Department of Electronic EngineeringNational Taipei University of Technology

•••••• T π

•

2/28 Department of Electronic Engineering, NTUT

• 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

( )

• ( ) ( ) ( ) ( )1 1 11 1 1

d t d t di tv 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 11 121 1 1

d t d t d tv t N N

dt dt dt

λ φ φ= = +

( )( )

( ) ( )( )

( )11 1 12 21 1

1 2

d t di t d t di tN N

di t dt di t dt

φ φ = +

( ) ( )1 21 12

di t di tL M

dt dt= +

( )1 tφ

+

−

+

−

1

( )11 tφ ( )12 tφ

( )1i t

( )1v t ( )2v t

( )2i t2

1 :

Department of Electronic Engineering, NTUT4/28

• 1

• 1 2

• 2

• 2 1

• :

( )( )

111 1

1

d tL N

di t

φ=

( )( )

1212 1

2

d tM N

di t

φ=

( ) ( ) ( )1 21 1 12

di t di tv t L M

dt dt= +

( )( )

2121 2

1

d tM N

di t

φ=

( )( )

222 2

2

d tL N

di t

φ=

( ) ( ) ( )1 22 21 2

di t di tv t M L

dt dt= +

( )1 tφ

+

−

+

−

1

( )11 tφ ( )12 tφ

( )1i t

( )1v t ( )2v t

( )2i t2

( )2 tφ

+

−

+

−

1

( )22 tφ( )21 tφ

( )1i t

( )1v t ( )2v t

( )2i t2

12 21M M M= =

Department of Electronic Engineering, NTUT5/28

•( )

• ( )

( )1i t ( )1 tφ

1 2

_

+

( )1 tφ ( )1 tφ′

( )1i t ( )2i t

( )2v t

1 2

Department of Electronic Engineering, NTUT6/28

1 ( )

• ( )

+

−

M

+

−

( )1i t

( )1v t ( )2v t1L 2L

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212 0

di tv t M

dt= =

:

( ) ( )222 2 0

di tv t L

dt= =

( ) ( )121

di tv t M

dt=

+

−

M

+

−

( )1i t

( )1v t ( )2v t1L 2L

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212 0

di tv t M

dt= =

:

( ) ( )222 2 0

di tv t L

dt= =

( ) ( )121

di tv t M

dt= −

( ) ( )12

di tv t M

dt=

( ) ( )12

di tv t M

dt= −

( ) ( ) ( )1 11 12v t v t v t= +

( ) ( )111 1

di tv t L

dt= =

( ) ( ) ( )2 22 21v t v t v t= +

( ) ( )121

di tv t M

dt= =

( ) ( ) ( )1 11 12v t v t v t= +

( ) ( )111 1

di tv t L

dt= =

( ) ( ) ( )2 22 21v t v t v t= +

( ) ( )121

di tv t M

dt= = −

Department of Electronic Engineering, NTUT7/28

1 ( )

+

−

M

+

−

( )1i t

( )1v t ( )2v t1L 2L

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212 0

di tv t M

dt= =

:

( ) ( )222 2 0

di tv t L

dt= =

( ) ( )121

di tv 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

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212 0

di tv t M

dt= =

:

( ) ( )222 2 0

di tv t L

dt= =

( ) ( )121

di tv t M

dt=

( ) ( ) ( )1 11 12v t v t v t= +

( ) ( ) ( )2 22 21v t v t v t= +

( ) ( )12

di tv t M

dt=

( ) ( )12

di tv t M

dt= −

( ) ( )111 1

di tv t L

dt= =

( ) ( )121

di tv t M

dt= = −

( ) ( )111 1

di tv t L

dt= =

( ) ( )121

di tv t M

dt= =

Department of Electronic Engineering, NTUT8/28

2 ( )

• v1(t) v2(t) i1(t) i2(t)

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212

di tv t M

dt= −

:

( ) ( )222 2

di tv t L

dt= −

( ) ( )121

di tv 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

( ) ( )111 1

di tv t L

dt=

:

( ) ( )212

di tv t M

dt= −

:

