# A Frequency-Tracking and Impedance-Matching Combined System for Robust Wireless Power Transfer

Authors: Yanting Luo; Yongmin Yang; Suiyu Chen; Xisen Wen
Journal: International Journal of Antennas and Propagation
 (2017)

---

## Abstract

One of the greatest challenges to power embedded devices using magnetically coupled resonant wireless power transfer (WPT) system is that the amount of power delivered to the load is very sensitive to load impedance variations. Previous adaptive impedance-matching (IM) technologies have drawbacks because adding IM networks, relay coils, or other compensating components in the receiver-side will significantly increase the receiver size. In this paper, a novel frequency-tracking and impedance-matching combined system is proposed to improve the robustness of wireless power transfer for embedded devices. The characteristics of the improved WPT system are investigated theoretically based on the two-port network model. Simulation and experimental studies are carried out to validate the proposed system. The results suggest that the frequency-tracking and impedance-matching combined WPT system can quickly find the best matching points and maintain high power transmission efficiency and output power when the load impedance changes.


---

## Body

## 1. Introduction

Magnetically coupled resonant wireless power transfer (WPT) is one of the most potential technologies to solve the energy problems for embedded devices. The built-in battery is not a desired power source because it always limits system size and its replacement needs disassembling the device for most embedded applications [1–4]. By making use of WPT technology, embedded devices can be powered effectively without changing batteries or increasing battery size, which contribute to realizing a compact-sized, long-life embedded system.One of the greatest challenges to power embedded devices using WPT technology is that the amount of power delivered to the load is very sensitive to load impedance variations [5–9]. For example, when an ultracapacitor is being charged, the load can be seen as an increasing instantaneous impedance. However, the ultracapacitor only receives the maximum power at a single optimal impedance [6]. At other impedances, the power delivered to the ultracapacitor decreases rapidly due to impedance mismatch. Moreover, in some applications, there is a need to power multiple embedded devices using one power transmitter [7–9]. Therefore, the total load impedance depends on how many devices are being powered simultaneously. But the output power of the WPT system only reaches the maximum value at an optimal load impedance as well. For these reasons, it is important to develop adaptive impedance-matching (IM) technology to quickly find the best impedance points [10].Many works have been done on the adaptive impedance-matching technology for the WPT system, including the following: (1) adjusting the source operating frequency until the output power of the WPT system is maximized [11–14], (2) adding variable impedance-matching networks in the WPT system to transform the input and output impedance to optimal values [15–18], (3) optimizing the mutual inductance between coils using multicoil topology [19–23], and (4) using a receiver-side switching converter to transform the actual load resistance into optimal load resistance [5, 24–26].For many WPT systems, the operating frequency is fixed or limited to a very narrow band. Meanwhile, in some cases, because the operating frequency may not be optimal depending on factors such as load variations, the frequency-tuning approach has been adopted and shown its advantages [11–14]. In [13], the source operating frequency is adjusted to compensate for changes in the input impedance of the WPT system until the imaginary part of the input impedance becomes zero. Their experimental results show that the output power of the WPT system can be improved by optimizing the operating frequency. This method is simple to implement but cannot work well in weakly coupled WPT systems.In [18], two variable IM networks are inserted in the transmitter circuit and the receiver circuit, respectively. By doing so, both the transmitter circuit and the receiver circuit can be adjusted to satisfy conjugate impedance-matching conditions through feedback control. Although this approach can achieve higher output power than frequency-tuning, it needs additional IM networks and control components in both the transmitter circuit and the receiver circuit, which will inevitably increase the system size.In [20], a WPT system topology which includes an excitation-coil, a transmitter coil, a receiver coil, and a pickup-coil is proposed for automatic impedance-matching purpose. An actuator is used to change the relative position of coils to match the impedance of the WPT system. However, using four-coil topology will also increase the system complexity and energy consumption.In [24], a novel GaN Buck converter is used to emulate a variable load resistor to adjust the WPT system for maximum efficiency or received power, thereby eliminating the requirements of using IM networks. In [5], a switching converter is added in the receiver-side to automatically transform the load resistance into a desired value. Experimental results demonstrate that the WPT system can obtain high efficiency under load resistance variations.In order to ensure a compact embedded system, the receiver should be designed as small as possible [27–29]. Thus, previous adaptive impedance-matching approaches have drawbacks because adding IM networks, relay coils, or switching converters with their control circuits in the receiver-side will significantly increase the system size. In this paper, a novel frequency-tracking and impedance-matching combined system is proposed to improve the robustness of wireless power transfer for embedded devices. To quickly find the best matching points, the source operating frequency is automatically optimized at first, and then a variable IM network in the transmitter circuit is used to match the system’s input impedance to the source impedance. Through this approach, both the power transmission efficiency and the output power of the WPT system can be significantly improved. The proposed system is validated through numerical simulations and experimental measurements.The rest of the paper is organized as follows. Section2 describes the traditional dual-side conjugated IM principle for the WPT system. Section 3 presents the frequency-tracking and impedance-matching combined system. The characteristics of the improved WPT system are analyzed based on the two-port network theory and the effects of load impedance change on the power transmission efficiency and output power are investigated through numerical simulations. Experimental measurements are evaluated in Section 4. As a comparison, the calculated and measured results of the original WPT system, the dual-side impedance-matching WPT system, and the frequency-tracking and impedance-matching combined WPT system are provided as well. Finally, conclusions are drawn in Section 5.


