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
KUTELA CorporationCopyright© 2014..KUTELA Corporation.All Rights Reserved 2110A6140
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
SOPTSingle Out Put Transformer
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⋅ = ⋅ = ≡ +
SOPTSingle Out Put Transformer
9
Coffee Break(cdef&HWef g(ef<hi?$j
1 1 1
2 2 2
1 1 1 1 1 1 1
2 2 2 1 2 2
cos( )
O W
E W
W O
W E
r r r
r r r
v v r i V t r i
v v r i r i
ω
= +
= +
′ = + = −
′ = − =
wx
yz
yz
|
10
11 2
1
21 2
2
( )
( )
L
L
v A A k Anv A k A An
= +
= − +
i i
i i
Voltage-Turn Relation(SOPT)
( )
11 1 1 1 0 1 1
1
22 2 2 2 2
2
1 20 1 1 2 2
1 2
cos( ) cos( )
cos( )
L
L
L
vv V t r i A t R AA nvv r i R A
A nv v A A t R A R An n
ω ωω
ω
ω ω
= − ⇒ = −
= ⇒ =
+ = − +
1 20 1 1 2 2
1 2
cos( ) 1v v A t R A R A Where kn n
ω+ = = − =
( only k=1)
( kCoupling constant)(0 k 1) (0 k 1)
SOPTSingle Out Put Transformer
11
[ ]
0 1 1 2 2
1
2 201 1 2 1 2 22 2
1 2 1 2
02 1 2 1 22 2
1 2 1 2
1 1 2 2
2 2 201 2 1 1 2 1 22 2
1 2 1 2
cos( ): cos( )
( ) cos( ) sin( )( ) ( )( ) cos( ) sin( )( ) ( )
(( ) ( )
A t R A R AInputVoltage V t
AA R R R R t R tR R R RAA R R t R R tR R R R
R A R AA R R R R R R R RR R R R
ω
ω
ω ω
ω ω
= −
= + + + + += − + ++ +
−= + + + ++ +
2 2 22 1 2 1 2
0
) cos( ) ( )sin( )cos( )
t R R R R t
A t
ω ω
ω
+ − =
Coffee break!"
OK
12
[ ]
1
2
2 20 21 1 2 1 2 2 0 12 2
21 2 1 2
1 2
0 22 1 2 1 2 0 22 2
21 2 1 2
1 2
1 2
: cos( )11 ( )
( ) cos( ) sin( ) sin( )1 1( ) ( ) 1 ( )
1( ) cos( ) sin( ) sin( )( ) ( ) 1 11 ( )
R
R
T
InputVoltage V t
A RA R R R R t R t A tR R R RR R
A RA R R t R R t A tR R R RR R
A A A A
ω
ω ω ω φ
ω ω ω φ
+ = + + + = + + + + +
= − + + = ++ + + += + =
0
0 2 0 1 10 0 02 2
2 1 11 2 1 2
1 2
1 2 1 122 2 2 22
2 21 2 1 2 2 1 1 1 21 2
11
sin( )1[ ] , [ ]1 1( ) ( ) 1 ( )
( ) ( ) 1
m T
m R R
LL
t Total Amper turnA R A nVWhere A Amper turn A A AtR rR R R R
R RV r n V
rr A rn r n r n nn A r rn
ω φ
ω ω
+ ⋅⋅ = = ≡ =+ + + +
⋅ ⋅= = + + + +
2
2
Solution of Analytical Equation where k=1(SOPT)
!
"""""""#$% &'()*+,-./01&23&456*7!
"n18*99*%":;; %
"<Ampere Turn)*"= &>*
"=? @A B0@
"CDEF$+!
SOPTSingle Out Put Transformer
00 0T mA A⇔ ≈ ⇔ ≈
3
13
1
22 1 2 1 2
1 0 0 12 2 21 2 1 2 1 2
2 1 2 2 22 0 02 2
1 2 1 2 1
1: cos( ) put
1 ( ) 1 ( )cos( ) sin( ) sin( )1 ( ) 1 ( ) 1 ( )1 ( ) cos( ) sin( )1 ( ) 1 ( ) 1 (
jj
R R
R R
InputVoltage V t Q RQ Q Q Q QA A t t A tQ Q Q Q Q QQ Q Q Q QA A t t AQ Q Q Q Q
ω
ω ω ω φ
ω ω
=
+ + += ⋅ + = + + + + + + + + += ⋅ + = + + + + + +
222
1 2 0
0 20 02 2 2
1 2 1 2 1 2
1 2 1 122 2 2 22
2 21 2 1 2 2 1 1 1 21 2
1 21
sin( ))
sin( )1[ ]
( ) ( ) 1 ( )
( ) ( ) 1
T m T
m R
LL
tQ
A A A A t Total Amper turnA RWhere A Amper Turn A
R R R R Q QV r n V
rr A r n r n r n nn A r rn
ω φ
ω φ
ω ω
+
= + = + ⋅⋅ = =+ + + +
⋅ ⋅= = + + + +
2
0 1 10
1 1
[ ]RA nVA ATR r
≡ =
Solution of Analytical Equation where k=1(SOPT)#$%
!
