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# 電路學 - [第八章] 磁耦合電路

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### 電路學 - [第八章] 磁耦合電路

1. 1. Department of Electronic Engineering National Taipei University of Technology
2. 2. • • • • • • T π • 2/28 Department of Electronic Engineering, NTUT
3. 3. • N1 (Flux Linkage) L1 ( ) Department of Electronic Engineering, NTUT3/28 ( ) ( ) ( )1 1 1 1 1t N t L i tλ φ= = + − ( )1i t ( )1v t ( )1 tφ
4. 4. ( ) • ( ) ( ) ( ) ( )1 1 1 1 1 1 d t d t di t v t N L dt dt dt λ φ = = = ( ) ( ) ( ) ( )1 1 1 1 11 12t N t N t tλ φ φ φ= = +   ( ) ( ) ( )1 11 12t t tφ φ φ= + ( ) ( ) ( ) ( )1 11 12 1 1 1 d t d t d t v t N N dt dt dt λ φ φ = = + ( ) ( ) ( ) ( ) ( ) ( )11 1 12 2 1 1 1 2 d t di t d t di t N N di t dt di t dt φ φ    = +        ( ) ( )1 2 1 12 di t di t L M dt dt = + ( )1 tφ + − + − 1 ( )11 tφ ( )12 tφ ( )1i t ( )1v t ( )2v t ( )2i t 2 1 : Department of Electronic Engineering, NTUT4/28
5. 5. • 1 • 1 2 • 2 • 2 1 • : ( ) ( ) 11 1 1 1 d t L N di t φ = ( ) ( ) 12 12 1 2 d t M N di t φ = ( ) ( ) ( )1 2 1 1 12 di t di t v t L M dt dt = + ( ) ( ) 21 21 2 1 d t M N di t φ = ( ) ( ) 22 2 2 2 d t L N di t φ = ( ) ( ) ( )1 2 2 21 2 di t di t v t M L dt dt = + ( )1 tφ + − + − 1 ( )11 tφ ( )12 tφ ( )1i t ( )1v t ( )2v t ( )2i t 2 ( )2 tφ + − + − 1 ( )22 tφ ( )21 tφ ( )1i t ( )1v t ( )2v t ( )2i t 2 12 21M M M= = Department of Electronic Engineering, NTUT5/28
6. 6. • ( ) • ( ) ( )1i t ( )1 tφ 1 2 _ + ( )1 tφ ( )1 tφ′ ( )1i t ( )2i t ( )2v t 1 2 Department of Electronic Engineering, NTUT6/28
7. 7. 1 ( ) • ( ) + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 0 di t v t M dt = = ( ) ( )2 22 2 0 di t v t L dt = = ( ) ( )1 21 di t v t M dt = + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 0 di t v t M dt = = ( ) ( )2 22 2 0 di t v t L dt = = ( ) ( )1 21 di t v t M dt = − ( ) ( )1 2 di t v t M dt = ( ) ( )1 2 di t v t M dt = − ( ) ( ) ( )1 11 12v t v t v t= + ( ) ( )1 11 1 di t v t L dt = = ( ) ( ) ( )2 22 21v t v t v t= + ( ) ( )1 21 di t v t M dt = = ( ) ( ) ( )1 11 12v t v t v t= + ( ) ( )1 11 1 di t v t L dt = = ( ) ( ) ( )2 22 21v t v t v t= + ( ) ( )1 21 di t v t M dt = = − Department of Electronic Engineering, NTUT7/28
8. 8. 1 ( ) + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 0 di t v t M dt = = ( ) ( )2 22 2 0 di t v t L dt = = ( ) ( )1 21 di t v t M dt = − ( ) ( ) ( )1 11 12v t v t v t= + ( ) ( ) ( )2 22 21v t v t v t= + + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 0 di t v t M dt = = ( ) ( )2 22 2 0 di t v t L dt = = ( ) ( )1 21 di t v t M dt = ( ) ( ) ( )1 11 12v t v t v t= + ( ) ( ) ( )2 22 21v t v t v t= + ( ) ( )1 2 di t v t M dt = ( ) ( )1 2 di t v t M dt = − ( ) ( )1 11 1 di t v t L dt = = ( ) ( )1 21 di t v t M dt = = − ( ) ( )1 11 1 di t v t L dt = = ( ) ( )1 21 di t v t M dt = = Department of Electronic Engineering, NTUT8/28
