This article has been accepted for publication in IEEE Transactions on Transportation Electrification. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TTE.2022.3229178
A Detuned S-S Compensated IPT System with
two Discrete frequencies for Maintaining Stable
Power Transfer versus Wide Coupling Variation
Bin Yang, Yiyang Li, Zeheng Zhang, Shuangjiang He, Yuner Peng, Yang Chen, Member, IEEE,
Zhengyou He, Senior Member, IEEE, Ruikun Mai, Senior Member, IEEE
Abstract—Stable power transmission is one of the key factors in
the inductive power transfer (IPT) system. However,
misalignment between the primary and secondary sides is almost
inevitable in practice, affecting the system performance due to the
coupling variation. With the widespread use of IPT technology, it
is desired to transfer power from the primary side to the secondary
side with a wide misalignment range as large as possible. To
address this issue, the design method of the detuned circuit is
widely applied in the IPT system. This paper analyzes the
characteristics of the transfer power of the detuned series–series
(S-S) topology with frequency variations and proposes a
maintaining stable power transfer method versus wide coupling
variation by adopting two discrete frequencies. Further, a design
step is given to obtain the system parameters. Theoretical and
experimental results are provided to demonstrate the
misalignment performance of the proposed method. The results
show that the coupling range is extended from [0.115-0.2] to
[0.115-0.27] with the 5.6% fluctuation of the output power, and the
corresponding efficiency varies from 91.46% to 95.52%.
Index Terms—inductive power transfer (IPT), coupling variation, fixed frequencies, parameters design, stable power transfer.
I. INTRODUCTION
I
NDUCTIVE power transfer (IPT) technique has drawn much
attention due to its attractive advantages, such as flexibility,
convenience, safety, and user-friendliness. It is widely
employed in many applications like consumer electronics [1],
underwater power supplies [2], electric vehicles [3], automatic
guided vehicles [4], and so on.
In practice, misalignment between the primary and
secondary sides is an inevitable issue, resulting in the
fluctuation of power transfer due to the coupling variation.
Stable power transfer is one of the critical indicators to evaluate
This work was supported in part by the National Natural Science Foundation of China under Grant 52207226 and 51977184, in part by the Natural
Science Foundation of Sichuan, China under Grant 23NSFSC4112, and in
part by the the Sichuan S&T Innovation Project under Grant 23MZGC0228.
(Corresponding author: Ruikun Mai)
Bin Yang, Yiyang Li, Zeheng Zhang, Yuner Peng, Yang Chen,
Zhengyou He, and Ruikun Mai are with the School of Electrical Engineering, Southwest Jiaotong University, Sichuan 611756, China (e-mail:
yb@my.swjtu.edu.cn; eayoung23@foxmail.com; hengs@my.swjtu.edu.cn;
pengyuner@my.swjtu.edu.cn; yangchen@swjtu.edu.cn; Hezy@home.
swjtu.edu.cn; mairk@swjtu.edu.cn).
Shuangjiang He is with with the Tangshan Institute, Southwest Jiaotong
University, Tangshan 063000, China (e-mail: hsjhsj@my.swjtu.edu.cn).
the reliability of the IPT system. Therefore, the IPT system is
required to have the ability of misalignment tolerance.
To keep the output power stable, a common method is to
introduce a control strategy into the IPT system. A dc-dc
converter can be cascaded on the primary or secondary side [5]
- [6] to adjust the power flow. But the dc-dc stage will introduce
extra loss and cost. The phase shift control [7] and the frequency
control [8] for the inverter also can be used to restrain the
fluctuation of power. For the phase shift control, the softswitching region is narrow with the shift of drive signals in
different bridge arms of the inverter. For the frequency control,
the frequency bifurcation phenomena may occur with a wide
range of variable coupling, which may decrease the system's
reliability[9]. Moreover, a variable inductor [10] is proposed to
deal with the misalignment issue of the IPT system. However,
the extra cost and the installation space are required. Although
the control strategy is an effective and active approach to
addressing the misalignment problem, an extensive range of
modulation depth is usually compromised. It may degrade the
stability and efficiency of the system.
To avoid the deep modulation index of the control strategy,
various works are proposed to resist the variation of coupling.
The related work can be roughly divided into three categories:
magnetic couplers, hybrid topologies, and detuned circuits.
Many magnetic couplers are designed to maintain stable
power transfer with a large misalignment region. Some
polarized coil structures like double-D [11] and bipolar pads [4]
are presented, which only can tolerate one-direction
misalignment because of the effect of non-orthogonal magnetic
fields. Asymmetric coil [12] and three-coil structure [13] are
designed to enhance the misalignment performance in two
directions. Aiming at three-dimension misalignment tolerance,
reconfigurable coils [14] - [16] with the help of multi-mode operation are reported. Still, multiple switches and coils are required, resulting in a complex multimodal discrete control and
high cost in the circuit.
To simplify control, many researchers advocate approaches
employing the circuit’s inherent misalignment characteristics to
address the power fluctuation. As a typical method, the hybrid
topology is popular. The main principle of the hybrid topology
is that two topologies have two adverse outputs relationship
with the coupling variation so that the total transfer power can
maintain relatively constant. For example, a hybrid IPT system
can be integrated by LCC-LCC and S-S compensation
topologies [17]-[18] or S-LCC and LCC-S topologies [19] -[21].
Since the unique feature of the hybrid topology merges with
two compensation topologies, the complexity and high cost are
inevitable.
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content may change prior to final publication. Citation information: DOI 10.1109/TTE.2022.3229178
Compared to hybrid topology, the detuned circuit is more
low-cost and straightforward. Many works have recorded the
misalignment performance of the different detuned circuits in
the past. The optimization of the compensation capacitors in a
multi-coil IPT system is reported in [22] - [23]. In order to
simplify the system, the detuned circuit with a two-coil IPT
system is more accepted. In [24] - [26], the detuned S-S, LCCS, and X-type topologies are proposed to resist coupling
variation. Besides, S-SP, S-CLC, and double-T topologies are
employed in [27] - [29], where the stable power transfer is
realized against coupling variations. Further, a family of
compensation topologies with secondary parallel compensation
is summarised to obtain strong misalignment tolerances by Mai
et al.[30]. As we know, the profile of transfer power vs coupling
variation (P-k profile) is determined once the impedance of the
compensation elements is designed. By changing the
impedance of the compensation elements, the P-k profile can be
altered so that the coupling range with stable transfer power can
be extended. For example, by combining the detuned S-S and
the detuned LCC-S topologies, a reconfigurable topology is
introduced to enlarge the coupling range using two created P-k
curves [31]. But extra elements have to be added.
In terms of the impedance adjustment of the compensation
elements, the frequency variation is a direct method. Compared
with the variable inductor[10], the reconfigurable coil [14] -[16],
and the reconfigurable topology [31] - [32], the frequency
regulation is not required extra reactive elements or switching
devices in the IPT system. Compared to continuous frequency
control, two separate frequencies switching is more convenient
and simple, and it can also avoid the frequency bifurcation
phenomena caused by the large coupling variation. In the past,
the output characteristics of the IPT systems versus the
frequency were analyzed, and some IPT systems with two
separate frequencies were proposed and applied for the constant
current and constant voltage charging [33] - [34]. However,
these works mainly focus on the case of fixed coupling
coefficient. For stable power transfer versus wide coupling
variation, this method has a lack of investigation. If this method
can be used in the IPT system to resist the coupling variations,
it means that the system can maintain stable power transfer with
a cost-effective and straightforward approach.
This paper systematically analyzes the transfer power variation of the detuned S-S topology versus different couplings and
frequencies. According to the transfer power profiles at the different frequencies, a method using discrete frequencies is proposed to extend the coupling range with stable power transfer.
Besides, a parameter design method is introduced to limit the
fluctuation of the transfer power within a certain range using
two different discrete frequencies. With the combination of
transfer power curves in two frequencies, the system can
achieve a relatively constant power transfer with a wide coupling variation. The rest of this work is expressed as follows. In
section II, the analysis of detuned S-S topology is described,
and the misalignment characteristics versus frequency are
grasped. In section III, a method of extending the coupling
range is proposed, and an example is designed. A 750W prototype was constructed to verify the feasibility of the proposed
method in section IV. The conclusion is drawn in section V,
finally.
II. MISALIGNMENT CHARACTERISTICS OF THE DETUNED S-S
TOPOLOGY
A. Misalignment characteristics with a fixed frequency
Q1
Cp
Q3
Cs
M
Is
Ip
A
E
Q2
Lp
Vp
B
Ls
Vs
Zin
Q4
D1
D3
C
Cf
D
Rac
io
R
D4
D2
+
uo
-
Fig. 1. An IPT system with S-S topology.
An IPT system with S-S topology is described in Fig. 1. The
inverter and rectifier are formed by MOSFETs (Q1~Q4) and diodes (D1 ~ D4), respectively. E (Vs) and Vp (uo) are the input and
output voltage of the inverter and rectifier. R is the resistance
load. And Rac is used to express the input ac load of the rectifier.
The operating angular frequency of the IPT system is defined
as ω. Cp and Cs are the compensated capacitor on the primary
and secondary sides. The primary self-inductance, secondary
self-inductance, and mutual inductance of the loosely coupled
transformer are expressed as Lp, Ls, and M, respectively. The
coupling coefficient k of the loosely coupled transformer is
given as
M
(1)
k=
Lp Ls
The reactance of Cp, Cs, Lp, and Ls can be written as
1

