Single Out Put Transformer Trance Equations...SOPT (2) : Model 2 2 0 [0] j j j j j L j r j r r r l R...

1

1

KUTELA CorporationCopyright© 2014..KUTELA Corporation.All Rights Reserved 2110A6140

Trance Equations

SOPT MOPT Transformer Loss Appendix

SOPTSingle Out Put TransformerMOPTMultiple Out Put Transformer

2

Single Out Put Transformer


r1()

V1(V) COS(t) v1(V) r2()

i1(A) i2(A)

v2(V)

n1 : n2L1(H) : L2(H)

CoreS(m^2) , l(m)

r1 r2

SOPTSingle Out Put Transformer

3

Model(Governing Equation of SOPT)(Analytical Equation of SOPT ) Voltage-Turn Relation(SOPT)Solution of Analytical Equation where k=1(SOPT)v1,v2 !"#!$%"AT&'Solution of Analytical Equation where k=1(SOPT)(!"#!$%")*+,-.+/0123

4567R2).89:i1;<4567R2.89:i2;<45Total Ampere TurnA=R1,R2.89;<Solution of Analytical Equation where k=1 Special Case(SOPT)Energy Conservation and Core Energy (SOPT):0>k>1Energy Conservation and Core Energy (SOPT):k=145R1,R2 ?@AP2B)&'45R1,R2 CD E?@(AP1B)&'45R1,R2CD2 ?@(APTB)&'Impedance Matching Condition(SOPT)Instantaneous Maximum Inductance Electric Energy , Magnetic Core Energy & Non Core Saturation Condition (SOPT) FG"HIJ2KL

JFG"HSOPTR1,R2EMNOP"FFG"HCurrent Transformer

4

r1()

V1(V) COS(t) v1(V) r2()

i1(A) i2(A)

v2(V)

n1 : n2L1(H) : L2(H)

CoreS(m^2) , l(m)

r1 r2

SOPT(1) Model

1[0]j j

jj

j j

Q RLQ r Rω

≡ =

2

7 60

10 1

1

0 1 10

1 1

1 1 2 2

[ ] 1 2)[ ] 4 10 1.26 10 [ / ]

[ ] , [ ] ( 1 2)

[0] 1 2) ,

[ ] ( )

j L j

rL

j j j

jj R

j

L L T

L Henry A n jSA Henry H ml

VA AmperTurn n A AmperTurn n i j orLr A n VR j AL R r

W b A n i n i A A wh

µ µ µ π

ω

ω

Φ

− −

= ⋅ =

= = × ≈ ×

≡ ≡ =

≡ = ≡ =

= + =

2

1

22 00

[ ]1 1[ ] [ ] , [ ] ( )2 2

T j jj

rT Tr T L L T

ere A AmperTurn n iA AH AT m B Wb m H A A S E J HB Sl A Al l S

µ µµ µ Φ=

=

= = = = = = =

∑

5

2

7 60

10 1

1

0 1 10

1 1

1 1 2 2

[ ] 1 2)[ ] 4 10 1.26 10 [ / ]

[ ] , [ ] ( 1 2)

[0] 1 2) ,

[ ] ( )

j L j

rL

j j j

jj R

j

L L T

L Henry A n jSA Henry H ml

VA AmperTurn n A AmperTurn n i j orLr A n VR j AL R r

Wb A n i n i A A wh

µ µ µ π

ω

ω

Φ

− −

= ⋅ =

= = × ≈ ×

≡ ≡ =

≡ = ≡ =

= + =

2

1

2 00

2

[ ]

[ ] [ ]1 1[ ] ( )2 2

T j jj

rT Tr T L

L T

ere A AmperTurn n i

A AH AT m B Wb m H A A Sl l SE J HB Sl A A

µ µµ µ Φ=

=

= = = = =

= =

∑

SOPT(2) : Model

2 20

[0] j j jj

j L j r j

r r r lR

L A n S nω ω µ µ ω⋅

≡ = =⋅ ⋅ ⋅ ⋅

10 1

1

[ ] VA AT nLω

≡ 6

Coffee Break !"#$%&'()*(+,-

./$Model012345678 9:;<=(>12?@AB#5

5QRST+!UVW

XY1 Z[\]

5^RST!UVW

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CDEFG&-DHFG IJ5

KLMNO678%PQR&9:;%SR&TL*U78%V@&HW9XY

ZZZZZZZ%[*#\](Configuration)%^_`\]a$5`abcFG"Hdefg!UVWhijkh!lVWhimn+

opE+ Z[\]

Top

Bottom

2

7

2

1 2

7 60

1 1 1 1

[ ] 1[ ] , 1[ ] 4 10 1.26 10 [ / ]

cos( ) 1)[ ]

1[ ]

j L j

m

rL

j j j

where L Henry A n j orL Henry k L L k

SA Henry H mlv V t r i v r i j

f f HzT Sec

µ µ µ π

ωω π

− −

= ⋅ =

= ≤ ≤

= = × ≈ ×

= − ⋅ = ⋅ ≠

=

1

2

, [ ] , [ ] , [ ]j

fV v Volt i Amper L Henry

πω

=

1 21 1 1 1

1 22 2 2

cos( )

0

m

m

di diV t r i L Ldt dt

di dir i L Ldt dt

ω = ⋅ + ⋅ +

= ⋅ + ⋅ +

(Governing Equation of SOPT)

qr

qst

v1

-v2


8

0 1 1 1 2

2 2 1 2

cos( )0

A t R A A kA

R A kA A

ω ω ω

ω

• •

• •

= + +

= + +

(Analytical Equation of SOPT )

uv2&'

10 1

1

2 2 22 0 11 1 1 1

0 21 1 1 0 1

220 1 1 1

0 0 21 1 1 1

1 2)

[ ]

[0] 1 2)

[ ]

1[ ] , [ ]

jj

jj

j

Lr

R L R

dAWhere A j ordtVA Amper Turn n L

rR j orLA VV V R V lA A WL n r Sn

A nV VA AT A A WR r L R

ω

ω

ωω µ µ ω

ωω

•

≡ =

⋅ =

= =

= = = =

≡ = = ⋅

11 1 1 1 0 1 1

1

22 2 2 2 2

2

cos( )L

L

vv V t r i A R AA nvv r i R A

A n

ωω

ω

= − ⇒ = −

= ⇒ =

(0b k b1)(0b k b1)

( kOCoupling constant)1 2[ ] 1 2)j j j TPut A Amper turn n i j or and A A A⋅ = ⋅ = ≡ +


9

Coffee Break(cdef&HWef g(ef<hi?$j

1 1 1

2 2 2

1 1 1 1 1 1 1

2 2 2 1 2 2

cos( )

O W

E W

W O

W E

r r r

r r r

v v r i V t r i

v v r i r i

ω

= +

= +

′ = + = −

′ = − =

wx

yz

yz

|

10

11 2

1

21 2

2

( )

( )

L

L

v A A k Anv A k A An

= +

= − +

i i

i i

Voltage-Turn Relation(SOPT)

( )

11 1 1 1 0 1 1

1

22 2 2 2 2

2

1 20 1 1 2 2

1 2

cos( ) cos( )

cos( )

L

L

L

vv V t r i A t R AA nvv r i R A

A nv v A A t R A R An n

ω ωω

ω

ω ω

= − ⇒ = −

= ⇒ =

+ = − +

1 20 1 1 2 2

1 2

cos( ) 1v v A t R A R A Where kn n

ω+ = = − =

( only k=1)

( kCoupling constant)(0 k 1) (0 k 1)


11

[ ]

0 1 1 2 2

1

2 201 1 2 1 2 22 2

1 2 1 2

02 1 2 1 22 2

1 2 1 2

1 1 2 2

2 2 201 2 1 1 2 1 22 2

1 2 1 2

cos( ): cos( )

( ) cos( ) sin( )( ) ( )( ) cos( ) sin( )( ) ( )

(( ) ( )

A t R A R AInputVoltage V t

AA R R R R t R tR R R RAA R R t R R tR R R R

R A R AA R R R R R R R RR R R R

ω

ω

ω ω

ω ω

= −

= + + + + += − + ++ +

−= + + + ++ +

2 2 22 1 2 1 2

0

) cos( ) ( )sin( )cos( )

t R R R R t

A t

ω ω

ω

+ − =

Coffee break!"