## 2. Traditional Dual-Side IM Principle

Figure1 shows the circuit model of a series-compensated WPT system, which consists of a power source, a coupled resonators-circuit, and a load. U
S, ω, and Z
S are the voltage, angular frequency, and internal impedance of the power source, respectively. L
T and L
R are the inductances of the transmitter coil and the receiver coil, respectively. R
T and R
R are the equivalent resistances of the transmitter coil and the receiver coil, respectively. M is the mutual inductance between coils. In order to achieve resonance, two lumped capacitors, C
T and C
R, are connected to the transmitter coil and the receiver coil, respectively. Z
i
n and Z
o
u
t are the input impedance and output impedance of the resonators-circuit, respectively. Z
L is the load impedance.Figure 1
Circuit model of a series-compensated WPT system.To simplify the analysis, the circuit model shown in Figure1 is seen as a two-port network. According to the maximum power transfer theorem [14, 30], if the two-port network satisfies the dual-side conjugate impedance-matching equation(1)
Z
i
n
=
Z
S
∗
,
Z
o
u
t
=
Z
L
∗the power delivered to the load will be maximized, where Z
S
∗ is the conjugate value of Z
S and Z
L
∗ is the conjugate value of Z
L.For a constant load impedanceZ
L, (1) can be easily satisfied by optimizing the parameters of the WPT system. However, if Z
L changes, the dual-side conjugate impedance-matching equation can only be satisfied at an optimal load impedance. At other load impedances, the output power of the WPT system will decrease rapidly due to impedance mismatching.In a practical WPT system, the load impedance may change due to many factors, like load temperature rise, operating mode switching, and so on [5–9]. The output power of the WPT system will become unstable if the load impedance changes and this will cause serious problems in most engineering applications. In order to maintain the maximum output power, the dual-side impedance-matching system is proposed in [14, 18]. In their systems, two controllable IM networks are added in the transmitter circuit and the receiver circuit, respectively, as shown in Figure 2. Thus, the input and output impedance of the WPT system can be automatically adjusted to satisfy (1) through scattering-parameters or reflected coefficient feedback control. Then, the output power of the WPT system can maintain the maximum value when the load impedance changes.Figure 2
Traditional dual-side impedance-matching system.However, the dual-side IM approach has some drawbacks, especially when it is used in the WPT system for embedded applications. Since spaces in most embedded devices are not available to accommodate too many circuits and components, the size of the power receiver should be as small as possible [27–29]. The dual-side IM approach needs additional variable IM networks and control components in the receiver circuit, which will obviously increase the receiver size and weight. Moreover, more IM components in the receiver means more energy consumption and less power delivered to the load. To overcome these drawbacks, a frequency-tracking and impedance-matching combined system is proposed in the next section.


## 3. The Proposed Frequency-Tracking and Impedance-Matching Combined WPT System

### 3.1. Modeling and Optimization

The proposed frequency-tracking and impedance-matching combined WPT system is shown in Figure3. It includes a frequency-adjustable power source, an IM network in the transmitter circuit, which contains two variable components, a coupled resonators-circuit, and a load. In the WPT system, the variable IM network is used to match the transmitter’s impedance to the source impedance through feedback control. The source operating frequency is adjusted to change the receiver’s impedance instead of using another variable IM network, which can simplify the structure of the receiver circuit.Figure 3
Proposed frequency-tracking and impedance-matching combined WPT system.The circuit model shown in Figure3 can be looked at as a cascade connection of the power source, the variable IM network, the coupled resonators-circuit, and the load. For simplicity, the characteristics of them are analyzed separately.