9. 9. 2 ( ) • v1(t) v2(t) i1(t) i2(t) ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 di t v t M dt = − ( ) ( )2 22 2 di t v t L dt = − ( ) ( )1 21 di t v t M dt = ( ) ( ) ( )1 11 12v t v t v t− = + ( ) ( ) ( )2 22 21v t v t v t= + + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( )2i t ( ) ( )1 11 1 di t v t L dt = ( ) ( )2 12 di t v t M dt = − ( ) ( )2 22 2 di t v t L dt = ( ) ( )1 21 di t v t M dt = − ( ) ( ) ( )1 11 12v t v t v t= + ( ) ( ) ( )2 22 21v t v t v t= + + − M + − ( )1i t ( )1v t ( )2v t1L 2L ( )2i t ( ) ( )1 2 1 di t di t L M dt dt = − ( ) ( )1 2 2 di t di t M L dt dt = − ( ) ( )1 2 1 di t di t L M dt dt = − ( ) ( )1 2 2 di t di t M L dt dt = − + Department of Electronic Engineering, NTUT9/28
10. 10. 3 − ( ) ( ) ( ) ( ) ( )1 2 1 1 1 di t di t di t v t L M L M dt dt dt = − = − ( ) ( ) ( ) ( ) ( )1 2 2 2 2 di t di t di t v t M L M L dt dt dt = − = − ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 di t v t v t v t L L M dt = − = + − 1 2 2aL L L M= + − + − + − + − M( )1i t ( )2i t( )i t ( )1v t ( )2v t1L 2L ( )v t ( )1i t ( )2i t +−+ − + − M( )1v t ( )2v t 1L 2L ( )v t • v1(t) v2(t) i1(t) i2(t) − + + − + − M( )1i t ( )2i t( )i t ( )1v t ( )2v t1L 2L ( )v t ( )1i t ( )2i t −++ − + − M( )1v t ( )2v t 1L 2L ( )v t ( ) ( ) ( ) ( ) ( )1 2 1 1 1 di t di t di t v t L M L M dt dt dt = − = − ( ) ( ) ( ) ( ) ( )1 2 2 2 2 di t di t di t v t M L L M dt dt dt = − + = − ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 di t v t v t v t L L M dt = + = + − 1 2 2aL L L M= + − Department of Electronic Engineering, NTUT10/28
11. 11. 3 − ( ) ( ) ( ) ( ) ( )1 2 1 1 1= di t di t di t v t L M L M dt dt dt + = + ( ) ( ) ( ) ( ) ( )1 2 2 2 2= di t di t di t v t M L L M dt dt dt + = + ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 di t v t v t v t L L M dt = + = + + 1 2 2aL L L M= + + + − + − + − M( )1i t ( )2i t( )i t ( )1v t ( )2v t1L 2L ( )v t ( )1i t ( )2i t + −+ − + − M( )1v t ( )2v t 1L 2L ( )v t ( ) ( ) ( ) ( ) ( )1 2 1 1 1= di t di t di t v t L M L M dt dt dt + = + ( ) ( ) ( ) ( ) ( )1 2 2 2 2= di t di t di t v t M L L M dt dt dt − + = + ( ) ( ) ( ) ( ) ( ) 1 2 1 2 2 di t v t v t v t L L M dt = − = + + 1 2 2aL L L M= + + − + + − + − M( )1i t ( )2i t( )i t ( )1v t ( )2v t1L 2L ( )v t ( )1i t ( )2i t − ++ − + − M( )1v t ( )2v t 1L 2L ( )v t Department of Electronic Engineering, NTUT11/28
12. 12. 4 − • ( ) ( ) ( )1 2 1 di t di t v t L M dt dt = + ( ) ( ) ( )1 2 2 di t di t v t M L dt dt = + ( ) ( )1 2 2 1 2 di t L M v t dt L L M − = − ( ) ( )2 1 2 1 2 di t L M v t dt L L M − = − ( ) ( ) ( )1 2i t i t i t= + ( ) ( ) ( ) ( )1 2 1 2 2 1 2 2di t di t di t L L M v t dt dt dt L L M + − = + = − ( ) ( )2 1 2 1 2 2 di tL L M v t L L M dt − = + − 2 1 2 1 2 2 a L L M L L L M − = + − M + − ( )i t ( )1i t ( )2i t 1L 2L( )v t , M + − ( )i t ( )1i t ( )2i t 1L 2L( )v t 2 1 2 1 2 2 a L L M L L L M − = + + 2 1 2 0L L M− ≥ 1 2M L L≤ Department of Electronic Engineering, NTUT12/28