OK

12

[ ]

1

2

2 20 21 1 2 1 2 2 0 12 2

21 2 1 2

1 2

0 22 1 2 1 2 0 22 2

21 2 1 2

1 2

1 2

: cos( )11 ( )

( ) cos( ) sin( ) sin( )1 1( ) ( ) 1 ( )

1( ) cos( ) sin( ) sin( )( ) ( ) 1 11 ( )

R

R

T

InputVoltage V t

A RA R R R R t R t A tR R R RR R

A RA R R t R R t A tR R R RR R

A A A A

ω

ω ω ω φ

ω ω ω φ

+ = + + + = + + + + +

= − + + = ++ + + += + =

0

0 2 0 1 10 0 02 2

2 1 11 2 1 2

1 2

1 2 1 122 2 2 22

2 21 2 1 2 2 1 1 1 21 2

11

sin( )1[ ] , [ ]1 1( ) ( ) 1 ( )

( ) ( ) 1

m T

m R R

LL

t Total Amper turnA R A nVWhere A Amper turn A A AtR rR R R R

R RV r n V

rr A rn r n r n nn A r rn

ω φ

ω ω

+ ⋅⋅ = = ≡ =+ + + +

⋅ ⋅= = + + + +

2

2

Solution of Analytical Equation where k=1(SOPT)

!

"""""""#$% &'()*+,-./01&23&456*7!

"n18*99*%":;; %

"<Ampere Turn)*"= &>*

"=? @A B0@

"CDEF$+!


00 0T mA A⇔ ≈ ⇔ ≈

3

13

1

22 1 2 1 2

1 0 0 12 2 21 2 1 2 1 2

2 1 2 2 22 0 02 2

1 2 1 2 1

1: cos( ) put

1 ( ) 1 ( )cos( ) sin( ) sin( )1 ( ) 1 ( ) 1 ( )1 ( ) cos( ) sin( )1 ( ) 1 ( ) 1 (

jj

R R

R R

InputVoltage V t Q RQ Q Q Q QA A t t A tQ Q Q Q Q QQ Q Q Q QA A t t AQ Q Q Q Q

ω

ω ω ω φ

ω ω

=

+ + += ⋅ + = + + + + + + + + += ⋅ + = + + + + + +

222

1 2 0

0 20 02 2 2

1 2 1 2 1 2

1 2 1 122 2 2 22

2 21 2 1 2 2 1 1 1 21 2

1 21

sin( ))

sin( )1[ ]

( ) ( ) 1 ( )

( ) ( ) 1

T m T

m R

LL

tQ

A A A A t Total Amper turnA RWhere A Amper Turn A

R R R R Q QV r n V

rr A r n r n r n nn A r rn

ω φ

ω φ

ω ω

+

= + = + ⋅⋅ = =+ + + +

⋅ ⋅= = + + + +

2

0 1 10

1 1

[ ]RA nVA ATR r

≡ =

Solution of Analytical Equation where k=1(SOPT)#$%

!

"""""""#$% &'()*+,-./01&23&456*7!

"n18*99*%":;; %

"<Ampere Turn)*"= &>*

"=? @A B0@

"CDEF$+!

1[0]j j

jj

j j

Q RLQ r Rω

≡ =

00 0T mA A⇔ ≈ ⇔ ≈


14

&'v1,v2( )A*+,

1 2 1 21 1 1 1 1 1 1 2

2 1 2 12 2 2 2 2 1 2

2

1 1 1 2 1 1 0

2 2 1 2 2

cos( )

0

( ) cos( )( )

m

m

j L j

L L T L m

L L

di di di diV t r i L L v L L Ldt dt dt dt

di di di dir i L L v L L Ldt dt dt dt

L A n

v A n A A A n A n A A t

v A n A A A n

ω

ω ω φ

= ⋅ + ⋅ + ⇒ = ⋅ + = ⋅ + + ⋅ ⇒ = − + ⋅

⇓ = ⋅= + = = += − + =

i i i

i i

2 0

1 2

cos( )

sin( )

T L m

T mo

A n A A t

where A A A A t

ω ω φ

ω φ

= − +

≡ + = +

i

-v1,v2./0123Amo./012- Amo4AT./056789v2:/056;

J Ln A ω

15

< )=>?@AB?CD56;E

1 2 0

10 22 2

2 21 1 21 2

1 21

7 60

0

sin( )[ ]

1

[ ] 4 10 1.26 10 [ / ]

0 0

T m T

m

L

rL

T m

A A A A t Total Amper turnVWhere A Amper turn

r n nn A r rnSwhere A H H ml

A A

ω φ

ω

µ µ µ π − −

= + = + ⋅⋅ =

+ + = = × ≈ ×

⇔ ≈ ⇔ ≈

!"#$%&'(

)*+,-./0%&'1

23r4AL54Am067(

Am00 16

< )=>?@AB?CD56;F

1 2 0

10 22 2

2 21 1 21 2

1 21

1 12 2 2

1 1 2 1 21 12 2

1 2 21 1

sin( )[ ]

1

1

T m T

m

L

L L

A A A A t Total Amper turnVWhere A Amper turn

r n nn A r rnV V

r n n r nn A A nr r rn n

ω φ

ω

ω ω

= + = + ⋅⋅ =

+ + ≈ = + +

22 22 2 1 2

1 2

1 Ln nwhere A r rω +

≪

2 [ ] [0]r WireMeanLengthmn CoreWindowAreaρ Ω α=

G &HI, JK L7MNO+

G LPQR8S9*

21 1 21 2

2 22 2 2 11

( )J wmJ wmJ

J wmJ

r l l Sr nS r l Sn n

αραα

= ⇒ =

R&T U VVWOX

""YZ[\P]K+

", HI&^SK+7

~8

SJ5J!lVWl J5Jy!k

!"#$%&'

17

< )=>?@AB?CD56;G

(((()%*+!,-./

21 1 21 2

2 22 2 2 11

( )J wmJ wmJ

J wmJ

r l l Sr nS r l Sn n

αραα

= ⇒ = ⇒

1 2 0

1 10

10 1

sin( )[ ] 12 2

T m T

mL

r

A A A A t Total Amper turnV VWhere A Amper turn SA n n l

ω φ

ω µ µ ω

= + = + ⋅

⋅ ≈ = ≪

!UVW

!lVW

2 01 11 1 1 1

210 1

, ,

rL L

r

Sr rR L A n A R SL l nl

µ µω µ µ ω

= = ⋅ = ⇒ = → R129

'_`l(m)

1 1r

SV nl

µ ω→ → → → →

18

0)12345R2)-62785i19:; R1101

0.1

0.01

0.001

2

21

2

1 2

2

1

11

1

2

1 1 2 2 21 1

11 ( )1 11 ( )

01

1( )L

RyR R

if Rthen y

Vand i r

if Rthen i V

r A n ω

+=

+ +

⇒

⇒

⇒

⇒+∞

⇒+

Short Open2 22 2

2 2

[0]L

r rRL A nω ω

= =

A1=n1i1=A0Ry1

0 1 10

1 1

[ ]RA nVA ATR r

≡ =

4

19Short Open

22

2

1 2

1

0 2 0

1 1 2 2

1

1 11 ( )

10R

RyR R

if Rthen A y Aso n i n i

=

+ +

⋅ ≅ ≅

≅ −

≪

R1 5 0.001 0.01 0.1 1 10

0)12345R29-12785i29:;