#### 3.1.1. Coupled Resonators-Circuit

Based on the two-port network theory [31–33], the transmission matrix equation can be used to describe the input and output characteristics of the resonators-circuit, as (2)
U
1
I
1
=
A
B
C
D
U
2
I
2
,where U
1 and I
1 are the input voltage and current at port 1, respectively. U
2 and I
2 are the input voltage and current at port 2, respectively. A, B, C, and D are the transmission parameters of the coupled resonators-circuit. For a series-compensated wireless power transfer system, A, B, C, and D can be calculated as follows: A
=
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
/
j
ω
M, B
=
[
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
+
ω
2
M
2
]
/
j
ω
M, C
=
1
/
j
ω
M, and D
=
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
/
j
ω
M.From the circuit model shown in Figure3, the input impedance at port 1 can be found:(3)
Z
i
n
=
U
1
I
1
=
A
Z
L
+
B
C
Z
L
+
D
=
R
i
n
+
j
X
i
n
,where R
i
n and X
i
n represent the real part and the imaginary part of Z
i
n, respectively. Substituting A, B, C, and D parameters into (3) yields(4)
R
i
n
=
R
T
R
R
+
R
L
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
+
R
R
ω
2
M
2
+
R
L
ω
2
M
2
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
,
X
i
n
=
ω
L
T
-
1
/
ω
C
T
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
-
ω
2
M
2
X
L
+
ω
L
R
-
1
/
ω
C
R
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the input active power at port 1 can be calculated by(5)
P
i
n
=
I
1
2
R
i
n
.Because the load in a practical system may not be a pure resistance [34, 35], the load impedance Z
L is assumed to be R
L
+
j
X
L, where R
L and X
L represent the resistance and reactance of the load, respectively. Then, the output active power at port 2 can be calculated by(6)
P
o
u
t
=
I
2
2
R
L
.In previous studies, two kinds of power transfer efficiencies are defined to analyze the performance of the WPT system [36–38]. The system efficiency η
S refers to the ratio of the output power P
o
u
t at port 2 and total input power P
S from the wall outlet. It considers both the power losses in the AC power source and the resonators-circuit. Therefore, the system efficiency η
S cannot exceed 50% if the conjugate impedance-matching theorem is adopted because the source internal resistance will consume a half of the total input power P
S. The power transmission efficiency η
T is the ratio of the output power P
o
u
t at port 2 and the input power at port 1 P
i
n. It only includes the power losses in the resonators-circuit. In this paper, because the goal is not to improve the system power efficiency, the power transmission efficiency η
T is used to characterize the WPT system. η
T can be calculated using (7)
η
T
=
P
o
u
t
P
i
n
.By substituting (5) and (6) into (7), the power transmission efficiency of the resonators-circuit can be found:(8)
η
T
=
ω
2
M
2
R
L
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.


#### 3.1.2. IM Network

As shown in Figure3, the IM network includes two compensating components. X
1 and X
2 are the values of them. The transmission matrix equation of the IM network can be expressed as(9)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
U
2
′
I
2
′
,where U
1
′ and I
1
′ are the input voltage and current at port 1′, respectively. U
2
′ and I
2
′ are the input voltage and current at port 2′, respectively. A
′, B
′, C
′, and D
′ are the transmission parameters of the IM network. Based on the structure of the IM network, A
′, B
′, C
′, and D
′ can be calculated as follows: A
′
=
1, B
′
=
j
X
2, C
′
=
1
/
j
X
1, and D
′
=
X
1
+
X
2
/
X
1.Then, the input impedance at port 1′ can be found:(10)
Z
i
n
′
=
U
1
′
I
1
′
=
A
′
Z
i
n
+
B
′
C
′
Z
i
n
+
D
′
=
R
i
n
′
+
j
X
i
n
′
,where R
i
n
′ and X
i
n
′ represent the real part and the imaginary part of Z
i
n
′, respectively. Substituting A
′, B
′, C
′, and D
′ parameters into (10) yields(11)
R
i
n
′
=
-
X
1
X
2
R
i
n
-
X
1
X
i
n
R
i
n
+
X
1
+
X
2
+
X
i
n
X
1
R
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
,
X
i
n
′
=
X
1
R
i
n
2
+
X
1
+
X
2
+
X
i
n
X
1
X
2
+
X
1
+
X
2
+
X
i
n
X
1
X
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
.In this paper, the ratio of the output active powerP
o
u
t
′ at port 2′ and input active power P
i
n
′ at port 1′ is defined as the power transmission efficiency of the IM network. It can be calculated by(12)
η
I
M
=
P
o
u
t
′
P
i
n
′
,where P
i
n
′
=
I
1
′
2
R
i
n
′ and P
o
u
t
′
=
I
2
′
2
R
i
n.Substituting (11) into (12) yields η
I
M
=
100
%. This is because the compensating components are assumed to be ideal energy-storage components without any energy losses. But, in fact, X
1 and X
2 also consume part of the total energy in the forms of heat and radiation [39, 40]. Therefore, the power transmission efficiency of the IM network cannot reach 100% in a practical system. To simplify the analysis, η
I
M is assumed to be 100
% in the theoretical analysis.