1

 X Cp = C
 X Cs =

Cs
(2)
,
p


X = L
X = L
p
s
 Ls
 Lp
Further, the equivalent capacitive or inductive resistances (Xp
and Xs) of the primary and secondary sides can be expressed as

 X p = X Lp − X Cp
(3)


 X s = X Cs − X Ls
Based on Kirchhoff’s voltage law, we have
j

( j Lp − C ) I p − j MI s = V p

p
(4)

− j MI + ( j L − j + R ) I = 0
p
s
ac
s

 Cs

Substituting (1), (2), and (3) into (4), the primary current Ip
and the secondary current Is can be solved as
Ip =
(
V p  2 k 2 L p Ls Rac + j ( X s 2 k 2 L p Ls − X p X s 2 − X p Rac 2 )
( X X − k L L ) + X R
L L V ( X R + j k L L − jX X )
( X X − k L L ) + X R
2
p
Is =
j k
2
2
s
p
2
p
s
p
p
ac
2
p
s
s
p
p
s
p
s
ac
p
2
p
)
2
2
2
2
2
(5)
s
2
ac
According to (5), the input impedance Zin and the input impedance angle θ can be calculated as

V p  2 k 2 L p Ls Rac
 2 k 2 L p Ls X s
=
+
j
(
X
−
)
 Z in =
p
Ip
X s 2 + Rac 2
X s 2 + Rac 2

(6)

X p X s 2 + X p Rac 2 X s

−
)
 = arctan(  2 k 2 L L R
Rac
p s ac

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Besides, the output power Po can be obtained from (5), i.e.
V p 2 2 k 2 Lp Ls Rac
2
(7)
Po = I s Rac =
2
( X p X s −  2 k 2 Lp Ls ) + X p 2 Rac 2
In terms of (7), the output power Po can be rewritten as
Vp 2 2 Lp Ls Rac
Po =
(8)
X p 2 X s 2 + X p 2 Rac 2
4 2
2
2
2
+  k Lp Ls − 2 X p X s Lp Ls
k2
Po
Pm ax
Pm in
B. Misalignment characteristics with variable frequencies
Based on the above analysis, the profile of output power Po
versus the coupling coefficient k can be depicted in Fig. 2 with
any fixed frequency. Namely, the detuned S-S topology can
achieve a stable power transfer under any fixed frequency with
proper parameters. However, if the parameters are fixed, the
coupling range with stable power transfer is variable with different operating frequencies, as shown in Fig. 3. Once β is determined, the extreme point (kPmax, Pmax) of the output power Po
can be used to roughly analyze the variation trend of the output
power profile. According to (9) and (10), two cases should be
discussed, i.e., Xp > 0 and Xp < 0.
Po
Pm ax1
0
km in
kPm ax
km ax
1 k
ω=ω1
The power fluctuation β1
The power fluctuation β2
Pm in 1
Pm ax2
Pm in 2
Fig. 2. The profile of the output power Po at a fixed frequency, where
Pmax (Pmin) is the maximum (minimum) output power and kpmax (kmin
and kmax) is the corresponding coupling coefficient.
The coupling range
0
ω=ω2
km in 1 kPm ax1
km ax1
The coupling range
km in 2 kPm ax2
km ax2
1 k
From (8), the output power Po versus k is a non-monotone
function. Since the coupling range belongs to (0, 1), the corresponding power curves can be depicted in Fig. 2. And the maximum output power Pmax and corresponding critical coupling
kPmax can be obtained by taking the derivative of Po equal to zero,
i.e., dPo/dk =0. So we have
1

