IEEE Distinguished Lecture
10:30 - 11:30 Wednesday, May 13, 2015
A Lucid View on What Role kQ Product
Plays in Electric- and/or MagneticCoupling Wireless Power Transfer Systems
Takashi Ohira
Toyohashi University of Technology
Special Acknowledgment to Professor Naoki Shinohara, Kyoto University
!
There must be a Principle in any “Transfer”
wave
[1]
information
[2]
energy
？
[1] Harald.T. Friis, “A note on a simple transmission formula,” Proc. IRE, vol.34, pp.254-256, May 1946.
[2] Claude. E. Shannon, “Communication in the presence of noise,” Proc. IRE, vol.37, pp.10-21, Jan. 1949.
Face a Basic Exercise with Pencil and Paper
電
源
i1
送
電
器
受
電
器
coupler #1
coupler #2
v1
source
i1 = 1+ j 0
P1 = ℜ {v i
*
11
−i2
}
v1 = z11i1 + z12i2
v2 = z21i1 + z22i2
v2
負
荷
load
i2 = α + j β
P2 = −ℜ {v i
*
2 2
Exercise
P2
= η (α , β )
P1
Given matrix Z known, find
maximum η in (α, β) plane
}
Let us climb Mt. η to reach its summit
Mathematical idea
ηmax
Gradient vanishes
at the hilltop
efficiency
contours
β
∂η ∂η
=
=0
∂α ∂β
Mt. η
効率
等高線
P2
η=
P1
substitute
α
P1
P2
∂P1
∂α
∂P2
∂α
P1
P2
= ∂P1
∂β
∂P2
∂β
=0
Pseudo-Jacobian
determinants
must be zero
Formulate P1 and P2 in terms of α and β
{ } = ℜ {( z i + z i ) i }
= ℜ { z + z (α + j β )}
= ℜ {r + j x + ( r + j x ) (α + j β )}
P1 = ℜ v i
*
11
11
11 1
*
1
12 2
12
11
11
12
12
= r11 + r12α − x12 β
i1 = 1+ j 0
i2 = α + j β
z11 = r11 + j x11
{ } = −ℜ {( z
P2 = −ℜ v i
*
2 2
{
v1 = z11i1 + z12i2
i + z22i2 ) i
*
2
21 1
}
(
= −ℜ ( r21 + j x21 ) (α − j β ) + ( r22 + j x22 ) α + β
(
= −r21α − x21β − r22 α + β
2
2
)
2
2
)}
Put P1 and P2 into Jacobian Equations
(
P2 = −r21α − x21β − r22 α + β
P1 = r11 + r12α − x12 β
P1
P2
∂P1
∂α
∂P2
∂α
P1
P2
= ∂P1
∂β
∂P2
∂β
2
=0
α β
1 α
2α 1 ⎛ α
1 ⎞ x21
+
+
=
+
+ −⎜ + ⎟
β=0
r21 x21 r22 r11 r21 r22 ⎝ r12 r22 ⎠ r11
2
(
r21 r11r22 + x
α=
2
21
)( r
21
(
+ 2r22α ) + r + x
−r11r22 − x +
2
21
(r r
2
21
11 22
−r
r +x
2
21
2
21
2
21
2
21
)r α
2
22
)( r r
2
11 22
=0
+x
2
21
)⋅r
21
r22
2
)
Solution (α, β ) for the Summit of Mt. η
α=
β=
−r11r22 − x +
2
21
(r r
11 22
−r
2
21
)( r r
11 22
2
r212 + x21
r11r22 − r212 −
+x
2
21
η(α,β)
ηmax
)⋅r
21
r22
β
2
2
r
r
−
r
r
r
+
x
( 11 22 21 )( 11 22 21 ) ⋅ x21
2
r212 + x21
r22
α
P2 −r21α − x21β − r22 (α + β
η= =
P1
r11 + r12α − x12 β
2
ηmax = 1−
2
z21
1+ 1+
R
2
2
2
)
z21 = r + x
2
21
2
21
R = r11r22 − r12 r21
Z = R + jX
Heuristic Introduction of θ
ηmax = 1−
= 1−
transfer
function
2
z21
1+ 1+
R
2
tan 2θ =
z21
R
loss
factor
2
1+ 1+ tan 2θ
1− cos 2θ
=
1+ cos 2θ
P2
P1
2
What an elegant …
∴ ηmax = tan θ
2
“efficiency tangent”
power
ratio
Example of tan2θ #1: coupled coils
⎡ r + jω L jω M ⎤
Z=⎢
⎥
j
ω
M
r
+
j
ω
L
⎣
⎦
coil #1
coil #2
⎡
⎤
⎡
⎤
r
0
L
M
=
+ jω
⎢⎣ 0 r ⎥⎦
⎢⎣ M L ⎥⎦
R =r
2
∴ tan 2θ =
z21 = ω M
z21
R
M
L
L
ωM M ωL
=
=
= kQ
r
L r
We find out double angle tangent physically means kQ
Example of tan2θ #2: capacitor pair
⎛1
⎞ ⎡ 1 0 ⎤ 1⎛ 1
⎞ ⎡ 1 −1 ⎤
Y = ⎜ + jω C1 ⎟
+ ⎜ + jω C2 ⎟
⎝ r1
⎠ ⎢⎣ 0 1 ⎥⎦ 2 ⎝ r2
⎠ ⎢⎣ −1 1 ⎥⎦
#2
#1
C2
tan 2θ =
y21
G
r1 1+ ω C r
=
2 ( r1 + r2 ) r2
2
2 2
2 2
C1
C1
C2
1
C2
lim tan 2θ = ω C2 r1 =
ω C1r1 = kQ
