Inductively Coupled Power Transfer (ICPT) technology allows electrical energy
to be transferred from a stationary primary source to one or more movable secondary
loads over relatively large air gaps (Si et al.,
2008). It is widely used in special fields, such as inflammable and explosive
areas and wet or undersea environment (Li et al.,
2013; Wu et al., 2011). Moreover, household
electric appliances, battery charging and other portable electronic devices
are also beginning to apply it (Sun et al., 2012;
Moradewicz and Kazmierkowski, 2010; Tian
et al., 2012; Boys et al., 2007;
Keeling et al., 2010).
It is well known that one of the major constraints of ICPT systems is the frequency
stability, particularly at soft-switching mode which can largely reduce the
loss and Electromagnetic Interference (EMI) (Hu et al.,
2000). Presently, several thesis have studied the frequency bifurcation
phenomena (Tang et al., 2009, 2011;
Si et al., 2008). It has been pointed out that
the zero phase frequency must be equal to the secondary resonant frequency so
that the maximum output power could be achieved (Wang et
al., 2004). Moreover, it has also been pointed out (Li
et al., 2012) that, once the frequency exceeds a certain range, ICPT
systems will have multiple operating conditions and system stability will be
affected. So, it is crucial to maintain the frequency stability (Boys,
Obviously, the common method of maintaining the frequency stability is to ensure
the reactive elements of the primary part and secondary part to be suitably
tuned (Sun et al., 2005). Although the ICPT
system is completely tuned, frequency bifurcation phenomena are still caused
by parameter variations, particularly small changes in load (Wang
et al., 2004). Current studies which assuming the coupling coefficient
to be constant mainly focus on the bifurcation caused by load variation. The
stable operation boundary condition of the system has been given (Wu
et al., 2004; Wei and Houjun, 2007), which
consists of the secondary quality factor and the coupling coefficient.
However, for the ICPT system, in many applications especially household appliances
in kitchen, the coupling coefficient which is determined by the geometrical
structure and performance parameters of the magnetic structure is not always
to be constant. It will change as the distance between the primary and secondary
side varies (Kurschner et al., 2011).
Few researches have been done on the frequency stability with the coupling
coefficient variation. It is pointed out that the efficient of the magnetically
coupled resonator changes as the coupling coefficient varies and the frequency
splitting phenomena occur (Sample et al., 2011),
however, it does not give the expression for the bifurcation boundary.
This study uses the typical voltage-fed ICPT system as an example. An equivalent
circuit model based on mutual induction model has been presented to investigate
the frequency bifurcation phenomena. The frequency stability with the coupling
coefficient variation has been discussed in detail and the expression for the
bifurcation boundary is derived. Simulations and experiments have been carried
out to verify the theoretical results.
VOLTAGE-FED ICPT SYSTEM TOPOLOGY
The ICPT system can be classified by voltage-fed system and current-fed system.
||Typical voltage-fed ICPT system, Ein: DC input
supply, S1-S4: Metal-oxide-semiconductor field-effect
transistor (MOSFETs), Cp: Primary resonant capacitance, Rp:
Resistance of primary inductor, Lp: Primary resonant inductance,
iLp: Current of the primary inductor, M: Mutual inductance, Ls:
Secondary resonant inductance, Rs: Resistance of secondary inductor,
Cs: Secondary resonant capacitance, iLs: Current of
the secondary inductor, RL: Resistance of the load, UL:
The voltage-fed system normally matches series tuned types of resonant tanks,
while current-fed system matches parallel tuned types of resonant tanks. This
study investigates the frequency bifurcation phenomena of a typical voltage-fed
resonant converter with the variety of the coupling coefficient. The main topology
of voltage-fed ICPT system can be shown in Fig. 1.
In voltage-fed ICPT system, the high-frequency inverter comprises four MOSFET
switches from S1 to S4. Two switch pairs (S1,
S4) and (S2, S3) operate in complementary mode
to invert the input DC voltage source Ein to square wave voltage
output. Hence, the series network consisting of a capacitor Cp in
series with an inductor Lp and a series inherent resistor Rp
will produce a high-frequency sinusoidal current with low distortion. Such current
provide the ZCS conditions for MOSFET switches.
In the secondary part, the pick up coil Ls receives the high-frequency
energy by the magnetic fields coupling. The inductor Ls is completely
tuned by the series capacitor Cs. Such resonant network converts
high-frequency AC voltage supply UL for load RL. Resistors
Rs is inherent resistance of the secondary inductor Ls.