A2=n2i2=A0Ry2

0 1 10

1 1

[ ]RA nVA ATR r

≡ = 2 2

2 22 2

[0]L

r rRL A nω ω

= = 20

R1101

0.1

0.01

0.001

2

1 2

1

0 0

1 1 2 2

11 11 ( )

10

T

R T

yR R


=

+ +

⋅ ≅ ≅

≅ −

≪

2[0] j jj

j L j

r rR

L A nω ω≡ =

R123 FG"H 1ZfgE+23

Short Open

2 1 2

1 2

0 ,When R Then A ABut A A

→ →

+ →

R2AT

!"#$%

&R1=R2'()

0)Total Ampere Turn5AT9

R1,R2-:;

AT

=A1+A2 =A0RyT

2 22 2

2 2

[0]L

r rRL A nω ω

= =

AT;<123v2;<123R13,x3

21

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22

[ ][ ]

01 2

2

02

2

1 2 0

cos( ) sin( )

cos( )

sin( )T

AA t R tRAA tR

A A A A t Total Amper turn

ω ω

ω

ω

= +

= −

= + = ⋅

Solution of Analytical Equation where k=1 Special Case(SOPT)

[ ]01 12

1

2

01 2 0 0 2

1

cos( ) sin( )10

sin( )1T m m

AA R t tR

AAA A A A t where AR

ω ω

ω φ

= ++

=

= + = + =+

<Case2 : R10 , R2pq <V1COS(t)

<Case3 : R1p0 , R2=q <V1COS(t)

[ ]1 0

2

1 2 0

sin( )0

sin( )T

A A tAA A A A t Total Amper turn

ω

ω

=

=

= + = ⋅

<Case1 : R10 , R2=q((<V1COS(t)SOPT


23

Solution of Analytical Equation where k=1 Special Case(SOPT)

[ ]

201 3

02 3

1 2 0

0 1 10 2 2 2

1 1

1 2 1 22

1 2 1

( 2)cos( ) sin( )42cos( ) sin( )4

sin( )

[ ]4 (2 )

T m

m

AA R t R tR RAA t R tR R

A A A A t Total Amper turn

A nVWhere A Amper turnR r L

r r r rImpedanceMachingCondition R L L n n

ω ω

ω ω

ω φ

ω

ω ω

= + + += − ++

= + = + ⋅

⋅ = =+ +≡ = ⇒ =

22

0.2and R ≤

<Case4 : R1= R2=R <V1COS(t) SOPTSingle Out Put Transformer

24

2 21 1 1 1 2 20 0 0

2 20 1 1 1 2 20 0 0

cos( )cos( )0 1

T T T

T T T

i V t dt r i dt r i dt

A A t dt R A dt R A dtWhere k

ω

ω

⋅ = ⋅ + ⋅

⋅ = ⋅ + ⋅

≤ ≤

∫ ∫ ∫∫ ∫ ∫

Energy Conservation and Core Energy (SOPT):0k1Universal Energy Conservation

2 2 221 1 2 2 1 2 1 2 1 1 2 2

2 22 1 2

0 2 22 1 1 2 1 2

1 2

1 1 1 1 1[ ] ( )2 2 2 2 21 1 1

1 12 2 ( ) ( )1 ( )

C L L T

L R

E J HBV L i L i L L i i A n i n i A A

V RA AL R R R R

R R

ωω ω ω

= = + + = + =

= ⋅ ⋅ = ⋅ ⋅+ ++ +

Instantaneous Magnetic Core Energy

[ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =


5

25

Strong Coupling Energy ConservationEnergy Conservation and Core Energy (SOPT):k=1

1 22 2

0 1 2 1 21 1 2 20

1 2 1 22 2

2 0 1 21 1 1 1 1 2 20

1 2 1 22

2 02 2 2 20

[ ] [ ] [ ]1 ( )cos( ) [0]2 ( ) ( )1 (1 )[0]2 ( ) ( )1

2

T

T LT T T

T L

T L

P W P W P WA A R R R RP i V t dt P where P

T R R R RA A R RP r i dt P where P

T R R R RA AP r i dt P wher

T

ωω

ω

ω

= ++ +′ ′≡ ⋅ = ⋅ ≡+ +

+′ ′≡ ⋅ = ⋅ ≡+ +

′≡ ⋅ = ⋅

∫∫∫

22 2 2

1 2 1 2

[0] ( ) ( )1

Re PR R R R

Where k

′ ≡+ +

=

1 222 2 2

01 1 1 2 1 2 1 2 1 21 1 2 2 2 20

1 1 2 1 2 1 2 1 222 2 2 2

2 01 1 2 1 21 1 1 2 20

1 1 2 1 2 1

[ ] [ ] [ ]( ) ( )cos( ) 2 ( ) ( ) 2 ( ) ( )

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

T

T LT

T L

E J E J E JA AV R R R R R R R R RE i V t dt T T

r R R R R R R R RA AV R R R RE r i dt T

r R R R R R

ωω

ω

= +

+ + + +≡ ⋅ = ⋅ ⋅ = ⋅ ⋅

+ + + +

+ +≡ ⋅ = ⋅ ⋅ = ⋅

+ +

∫∫

2 22 1 2

222 01 1 2 2

2 2 2 2 2 2 201 1 2 1 2 1 2 1 2

) ( )

2 ( ) ( ) 2 ( ) ( )1

T L

TR R R

A AV R R RE r i dt T Tr R R R R R R R R

Where k

ω

⋅+ +

≡ ⋅ = ⋅ ⋅ = ⋅ ⋅+ + + +

=

∫



26R2[0]

P2r[0]

22 2 2

1 2 1 2

[0] ( ) ( )RP

R R R R′ ≡

+ +

R1=0.05

0.1

0.2

0.51

R12323]R2=R1E29]R13%]

0)R1,R2D12stuvDwxIy7z5P2r)|

20

2 2LA AP ω′ ×

Short

27

P1r[0]

R2[0]

21 2

1 2 21 2 1 2

(1 )[0] ( ) ( )R RP

R R R R+′ ≡+ +

R1=0.05

0.1

0.2

0.5

1

R2 R2 !)"#R1 $%&

0)R1,R2D7W~zuv62stuv%Iy7z(P1r)|

20

1 2LA AP ω′ ×

Short 28R2[0]

PTr[0]

R1=0.05

0.1

0.2

0.5

1

21 2 1 2

2 21 2 1 2

[0] ( ) ( )TR R R RP

R R R R+ +′ ≡+ +

R2 'R2 !)"#R1 $%&

0)R1,R2D6222stuvD7W7zDwxIy7z(PTr)|

20

2L

TA AP ω′ ×

Short

29

Impedance Matching Condition(SOPT)

22 1 1 2

2 2 2 2 201 1 2 1 22 ( ) ( )

1

T V R RE r i dt Tr R R R R

Where k

≡ ⋅ = ⋅ ⋅+ +

=

∫

!"#$%&'()

*+

%vFG"Hh !"¡¢"£"H;¤¥(¦

§¨xh§¨xh©ª;]FG"H.«¬+®vª

2 2 1 2 1 21 2 2 2

1 2 1 2 1 2

221

2 221

22 1

2

2

1

1

1

0 0

1 42 4

0.2 48

E E r r r rand R R then thenR R L L n nVE E

r rImpedance maching conditio

Rr RVR R E

L

rn

ω

ω ω

∂ ∂= = ⇒ = = =∂ ∂ ⋅ << +

⇒ ≤ <

⇒

< =

2

0.2L ω≤


V¯2°±²³

30

Coffee BreakVHF-TV1ch(93MHz)()VHF-TV12ch(219MHz)*+,-+.#300/ 75/Impedance 01234562789:;<6-$=>?$

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11 12 29 6

1 1

22 1 22 29 6

2 2

300 0.2 730.49 10 2 93 10

75 370.49 10 2 93 10

L

L

rR nA n n

rR R nA n n

ω π

ω π

−

−

= = ≤ ⇒ =× × × × ×

= = = ⇒ =× × × × ×

300Ë5n1=73Ì 75Ë5n2=37Ì

6

31

Instantaneous Maximum Inductance Electric Energy , Magnetic Core Energy & Non Core Saturation Condition (SOPT)(1)