r2 →∞
2
2C1
Double angle tangent implies kQ again
kQ formulas for typical schemes
ccoouupplliinngg ccaappaacciittiivvee　iinndduuccttiivvee
ssttrruucc-ttuurree
v1
v2
i1
i2
C2
eeqquuii-vvaalleenntt
C1
M
C1
L
L
C2
kkQQ
C2
1
=
⋅ ω C1r1 = ω C2 r1
2C1
2
M ωL ωM
=
⋅
=
L r
r
Graphical Way to Understand General Relation
kQ = tan 2θ
2
ηmax = tan θ
square
efficiency angle
π
0 ≤θ ≤
4
Recognize tan2θ as general extension of kQ
How to Estimate kQ and ηmax
R
⎡ z11 z12 ⎤ ⎡ r11 r12 ⎤
Z=⎢
=⎢
+
⎥
⎥
z
z
r
r
21
22
21
22 ⎦
⎣
⎦ ⎣
kQ =
z21
r11r22 − r12 r21
kQ = tan2θ
ηmax = tan θ
2
R
X
⎡ x11 x12 ⎤
j⎢
⎥
x
x
⎣ 21 22 ⎦
How to Accurately Estimate Two-Port Z
#1
#1
#1
#2
#2
#2
any type of coupling structure
VNA
measurement
EM field
simulation
HFSS, CST,
WiPlD, Feko, …
⎡ s11 s12 ⎤
S=⎢
⎥
s
s
⎣ 21 22 ⎦
Rohde&Schwarz,
Keysite, Anritsu, …
matrix transform
Z = 50 ( I + S ) ( I − S )
⎡ z11 z12 ⎤
Z=⎢
⎥
z
z
⎣ 21 22 ⎦
I=⎡ 1 0 ⎤
⎢⎣ 0 1 ⎥⎦
identity
−1
Mutual translation between kQ and ηmax
direct expression
angular expression
2 ηmax
kQ =
1− ηmax
ηmax = 1−
kQ = tan 2θ
ηmax = tan θ
2
2
2
1+ 1+ k Q
2
Required kQ square to achieve typical goals of maximum power efficiency
2
kQ
2
ηmax [%]
0
8
0
50
!
15!
!
60!
80
360 9800
80
90
98
∞
100
Proof-of-Theory Demonstration #1
Electric-coupled V-WPT miniature EV
RF
rectifier
RF
inverter
back side
printed electrodes
metal
leaf
quite lightweight
free of batteries onboard
Via-Wheel
Power Transfer
!
S Parameter Measurement of V-WPT System
V-WPT Measured RF Power Transfer Efficiency
thanks to the theory
%
100!
η
ηmaxmax
efficiency
ηmax S21! η
90!
of V-WPT alone
80!
70!
η
of V-WPT
with
S21 with
!
matching
circuitsmatching
input
& output
60!
50!
40!
30!
20!
10!
0!
49!
50!
51!
52!
53!
54!
[MHz]
55!
56!
MHz
57!
RF-to-RF efficiency exceeds 77 ％
Proof-of-Theory Demonstration #2
Experiment on Actual Concrete and Wheels
tread rubber
metal hub
steel belt
bulb
air
output
matching
10 cm thick
concrete
steel mesh
input
matching
RF power supply
A 60W Incandescent Bulb Lit by V-WPT
A 60W Incandescent Bulb Lit by V-WPT
thanks to the theory
100
ηmax
ηmax of
of V-WPT
V-WPTalone
alone
90
80
efficiency η [%]
70
η of V-WPT with
input & output matching
60
50
40
30
20
10
0
6
6.2
6.4
6.6
6.8
7
7.2
Frequency[MHz]
7.4
7.6
7.8
8
Proof-of-Theory Demonstration #3
under construction
iron plates
track
no battery
onboard
13.56 MHz
500 W
Power Supply
floor completed
Proof-of-Theory Demonstration #3
motor runs without
battery onboard
not charging
but powering
not inductive
but capacitive
Conclusion
We introduced θ as an essential performance index of WPT
You can simply estimate θ from matrix Z of the system
Criterion tan2θ behaves as kQ in arbitrary coupling scheme
!
!
!
!
!
Z = 50 ( I + S ) ( I − S )
= R + jX
−1
kQ =
z21
R
kQ
θ
θ
ηmax
Index θ keeps its invariance against source or load properties
It works as a versatile pilotage in WPT structure optimization
[3] T. Ohira,“Maximum available efficiency formulation based on a black-box model of linear
two-port power transfer systems,” IEICE Electronics Express, vol.11, no.13, July 2014.