( ) ( )222 2

di tv t L

dt=

( ) ( )121

di tv 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 21

di t di tL M

dt dt= −

( ) ( )1 22

di t di tM L

dt dt= −

( ) ( )1 21

di t di tL M

dt dt= −

( ) ( )1 22

di t di tM L

dt dt= − +

Department of Electronic Engineering, NTUT9/28

3 −

( ) ( ) ( ) ( ) ( )1 21 1 1

di t di t di tv t L M L M

dt dt dt= − = −

( ) ( ) ( ) ( ) ( )1 22 2 2

di t di t di tv t M L M L

dt dt dt= − = −

( ) ( ) ( ) ( ) ( )1 2 1 2 2

di tv 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 21 1 1

di t di t di tv t L M L M

dt dt dt= − = −

( ) ( ) ( ) ( ) ( )1 22 2 2

di t di t di tv t M L L M

dt dt dt= − + = −

( ) ( ) ( ) ( ) ( )1 2 1 2 2

di tv t v t v t L L M

dt= + = + −

1 2 2aL L L M= + −

Department of Electronic Engineering, NTUT10/28

3 −

( ) ( ) ( ) ( ) ( )1 21 1 1=

di t di t di tv t L M L M

dt dt dt+ = +

( ) ( ) ( ) ( ) ( )1 22 2 2=

di t di t di tv t M L L M

dt dt dt+ = +

( ) ( ) ( ) ( ) ( )1 2 1 2 2

di tv 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 21 1 1=

di t di t di tv t L M L M

dt dt dt+ = +

( ) ( ) ( ) ( ) ( )1 22 2 2=

di t di t di tv t M L L M

dt dt dt− + = +

( ) ( ) ( ) ( ) ( )1 2 1 2 2

di tv 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

4 −

•

( ) ( ) ( )1 21

di t di tv t L M

dt dt= + ( ) ( ) ( )1 2

2

di t di tv t M L

dt dt= +

( ) ( )1 22

1 2

di t L Mv t

dt L L M

−=−

( ) ( )2 12

1 2

di t L Mv t

dt L L M

−=−

( ) ( ) ( )1 2i t i t i t= + ( ) ( ) ( ) ( )1 2 1 22

1 2

2di t di t di t L L Mv t

dt dt dt L L M

+ −= + =−

( ) ( )21 2

1 2 2

di tL L Mv t

L L M dt

−=+ −

21 2

1 2 2a

L L ML

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

21 2

1 2 2a

L L ML

L L M

−=+ +

21 2 0L L M− ≥

1 2M L L≤

Department of Electronic Engineering, NTUT12/28

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 2I V V

j jω ω= +

1 1 2

1 1

2 2I V V

j jω ω= +

( ) 1 2 11 2 2 sj V j V I Iω ω+ − = −

1 2 2

12 2

2j V j V Iω ω − + + = −

1 2

1 11 2 2

2 2 sj V j V Ij j

ω ωω ω

+ + + − =

1 2

1 1 32 2 0

2 2 2j V j V

j jω ω

ω ω

− + + + =

KCL:I1 I2

1 2

3 51 1

2 2j V j V

+ − =

1 2

5 1 1+ 0

2 2 2j V j V − + =

2

100.408 78.2

24 5

jV

j= = ∠

+

( ) ( )( )2 0.408cos 78.2 Vv t t= +

1 0sI = ∠ 1ω =( ) cossi t t= , ,

Department of Electronic Engineering, NTUT13/28

(Ι)

• 1(Unity Coupled Coil)

•

+

−

+

−

( )1i t

1N

( )1v t

( )2v t

2N

( )2i t

( )1 tφ

Department of Electronic Engineering, NTUT14/28

(II)

•

• (Turns Ratio)

a 1 ; a 1

2

1

Na

N=

( ) ( ) ( ) ( )22 1 2 1

1

Nv t v t v t av t

N= ⇒ =

( ) ( )22 1

1

Nv t v t

N= ( ) ( )1 2i t ai t= −

( )

+

−

+

−

( )1v t ( )2v t

( )1i t ( )2i t1 2:N N

−+

+

−

+

−

( )1i t ( )2i t

( )1v t ( )2v t( ) 22

1

Ni t

N( ) 2

11

Nv t

N

Department of Electronic Engineering, NTUT15/28

(III)

•

( ) ( )22 1

1

Nv t v t

N= − ( ) ( )2

1 21

Ni t i t

N=

( )

+

−

+

−

( )1v t ( )2v t

( )1i t ( )2i t1 2:N N

−+

+

−

+

−

( )1i t ( )2i t

( )1v t ( )2v t( ) 22

1

Ni t

N( ) 2

11

Nv t

N−

Department of Electronic Engineering, NTUT16/28

(I)

Department of Electronic Engineering, NTUT

(a)

+

−

+−

+

−

( )1i t ( )2i t1 2:N NsR

( )1v t ( )2v t LR( )sv t+−

sR1 2:N N

( )sv t

2

1

2L

NR

N

LR

(b)

+−

sR

( )sv t2

1

2L

NR

N

(c)