2 2 221 1 2 2 1 2 1 2 1 1 2 2

2 22 1 2

0 2 22 1 1 2 1 2

1 2

1 1 1 1[ ] ( )2 2 2 21 1 1

1 12 2 ( ) ( )1 ( )

m L L T

L R

E J L i L i L L i i A n i n i A A

V RA AL R R R R

R R

ωω ω ω

= + + = + =

= ⋅ ⋅ = ⋅ ⋅+ ++ +

21 2

2 2 21 1 2 1 221 2

2 2 21 1 2 1 2

( ) ( )12 ( ) ( )

V REL R R R RV RPL R R R R

π

ω ω

ω

= ⋅ ⋅+ +

= ⋅ ⋅+ +

@ABCDEFG+HI:JKL@MNOPL

Instantaneous Maximum Inductance Electric Energy )Em[J]SOPTSingle Out Put Transformer

32

Instantaneous Maximum Saturation Core Energy

,-./0 1 2!3$4,-./0

21 1[ ] 2 2m L T m m CE J A A H B V= ≤


Magnetic Core Energy & Non Core Saturation Condition

H

B

Hm

Bm

33

0 0 0 0

2 0 1 10 02

1 1

20 2

2 21 2

1

or ( )1 1 ( ) [ ]2 21

1 1 ( )2 21

2 (

m m r T r m m

L R m m C RT

L R m m CT

A H l A l A l B GeneralCondition AmperTurenA nVA A H B V GeneralCondition where A AtR rQ

A A H B V f GeneralCondition WT QV RL

µ µ µ µ

ω ω

≤ ⋅ = ≤

⋅ ≤ ≡ =+

⋅ ≤+

⋅ ⋅

2 21 2 1 2

1 10 1

1 12

11 2

1

1 2

1 ( )) ( ) 2( )

[ ] 0.2( )1 1

[ /

m m C

m m

m m C

T

m

H B V GeneralConditionR R R RV VA n H l B S where NormalTranceCondition EqualAmperTurnL n

V H B V f Hz where R R PowerLWhere Q R R

H A T m

ωω

π ω

≤+ +

≡ ≤ ⋅ ⇔ ≤ ⋅ ⋅

⋅ ≤ = ≤

≡ +

−

3

] :[ ] :[ ] :

m

C

MaximumMgneticFiledStrengthB Tesla MaximumMgneticFluxDensityV m MagneticCoreVolume

atR150.2 and 6789

:;<=>6789?2!@ABCDEFG67HIJDHKLKCMN2!ABCDO

67HKLKPGKQRI2!$4STDUDVWXY*Z$[\$]^H_U


34

1 2

1 2

0 1v v kn n

+ = ⇔ =

⇔

ÍÎ

(!"#!$%"/0

Impedance MatchingÇÈ

!"#$%&'()

$`aH @$bHcQV

1 2 1 1 2 2

1

0 0TA A A n i n i

n

= + ≈ ⇔ + ≈

⇔

⇔

⇔

!"#

1 2 1 21 2 2 2

1 2 1 2

2 1

0.2 0.2r r r rR RL L n n

R Rω ω

= ≤ ⇔ = ≤ ⇔ =

⇔

⇔

35

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37

R1à1

No UseOP"F

FG"H

AT=Am0á0)

R1á1

FG"H

(AT=Am0=A0)

R1=R2ImpedanceMatchingTransformer

R1 â0.2R2à1R2á1R2â0.2

R1(=r1/L1©)

R2(=r2/L2©)

SOPTR1,R2

FG"H(+§¨xh§ª;h©¨x]

%vFG"Hh !"¡¢"£"H;¤¥(+§¨xh§¨xh©ª;]

OP"FFG"H+§ª;h§¨xh©ª;]


38

Current Transformer(1)ã::X°w1h äåæEÌjçè1Zh

ã::wxE¸9FG"H

[ ]

2

2 20 21 1 2 1 2 2 0 12 2

21 2 1 2

1 2

0 22 1 2 1 2 0 22 2

21 2 1 2

1 2

1 2 0

11 ( )( ) cos( ) sin( ) sin( )1 1( ) ( ) 1 ( )

1( ) cos( ) sin( ) sin( )( ) ( ) 1 11 ( )

sin( )

R

R

T m T

A RA R R R R t R t A tR R R RR R

A RA R R t R R t A tR R R RR R

A A A A t Total Am

ω ω ω φ

ω ω ω φ

ω φ

+ = + + + = + + + + +

= − + + = ++ + + +

= + = +

0 20 02 2

21 2 1 2

1 2

0 1 10

1 1

1[ ] 1 1( ) ( ) 1 ( )

[ ]

m R

R

per turn

A RWhere A Amper turn AR R R R

R RA nVA ATR r

⋅

⋅ = =+ + + +

≡ =

FG"H(+§¨xh§ª;h©¨x]

%vFG"Hh !"¡¢"£"H;¤¥(+§¨xh§¨xh©ª;]

OP"FFG"H+§ª;h§¨xh©ª;]

39

Current Transformer(2) 11hAm0á0EéÄÅA1+A2á0 n1i1áên2i22OP"FFG"H.29]ë1ìííîíêííîíííîíï ìííîíáííîííï‘OK’ ìíîíêíîííîíï ìíîíðíîíï‘NO GOOD’

E+ !"#!$%"+[\íîíEé9hñ+[!"#!$%"ò9

óñ+ôõ2[!"#!$%"+ò23

OP"FFG"H/0ö÷øMÇÈ

Am0á0 A1ùAm0 and A2ùAm0à1

1 2 0

10 0 1 1 2

2 1

1 2

2

2121 0 1 2

2 0 2

1 2

22 0 2

2

1 2

sin( )1[ ] ( ), , 11 11 ( )

11 ( ) 1sin( ) 1 ( ) 1 11 11 ( )

1sin(1 11 ( )

T m T

m R

Rm

R

A A A A t Total Amper TurnVWhere A Amper Turn A n R Rr

R R

ARA A t RA RR R

RA A tR R

ω φ

ω φ

ω φ

= + = + ⋅

⋅ =

+ +

+

= + = ++ +

= +

+ +

≪

≫ ≪

22

0 2

1) 1 1m

A RA R= ≫ ≪

n1,R2úÄÅ Z/0»9 40

2 21 1 2 2 2 2

2 2

1L

r rn i n i RL A nω ω

= − ⇒ = = ≪

Current Transformer()

!!! r2=0(22=short9R2$$M

F¹£!.1000ÌXÍ

n2=1000

+

F¹£!

Xûü

n1=1

7

78

RXXYRZ

)r2qDÊ((R21D

)r20DÊ((R21DefKL1000gQFG 6K@hiV

jkl?Kef

W 67mnomPMef

pfqr

41

Ref)http://www.u-rd.com/

Current Transformer(")T

878

ýwxþ

42

Coffee Break#$%

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ã:°

67

à°w+äçèEé9]««.(AC100V)ê(5V,1A)FG"Hé9]«ÄXv3çèã::°X].ýwx:X:1hX

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43

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44

Multiple Out Put TransformerMOPTMultiple Out Put Transformer

!"#$%&


r1 y$z| 62%/$rzr j yj~z|j62%/$rz

r1()

V1(V)'COS((t) v1(V)

r j(Ë)i1(A) i j(A)

v j(V)

n1) : nj)j=1*N+1L1(H) : Lj(H)

Core$>S(m^2) ,4>l(m)

r j+1(Ë)i j+1(A)

v j+1(V)

45

2

<Model<(Governing Equation of MOPT: Coupling Constant k=1)<(Analytical Equation of MOPT )Coupling Constant k=1 ) <Solution of Analytical Equation where k=1(MOPT)<(Equation of MOPT))7<Energy Conservation and Core Energy (MOPT)<Voltage-Turn Relation(MOPT)<Impedance Matching Condition(MOPT)<Instantaneous Maximum Inductance Electric Energy , Magnetic Core Energy (& Non Core Saturation Condition (MOPT)