2
2 4
X
(
X
+
R
)

p
s
ac
,Xp 0
k P max =

L
L
p s

(9)

1

2
2 4
− X p ( X s + Rac )
k
=
,Xp 0
 P max
 Lp Ls

Then, substituting (9) into (7), the maximum output power
Pmax can be obtained as

V p 2 Rac
, Xp  0
 Pmax =
2 X p ( X s 2 + Rac 2 − X s )

(10)

V p 2 Rac

, Xp  0
 Pmax = −
2 X p ( X s 2 + Rac 2 + X s )

A variable β is defined to express the allowable power fluctuation, namely,
P −P
 = max min
(11)
Pmax + Pmin
Fig. 3. The profile of the output power Po with different frequencies,
where Pmax1 (Pmax2) is the maximum output power, kpmax1 (kpmax2) is the
corresponding critical coupling coefficient, Pmin1 (Pmin2) is the minimum output power, β1 (β2) is the power fluctuation, and the corresponding coupling range is from kmin1 (kmin2) to kmax1 (kmax2) when the
angular frequency is ω1 (ω2).
Then, the coupling range with stable power transfer [kmin ~
kmax] can be obtained by substituting (7) into (11), as

V p 2 Rac + 2 X p X s Pmax (1 −  ) / (1 +  ) − A
kmin =
2 2 L p Ls Pmax (1 −  ) / (1 +  )


(12)

V p 2 Rac + 2 X p X s Pmax (1 −  ) / (1 +  ) + A

kmax =
2 2 L p Ls Pmax (1 −  ) / (1 +  )


where
4V 2 X X R P (1 −  ) 4 Rac 2 X p 2 Pmax 2 (1 −  ) 2
A = Rac 2V p 4 + p p s ac max
−
(1 +  )
(1 +  ) 2
B = (2 Lp − 3 X p )( Rac 2 + X s 2 ) + X p X s (2 Ls − X s ) (16)
a) Xp > 0
A variable ωp is defined to represent the primary resonant
frequency, which satisfies
1
(13)
p =
Lp C p
Combining (3) and (13), when Xp > 0, the operating frequency of the IPT system will satisfy (14), namely, the system
angular frequency ω∈(ωp, +∞).
1
(14)
  p =
Lp C p
1) The variation trend of kPmax versus frequency
Substituting (2) and (3) into (9), the derivative of kPmax can
be expressed as
1
B( Rac 2 + X s 2 ) 4
d
k P max =
2
d
2 X p Lp Ls ( Rac 2 + X s 2 )
(15)
where
From (15), the monotonicity of kPmax depends on the sign of
B. Let the derivative of kPmax equal to zero, namely,
d
k P max = 0
(17)
d
Combining (15), (16) and (17), we can obtain the extreme
frequency ωefk,
4 X p X s 2 + 3 X p Rac 2
(18)
efk =
2 L p ( X s 2 + Rac 2 ) + 2 X p X s Ls
The system angular frequency satisfies ω∈[ωp, +∞] when
Xp > 0, so two cases should be discussed, as follows.
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Case aⅠ: ωefk ≤ ωp or ωefk is not an extreme value. For this
case, kPmax is monotonous with frequency ω because the system
angular frequency ω > ωp. And the monotonicity of kPmax can
be estimated by substituting any angular frequency ω∈[ωp, +∞]
into (16). For example, if ω approaches ωp, B will be greater
than zero, as
(19)
lim B = 2 Lp ( Rac 2 + X s 2 )  0
 → p
In combining (15) and (19), we have
d
(20)
lim
k P max  0
 → p d 
Substituting (3) into (9), when the angular frequency ω is
close to ωp or +∞, the critical coupling kPmax will equal to
 lim kP max =0
 → p
(21)

k P max =1
lim
→+
kpm ax
1
ω
ωp
0
coupling range (0, 1), which is impossible and will not be discussed in the subsequent analysis.
In brief, the critical coupling kPmax will be monotonically increasing with the angular frequency ω based on the analysis of
Case aⅠ and Case aⅡ when it satisfies 0<kPmax<1.
2) The variation trend of Pmax versus frequency
Substituting (2) and (3) into (10), the derivative of Pmax can
be given as
CVp 2 Rac
d
Pmax =
(24)
d
2 X p 2 Rac 2 + X s 2 ( Rac 2 + X s 2 − X s )
where
C = X p ( Rac 2 + X s 2 − X s ) + 2 ( X p Ls − Lp Rac 2 + X s 2 )
(25)
The sign of C will determine the monotonicity of the maximum output power Pmax according to (24). Similarly, set the derivative of Pmax equal to zero, as
d
(26)
Pmax = 0
d
And combining (24), (25) and (26), the expression of extreme frequency ωefP of Pmax can be solved as
Fig. 4. The profile of the critical coupling kPmax versus the angular frequency ω in Case aⅠ.
Combining (20) and (21), the profile of the critical coupling
kPmax versus the angular frequency ω can be roughly depicted in
Fig. 4. We can see that kPmax increases with the angular frequency ω, and it locates in the coupling range (0, 1), which
means that there is an extreme point (kPmax, Pmax) when ω > ωp.
Case aⅡ: ωefk is an extreme value and satisfies ωefk > ωp.
From (18), we have
 d
 d  k P max  0,   ( p , efk ]
(22)