13. 13. 5 • is(t) = cost v2(t) ( ) + − ( )si t ( )2v t 2 F 1 Ω 3 H 1 H 2 Ω ( )2i t ( )1i t ( )1v t 1 HM = 1 1 23V j I j Iω ω= − 2 1 2V j I j Iω ω= − − 2 1 2 1 3 2 2 I V V j jω ω = + 1 1 2 1 1 2 2 I V V j jω ω = + ( ) 1 2 11 2 2 sj V j V I Iω ω+ − = − 1 2 2 1 2 2 2 j V j V Iω ω   − + + = −    1 2 1 1 1 2 2 2 2 sj V j V I j j ω ω ω ω     + + + − =        1 2 1 1 3 2 2 0 2 2 2 j V j V j j ω ω ω ω     − + + + =        KCL: I1 I2 1 2 3 5 1 1 2 2 j V j V   + − =    1 2 5 1 1 + 0 2 2 2 j V j V   − + =    2 10 0.408 78.2 24 5 j V j = = ∠ + ( ) ( )( )2 0.408cos 78.2 Vv t t= + 1 0sI = ∠ 1ω =( ) cossi t t= , , Department of Electronic Engineering, NTUT13/28
14. 14. (Ι) • 1 (Unity Coupled Coil) • + − + − ( )1i t 1N ( )1v t ( )2v t 2N ( )2i t ( )1 tφ Department of Electronic Engineering, NTUT14/28
15. 15. (II) • • (Turns Ratio) a 1 ; a 1 2 1 N a N = ( ) ( ) ( ) ( )2 2 1 2 1 1 N v t v t v t av t N = ⇒ = ( ) ( )2 2 1 1 N v t v t N = ( ) ( )1 2i t ai t= − + − + − ( )1v t ( )2v t ( )1i t ( )2i t 1 2:N N − + + − + − ( )1i t ( )2i t ( )1v t ( )2v t( ) 2 2 1 N i t N ( ) 2 1 1 N v t N Department of Electronic Engineering, NTUT15/28
16. 16. (III) • ( ) ( )2 2 1 1 N v t v t N = − ( ) ( )2 1 2 1 N i t i t N = + − + − ( )1v t ( )2v t ( )1i t ( )2i t 1 2:N N − + + − + − ( )1i t ( )2i t ( )1v t ( )2v t( ) 2 2 1 N i t N ( ) 2 1 1 N v t N − Department of Electronic Engineering, NTUT16/28
17. 17. (I) Department of Electronic Engineering, NTUT (a) + − + − + − ( )1i t ( )2i t 1 2:N NsR ( )1v t ( )2v t LR( )sv t + − sR 1 2:N N ( )sv t 2 1 2 L N R N       LR (b) + − sR ( )sv t 2 1 2 L N R N       (c) • RL 2 1 2 N N       17/28
18. 18. (II) • L • C RLC + − + − ( )1v t sR ( )sv t ( )1i t + − ( )2v t LZ ( )2i t 1 2:N N Department of Electronic Engineering, NTUT 2 1 2 N L N       2 2 1 N C N       18/28
19. 19. (III) • (a) (b) • 2 2 1 N N       + − a b (b) + − a b (a) 1 2:N NsR ( )sv t 2 2 1 s N R N       ( ) 2 1 s N v t N Department of Electronic Engineering, NTUT19/28
20. 20. 6 • Vs I1 RL V0 I1 I2 I1 − + − + + − sV 1R 1I :1a 2I LR 0V 1V 2V 2R ( )1 2 1 2 2 1sR R I R I V V+ − = − ( )2 2 2 2LR I R R V− + + = 2 1 1 V V a = 2 1I aI=, ( ) 2 1 1 1 1 0La R aR I V a − + − =   ( )1 2 1 11 sR a R I V V+ − + =   ( ) ( )1 2 2 1 2 A 1 s L V I R a R a R = + − + ( ) ( )0 2 1 1 2 V 1 2 2 L L L s L aR V I R aI R V R a R a R = = = + − + Department of Electronic Engineering, NTUT20/28
21. 21. 7 • a-b 1. (a) b + − aa b (a) b + − a b 2. (b) 3. a-b (c) a ( ) 3sinsv t t= 25 Ω 4 Ω 1: 5 12sint 4 Ω 1: 5 25 Ω 100 Ω 25 Ω 60sint 125 Ω 60sint (b) (c) Department of Electronic Engineering, NTUT21/28