46

2

7 60

10 1

12

20 1 1 10 0 2

1 1 1 1

[ ] 1 )[ ] 4 10 1.26 10 [ / ]

[ ] , [ ] 1 )

1[0] 1 ) , [ ]

j L j

rL

j j j

jj R L R

j

L Henry A n j NSA Henry H ml

VA AmperTurn n A AmperTurn n i jLr A nV VR j A A A WL R r L R

µ µ µ π

ω

ωω ω

− −

= ⋅ =

= = × ≈ ×

≡ ≡ =

≡ = ≡ = = ⋅

Г

Г

Г

r1 y$z| 62%/$rzr j yj~z|j62%/$rz

)MOPTModel:N)+,-./0123N+1)+. Coupling Const.K=1

r1()

V1(V)'COS((t) v1(V)

r j(Ë)i1(A) i j(A)

v j(V)

n1) : nj)j=1*N+1L1(H) : Lj(H)

Core$>S(m^2) ,4>l(m)

r j+1(Ë)i j+1(A)

v j+1(V)

1[0]j j

jj

j j

Q RLQ r Rω

≡ =

47

2

7 60

1 1 1 1

1 1

[ ][ ] 4 10 1.26 10 [ / ]

cos( ) 1)[ ] :

1 2

, [ ] , [ ] ,

j L j

rL

j j j

where L Henry A nSA Henry H ml

v V t r i v r i jf f Hz

T fV v Volt i Amper L

µ µ µ π

ωω π

πω

− −

= ⋅

= = × ≈ ×

= − ⋅ = ⋅ ≠

= =

[ ]Henry

31 21 1 1 1 1 2 1 3

31 22 2 2 1 2 2 3

31 23 3 3 1 3 2 3

1

1

cos( )

0

0

0N j

k k k jj

didi diV t r i L L L L Ldt dt dt

didi dir i L L L L Ldt dt dt

didi dir i L L L L Ldt dt dt

dir i L L

dt

ω

+

=

= ⋅ + ⋅ + + +

= ⋅ + + ⋅ + +

= ⋅ + + + ⋅ +

= ⋅ +∑

*+(Governing Equation of MOPT: Coupling Constant k=1)

v1

-v2

-v3

-vk

48

2 10 1 1 1 2 3 1 0 1 1 0 1

1

2 2 1 2 3 22

3 3 1 2 3 33

1

1

1c o s ( ) ( c o s ( ) ) c o s ( )

10

10

10

T T

T

T

Nk k j k T

j k

AA t R A A A A A A t A R A A A t ARR A A A A A ARR A A A A A ARR A A A AR

ω ω ω ω ω ωω ω

ωω

ωω

ωω

• • • • •

• • • •

• • • •

+ • •

=

= + + + + ⇒ = − ⇒ = −

−= + + + + ⇒ =

−= + + + + ⇒ =

−= + ⇒ =∑

2

1 10

1 1 1

12

0 11

10 1

1

, c o s ( )

c o s ( )

[ ]

[ ]

[ 0

kk k T

N Nj Tj T j T j T T

j j

NT T

j jj

j j j

j

AR A A

d A A QW h e r e A A A A A A t Ad t RA AR A A A t

VA A m p e r T u r n n LA A m p e r T u rn n i

R

ω

ωω

ωω

ω

•

+ +• • • •

= =

•+

=

⇒ = −

≡ ≡ ≡ = −

= −

⋅ =

⋅ = ⋅

∑ ∑

∑

1

1

1 1 1 1 1 0 1 1

2 2 22 0 11 1 1 1

0 21 1 1 0 1

20 1 10 0

1 1

1]

c o s ( ) c o s ( )1

[ ]

[ ] ,

NjT

jj j

L

j j j L j j j

Lr

R L R

r QL Rv V t r i A n A t R Av r i A n R A w h e r e j

A VV V R V lA A WL n r S nA n VA A T A AR r

ωω ω ω ω

ω

ωω µ µ ω

ω

+

=

= ≡

= − ⋅ = −

= ⋅ = ≠

= = = =

≡ = =

∑

21

21 1

1 [ ]V WL Rω⋅

,-(Analytical Equation of MOPTCoupling Constant k=1)

uv2&'

9

49

2 20 0

2

20 11 1 1 02 2

1

02

co s ( ) s in ( )0 , ( ) 0 , ,

1 )( ) co s ( ) s in ( ) s in ( )(1 ) 1

((1 )

j j jT T

j j j k j k j j T T

TR

T T RT T

RTj

j j T

P u t A C t S tthen A A d t A A A A d t A A A A

QA RA R R Q Q t t A tR Q QAAA QR R Q

ω ωω ω

ω ω ω φ

ω

• • • • • • •

•

= += + = = − = −

−= + − + = ++ +

−= − = +

∫ ∫

Г

1

1

10

021

220 0 0 1 1 1

0 0 0 22 11 1 1 12

1

co s ( ) s in ( )) 1

1

co s ( ) s in ( ) s in ( )(1 )1[ ] [ ] , [ ]

11 1 ( )

T

NT

j jN

RT j T m

j T

R Rm R L RN

Tj j

T jj

t t w he r e j

Q RAA A t Q t A tQ

A A A n V VA Am pe r T u rn w h ere A A T A A WR r L RQR

A A

ω ω

ω ω ω φ

ω ω

+

=

+

=

+

=

• •

=

− ≠

≡

= = + = ++⋅ = = ≡ = = ⋅

+ +

=

∑∑

∑

10

2 01

122 2

0 20 01

12 2

0 1 0 1 20 01

21

2 2 11 1 00

co s ( ) s in ( ) , 0(1 )1( ) ( ) (1 )

( ) co s ( ) 2 (1 )11 ( )

2

N TRT T T

TNT T

j T Rj TNT T T

j j Rj T

TTR

A Q t t A A d tQA d t A d t A Q

T QR A d t A A t d t A R Q

R Q RTR A d t A

ω ω ω

π ω

ω

+ •

+ • •

=

•+

=

= − =+= = ⋅ ⋅ +

= = ⋅ ⋅ − + + − = ⋅ ⋅

∑ ∫∑∫ ∫∑∫ ∫

∫

22

1 1 12 0

22 2 2

0 20 0

, 211 1 1 ,2 21

TL

TT TL

j j R j j jj T

AP R A d tQT AR A d t A w he re j P R A d tR Q

ωπ

ωπ

=+= ⋅ ⋅ ⋅ ≠ =+

∫∫ ∫

Solution of Analytical Equation where k=1(MOPT)

22

02TL

j j jAP R A dtω

π= ∫

u$=LEFGvDV

1[0]j j

jj

j j

Q RLQ r Rω

≡ =

50

Solution of Analytical Equation where k=1(MOPT).