 d k
 0,   (efk , +)
 d  P max
kpm ax
1
0
efP =
X p ( Rac 2 + X s 2 − X s )
(27)
2 Lp Rac 2 + X s 2 − 2 X p Ls
Assuming the critical coupling kPmax satisfies 0<kPmax<1 versus frequency, the monotonicity of the maximum output power
Pmax can be gained similarly, as follows.
Case aⅢ: ωefP ≤ ωp or ωefP is not an extreme value. We can
evaluate the monotonicity of the maximum output power Pmax
by (28), i.e.,
lim C = −2 Lp Rac 2 + X s 2  0
(28)
 → p
According to (28), the maximum output power Pmax will decrease with the angular frequency ω. And the corresponding
profile can be roughly given in Fig. 6.
Pm ax
ωp
ωck
ωefk
ω
0
Fig. 5. The profile of the critical coupling kPmax versus the angular frequency ω in Case aⅡ.
Combining (21) and (22), the profile of the critical coupling
kPmax can be roughly drawn as Fig. 5. According to (9), when
kPmax=1, the corresponding angular frequency ωck can be expressed as
Fig. 6. The profile of the maximum output power Pmax versus the angular frequency ω in Case aⅢ.
Pm ax
1
ck =
X p ( X s 2 + Rac 2 ) 4
Lp Ls
0
(23)
From Fig. 5, It can be seen that the critical coupling kPmax
increases with the angular frequency ω when the angular frequency satisfies ωck>ω>ωp. Once the angular frequency is
more than ωck, the critical coupling kPmax will be beyond the
ω
ωp
ωp
ωefP
ω
Fig. 7. The profile of the maximum output power Pmax versus the angular frequency ω in Case aⅣ.
Case aⅣ: ωefP is an extreme value and ωefP > ωp. Obviously,
this case indicates that there is a non-monotonic region of the
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maximum output power Pmax versus the angular frequency ω,
such as Fig. 7.
Briefly, the variation of the maximum output power Pmax depends on the primary resonant frequency ωp and the extreme
frequency ωefP of Pmax. If the ωefP ≤ ωp or ωefP is not an extreme
TABLE I
Monotonic
The side view of Po
8
Normalized transfer power Po
Condition
THE OUTUT POWER Po AT DIFFERENT CASE
The output power Po versus k and ω
6
4
2
0
Fig. 8. The output power Po versus k and ω in the case
of monotone .
Xp > 0
ω=1.05ωp
ω=1.8ωp
ω=1.1ωp
ω=2ωp
ω=1.2ωp
ω=2.5ωp
ω=1.4ωp
ω=3ωp
ω=1.6ωp
ω=5ωp
The profile of the maximum
output power Pmax
0
0.2
Normalized transfer power Po
Monotonic
1.5
Non-monotonic
1
0.5
0
0
0.2
0.4
0.6
The coupling coefficient k
0.8
1
1.5
ω=0.99ωp
ω=0.97ωp
ω=0.95ωp
1
0
ω=0.9ωp
ω=0.85ωp
ω=0.8ωp
ω=0.7ωp
The profile of the maximum
output power Pmax
0.5
0.2
1.6
1.5
0.4
0.6
0.8
The coupling coefficient k
1
ω=0.98ωp
ω=0.8ωp
ω=0.95ωp
ω=0.75ωp
ω=0.9ωp
ω=0.7ωp
ω=0.85ωp
ω=0.65ωp
The profile of the maximum
output power Pmax
1
0.5
0
Fig. 14. The output power Po versus k and ω in the
case of non-monotone when Xp < 0.
ω=2ωp
ω=2.5ωp
ω=3ωp
ω=5ωp
Fig. 13. The side view of Po in the case of monotone
when Xp < 0.
Normalized transfer power Po
Xp < 0
ω=1.2ωp
ω=1.02ωp
ω=1.4ωp
ω=1.05ωp
ω=1.6ωp
ω=1.1ωp
ω=1.8ωp
The profile of the maximum
output power Pmax
2
0
Fig. 12. The output power Po versus k and ω in the
case of monotone when Xp < 0.
1
Fig. 11. The side view of Po in the case of non-monotone when Xp > 0.
Normalized transfer power Po
Fig. 10. The output power Po versus k and ω in the
case of non-monotone.
0.8
Fig. 9. The side view of Po in the case of monotone
when Xp > 0.
2.5
Non-monotonic
0.4
0.6
The coupling coefficient k
0
0.2
0.4
0.6
The coupling coefficient k
0.8
1
Fig. 15. The side view of Po in the case of non-monotone when Xp < 0.
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value, the maximum output power Pmax will decrease with the
angular frequency ω. Otherwise, it is non-monotonic.
Based on the above analysis, the variation of the output
power Po versus the coupling coefficient k and the angular frequency ω can be summarized in the second and third rows of
TABLE I. The parameters used for the simulation are Lp =
100μH, Ls = 100μH, Cp = 10nF, R = 35Ω, and Cs =
8nF/20nF/15nF/60nF in four cases. For intuitive presentation,
the side view of the hook face of the output power Po is also
given in the fourth column of TABLE I. Some conclusions can
be obtained as follows,
(1) The coupling range gradually moves to the right as the
frequency increases when Xp > 0.
(2) The maximum output power Pmax has two variation trends
versus frequency: monotonic in Case aⅢ and non-monotonic
in Case aⅣ.
From the above conclusions, the non-monotonic case (as
shown in Fig. 10 or Fig. 11) is employed to extend the coupling
region with stable transfer power with two discrete frequencies
(ω1 and ω2) [31]. As depicted in Fig. 16, the two transfer power
curves have the same maximum output power Pmax and power
fluctuation β. The two curves intersect at the point (kmax1, Pmin)
or (kmin2, Pmin). In the misalignment condition, once the coupling
is lower (more) than kmax1 or kmin2, the system operating frequency will be switched from ω2 (ω1) to ω1 (ω2) to suppress the
fluctuation of output power.
ω=ω1
ω=ω2
Po
Pm ax1 (Pm ax2)
The power fluctuation β1=β2
The profile of the
maximum output
power Pmax
Pm in 1 (Pm in 2)
The coupling range
0
The coupling range
km in 1 kPm ax1 km ax1 (km in 2) kPm ax2
km ax2
1 k
Fig. 16. The profile of the output power Po with two different frequencies, where Pmax1 (Pmax2) is the maximum output power, kpmax1 (kpmax2)
is the corresponding critical coupling coefficient, Pmin1 (Pmin2) is the
minimum output power, β1 (β2) is the power fluctuation, and the corresponding coupling range is from kmin1 (kmin2) to kmax1 (kmax2) when the
angular frequency is ω1 (ω2).
b) Xp < 0
Based on (3) and (13), the system angular frequency satisfies
ω∈(0, ωp) when Xp < 0. Likewise, the variation trends of kPmax
and Pmax can be obtained, and the corresponding profiles of output power Po can be given in the fourth and fifth rows of
TABLE I. From Fig. 13 and Fig. 15, the variation of kPmax is
opposite with Xp > 0 (i.e., the extreme (kPmax, Pmax) of the Po has
a movement to the left versus frequency), while Pmax also has
monotonic and non-monotonic cases. Similarly, we also obtain
two transfer power curves as shown in Fig. 16, so that the IPT
system has a larger misalignment range by employing two discrete frequencies.
III. SYSTEM DESIGN
A. Parameter design
To extend the coupling variation range, some conditions
should be satisfied as
 Pmax1 = Pmax 2