"""""""#$% &'()*+,-./01&23&456*7!
"n18*99*%":;; %
"<Ampere Turn)*"= &>*
"=? @A B0@
"CDEF$+!
1[0]j j
jj
j j
Q RLQ r Rω
≡ =
00 0T mA A⇔ ≈ ⇔ ≈
SOPTSingle Out Put Transformer
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
if Rthen A y Aso n i n i
=
+ +
⋅ ≅ ≅
≅ −
≪
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
Coffee Break
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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
Single Out Put Transformer
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= =
SOPTSingle Out Put Transformer
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
⋅+ +
≡ ⋅ = ⋅ ⋅ = ⋅ ⋅+ + + +
=
∫
[ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =
SOPTSingle Out Put Transformer
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≤
SOPTSingle Out Put Transformer
V¯2°±²³
30
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31
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32
Instantaneous Maximum Saturation Core Energy
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Instantaneous Maximum Inductance Electric Energy , Magnetic Core Energy & Non Core Saturation Condition (SOPT)(2)
Magnetic Core Energy & Non Core Saturation Condition
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ωω
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34
1 2
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R
R
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ω ω ω φ
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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
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r1()
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r j(Ë)i1(A) i j(A)
v j(V)
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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
+ +
= =
+ +
= =
≡ ⋅ = = =
≡ ⋅ = = =
∑ ∑ ∑∫ ∫ ∫∑ ∑ ∑∫ ∫ ∫
[ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =
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
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V m l l l l S= ⋅ ⋅ = ⋅
234
534
-
[ ] ( )[ ] 2 ( )
WM IN CW
WM CW
l m D ll m a b l
π
π
= +
= ⋅ + + ⋅
CRF[1 m]Core Form Factor
C CS l=
CLF[1 m]Coil Form Factor
WM CWl Sα=
WireCoil.pdf
W NW
CL CC
CC WC
Wire TransS SS SS S
→
→
→
→
16
91
2
2 21 21 2 1 1 2 2
1 2
2 1 21 2 1 2
1 2
1 2
[ ] ( )
( ) ( )
0 ( ) ( )
WMWL
CW CH
WM WMT TWL
WC WC
WM WMT TWL
WC WC
lP W Al l
l lA A A P A AS S
l lA A A P AS S
where A A A
ρα
ρ α ρ α
ρ α α
=
= + ⇒ = + ≈ ⇒ ≈ − ⇒ = +
= =
2 0 0 0TWLand A A P= ⇒ ≈ ⇒ ≈
=
¿¶À¢S ¡¶À¢S
b$´µ¶·¸q®¯°q¹ºh»¼upqGW
X$´½¾h%P´µ<WMNh¿ÀNvC§5W¿À"
( )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
!"#$%&'( )
[ ] ( )[ ] 2 ( )
WM IN CW
WM CW
l m D ll m a b l
π
π
= +
= ⋅ + + ⋅
2 21 21 2 1 1 2 2
1 2
2 1 21 2 1 2
1 2
1 2
( ) ( )
0 ( ) ( )
WM WMT TWL
WC WC
WM WMT TWL
WC WC
l lA A A P A AS S
l lA A A P AS S
where A A A
ρ α ρ α
ρ α α
= + ⇒ = + ≈ ⇒ ≈ − ⇒ = +
= =
SWC252¶ º?*SWC151¶ º?*
=
93
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Xy<"ÕÖ%¸q®¯°q¹ºh»¼u%u
XcÑËÌ$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!"
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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
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NOFPQ>?M
95
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[ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =
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
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WXYBCDEPZ
>?A[ >\PQ] LM
17
97
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`Core*=b+`*+cdef*Hysteresis Loss0PHL)`*+ghi(Eddy Current Loss:PEL)`*1+b(Total Iron Loss0PTL)`*j+klm
```nbo4
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`*+SOPT```n%
`*+MOPT1`hYTor*s_8t_uvhw+
98
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áâãÂkä-3Ó,
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99
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+: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
→
→
→
→
¡' S-=¢S - £¤5Ref5WireCore
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+ =∫
[ sec] [ ], [sec] 1 2rad f Hz T fω π π ω= =
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
if Rthen A y Aso n i n i
=
+ +
⋅ ≅ ≅
≅ −
≪
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 Loss[ ][ ] [ ] [ ] 1[ ]
Hysteresis const. ( 1.6) :Steinmetz const.
hysn
mhys hys
hys
PB TW JP k f Hz Tm m
where k n
= ⋅ ⋅ ≈