22. 22. (Primary Winding) (Secondary Winding) + − + − M1R 2R sV 1I 2I1L 2L LVLZ ( )1 1 1 2 sR sL I sMI V+ − = ( ) 2 2 1 L s R sI Z I V s + + = ∆ ( )2 s sM I V s = ∆ ( ) ( ) 2 2 1 1 1 2 2 s i L V s M Z s R sL I R sL Z = = + − + + ( ) 2 1 2 2 i L I sM H s I R sL Z = = + + ( ) ( ) L L v s V sMZ H s V s = = ∆ ( ) ( ) ( ) ( )2 2 1 2 1 2 2 1 1 2L Ls L L M s R L R Z L s R R Z∆ = − + + + + +   ( )2 2 2 1 0LR sL I sMI V+ − + = , Department of Electronic Engineering, NTUT22/28
23. 23. • (–M) M ( ) ( ) 2 2 1 1 1 2 2 s i L V s M Z s R sL I R sL Z = = + − + + ( ) ( ) ( )1i fZ s Z s Z s= + ( ) 2 2 2 2 f L s M Z s R sL Z = − + + ( )1 1 1Z s R sL= + s jω= L L LZ R jX= + ( ) ( ) ( ) 2 2 2 2 f f f L L M Z j R jX R R j L X ω ω ω = = + + + + ( ) ( ) ( ) 2 2 2 2 2 2 2 L f L L M R R R R R L X ω ω + = + + + ( ) ( ) ( ) 2 2 2 2 2 2 2 L f L L M L X X R R L X ω ω ω − + = + + +, Department of Electronic Engineering, NTUT23/28
24. 24. 8 • (a) (a) I1 (b) (b) (a) (b) + − (a) 100 0° ∠ 1I j j2j 1 + − 2j 1I 100 0° ∠ 1 2 1 2 j− fZ ( ) 2 2 2 2 f L s M Z s R sL Z = − + + ( ) 2 2 2 1 1 1 1 2 2 f M Z j j L R j ω ω = = = − + + ( ) ( )2 1 1 1 1 100 0 20 60 A 63.2 71.6 A 2 2 I j j j I   = ∠ + − = − = = ∠ −    ( ) 2 2 1 f M Z j j L j ω ω = = = − ( 1j )( )2iZ j j j= − = Ω (b) Department of Electronic Engineering, NTUT24/28
25. 25. T • T π (c) T T + − + − (a) ( )1i t ( )2i t ( )1v t ( )2v t M 1L 2L + − + − + − + − ( )1i t M ( )2i t ( )1v t ( )2v t( )2di t M dt ( )1di t M dt + − + − ( )2i t( )1i t( )1v t ( )2v tM 1L M− 2L M− ( ) ( ) ( )1 2 1 1 di t di t v t L M dt dt = + ( ) ( ) ( )1 2 2 2 di t di t v t M L dt dt = + ( ) ( ) ( ) ( ) ( )1 1 1 1 2 di t d v t L M M i t i t dt dt = − + +   ( ) ( ) ( ) ( ) ( )1 2 2 1 2 di td v t M i t i t L M dt dt = + + −   (b) (c) Department of Electronic Engineering, NTUT25/28
26. 26. 9 • (a) ω = 500 / 0.6 T I1 V2 + − + − M (a) ( )100 0 VsV ° = ∠ 1 40 mHL = 2 70 mHL = 1 15R = Ω 2 10R = Ω 1I 2V30LR = Ω 2 70mHL =1 40mHL = 0.6k = ( )( )0.6 40 70 31.7mHM = = ( ) ( )1 1 2 2 2 2 sR j L M j M R I j M R I Vω ω ω+ − + + − + =   ( ) ( )2 2 2 2 2 0j M R R j M j L M R Iω ω ω− + + + + − + =   ( ) ( )1 225 20 10 15.9 100j I j I+ − + = ( ) ( )1 210 15.9 40 35 0j I j I− + + + = ( )2 2 39.3 13.70 VLV I R= = ∠− , , + − (b) 30LR = Ω 31.7 mHM = 2 10R = Ω 1I 2I 1 15R = Ω ( )100 0 VsV ° = ∠ ( )1 8.3 mH L M− = ( )2 38.3 mH L M− = + − 2V ( )1 3.72 30.3 AI = ∠ − ( )2 1.31 13.70 AI = ∠ −, Department of Electronic Engineering, NTUT26/28
27. 27. π • π Y ∆ T π sL R Y ∆ T + − + − ( )2i t( )1i t( )1v t ( )2v t3L 1L 2L + − + − ( )1i t ( )2i tBL ( )1v t ( )2v tAL CL 1 2 2 3 3 1 2 A R R R R R R R R + + = 1 2 2 3 3 1 3 B R R R R R R R R + + = 1 2 2 3 3 1 1 C R R R R R R R R + + = 1 1L L M= − 2 2L L M= − 3L M= 2 1 2 2 A L L M L L M − = − 2 1 2 B L L M L M − = 2 1 2 1 C L L M L L M − = − , , , , , , Y ∆ T π Department of Electronic Engineering, NTUT27/28
28. 28. • • • • AC Department of Electronic Engineering, NTUT28/28