(29)
 1 =  2
k
 max1 = kmin 2
When ω = ω1 (ω2), the maximum and minimum transmission
power Pmax1 (Pmax2) and Pmin1 (Pmin2) can be expressed from (7)
and (10), i.e.,

V p 2 Rac
 Pmax1 =
2 X p1 ( X s12 + Rac 2 − X s1 )

, Xp 0

V p 2 Rac

P
=
 max 2
2 X p 2 ( X s 2 2 + Rac 2 − X s 2 )

(30)

V p 2 Rac
 Pmax1 = −
2 X p1 ( X s12 + Rac 2 + X s1 )

, Xp 0

V p 2 Rac

 Pmax 2 = −
2 X p 2 ( X s 2 2 + Rac 2 + X s 2 )


V p 212 kmax12 Lp Ls Rac
P
=
 min1
2

( X p1 X s1 − 12kmax12 Lp Ls ) + X p12 Rac 2

V p 22 2 kmin 2 2 Lp Ls Rac

P
=
min
2
2

( X p 2 X s 2 − 22kmin 22 Lp Ls ) + X p 22 Rac 2

(31)
where Xp1 (Xp2) and Xs1 (Xs2) are the corresponding equivalent
reactive resistances when ω = ω1 (ω2). Combining (2) and (3),
Xp1 (Xp2) and Xs1 (Xs2) can be expressed as
1
1


 X p1 = 1 Lp −  C
 X p 2 = 2 L p −  C


1 p
2 p
(32)


1
1
X =  L −
X =  L −
s1
1 s
s2
2 s


1Cs
2 C s


A value ko is defined to express the same coupling when ω =
ω1 (ω2), namely,
(33)
ko = kmax1 = kmin 2
The primary inductively tuned is easier to obtain the inductive input impedance, which contributes to realizing the zero
voltage switching (ZVS) condition [24]. Hence, the case of Xp
< 0 is excluded in the design part of this work. Next, the constraints can be given and simplified as (34) and (35) by substituting (30), (31) and (33) into (29).
X p1 ( X s12 + Rac 2 − X s1 ) = X p 2 ( X s 2 2 + Rac 2 − X s 2 ), X p  0 (34)
(X
X s1 − 12 ko 2 Lp Ls ) + X p12 Rac 2 = ( X p 2 X s 2 − 2 2 ko 2 L p Ls ) + X p 2 2 Rac 2
2
p1
2
(35)
Besides, based on the analysis of kpmax and Pmax variation
trend versus frequency when Xp > 0, the system should also satisfy as
in Case aⅠ
efP  P

