APPENDICES
I. Dynamic Current Sharing for Paralleled Transformers
(i) Single Forward Diode (Fig. 3.2)
Let: L p1 + Ls1 = L + ∆L, R p1 + Rs1 = R + ∆R, L p2 + Ls2 = L − ∆L, R p2 + Rs2 = R − ∆R .
Assume: im1 = im 2 = 0 and model the diodes have a body resistance of Rb
(a) t<tr:
Vin
di
= ( L + ∆L ) 1 + ( R + ∆R )i1 + R D (i1 + i2 ) − L f

n
dt

di2
Vin
= ( L − ∆L )
+ ( R − ∆ R )i2 + R D (i1 + i2 ) − L f

n
dt
di f
dt
di f
dt
− R f i f − Rb i f
(I-i)
− R f i f − Rb i f
Therefore:
[
]
[
] [
]
]
 s( L + L + ∆L ) + R + R + 2 R + ∆ R i + ( sL + R + 2 R )i = V / n + ( R + R ) I / s
f
f
D
1
f
f
D 2
in
f
D o


( sL f + R f + 2 R D ) i1 + s( L + L f − ∆L) + R + R f + 2 R D − ∆R i2 = Vin / n + ( R f + R D ) Io / s

[
Neglect ∆2 terms, the s-domain solutions are:
Appendices
185
i1, 2 ( s) =
1
L + 2L f

Vin
+ ( R f + R D ) Io 
R
∆L
∆ R  n

(s + ) m
(s +
) 


L
L
s
∆L 





 i1, 2 (0 )

+
R
 R + 2R f + 4 RD 
R
s+
( s + )
s +
L
L + 2L f
L




Lf
∆L
∆L ∆ R
(
)(
)
(
)
R
+
R
+
R
R
2
1
m
m
∆
−
f
D
Io
L
L
L
R
−
±
L + 2Lf
( L + 2 L f )  R + 2R f + 4 RD 
 R + 2 R f + 4RD 
R
R
( s + )
( s + )
s +
s +
L
L + 2L f
L + 2L f
L








Ri1, 2 (0 )
Let τ1 =
L + 2Lf
R + 2 R f + 4RD
,τ2 =
L
. Since during this time interval: t << τ1 , τ 2 ,
R
I  ∆L  t

i1,2 = i1,2 ( 0 ) + o 1 m
)
;


2 
L 
 tr

t
i f = Io (1 − );

tr

(I-ii)

( L + ∆ L ) di1 + ( R + ∆R ) i1 = ( L − ∆L ) di2 + ( R − ∆R )i2

dt
dt

i1 + i2 = 0
(I-iii)
 2 Vin 
where tr = Io / 
L + L n 
.


f
(b) ∆Ts > t > tr (if = 0):
i1,2 ( s ) =
Appendices
R m ∆R Io
L
+
i (t )
sL + R 2 s sL + R 1,2 r
186
R
∴
R
- ( t − tr )
- ( t − tr )
∆ R Io
i1,2 ( t ) = (1 m
) (1 − e L
) + i1,2 ( tr )e L
.
R 2
(I-iv)
(c) ∆Ts + tf > t > ∆Ts (secondary voltage changes polarity, similar to stage (a)):

I  ∆L t − DTs
)
;
i1,2 ( t ) = i1,2 ( DTs ) − o 1 m

2 
L 

 tf

t
i f = Io ;
tf


(I-v)
 2 Vr 
where t f = Io / 
.

L + L f n 

(d) Ts > t > ∆Ts + tf (similar to stage (b)):
i1, 2 ( t ) = i1,2 ( DTs + t f
R
- ( t − DTs − t f )
)e L
(I-vi)
Considering the current continuity among stages;
∆L
I
i1,2 ( tr ) = i1, 2 ( 0 ) + o (1 m
),
2
L
i1,2 ( DTs + t f ) = i1,2 ( DTs ) −
∴
Appendices
R
∆L
Io
(1 m
),
2
L
i1,2 ( 0 ) = m
R
− DTs
− DTs
∆R
I
i1,2 ( DTs ) = o (1 m
)(1 − e L
) + i1,2 ( tr )e L
2
R
i1, 2 ( 0 ) = i1,2 ( DTs + t f )e
Io ∆R ∆ L
(
−
)
2 R
L
1− e
R
D' Ts
eL
−
−
R
D' Ts
L
(I-vii)
R
DTs
L
−e
−
R
DTs
L
(I-viii)
187
Since tr and tf are very brief compared with switching period, therefore, the average current of
each module can be approximated by:
Ts
DTs

 ∫ i1,2 ( t ) dt + ∫ i1, 2 ( t ) dt
DTs + t f


 tr

R
− DTs
Io
Io
∆R
∆R L ⋅ f s
(I-ix)
= (1 m
) D − (1 m
)
(1 − e L
)
R
R
R
2
2
R
R
R
D ' Ts L ⋅ f s
− DTs
− D' Ts
Io
∆L  L ⋅ f s