#### 3.1.3. Overall WPT System

By substitutingU
2
′
=
U
1 and I
2
′
=
-
I
1 into (2), the transmission matrix equation of the overall WPT system can be found:(13)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
A
B
C
D
U
2
I
2
.Thus, the power transmission efficiency of the overall WPT system can be calculated as(14)
η
t
o
t
a
l
=
P
o
u
t
P
i
n
′
=
P
o
u
t
′
P
i
n
′
×
P
o
u
t
P
i
n
=
η
I
M
η
T
.Substituting (7) and (12) into (14) yields(15)
η
t
o
t
a
l
=
ω
2
M
2
R
L
η
I
M
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the output active powerP
o
u
t of the overall WPT system can be expressed as(16)
P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l
.From here it can be seen that the output powerP
o
u
t is the function of the input power P
i
n
′ and the power transmission efficiency η
t
o
t
a
l. To maximize the output power of the WPT system, both P
i
n
′ and η
t
o
t
a
l should be improved.


#### 3.1.4. Efficiency and Output Power Optimization

According to (15), the operating frequency can be adjusted to increase the power transmission efficiency η
t
o
t
a
l. By solving “d
η
t
o
t
a
l
/
d
f
=
0,” the optimal frequencies to maximize η
t
o
t
a
l can be found:(17)
f
o
p
t
=
C
R
+
C
L
π
4
L
R
C
R
C
L
2
+
4
L
R
C
R
2
C
L
-
2
C
R
2
C
L
2
R
R
+
R
L
2
,
for Capacitive Load,
1
π
4
L
R
C
R
+
4
L
L
C
R
-
2
C
R
2
R
R
+
R
L
2
,
for Inductive Load,where C
L
=
-
1
/
ω
X
L represents the equivalent capacitance of a capacitive load and L
L
=
X
L
/
ω represents the equivalent inductance of inductive load. The operating frequency is f
=
ω
/
2
π.To maximize the input active powerP
i
n
′, the IM network is used to conjugately match the input impedance Z
i
n
′ to the source impedance Z
S. Then the output power of the overall WPT system can be maximized because P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l. By introducing (10) into Z
i
n
′
=
Z
S
∗, the optimal values of X
1 and X
2 can be obtained:(18)
X
o
p
t
,
1
=
±
R
S
R
i
n
R
S
R
i
n
-
R
i
n
2
,
X
o
p
t
,
2
=
-
X
i
n
±
R
S
R
i
n
-
R
i
n
2
.The compensating componentsX
1 and X
2 can be realized using variable capacitors/inductances. The optimal values of the capacitors/inductances can be calculated by(19)
L
o
p
t
,
i
=
X
o
p
t
,
i
2
π
f
o
p
t
if
X
o
p
t
,
i
>
0
,
C
o
p
t
,
i
=
-
1
2
π
f
o
p
t
X
o
p
t
,
i
if
X
o
p
t
,
i
<
0
,
i
=
1,2
.According to the analysis, when the load impedance changes, the power transmission efficiency and the output power of the WPT system can be optimized following these steps: (1) adjusting the source operating frequencyf to the optimal frequency f
o
p
t at first and (2) adjusting the variable components, X
1 and X
2, to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