2

11 0 12

0 0( 1 ) 2 2

10

021

00 2

10

1

1 )s in ( )1

( c o s( ) s in ( ) ) s in ( )(1 ) 1

c o s ( ) s in ( ) s in ( )(1 )[ ]

11 ,

T

RT

R Rj T J

j T j TN

RT j T m T

j T

Rm

TN

T Rj j

Q RA A tQA AA Q t t tR Q R Q

AA A t Q t A tQAA Am p e r tu rn

Qw h ere Q AR

ω φ

ω ω ω φ

ω ω ω φ

≠

+

=

+

=

−

= ++

−= − = +

+ +

= = + = ++

⋅ =+

≡

∑

∑

Г

0 1 1

1 12

2 10 2

1 1

[ ]

1 [ ]L R

A n V A tR rVA A WL Rω ω

≡ =

= ⋅

1 111 1 1 1 1 1

1 1 1

O WO W O W

J JE JWJ JE JW J JE JW

J J J

r rrr r r R R RL L L

r r rr r r R R RL L L

ω ω ω

ω ω ω

= + ⇒ ≡ = ≡ + ≡ = + ⇒ ≡ = ≡ + ≡

51

Solution of Analytical Equation where k=1(MOPT)Ampere turn1

00 2

1

1 12

1

1 12

11

1

11 1 1 1

21 11 1 1

1

1 1

1

1

s in ( ) s in ( )1

1 s in ( )1

1 s in ( )1

1 1

1

1

0

0 1

NR

T j m T Tj T

TT

TN

j

j jN j

T N

T

L

Njj j L jj j j j

AA A A t tQ

n V tr Qn V tr L

rL n V n V

n VA r nA

A Lr r r A nr r

ω φ ω φ

ω φ

ω φω

ω

ωω ω

+

=

+

=

+

+ +=

= =

= = + = ++

= ++

= + +

≈ ⇒ >> ⇒ ⋅ = ⋅ < <⋅

∴ ≈ ⇔ ⋅⋅

∑

∑

∑∑ ∑

21

1

1N

j

j jr+

=

< < ∑

Ampere turn !"#$

2 1j CWj

j j WMj

n Sr lρ α

= ⋅

52

11 1 1 1 0 1 1

1

22 2 2 2 2

2

2

cos( )L

L

jj j j j j

L

vv V t r i A R AA nvv r i R A

A nv

v r i R AA n

ωω

ω

ω

= − ⇒ = −

= ⇒ =

= ⇒ =

*+(Equation of MOPT)/

( )

11 1 1 1 0 1 1

1

22 2 2 2 2

2

2

1 20 1 1 2 2

1 2

1

0 1 11 2

cos( ) cos( )

cos( )

1 cos( )

L

L

jj j j j j

L

jL j j

jN Nj

j jj jL j

vv V t ri A t R AA nvv r i R A

A nv

v r i R AA n

vv v A A t R A R A R An n n

vA t R A R A

A n

ω ωω

ω

ω

ω ω

ωω

+

= =

= − ⇒ = −

= ⇒ =

= ⇒ =

+ + + = − + + +

= − +∑ ∑

iii iii

(0% k %1)

(0% k %1)

( kCoupling constant)

31 21 1 1 2 1 3 1 1 2 3

31 22 2 1 2 2 3 2 1 2 3

1 1

1 1

( )

( )

L

L

N Nk

j j k L j kk k

didi div L L L L L A n A A Adt dt dt

didi div L L L L L A n A A Adt dt dt

div L L A n Adt

+ +

= =

= ⋅ + + + ⇒ + + +

− = + ⋅ + + ⇒ + + +

− = ⇒∑ ∑

i i i

i i i

i

( only k=1)

53

12

1 10 01

12

0 10 01

cos( ) ( )

cos( ) ( )

NT Tj j

jNT T

j jj

i V t dt r i dt

A A t dt R A dt

ω

ω

+

=

+

=

⋅ =

⋅ =

∑∫ ∫∑∫ ∫

Energy Conservation and Core Energy(MOPT)

Universal Energy Conservation

1 1 12 22

1 , 1 1

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

C j j j k j k L j j L Tj j k j

E HBV L i L L i i A n i A A+ + +

= = =

= = + = =∑ ∑ ∑Instantaneous Magnetic Core Energy

2 2

0 02

2 2

0 0

[ ] 11[ ] 120 1

T Tj j j L j j

T Tj j j L j j

E J r i dt A R A dt where j N

P W r i dt A R A dt where j NTWhere k

ω

ω

π

= = = +

= = = +

≤ ≤

∫ ∫∫ ∫

1 12 2

1 10 0 01 1

21 12 2

1 10 0 01 1

[ ] cos( ) ( ) ( )

1 1[ ] cos( ) ( ) ( )2

N NT T TT j j j L j j

j j jN NT T T

T j j j L j jj j j

E J i V t dt E r i dt A R A dt

P W i V t dt P r i dt A R A dtT T

ω ω

ωω

π

+ +

= =

+ +

= =

≡ ⋅ = = =

≡ ⋅ = = =

∑ ∑ ∑∫ ∫ ∫∑ ∑ ∑∫ ∫ ∫


54

11 2 3

1

21 2 3

2

1

1

1 1

1 1

( )

( )

( 1)

L

L

NjL kkj

N NjL k

j kj

v A A A Anv A A A Anv

A An

vN A A

n

• • •

• • •

+ •

=

+ +

= =

= + + +

= − + + +

= −

= − −

∑∑ ∑ i

Voltage-Turn Relation(MOPT)

1

1

2 1j

j

vv j Nn n+ = = +

( )

11 1 1 1 0 1 1

1

22 2 2 2 2

2

2

1 20 1 1 2 2

1 2

1

0 1 11 2

cos( ) cos( )

cos( )

1 cos( )

L

L

jj j j j j

L

jL j j

jN N

jj j

j jL j

vv V t ri A t R AA nvv r i R A

A nv

v r i R AA n

vv v A A t R A R A R An n n

vA t R A R A

A n

ω ωω

ω

ω

ω ω

ωω

+

= =

= − ⇒ = −

= ⇒ =

= ⇒ =

+ + + = − + + +

= − +∑ ∑

iii iii

( only k=1)

( only k=1)

1

0 1 11 2

1 1

1 1

1

0 1 12 1

1 cos( )

1 ( 1)

cos( ) ( 1)

N Njj j

j jL j

N Nj kj kL j

N Nk

j jj k

vA t R A R A

A n

v ANA n

AA t R A R A N

ωω

ω ω

ωω

+

= =

+ +

= =

+

= =

= − +

= − −

= − − −

∑ ∑

∑ ∑

∑ ∑

i

i

( only k=1)

(0 k 1)

( kCoupling constant)

10 1

1

[ ] VA At nLω

≡

10

55

21

2 2 210 1 0 1 02 2 2

21

2 2

2

12

11 ( ) 1 112 2 2(1 ) 1 (1 )

, 1

11 ( )1

T NT

T L R L R L RJ jT T T

j jk kj k j k

j k j k

T

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( )22 2 2 21 2 01 1 2 2 1 2 2 22

1 2 1 2

1( ) ( ) (1 )2 1 ( )WM WM L R

TWL W WWC WC

l l A AP A A R Q R QS S Q Qωρ α ρ α= + = ⋅ + ++ + 92

!"#$%&'( )

[ ] ( )[ ] 2 ( )

WM IN CW

WM CW

l m D ll m a b l

π

π

= +

= ⋅ + + ⋅

2 21 21 2 1 1 2 2

1 2

2 1 21 2 1 2

1 2

1 2

( ) ( )

0 ( ) ( )

WM WMT TWL

WC WC

WM WMT TWL

WC WC

l lA A A P A AS S

l lA A A P AS S

where A A A

ρ α ρ α

ρ α α

= + ⇒ = + ≈ ⇒ ≈ − ⇒ = +

= =

SWC252¶ º?*SWC151¶ º?*

=

93

XÁÂP¥¥ÃÄ¥ÅÆÇ$´µÈP¥ÃPÁpqG]ÉÊ

XXbËÌPpqGP´P¥¥¥$´P¥¥ÍÎÏ<Ð~

XXbËÌ$pqGP´P¥¥¥¥$´P¥¥¥ÍÎÏ<Ð~

XcÑËÌ$pqGWW]Ò~Ð~RËÌPpqG=S¸Ó~Ô§[

Xy<"ÕÖ%¸q®¯°q¹ºh»¼u%u

XcÑËÌ$pqGÐ[WP´$´%¸<×

$´µ<P¥½¾%!Ø"Ù

$´µPÁMNhNv

ËÌPpqGWP¥¥ÁÚËÌ$pqGWP¥¥¥ÁÚ

ËÌ$pqGWËÌPpqGP¥¥ÛC§5%"

q®¯°q¹º»¼#Ü Ð[:Ý"~Þ§[u

!!"hÐ[:hÝ"ßvwq®¯°q¹ºW»¼!"