M is the mutual inductance between Lp and Ls.
The equivalent circuit of the coupling topology based on the mutual induction
model is shown in Fig. 2.
|| Equivalent coupling circuit, Vi: Output voltage
of inverter, Zi: The total impedance of the primary part, Zr:
The reflected impedance, Zs: The impedance of the secondary
The impedance of the secondary is calculated as a lumped impedance whose value
depends on the secondary compensation as given by:
The reflected impedance Zr which is dependent on the transformer
coupling and operating frequency is given by:
The total impedance of the primary part is:
Substituting Eq. 1 and 2 into Eq.
3, the impedance seen by the power supply can be given as:
where, the operators Re and Im represent the real and
imaginary components of the input impedance:
As the aim of designing the ICPT system is to deliver the power to the load,
the resonant frequency of the secondary part should be chosen as the operating
frequency of the system. The operating angular frequency ω0
of the system is given by:
where, the resonant angular frequency ω0 is defined as ω0
= 2π f0 and the f0 is the operating frequency of
the resonant circuit.
In addition, in order to analyze the bifurcation phenomena, the expression
of the coupling coefficient k is given by:
where, the M is the mutual inductance between primary coil and secondary coil.
Here, the operating frequency (ω) is normalized using (ω0)
from Eq. 7 as:
When the whole resonant network is completely tuned, the primary impedance
Substituting Eq. 6-9 into Eq.
10, the imaginary components of the primary impedance is derived as:
Normally, if Eq. 11 has only one root (γ = 1), the
system has only one zero phase angle resonant frequency. Apart from γ =
1, the Eq. 11 has the other sub-equation which is given as:
And the discriminant of Eq.12 is given by:
As is known, the ideal root of the Eq. 11 is γ = 1,
to ensure the secondary resonant frequency is the only zero phase angle frequency,
the Eq.12 must have no root (Δ<0) or the roots are
|| Relationship between Γ and the load RL
At first, letting the discriminant be zero and the coupling coefficient is
Then, the Γ is define as shown below and the relationship between Γ
and the load RL is shown in Fig. 3.
Finally, the bifurcate on boundary of the coupling coefficient k are discussed
in three circumstances:
||It can be seen in Fig. 3 that when the load
then Γ∈ (0, 1), in order to ensure the discriminant less than
zero (Δ<0), the coupling coefficient k should be
||Similarly, when the load RL satisfy
then Γ∈(0, 1], in order to ensure the discriminant less than zero
(Δ<0), the coupling coefficient k should be .
However, the roots are invalid (Δ≥0) when the coupling coefficient
k satisfy .
Therefore, the coupling coefficient should be k∈ [0, 1]
||When the load RL satisfy RL≥ (2ω0Ls-Rs)
then Γ∈ (-∞, 0), all the roots are invalid (Δ≥0),
thus the coupling coefficient should be k∈[0, 1]
In a word, to ensure the secondary resonant frequency is the only zero phase
angle frequency, the boundary of the coupling coefficient k is given as:
The frequency bifurcation region is:
In order to validate the bifurcation boundary of the system with the changes
of the coupling coefficient, numerical verification with MATLAB has been carried
On the basis of the theory analysis discussed above, it can easily be realized
the bifurcation boundary according to the parameter shown in Table
It can be seen from Fig. 4a that the imaginary of the primary
impedance has more than one zero-crossing point with increasing the coupling
coefficient (k) at the load RL1 = 2Ω, whereas the Fig.
4b shows that the imaginary of the primary impedance has only one zero-crossing
point over a range of the coupling coefficient (k) from 0 to 1 at the load RL2
= 12Ω. The key parameters can be calculated as
and Γ = 0.076:
||According to Eq. 16, the bifurcation boundary
of the coupling coefficient k can be calculated as 0≤k≤0.2758 at the
load RL1 = 2Ω
||According to Eq. 17, the bifurcation boundary of the
coupling coefficient k can be calculated 0≤k≤1 at the load RL2
On the basis of the conclusion discussed above, the imaginary of the primary
impedance with different coupling coefficient is shown in Fig.
