 0 [2] + M M 8.32 M M I 1 MI 2 V 1 L 1 L 2 V 2 I 1 M I 2 V 1 L 1 L 2 V 2 V 1 = jωl 1 I 1 + jω( M)I 2 V 2 = jω( M)I 1 + jωl 2 I 2 V 1 = jωl 1 I 1 + jωm I 2 V 2 = jωm I 1 + jωl 2 I 2 M = M < 0 8.32 (5) [9] 1. Arbirarily selec one erminal say, he D erminal of one coil and mark i wih a do. 2. Assign a curren ino he doed erminal and label i i D. 3. Use he righ-hand rule o deermine he direcion of he magneic field esablished by i D inside he coupled coils and label his field ϕ D. 4. Arbirarily pick one erminal of he second coil say, erminal A and assign a curren ino his erminal, showing he curren as i A. 5. Use he righ-hand rule o deermine he direcion of he flux esablished by i A inside he coupled coils and label his flux ϕ A. 6. Compare he direcions of he wo fluxes ϕ D and ϕ A. If he fluxes have he same reference direcion, place a do on he erminal of he second coil where he es curren (i A ) eners. (In he Figure, he fluxes ϕ D and ϕ A have he same reference direcion, and herefore a do goes on erminal A.) If he fluxes have differen reference direcions, place a do on he erminal of he second coil where he es curren leaves.
19 8.33 Nilsson Riedel [9] I B 8.34 8.34
20 8 () [1] ( 1) 8.35 v 2 v 2 L 2 di 2 / L k di k / (k = 1,2) 1. (di 1 / > 0) 2. 3. 4. 5. ±Mdi 1 / 6. ( ) v 2 > 0 7. 8. 9. 10. v 2 v 2 > 0 v 2 +Mdi 1 / [2] ( 2) 8.35 v 2 v 2 v 1 i 1 Φ i 2 v 2 v 1 i 1 Φ i 2 v 2 i i 1 v 2 M di 1 L di 2 = + 2 i i 1 v 2 M di 1 L di 2 = + 2 8.35 ( 1) 8.36 ( 2)
21 1. (di 1 / > 0) 2. 3. 4. 5. ±Mdi 1 / 6. ( ) v 2 > 0 7. 8. 9. 10. v 2 v 2 > 0 v 2 Mdi 1 /
22 8 () A. 8.37 V 1 V 2 I 1 I 2 jωm j1 Ω 4 Ω I 1 I 2 6 90 V V 1 j8 Ω j5 Ω V 2 10 Ω 8.37. (8.74) (10 + j5) I 1 = I 2 = (5 j10)i 2. (8.75) j (8.75) (8.73) j6 = (4 + j8)(5 j10)i 2 ji 2 = (100 j)i 2 100I 2. 100 1 j *8 I 2 = j6 100 = j0.06 = 0.0600 90 A 60.0 90 ma. (8.75) V 1 = j8i 1 j1i 2, (8.69) V 2 = j1i 1 + j5i 2. (8.70) B. 8.37 V 1 V 2 I 1 I 2 3 (8.69), (8.70) 4 4 I 1 = (5 j10) j0.06 = 0.6 + j0.3 = 0.6708 26.57 = 0.671 26.6 A 671 26.6 ma. V 2 V 2 = 10I 2 = j0.6 = 0.600 90.0 V. V 1 (8.71) V 1 = j6 4I 1 = j6 4 (0.6 + j0.3) = 2.4 + j4.8 = 5.366 116.5 = 5.37 117 V. j6 V 1 = 4I 1, (8.71) V 2 = 10( I 2 ). (8.72) (8.71) (8.69) V 1 (8.72) (8.70) V 2 j6 = (4 + j8)i 1 ji 2, (8.73) 0 = ji 1 + (10 + j5)i 2. (8.74) *8 A + B B A 1/100 A + B = A
23 [1] hp://g-gauge.world.coocan.jp/ransfmr.hm [2] hp://blog.livedoor.jp/2000k/archives/50603462.hml [3] hp://www.meppi.com/transformerfacory/pages/defaul.aspx [4] J. W. Nilsson and S. A. Riedel: Elecric circuis 10h Ediion (Pearson Educaion, Harlow, 2015) p.212. [5] hp://www.cp-ip.or.jp/ ishida96/ih-aichi/isan_wo_aruku/1996/1996-06_yosamisoshinsho.hml [6] hp://www.marinloganowners.com/forum/showhread.php?4263-build-my-own-crossover [7] hp://hoplaza.soundhouse.co.jp/how_o/ligh/choukou/index.asp [8], : (,, 1970) pp. 281-282. [9] J. W. Nilsson and S. A. Riedel: Elecric circuis 10h Ediion (Pearson Educaion, Harlow, 2015) p.213.