àáâ&P´$´Ð[Í^_ã8(WÀuvwxä

Coffee Break5(SOPT)=B*]¢B*#D]=S]¢S#D8Á¡¶kÂÃ#$feÄ

K\ÅXÆj$8Ç

=*+,-./0%*1+0234

94

2 1 21 2 1 2

1 2

1 2

0 ( ) ( )WM WMT TWL

WC WC

l lA A A P AS S

where A A A

ρ α α ≈ ⇒ ≈ − ⇒ = +

= =

Coffee Break: !

56789#$:;<=5

>?*@>?+A A8BCDEFGHIJKLM

NOFPQ>?M

95

="


1 12 2

1 1[ ] ( )

N NWMJ

TWL JW J J JJ J WCJ

lP W r i AS

ρ α+ +

= =

= ⋅ =∑ ∑1 1

2 2

0 01 1

2 120

12 221

1 1[ ] ( )

1 11 )2 1

N NT TWMJTWL JW J J j

J J WCJN

L R JWW T

JT J

lP W r i dt A dtT T SA A RR Q RQ R

ρ α

ω

+ +

= =

+

=

= ⋅ = = ⋅ + − + +

∑ ∑∫ ∫∑

BCRST

BCUVT

212 1

0 21 1 1

1 1, [ ]N

T L Rj j

VQ A A WR L Rωω

+

=

≡ = ⋅∑

2( )WMJJW J J

WCJ

lr nS

ρ α=2 WC

TotalWM

SA K S Tl

∆ ρα≤ ⋅ ⋅ ⋅

T MOPT-Multiple Out Put Transformer

96

Coffee Break:MOPT# $%&!