L
L
)
+ i1, 2 ( 0 ) + (1 m
)
(1 − e
) + i1,2 (0 )e
(1 − e L
L
R
R
2


I1,2 =
1
Ts
I1, 2 =
∆R
Io
(1 m
)D
2
R
(I-x)
I1 R − ∆R R2
=
=
I2 R + ∆R R1
(I-xi)
(ii) Separate Forward Diodes (Fig. 3.4)
(a) t < tr:
Vin
di1
= ( L + ∆ L)
+ ( R + ∆R )i1 + R D i1 − L f

n
dt

di2
Vin
=
(
L
−
∆
L
)
+ ( R − ∆ R ) i2 + R D i2 − L f

n
dt
i1,2 =
Appendices
1
L + 2L f
Vin / n + ( R f + R D ) I o
 R + 2R f + 3R D 
R + RD
s s +
)
( s +
L + 2L f
L




di f
dt
di f
dt
− R f i f − R Di f ;
(I-xii)
− R f i f − R Di f ;
R + RD
∆L
∆R 

⋅( s +
)m
(s +
) ,
L
L
∆L 


188
where tr =

Io  ∆ L  t
)
;
i1, 2 = 1 m
2 
L 

 tr

i = I 1 − t 
;
f
o

 tr 

(I-xiii)
di2
di1

+ ( R + ∆ R + R D )i1 = ( L − ∆L )
+ ( R − ∆R + R D )i2
( L + ∆L )
dt
dt

i + i = I
1 2
o
(I-xiv)
( L + 2 L f ) Io
.
V in
2
n
(b) ∆Ts > t > tr:
∴
Io
∆R
(1 − e
i1,2 ( t ) = (1 m
)
R + RD 2
R + RD
( t − tr )
L
)
+ i1,2 ( t r ) e
-
R + RD
( t − tr )
L
(I-xv)
(c) ∆Ts + tf > t > ∆Ts:
i1,2 ( t ) = i1,2 ( DTs ) −
Io  ∆L t − DTs
1m
)
,
2 
L 

 tf
(I-xvi)
 2 Vr 
.
where t f = Io / 


L + L f n 
(d) Ts > t > ∆Ts + tf (similar to stage (II)):
i1,2 ( t ) = 0
Appendices
(I-xvii)
189
The average currents of the transformers are:
I 1,2 =
1 DTs
∫i ( t ) dt
Ts tr 1,2
R + RD
R+ RD
Io
−
−
L ⋅f s
DTs  I o
DTs
∆R 
∆L L ⋅ f s
L


=
(1 m
) D−
(1 − e
) + (1 m
)
(1 − e L
)
R + RD 
R
2
2
L R + RD



I
= o
2
D ( R + RD ) 

L
⋅
f
R
R
L
∆
∆
∆
s
(1 m
−
)D ±
(
)( 1 − e L⋅f s ) .


R + RD
R + RD R + R D
L


(I-xviii)
Not only that separate diodes reduces the resistance deviation, but also they eliminate the
oscillation
between
modules
during
the
off-time.
Therefore
under
the
condition
R + R D >> L ⋅ f s , Eq. (I-xviii) is reduced to
I1, 2 =
Appendices
Io
∆R
(1 m
)D .
2
R + RD
(I-xix)
190
II. Electro-Thermal Analysis of Current Sharing
Taking linear approximation, the i-v characteristic of a diode can be expressed as:
i = AT 2 e − qΦm / kT ( e qv / kT − 1) ≅ B(1 + K∆T ) e qv / kTo = C( ∆T )e qv / kTo .
(II-i)
With the same voltage drop, v, across two paralleled Schottkies, the current distribution is
I = I1 + I2 = ( C1 + C2 ) e qv / kTo .
C1

I1 =
I;


C1 + C2

C1
I2 =
I.

C1 + C2

∴
(II-ii)
Considering the dispersion of the device parameter between the two Schottkies, it can be
assumed that
δ
∆B

C1 = ( Bo + 2 )(1 + K∆T1 ) = Bo (1 + 2 )( 1 + K∆T1 );

δ
∆B
C2 = ( Bo −
)(1 + K∆T2 ) = Bo (1 − )( 1 + K∆T2 ).

2
2
(II-iii)
Thus, the current sharing unbalance is
δ


δ + K ( ∆ T1 − ∆T2 ) + ( ∆ T1 + ∆T2 )
I1 − I2 C1 − C2
2


∆≡
=
=
.
δ
I
C1 + C2


2 + K ( ∆T1 + ∆T2 ) + ( ∆T1 − ∆T2 )
2


Appendices
(II-iv)
191
The electro-thermal relation ship can be derived by solving the thermal network, shown in Fig.
3.5:
P1 − P2
;
Rc
2+
Rb
Rb
∆T1 = P1 Ra + ( P1 − x ) Rb = P1 ( Ra + Rb ) −
( P − P );
Rc 1 2
2+
Rb
Rb ( P1 − x ) − Rb ( P2 + x ) = Rc x
∆T2 = P2 ( Ra + Rb ) +
⇒
x=
Rb
( P − P ).
Rc 1 2
2+
Rb
(II-v)
The power loss of each individual diode is the product of its forward voltage drop and its
conduction current:
C1

P
=
vI
=
vI;
1
1


C1 + C2

C2
P2 = vI2 =
vI.