(36)
or
     in Case aⅡ
efP
P
 ck
With two fixed frequencies, once the circuit parameters are
designed to satisfy the constraints, the coupling range will be
extended. A design procedure is given to explain further, as follows.
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The load R, the self-inductances Lp and Ls of the loosely coupled transfer, the critical coupling ko, the transfer power fluctuation β, and the operable frequency range [fl, fu] should be predetermined. Then, the initial value of ω1 and ω2 is got to equal
the lower limit of the predetermined frequency range, i.e., ω1 =
ω2 = 2πfl. Substituting R, Lp, Ls, ω1, ko, ω2 into (34) and (35), Cp
and Cs can be solved. Then, the calculated Cp and Cs can be
substituted into (36) to judge if the constraint is satisfied. If yes,
the results (Cp, Cs, ω1, and ω2) will be substituted into (7) and
(11) to calculate the power fluctuation. If not, ω1 and ω2 will be
adjusted to repeat the above procedures. Finally, the results satisfying constraints will be recorded. There may be multiple solutions for the calculated results, and we can use (6) to select
the solution helping the inductive input impedance. Of course,
the results could also be no solution, which means the frequency
range [fl, fu] is unreasonably chosen. And we can reselect the
frequency range [fl, fu] to repeat the design procedures. The detailed design steps can be illustrated in Fig. 17.
Start
Determine R, Lp, Ls, ko, β and the frequency
range [fl, fu]
Select the initial value of ω1 (ω1=2πfl)
Select the initial value of ω2 (ω2=2πfl)
ω1=ω1+2πΔ
No
ω2
2πfu
ω2=ω2+2πΔ
Yes
Substitute R, Lp, Ls, ko, ω1 and ω2 into (34) and
(35) to solve Cp, Cs
Is there a solution
satisfying (36)?
be given in Fig. 18. For readability, the output power curve versus the coupling with different frequencies can be depicted in
Fig. 19 according to the side views of Fig. 18. It can be observed
that the coupling range of the proposed IPT system with two
fixed frequencies can be extended from [0.115-0.2] to [0.1150.27]. Additionally, the output power varies between 670W to
750W. However, the output power can still be extended with
the increase of the discrete frequencies from Fig. 19. It is noted
that the coupling extension is the most significant when the IPT
system uses two discrete frequencies (236kHz and 257kHz) because the non-monotonic region of the output power curve is
used. Subsequently, the coupling range is slightly increased by
about 0.012 when the system increases by one discrete frequency because only the monotonic region of the output power
curve is employed. Additionally, more discrete frequencies may
lead to more complex system operations and more complicated
implementation. Moreover, there are many power step change
points (like a step from 750W to 670W) with the increase of the
coupling k when the IPT system uses multiple discrete frequencies (more than three discrete frequencies), which is not conducive to the smooth transition of power when the frequency is
switched [36]. It is undeniable that the coupling extension range
can be extended further with the use of more discrete frequencies for applications that do not care about the system complexity, operability, frequency range, smooth power transition, and
other aspects. Considering the above factors, only two discrete
frequencies were selected to verify the theoretical analysis for
this work.
No
Yes
No
ω1
2πfu
Yes
Employ (7) and (11) to calculate the power
fluctuation
Power fluctuation <β
No
Yes
Record results (Cp, Cs, ω1 and ω2) and mark it as
Case Nn (n=1,2,...N)
Fig. 18. The output power Po versus k and f.
Use (6) to select the solution satisfying the
inductive input impedance
1000
f =236kHz
f =257kHz
The output power Po (W)
Theoretically, the wider frequency range [fl, fu] will be more
helpful for the parameters design. As a compromise, the design
process will take more time. As an example, this paper uses
[200kHz, 300kHz] to design. The predetermined parameters are
R = 35Ω, Lp = 98.87μH, Ls = 99.03μH, ko = 0.2, and β = 5%.
According to the design procedure in Fig. 17, a set of theoretical
results can be obtained as Cp = 5.25nF, Cs = 4.42nF, ω1 =
2×π×236kHz, ω2 = 2×π×257kHz. Then, the variation of the output power versus the coupling coefficient and the frequency can
f =281kHz
f =284kHz
The profile of the maximum
output power Pmax
Stop
Fig. 17. The design steps, where Δ means the step length of the frequency, Case Nn represents all the results (Cp and Cs) satisfying the
constraints.
f =275kHz
f =278kHz
800
750
670
600
400
200
0
0.1
0.115
0.14
0.27 0.283
0.295
0.308
0.32
0.18 0.22 0.26
0.3
0.34
Coupling coefficient k
0.2
0.38 0.4
Fig. 19. The side view of Fig. 18.
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Q1
A
E
Cp
Q3
B
Q4
Q2
Ip
Lp
Vp
Is
Ls
Vs
Zin
Driver
Driving Signal
Q1, Q2, Q3, Q4
Rac
Communication
D1
D3
C
Cf
D
D2
D4
R
+
uo
-
io
Voltage/current Sensor
Actual power Po
ka
+
ko Minimum power Pmin
(a)
Determine the minimum power Pmin
and the operating frequency ω1 or ω2
Measure uo and io to obtain the actual
power Po
Po < Pmin
No
Yes
Switching frequency
from ω1 (ω2) to ω2 (ω1)
Po < Pmin
0.4
Air gap=6cm
Air gap=8cm
Coupling coefficient k
0.35
0.3
Air gap=10cm
Air gap=12cm
Air gap=14cm
Air gap=16cm
0.27
0.25
0.2
0.15
0.1
Cs
M
shown in Fig. 22. The experimental parameters are listed in Table II. Compared with the calculation results, the maximum errors of Cp, Cs are lower than 1.5%, respectively, which is acceptable.
Operation region
B. Control method
Aiming at the transition of two frequencies (ω1 and ω2), the
identification of the critical coupling ko (kmax1 or kmin2) is a key.
In this paper, the output power Po is employed to select the operating mode, which can be obtained by measuring the voltage
uo and current io of the resistance load R. Once the output power
Po is lower than the minimum output power Pmin, the frequency
will be switching from ω1 (ω2) to ω2(ω1). If the output power
Po is still below Pmin after the frequency switch, it means that
the coupling k is out of design range [kmin1, kmax2], and the IPT
system will stop operation and enter a standby state. One possible control diagram is depicted in Fig. 20.
Maintain present
frequency
No
Yes
Enter standby state
(b)
Fig. 20. (a) Control diagram of mode switching. (b) Flowchart for the
control.
IV. EXPERIMENTAL RESULTS
A. Theoretical and experimental results
0.05
0.115
0
2
4
6
8
10
12
Horizontal misalignment (cm)
14
16
Fig. 22. The coupling coefficient variations with different air gaps and
horizontal misalignments.
Table II
SYSTEM PARAMETER VALUES IN EXPERIMENT
Parameter
Design value
Parameter
Design value
E
200V
R
35Ω
Pmax
750W
Pmin
670W
Lp
98.87μH
Ls
99.03μH
Cp
5.17nF
Cs
4.47nF
ω1
2×π×236kHz
ω2
2×π×257kHz
ko
0.2
β
5.6%
The transfer power of the designed IPT system is shown in
Fig. 23. The measured results are almost consistent with the theoretical results. Moreover, it is observed that the coupling range
of the proposed IPT system with two fixed frequencies can be
extended from [0.115-0.2] to [0.115-0.27]. Additionally, the
output power varies from 670W to 750W, indicating that the
fluctuation of the output power is only 5.6% based on (11).