• RL

2

1

2

N

N

17/28

(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

NL

N

2

2

1

NC

N

18/28

(III)

• (a) (b)

•2

2

1

N

N

+−

a

b(b)

+−

a

b(a)

1 2:N NsR

( )sv t

2

2

1s

NR

N

( ) 2

1s

Nv t

N

Department of Electronic Engineering, NTUT19/28

6

• Vs I1

RL V0

I1 I2

I1

−

+−

+ +−

sV

1R

1I

:1a

2I

LR0V

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

1V V

a= 2 1I aI=,

( ) 2 1 1

11 0La R aR I V

a− + − =

( )1 2 1 11 sR a R I V V+ − + =

( )( )1 2 2

1 2

A1

s

L

VI

R a R a R=

+ − +

( ) ( )0 2 11 2

V1 2 2

LL L s

L

aRV I R aI R V

R a R a R= = =

+ − +

Department of Electronic Engineering, NTUT20/28

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

(Primary Winding) (Secondary Winding)+−

+

−

M1R 2R

sV 1I 2I1L 2LLVLZ

( )1 1 1 2 sR sL I sMI V+ − = ( )2 2

1L

s

R sI ZI V

s

+ +=∆ ( )2 s

sMI V

s=

∆

( ) ( )2 2

1 11 2 2

si

L

V s MZ s R sL

I R sL Z= = + −

+ +

( ) 2

1 2 2i

L

I sMH s

I R sL Z= =

+ +

( ) ( )L L

vs

V sMZH s

V s= =

∆

( ) ( ) ( ) ( )2 21 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

•

(–M) M

( ) ( )2 2

1 11 2 2

si

L

V s MZ s R sL

I R sL Z= = + −

+ +

( ) ( ) ( )1i fZ s Z s Z s= +

( )2 2

2 2f

L

s MZ s

R sL Z= −

+ +( )1 1 1Z s R sL= +

s jω= L L LZ R jX= +

( ) ( ) ( )2 2

2 2f f f

L L

MZ j R jX

R R j L X

ωωω

= = ++ + +

( )( ) ( )

2 22

2 2

2 2

Lf

L L

M R RR

R R L X

ωω

+=

+ + +( )

( ) ( )

2 22

2 2

2 2

Lf

L L

M L XX

R R L X

ω ωω

− +=

+ + +,

Department of Electronic Engineering, NTUT23/28

8

• (a) (a)I1 (b)

(b)

(a)

(b)

+−

(a)

100 0°∠

1Ij

j2j 1 +−

2j1I

100 0°∠

1

2

1

2j−

fZ

( )2 2

2 2f

L

s MZ s

R sL Z= −

+ +( )2

2 2

1 1 1

1 2 2f

MZ j

j L R j

ωω

= = = −+ +

( ) ( )21 1

1 1100 0 20 60 A 63.2 71.6 A

2 2I j j j I = ∠ + − = − = = ∠ −

( )2

2

1f

MZ j

j L j

ωω

= = = −

( 1j )( )2iZ j j j= − = Ω

(b)

Department of Electronic Engineering, NTUT24/28

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 tM

dt

( )1di tM

dt

+

−

+

−

( )2i t( )1i t( )1v t ( )2v tM

1L M− 2L M−

( ) ( ) ( )1 21 1

di t di tv t L M

dt dt= +

( ) ( ) ( )1 22 2

di t di tv t M L

dt dt= +

( ) ( ) ( ) ( ) ( )11 1 1 2

di t dv t L M M i t i t

dt dt= − + +

( ) ( ) ( ) ( ) ( )12 2 1 2

di tdv t M i t i t L M

dt dt= + + −

(b)

(c)

Department of Electronic Engineering, NTUT25/28

9

• (a) ω = 500 /0.6 T I1

V2

+

−

+

−

M

(a)

( )100 0 VsV °= ∠

1 40 mHL = 2 70 mHL =

1 15 R = Ω

2 10 R = Ω

1I

2V30 LR = 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)

30 LR = Ω

31.7 mHM =

2 10 R = Ω

1I 2I

1 15 R = 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

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

2A

R R R R R RR

R

+ += 1 2 2 3 3 1

3B

R R R R R RR

R

+ += 1 2 2 3 3 1

1C

R R R R R RR

R

+ +=

1 1L L M= − 2 2L L M= − 3L M=2

1 2

2A

L L ML

L M

−=−

21 2

B

L L ML

M

−=2

1 2

1C

L L ML

L M

−=−

, ,

, , , ,

Y ∆

T π

Department of Electronic Engineering, NTUT27/28

•

•

•

• AC

Department of Electronic Engineering, NTUT28/28