C1 + C2

(II-vi)
Combining Eqs. (II-v) and (II-vi), it can be derived that
∆T1 + ∆T2 = ( Ra + Rb )( P1 + P2 ) = ( Ra + Rb ) vI


2 Rb
2 Rb
)( P1 − P2 ) = ( Ra + Rb −
) v∆ I .
∆T1 − ∆T2 = ( Ra + Rb −
Rc
Rc

2+
2+

Rb
Rb

(II-vii)
Combing Eqs. (II-iv) and (II-vii) gives the current sharing unbalance due to the electro-thermal
interaction:
Appendices
192



δ
2 Rb
δ + KvI  ( Ra + Rb ) + ( Ra + Rb −
)∆ 
R
2
δ
2+ c 
( 2 + α ) + β∆


R
b


2
=
∆=
,
δ
δ


 (2 + α ) 2 + β 2 ∆

2 Rb
δ
2 + KvI ( Ra + Rb ) + ( Ra + Rb −
)∆ 
Rc 
2

2+

Rb 


(II-viii)
α = KvI ( Ra + Rb )


2 Rb
where β = KvI ( Ra + Rb −
) . Current unbalance, D, can be determined by solving the
R

2+ c
Rb


2nd-order Eq. (II-viii)
− ( 2 + α − β ) + ( 2 + α − β ) 2 + ( 2 + α ) βδ 2
(2 + α ) δ
1 1 ( 2 + α ) βδ 2
∆=
≈
=
βδ
βδ 2 ( 2 + α − β ) ( 2 + α − β ) 2
∴


 KvI( R + R ) 
a
b 
1 +
2
δ
∆=
KvIRb
2


 1+
Rc
2+


Rb


(II-ix)
(II-x)
By a uniform conduction approximation, the thermal coupling resistance can be calculated as
Rc =
Appendices
1 d
,
K ht w ⋅t
(II-xi)
193
where Kth is the thermal conductivity of the heatsink material, w and t are the width and thickness
of the heatsink respectively, and d is the distance between the two paralleled diodes. Therefore,
Eq. (II-x) can be written as


 KvI ( R + R ) 
a
b 
1 +
2
δ ,
∆=
KvIRb
2


1 +
d
2+


K th wtRb 

(II-xii)
i.e., the current distribution unbalance expressed in terms of the diode separation. Eqs. (II-x) and
(II-xii) are plotted in Fig. 3.6 to graphically show the effects of the thermal coupling (or the diode
separation) on the current unbalance.
Appendices
194
III. Partial Element Equivalent Circuit Method
The loop inductance of the rectangular loop formed by four traces, shown in Fig. III.1(a),
is by definition as
Lloop =
1 r r
B ⋅ds ,
s
I ∫∫
(III-i)
r
where I is the conduction current, B is the magnetic flux intensity, and s is the loop area. Using
r
r
r r
r r
the Maxwell equation, B = ∇ × A , and the Stoke’s theorem, ∫∫s ∇ × X ⋅ds = ∫c X ⋅dl , Lloop can be
(
)
rewritten as
Lloop =
1 r r 1  4 r r
A ⋅dl = ∑ ∫A ⋅dl .
c
l
I∫
I


n=1 n
(III-ii)
r
where A is the magnetic potential, and can be decomposed into the contribution from each
traces, i.e.,
r
A=
4
∑
m =1
r
Alm ,
(III-iii)
r
where Al n is the magnetic potential generated by the current flowing through ln. Then, Eqs. ((IIIii) and (III-iii) give
Lloop
Appendices
r
r
14 4
= ∑ ∑ ∫ Al m ⋅dl .
l
I

n=1 m=1 n

(III-iv)
195
l2
l1
l3
l4
(a)
L2
R2
R1
L1
R3
M13
L3
M 24
L4
R4
(b)
Figure III.1. A rectangular shaped loop described with PEEC method: (a) loop layout; (b)
equivalent circuit representation.
Appendices
196
Let Ln =
r
r
1 r
1 r
Al n ⋅dl , and M mn = ∫ Al n ⋅dl (m ≠ n), Eq. (III-iv) becomes
∫
I ln
I lm
Lloop


4
4 4


1
= ∑ Ln + ∑ ∑ M mn .
I n=1

n= 1m =1
m ≠n


(III-v)
Therefore, the loop inductance can be expressed as the combination of the trace partial selfinductance Ln, and partial mutual-inductance, Mmn (Mmn = Mnm). The representation of using
equivalent circuit is shown in Fig. III.1(b). It should be noted that in this rectangular loop
example, only the paralleled traces have mutual inductance between them, i.e., M12 = M14 = M23 =
0, because perpendicular traces have null induced fields.
Appendices
197