Comparing the theoretical fluctuation of power (5%), the deviation is only 0.6%. Minor differences mainly come from the
tolerance of parameters (Cp and Cs)
The measured efficiency is given in Fig. 24. The efficiency
increases from 91.46% to 95.52% with the coupling coefficient
variation (0.115~0.27), which demonstrates the proposed
method has a good misalignment performance. The equivalent
impedance of the system is different with two fixed frequencies,
resulting in a different profile of efficiency. Thus, when the frequency changes from 236kHz to 257kHz at k = 0.2, the efficiency has a small step change (step from 94.95% to 94.41%).
The variation is only 0.54%, which is acceptable.
800
750
Fig. 21. Experimental porotype
The transfer power PSS
670
600
400
200
To demonstrate the validity of the proposed method, a 750W experimental prototype was built, as shown in Fig. 21. The
coupling coefficients of the loosely coupled transformer with
different air gaps and horizontal misalignments are measured
0.115
100
0.1
0.14
Proposed method
Measured results
Theoretical results
(ω1=2×π×236kHz)
(ω1=2×π×236kHz)
Theoretical results
Measured results
(ω2=2×π×257kHz)
(ω2=2×π×257kHz)
Operation region
0.27
0.18
0.22
Coupling coefficient k
0.26
0.3
Fig. 23. The measured and theoretical transfer power against the coupling variation.
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0.96
Efficiency ŋ
0.94
0.92
Proposed method
ω1=2×π×236kHz
ω2=2×π×257kHz
0.9
0.88
Operation region
0.115
0.86
0.1
0.14
0.27
0.18
0.22
Coupling coefficient k
0.26
0.3
Fig. 24. The measured efficiency against the coupling variation.
Proposed method
ω1=2×π×257kHz
ω2=2×π×236kHz
70
60
Fig. 26. The experimental waveforms with ω1=2×π×236kHz at (a) k
=0.15 and (b) k = 0.2, and ω2=2×π×257kHz at (c) k =0.2 and (d) k =
0.25.
50
800
750
30
20
0.1
0.115
0.14
Operation region
0.18
0.22
Coupling coefficient k
0.27
0.26
0.3
(a)
Input impedance angle θ (°)
90
80
Proposed method
ω1=2×π×257kHz
ω2=2×π×236kHz
60
The output power Po (W)
40
670
600
0.115
40
0.14
96
20
0
0.1
Proposed method
Traditional detuned circuit (ω1=2×π×236kHz)
Traditional detuned circuit (ω2=2×π×257kHz)
Phase shift control / continuous frequency control
200
100
0.1
0.115
0.14
Operation region
0.18
0.22
Coupling coefficient k
0.27
0.26
0.3
(b)
Fig. 25. The variation of (a) input impedance Zin and (b) input impedance angle θ.
Substituting the parameters of Table II into (6), the calculated input impedance Zin and the input impedance angle θ are
obtained in Fig. 25. With the coupling k increasing from 0.115
to 0.2, the input impedance Zin and the input impedance angle θ
are varied from 22.29Ω and 64.37° to 37.80Ω and 39.50° when
the frequency operates in ω1 = 2×π×236kHz, respectively.
When the frequency operates in ω2 = 2×π×257kHz, Zin increases from 34.65Ω to 45.44Ω while θ drops from 45.70°to
13.89°with the rise of k from 0.2 to 0.27. It can be seen that the
inductive input impedance is achieved in the whole operation
region, which can help the inverter to realize soft-switching operation. Fig. 26 (a) and (b) show the experimental waveforms
of the output current/voltage of the inverter and input current/voltage of the rectifier at k = 0.15 and 0.2 with
ω1=2×π×236kHz. The input impedance exhibits slightly inductive, which verifies the calculation results. A similar situation
exists in ω2=2×π×257kHz. Some representative waveforms are
given at k = 0.2 and 0.25, as shown in Fig. 26 (c) and (d).
0.27
Operation region
400
Efficiency ŋ(%)
Input impedance Zin ( )
80
0.18
0.22
Coupling coefficient k
(a)
0.26
0.3
92
88
84
80
0.1
Proposed method
Traditional detuned circuit (ω1=2×π×236kHz)
Traditional detuned circuit (ω2=2×π×257kHz)
Phase shift control
Continuous frequency control
0.115
0.14
0.27
Operation region
0.18
0.22
Coupling coefficient k
(b)
0.26
0.3
Fig. 27 The experimental results with the coupling variation. (a)The
measured output power. (b)The measured efficiency.
In order to highlight the superiority of the proposed method,
some experiments of the traditional representative method are
implemented to compare with this work, such as continuous frequency control, phase shift control, and traditional detuned circuit. Considering the fairness of the comparison, the same circuit parameters are used in the analysis and experiment. The
measured results are given in Fig. 26. The detailed data are
listed in Table III. Compared to the traditional detuned circuit,
the output fluctuations are almost identical, but the coupling
range is larger. Compared to the conventional continuous frequency control, this work has a narrower frequency range and
higher system efficiency, even though the power tracking performance is inferior to the continuous frequency tuning method.
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Table III
Method
Output fluctuation
Frequency range
Couping range
System efficiency
Control complexity
EXPERIMENTAL RESULTS
Continuous frequency
Phase shift
Detuned circuit
control
control
<1%
<1%
5.6%
5.6%
[220.9kHz, 234.8kHz]
and
[265.8kHz, 270.5kHz]
220.9kHz
236kHz
257kHz
(large frequency
range)
0.115-0.27
0.115-0.27
0.115-0.2
0.2-0.27
80.18%91.46%94.41%91.45%-94.96%
94.82%
94.95%
95.52%
Complex
Complex
Simple
Simple
Compared to the phase shift control, the efficiency of the proposed method is much higher than that of the phase shift control
with the decrease of coupling. Although it has a 5.6% power
fluctuation, this slight fluctuation is acceptable in practice [24][31]. Moreover, it is well known that conventional control
methods (like continuous frequency control [6], and phase shift
control [7]) need many computing resources because extra proportional integral controllers and the corresponding calculation
are required. The proposed method is more convenient and simpler than the conventional control method because it only requires a trigger signal to change system frequency.
B. Comparison and Discussion
Many excellent works with high misalignment tolerance are
resulted to compare this work, as Table IV. Compared with [17],
This work
5.6%
236kHz and 257kHz
(only two discrete
frequencies)
0.115-0.27
91.46%-95.52%
Medium
[18], and [28]-[30], the output fluctuations are almost the same,
but the total number of the components (coils, switches, inductors, and capacitors) is smaller, and the coupling variation is
larger. Compared with methods [19]-[21] and [31], the component number of the proposed method is much smaller, although
the coupling variation is a little bit lower by 25% with the approximate output fluctuation. Compared with the method [24],
the component counts are almost the same, and the output fluctuation is suppressed by nearly half, although the coupling variation is slightly lower by 15%. Compared with methods [25][27], the proposed method can operate with smaller output fluctuation against larger coupling variation, and the cost of the
components is a little smaller. Compared with the method [32],
the coupling variation is lower by 175%, but the power fluctuation is less than a third of that.
Table IV
SYSTEM PARAMETER VALUES IN EXPERIMENT
Operation
Coupling
Fluctuation
frequency
variation
0.16-0.33
85kHz
5%
(206%)
0.15-0.35
85kHz
5%
(233%)