WXYBCDEPZ

>?A[ >\PQ] LM

17

97

^

_`Ta

`Core*=b+`*+cdef*Hysteresis Loss0PHL)`*+ghi(Eddy Current Loss:PEL)`*1+b(Total Iron Loss0PTL)`*j+klm

```nbo4

`*=+`*_+pq

`*+SOPT```n%

`*+MOPT1`hYTor*s_8t_uvhw+

98

ÈÉÊËÌÍÎÏÐÑ8GÒ%§ÓÔÕÖ×rØÙÚÛ Ü:ÝÞpßà

áâãÂkä-3Ó,

'()*+,-./012034,)5

7%LMN5$´½¾

22%åæ§5

64%§5$´½¾ç

7%FGHIJG5

×#5>èéÉè

+6789!

99

:;<

+:Transformer Loss

100

Appendix

101

bPêë¥ìPêê¥íî&TïðqHGpBî(

bTïðqNoMä or ñTGòqóôñGdNoMäX&HGpW:õhö÷àøTïðqù<RhñTGòqóôñG<ú(

bTïðqWdNûüýþ"~NoM<R

X&$dNkMBMode Competition(bcBdNWMefghijQSkM1?MelmnoM!

XpqG<=SrMestMe<fguvwxyWzvRh

XpqG|8W~NoM%QRpqGWDR

b!!dNpqG|8#WMNhQyhS

XvN#"R_pqG<R

bv%3dNk7<=MékÑdNoM

2h R

bBî<!RG¯q°G¯qñG°

A''''''åæç±''''''èVWfeÄ''''''éxê

Aêëâ-ìí`îï5ð~8Gñ`ò

,=>

102

1 2 1 2

2 2 2

2

[0] : Number of Turns ( 1[ ] : Electric Current in Wire

[ ] : Diameter of Net Wire[ ] : Net Wire CrossSectional Area 4)[ ] : Coil Cro

NW

CC WM CW

n m m m mi A

d mS m d or d

S m l lπ

= ×

=

= ×

≫

2

2

ssSectional Area[ ] : Coil Longitudinal CrossSectinal Area[ ] 2 ( ) : Coil Total Surface Area

[ ]: Coil Hight[ ]: Coil Width

[ ] :Wire M

WC CW CH

CTS WM CH CW

CH

CW

WM

S m l lS m l l l

l ml m

l m

= ×

= +

!

ean Length[ ] :Wire Length[0] 1: Defiend by

[ ] : Resistance of Wire[ ]: Wire Electric Resistivity

[ ] :Wire Loss Electric Power[ ] :

W WM

WC NW

WL

IN

l m l nS S n

rmP WD m

α αΩ

ρ Ω

= ×

≥ = ⋅ ⋅

!

"#$

%&'

(&')

*+,

-./ Inner Diameter of Circular Coil[ ] [ ] : Inner Rectangular Coil Lengtha m b m×0./

,?@:Bx*n+yy

[ ] ( )[ ] 2 ( )

WM IN CW

WM CW

l m D ll m a b l

π

π

= +

= ⋅ + + ⋅

CLF[1 m]Coil Form Factor

WM WCl Sα=

CRF[1 m]Core Form Factor

l S=

W NW

CL CC

CC WC

Wire TransS SS SS S

→

→

→

→


18

103

222 22 2

1 0 1 1 02 201 2 1 2

22 22 2

2 0 2 2 0 22 01 21 2

22 2 21 1 1 2

1 1 1 1 1 1 1 01 1 1

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

T

R R

T

R R

WM WM WMWL WL R

WC WC WC

Q QA A t A dt AQ Q T Q QQ QA A t A dt AT Q QQ Ql l l QP A P A dt AS S T S

ω φ

ω φ

ρ α ρ α ρ α

+ += + ⇒ =+ + + += + ⇒ = + ++ +

+= ⇒ = = +

∫∫

( )

201 2

222 2

2 2 0 22 1 2

22 20 1 2

1 2 1 2 2 221 2 1 2

22 20

1 2 1 2 2 221 2

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

T

WMWL R

WC

R WM WMTWL WL WL

WC WC

L RTWL WL WL W W

Q Ql QP AS Q Q

A l lP P P Q QQ Q S SA AP P P R Q R QQ Q

ρ α

ρ α α

ω

+= + +

= + = + + + + = + = + ++ +

∫

20

1 1sin ( ) 2T t dt

Tω φ+ =∫


1 2

1 1 1J T

jQ QR R R≡ ≡ +

0

( )( )WMJJW J

r WCJ

l lRS S

ρ αµ µ ω

=

22 0

0

( ) ( )( )JW JW WMJ JW WMJJW J J

J L WCJ r WCJL Jr J

r r l r l lR SL A S S SA n nl

ρ ρα αω ω µ µ ωω µ µ ω

= = = = =

2( )WMJJW J J

WCJ

lr nS

ρ α=

( )WMJ JW LJ

WCJ

l R AS

ωαρ

=

2 120

12 221

1 11 )2 1N

L R JWTWL W T

JT J

A A RP R Q RQ Rω +

=

= + − + + ∑

óôõ/5¢S-ö÷øù

104

( )2

2

0

2 1 1 12 2 2

20 0 01 1 1

2( )2 2 2

TJW L JW j

N N NT T TJW WMJTWL L JW j j J j

J J J WCJJ

P A R A dt

r lP A R A dt A dt A dtSn

ωπ

ω ω ω ρ απ π π

+ + +

= = =

= ⇒ = = =

∫∑ ∑ ∑∫ ∫ ∫

0

( )( )WMJJW J

r WCJ

l lRS S

ρ αµ µ ω

=

2( )WMJJW J J

WCJ

lr nS

ρ α=

( )12

1( )

NWMJ

TWL J JJ WCJ

lP AS

ρ α+

=

= ∑

1 111 1 1 1 1 1

1 1 1

O WO W O W

J JE JWJ JE JW J JE JW

J J J

r rrr r r R R RL L L

r r rr r r R R RL L L

ω ω ω

ω ω ω

= + ⇒ ≡ = ≡ + ≡ = + ⇒ ≡ = ≡ + ≡

22 2

0 0

1[ ] 2T T

j j j L j jP W r i dt A R A dtT

ω

π= =∫ ∫

22

02TL

j j jAP R A dtω

π= ∫

óôõ/5¢S-M÷øù 1

105

( ) 22 1 12 2012 201 21

2 22 21 1

1 0 1 1 02 20

2 20( 1 ) 0 22 0

1 11 )2 2 11 1) )

s in ( )1 11s in ( ) (1

N NT L R JWTW L L JW j W T

J JT J

T TTR R

T TTR

j J J Rjj T

A A RP A R A d t R Q RQ R

Q QR RA A t A d t AQ QAA t A d t A RR Q

ωωπ

πω φ ωπω φ ω

+ +

= =

≠

= = + − + + − −

= + ⇒ = ⋅ ⋅+ += + ⇒ = ⋅

+

∑ ∑∫

∫∫

Г Г

2

10 1 1

01 1 1

22 1

0 21 1

1 )1 , [ ]

1 [ ]

TN

T Rj j

L R

QA n Vw h e r e Q A A TR R r

VA A WL Rω ω

+

=

+≡ ≡ =

= ⋅

∑

20sin ( ) 2T Tt dtω φ+ =∫ [ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =

22 0

0

( ) ( )( )JW JW WMJ JW WMJJW J J

J L WCJ r WCJL Jr J

r r l r l lR SL A S S SA n nl

ρ ρα αω ω µ µ ωω µ µ ω

= = = = =

óôõ/5¢S-M÷øù 2

106

2

11 0 12

0 0( 1 ) 2 2

10

021

00 2

10

1

1 )s in ( )1

( c o s ( ) s in ( )) s in ( )(1 ) 1

c o s ( ) s in ( ) s in ( )(1 )[ ]

11 ,

T

RT

R Rj T J

j T j TN

RT j T m T

j T

Rm

TN

T Rj j

Q RA A tQA AA Q t t tR Q R Q

AA A t Q t A tQAA A m p e r T u r n

Qw h e r e Q AR

ω φ

ω ω ω φ

ω ω ω φ

≠

+

=

+

=

−

= ++

−= − = +

+ +

= = + = ++

⋅ =+

≡

∑

∑

Г

0 1 1

1 12

2 10 2

1 1

[ ]

1 [ ]L R

A n V A TR rVA A WL Rω ω

≡ =

= ⋅

12

1[ ] ( )

NWMJ

TWL J JJ WCJ

lP W AS

ρ α+

=

= ∑

12

01

[ ] ( )2N TWMJ

TWL J jJ WCJ

lP W A dtS

ω ρ απ

+

=

= ∑ ∫

óôõ/5¢S-M÷øù é

107

0 10 1

1

0 1 10

1 12

1 2 01

0 20 2 2

1 2 1 2

0 21 2

1 2 122 22

1 2 1 2 2 1

1

1

[ ] [ ]

[ ] [ ]

[ ] sin( )

[ ]( ) ( )

11 ( )

( ) ( )

rL

R j j j

T j j T m Tj

m

R

L

S VA H A AT nl LA nVA AT A AT n iR r

A AT n i A A A A t

A RA ATR R R R

AQ QV r n

r r A rn r nV

rn

µ µω

ω φ

ω

=

= ≡

≡ = ≡

= = + = +

= + += + +

⋅ ⋅=+ +

=

∑

22 22 21 1 2

21 21

1 Ln nA r rn ω + +

A

0 0 2

0 12

1

1[ ]1

111 ( )

m RT

R N

j j

A AT AQ

AR

+

=

=+

=

+ ∑

MOPT

SOPT

108

12

1 10 01

12

0 10 01

2 2

0 02

2 2

0 0

1 1

cos( ) ( )

cos( ) ( )

[ ] 11[ ] 120 1

[ ] cos( )

NT Tj j

jNT T

j jj

T Tj j j L j j

T Tj j j L j j

T

i V t dt r i dt

A A t dt R A dt

E J r i dt A R A dt where j N

P W r i dt A R A dt where j NTWhere k

E J i V t

ω

ω

ω

ω

π

ω

+

=

+

=

⋅ =

⋅ =

= = = +

= = = +

≤ ≤

≡ ⋅

∑∫ ∫∑∫ ∫

∫ ∫∫ ∫

1 12 2

0 0 01 1

21 12 2

1 10 0 01 1

( ) ( )

1 1[ ] cos( ) ( ) ( )2

N NT T Tj j j L j j

j j jN NT T T

T j j j L j jj j j

dt E r i dt A R A dt

P W i V t dt P r i dt A R A dtT T

ω

ωω

π

+ +

= =

+ +

= =

= = =

≡ ⋅ = = =

∑ ∑ ∑∫ ∫ ∫∑ ∑ ∑∫ ∫ ∫

Energy Conservation and Core Energy(MOPT)

Universal Energy Conservation

Strong Coupling Energy Conservation

1 1 12 22

1 , 1 1

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

m m m C j j j k j k L j j L Tj j k j

E H B V L i L L i i A n i A A+ + +

= = =

= = + = =∑ ∑ ∑Instantaneous Magnetic Core Energy

19

109

R1

101

0.1

0.01

0.001

2

1 2

1

0 0

1 1 2 2

11 11 ( )

10

T

R T

yR R


=

+ +

⋅ ≅ ≅

≅ −

≪

Total Ampere TurnAT

R1,R2

2[0] j jj

j L j

r rR

L A nω ω≡ =

R1¯"D#$ feÄ 1P|úz#$

AT

=A1+A2 =A0RyT

Short Open2 22 2

2 2L

r rRL A nω ω

= =

2 1 2

1 2

0 ,When R Then A ABut A A

→ →

+ →

R2 AT !

"#$%&'

()*+,-$.#!

110

1 11 2

1

11 1 1 12

1

1 2

2 2 2 1 2

1) 0.2 0.5 Given

2) Given ( 0) is fixed

3) Impedance Maching Condition

4) Impedance Maching Condition12

O W

L

OO W O

L

W E W

r rR A nrR R R RA n

R R

R R R R and P

ω

ω

+≡ ≅

≡ ⇒ = − ≥

=

= ⇒ =

∼

2

2 2321 1 1 1

1 2 12 21 1

221

2 1 2 2 221 1 11

1

12

5) Non Core Saturation Condition(1 ) (1 ) (1 )

6)1 1 Where 4 44

E

m m C O m m OO

EM

EM m m C W EO W

P

H B V r R H B A r RR k RV VPVP H B V R R and R Rr r RR

R R

ω ω

ω

=

⋅ ⋅ + ⋅ ⋅ +≤ ≤ ≡ +

= ⋅ ⋅ ≤ = =++

1 1 2

1 1 2 2 1

2), 5) ,( , ) ( , ) , 2)

O W W

W W W W

R RR R x y R R x y x y n= =

! "#$$ %#

111

W

CmmEM

WEM

CmmmoLmoL

RVBHP

RRPP

VBHAAAAR

P

22

2222

22

22

8

2,21

21

21,2

×=

==⇓

≤=

ω

ω

112

Transformer LossCore=

=!"#$%&'()*

"+,-.

2

3

2

3 2

8.4 [1 ] [ ] [ ]( [ ] [ ])[ ]8.4 [1 ] [ ] [ ]( [ ] [ ])[ ] [ ][ ]

eddy

meddy

meddy

Pm t m w m f Hz B TWP where t wmm t m w m f Hz B T Volt SecWP Tkg kg m m

σ Ωπ

σ Ωπρ

⋅=⋅ ⋅ = =

≪

!

"#

3 3


Hysteresis const. ( 1.6) :Steinmetz const.

hysn

mhys hys

hys

PB TW JP k f Hz Tm m

where k n

= ⋅ ⋅ ≈

Single Out Put Transformer Trance Equations...SOPT (2) : Model 2 2 0 [0] j j j j j L j r j r r r l R...

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Transcript of Single Out Put Transformer Trance Equations...SOPT (2) : Model 2 2 0 [0] j j j j j L j r j r r r l R...