5 in the form of a two-dimensional graphic representation.
|| Imaginary 3-D plot primary impedance with different coupling
coefficient k (a) RL1 = 2Ω and (b) RL2 = 12Ω
|| Imaginary sketch of the primary impedance for different k
(a) RL1 = 2Ω, fmin: Lower zero phase angle resonant
frequency, f0: Nominal zero phase angle resonant frequency, fmax:
Higher zero phase angle resonant frequency, kc: Critical coupling
coefficient and (b) RL2 = 12Ω
It can be seen from Fig. 5a that there is only one operating
frequency f0 which is the ideal operating point of the system when
the coupling coefficient k is >0.2758. Once the k is slightly larger than
the critical value kc (kc = 0.2758) there are three zero
phase angle resonant frequencies fmin, f0, fmax
in the primary resonant network, where fmin<f0<fmax.
Fig. 5b shows that there is only one operating frequency f0
(f0 = 20 kHz) at the load RL2 = 12Ω (RL2>10.665Ω)
with the varying coupling coefficient (k).
To verify the bifurcation phenomena of ICPT system varying with coupling coefficient,
a voltage-fed ICPT system shown in Fig. 1 has been implemented
in a laboratory scale. The inverter consists of four MOSFETs (International
Rectifier IRFB4310).The zero-crossing points of primary inductance current are
detected by the current sense transformers (Talema AS-103). Phase-locked loop
(CD4046) is selected to achieve the ZCS operation for the primary converter
with variable frequency. The parameters of the circuit are as shown in Table
1. And the experimental results of the frequency of the voltage-fed ICPT
system are shown below.
The experimental waveforms show that the system operate at ZCS condition for
the input inverter voltage vi is in phase with iLp, which
proves that the these frequency are the zero phase frequency of the primary
part of the system.
As discussed earlier, the bifurcation boundary of the coupling coefficient
is 0≤k≤0.2758 at the load RL1 = 2Ω. It can be seen from
Fig. 6a that there exactly is the only one zero phase frequency
f0 (f0 = 19.8 kHz) at the coupling coefficient k = 0.2;
as the coefficient k varies from 0.1-0.4, it can be seen in Fig.
6b-d that there are three zero phase frequency of the
primary current (17.8, 19.8 and 24.4 kHz, respectively).
|| Parameters of ICPT system
|| Steady-state waveforms of voltage-fed ICPT system for RL1
= 2Ω (CH1: Output voltage of inverter Vi-10V/div; CH2: Current
of the primary inductor iLp-2A/div) (a) f0 = 19.8
kHz, k = 0.2, (b) fmin = 17.8 kHz, k = 0.4, (c) f0
= 19.8 kHz, k = 0.4 and (d) fmax = 24.4 kHz, k = 0.4
|| Steady-state waveforms of voltage-fed ICPT system for RL2
= 12Ω (CH1: Output voltage of inverter Vi -10V/div; CH2:
Current of the primary inductor iLp-2A/div) (a) f0
= 19.8 kHz, k = 0.2, (b) f0 = 19.8 kHz, k = 0.5 and (c) f0
= 19.8 kHz, k = 0.8
|| Practical results of the frequency of the voltage-fed ICPT
system varying with coupling coefficient and ideal theoretical results (a)
RL1 = 2Ω and (b) RL2 = 12Ω
However, at the light load (RL2 = 12Ω) as shown in Fig.
7a-c, there is also exactly the only one zero phase frequency
f0 (f0 = 19.8 kHz) over a range of coefficient k from
As a comparison, a theoretical curve based on ideal component parameters is
also plotted in Fig. 8a-b.
As shown in Fig. 8, the measured results are in good agreement
with the theoretical curves. It also can be seen that the experimental curve
deviates by about 0.2 kHz from the theoretical value. This is caused by deviations
in actual component values from the nominal values. The theoretical results
are based on the simplifications in the model that assumes sinusoidal voltages
and currents and ignores losses in the capacitors and electromagnetic structure.
Therefore, this error in practice doesnt effect the analyzed results above
Aimed at the frequency stability of the voltage-fed ICPT system with the coupling
coefficient variation into consideration, the expression of the frequency bifurcation
boundary has been analyzed and given based on mutual induction model. It is
found that the bifurcation region of the zero phase resonant frequency is determined
by the value range of the coupling coefficient if the load satisfies certain
condition. Once the coupling coefficient exceeds a certain range, the system
will have more than one zero phase frequency and the stability of the system
will be affected a lot. Simulations and experiments have been carried out to
validate the theoretical results and it could be useful for guiding the design
of the ICPT system especially in applications which have the requirement of
the position flexibility.
This study is financially supported by National Natural Science Foundation
of China (No.51277192, No.51207173) and Fundamental Research Funds for the Central
Universities (No. CDJXS10170001). I also would like to give my special thanks
to the anonymous reviewers for their contributions to this study.