0.1-0.26
85kHz
5%
(260%)
0.09-0.23
85kHz
5%
(255%)
0.1-0.25
85kHz
5%
(250%)
0.08-0.2
200kHz
11.1%
(250%)
0.16-0.32
140kHz
11.1%
(200%)
0.14-0.28
200kHz
11.1%
(200%)
Ref.
Number of
coil/switch
Numbers of
inductor/capacitor
[17]
4/0
2/6
[18]
4/0
0/6
[19]
4/2
4/8
[20]
4/0
1/6
[21]
4/0
2/6
[24]
2/0
0/2
[25]
2/0
1/3
[26]
2/0
2/3
[27]
2/0
0/3
85kHz
[28]
2/0
1/3
85kHz
[29]
2/0
3/3
140kHz
[30]
2/0
0/3
200kHz
[31]
2/1
1/3
250kHz
[32]
2/1
0/2
250kHz
This
work
2/0
0/2
236kHz
257kHz
0.13-0.17
(131%)
0.2-0.4
(200%)
0.18-0.32
(178%)
0.21-0.355
(168%)
0.1-0.25
(250%)
0.1-0.4
(400%)
0.115-0.27
(235%)
18.9%
6.61%
5.8%
5.78%
5%
17.5%
5.6%
Output characteristic
Constant
power
Constant
power
Constant current/voltage
Constant voltage
Constant voltage
Constant
power
Constant
power
Constant
power
Constant
voltage
Constant voltage
Constant
current
Constant current/voltage
Constant
power
Constant
power
Constant
power
Efficiency
Max.
power
89.2%-91.6%
3.3kW
85%-94%
3.3kW
75.1%-93.9%
1kW
92%-94.5%
3.5kW
88%-93%
3.5kW
66%-73%
70W
88.5%-92.6%
450W
83.5%-87.5%
90W
N/A
(overall efficiency:
94.1%)
200W
61.1%-88.7%
110W
90.4%-96%
10W
86%-94.8%
300W
85.8%-91.7%
400W
87.5%-95.6%
400W
91.46%95.52%
750W
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This article has been accepted for publication in IEEE Transactions on Transportation Electrification. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TTE.2022.3229178
Besides, it should be noticed that the frequency switch will
inevitably intensify the detuning degree of the secondary side
according to (2) and (3), leading to the degradation of efficiency [35]. And the detuned topology can result in much reactive power. Therefore, the proposed method is similar to [24][26], [31], which is more suitable for low-power applications
with a relatively constant load, where steady and sufficient
transfer power is the top priority with large coupling variations[26], such as lighting equipment[31], kitchen appliances[37], etc.
[8]
[9]
[10]
[11]
V. CONCLUSION
In summary, the detuned S-S compensated IPT system can
achieve stable power transfer within a certain coupling range,
which is changed with the frequency variation. This work
analyzes and discusses the monotone features of the critical
coupling kPmax and the maximum output power Pmax versus the
frequency. And then, employing the variation of the extreme
point (kPmax, Pmax), the misalignment character is evaluated, and
the profile of maximum transfer power is depicted with variable
frequency. From the profile of maximum transfer power, there
are two transfer power curves with the same maximum output
power at two different frequencies if the parameters are reasonably designed. By properly choosing the system operation frequency, the two transfer power curves can be linked together so
that the coupling range with desired power is extended. A set of
parameters is designed to verify the feasibility of the proposed
method. Experimental results show that the power fluctuation
of the proposed method is only 5.6%, which is almost consistent
with the theoretical results. Furthermore, the coupling variation
is extended from 174% to 235%, verifying the feasibility of the
proposed method. The system efficiency varies from 91.46% to
95.52% over 235% coupling variation, and ZVS can be
achieved, which illustrates the proposed method has a good
misalignment performance. Additionally, the proposed method
is more suitable for low-power applications with a relatively
constant resistive load. In the future, the case of inductive/capacitive loads and load variation will be considered.
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8.
Zeheng Zhang received the B.S. degree in electrical
engineering and automation from the School of International Energy, Jinan University, Guangzhou, China,
in 2020. He is currently working toward the B.Sc. degree with the School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China.
His research interest includes wireless power transfer.
Shuangjiang He received the B.S. degree in electrical
engineering and automation from the School of Electrical Engineering, Changsha University of Science
and Technology, Changsha, China, in 2019. He is currently working toward the B.Sc. degree with the
School of Tangshan Graduate, Southwest Jiaotong
University, Tangshan, China.
His research interest includes wireless power transfer.
Yuner Peng (Student Member, IEEE) received the
B.S. degree in electrical engineering and automation in
2020 from the School of Electrical Engineering,
Southwest Jiaotong University, Chengdu, China,
where she is currently working toward the Ph.D. degree with the School of Electrical Engineering.
Her research interests includes wireless power
transfer.
Yang Chen (Member, IEEE) received the B.Sc. degree
in electrical engineering and automation and the Ph.D.
degree in electrical engineering from Southwest Jiaotong University, Chengdu, China, in 2015 and 2020, respectively.
From December 2018 to December 2019, he was a
joint Ph.D. student founded by the China Scholarship
Council with the Future Energy Electronics Center,
Virginia Tech, Blacksburg, VA, USA. He is currently
a Postdoctoral Researcher with Southwest Jiaotong
University, Chengdu, China. His research interests include wireless power transfer.
Bin Yang (Student Member, IEEE) received the B.S.
degree in electrical engineering and automation from
the School of Electrical and Automation Engineering,
East China Jiaotong University, Nanchang, China, in
2017. He is currently working toward the Ph.D. degree
with the School of Electrical Engineering, Southwest
Jiaotong University, Chengdu, China.
His research interests include wireless power transfer, especially on misalignment tolerance improvement.
Zhengyou He (Senior Member, IEEE) received the
B.Sc. and M. Sc. degrees in computational mechanics
from Chongqing University, Chongqing, China, in
1992 and 1995, respectively, and the Ph.D. degree
from the School of Electrical Engineering, Southwest
Jiaotong University, Chengdu, China, in 2001.
He is currently a Professor with the School of Electrical Engineering, Southwest Jiaotong University. His
research interests include signal process and infor-
mation theory applied to electrical power system,
and the application of wavelet transforms in
power system.
Yiyang Li received the B.S. degree in electrical engineering and automation from the School of Electrical
and Automation Engineering, East China Jiaotong University, Nanchang, China, in 2020. He is currently
working toward the B.Sc. degree with the School of
Electrical Engineering, Southwest Jiaotong University,
Chengdu, China.
His research interest includes wireless power transfer.
Muikun Mai (Senior Member, IEEE) received the
B.Sc. and Ph.D. degrees in electrical engineering from
the School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China, in 2004 and 2010, respectively.
He is currently a Professor with the School of Electrical Engineering, Southwest Jiaotong University,
Chengdu, China. His research interests include wireless
power transfer and its application in railway systems,
power system stability and control.
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