### 3.2. Numerical Simulations

In this section, the impacts of load impedance change on the power transmission efficiencyη
t
o
t
a
l and output power P
o
u
t are investigated through numerical simulations. Two different situations, the load reactance change and the load resistance change, are considered separately. The parameters used for calculations are given in Table 1.Table 1
Parameters used for calculations.
Parameters
Values
Source resistanceR
S
50 Ω
Lumped capacitorsC
T, C
R
216 pF
Coil resistanceR
T, R
R
2 Ω
Coil inductanceL
T, L
R
29 uH
Coil frequencyf
T, f
R
2 MHz
Coupling coefficientk
0.13
Load resistanceR
L
10~500 Ω
Load reactanceX
L
−500~500 Ω


#### 3.2.1. Power Transmission Efficiency

When the load reactanceX
L increases from −500 Ω to 500 Ω, the power transmission efficiency of the WPT system can be obtained using (15). Figure 4(a) shows the calculated results of power transmission efficiency η
t
o
t
a
l versus load reactance X
L and source operating frequency f. When f is fixed at 2 MHz, it can be seen from Figure 4(a) that η
t
o
t
a
l increases firstly and then decreases with the increase of X
L, and the maximum power transmission efficiency can only be achieved at a single load reactance X
L
=
0
Ω.Figure 4
(a) Calculatedη
t
o
t
a
l versus X
L and f. (b) Calculated η
t
o
t
a
l versus R
L and f.
(a)
(b)Figure4(b) shows the calculated results of η
t
o
t
a
l versus load resistance R
L and source operating frequency f, where the load resistance R
L increases from 10 Ω to 500 Ω. From Figure 4(b), the similar trend can be found: when f is fixed at 2 MHz, η
t
o
t
a
l increases firstly and then decreases with the increase of R
L, and the maximum power transmission efficiency can only be achieved at an optimal load resistance R
L
=
50
Ω.Based on the theoretical analysis, the source operating frequency is adjusted to improve the power transmission efficiency firstly when the load impedance changes. The optimal frequenciesf
o
p
t for maximizing η
t
o
t
a
l can be calculated using (17). They are marked with blue dashed line in Figure 4. It can be observed that the WPT system can maintain high efficiency when the source frequency is adjusted along the optimal frequency line.


#### 3.2.2. Output Power

SettingX
L
=
-
500
,
-
250
,
250
,
500
Ω and R
L
=
100
,
200,300,400
Ω, respectively, yields corresponding optimal frequencies f
o
p
t
≈
3.11, 2.64, 1.57, and 1.32 MHz and f
o
p
t
≈
2.05, 2.18, 2.47, and 3.19 MHz. To increase the output power of the WPT system, the variable components X
1 and X
2 are used to match the impedance of the WPT system at the optimal frequencies.Figures5 and 6 show the calculated results of output power P
o
u
t versus X
1 and X
2. The best matching points (X
o
p
t
,
1, X
o
p
t
,
2) for maximizing P
o
u
t can be found using (18). They are marked with black points in Figures 5 and 6. It can be seen that P
o
u
t will reach the maximum value if X
1 and X
2 are adjusted to the best matching points.Figure 5
CalculatedP
o
u
t versus X
1 and X
2. The load resistance R
L
=
100
Ω.
(a)
X
L
=
-
500
Ω and f
o
p
t
=
3.11
M
H
z
(b)
X
L
=
-
250
Ω and f
o
p
t
=
2.64
M
H
z
(c)
X
L
=
250
Ω and f
o
p
t
=
1.57
M
H
z
(d)
X
L
=
500
Ω and f
o
p
t
=
1.32
M
H
zFigure 6
CalculatedP
L versus X
1 and X
2. The load reactance X
L
=
0
Ω.
(a)
R
L
=
100
Ω and f
o
p
t
=
2.05
M
H
z
(b)
R
L
=
200
Ω and f
o
p
t
=
2.18
M
H
z
(c)
R
L
=
300
Ω and f
o
p
t
=
2.47
M
H
z
(d)
R
L
=
400
Ω and f
o
p
t
=
3.19
M
H
zFrom Figures5 and 6, it can be seen that the numerical simulation results agree with the theoretical analysis well. The results suggest that the power transmission efficiency of the WPT system can be significantly improved by changing the source operating frequency f to the optimal frequency f
o
p
t, and the output power can achieve high value by adjusting the IM network to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


## 3.1. Modeling and Optimization

The proposed frequency-tracking and impedance-matching combined WPT system is shown in Figure3. It includes a frequency-adjustable power source, an IM network in the transmitter circuit, which contains two variable components, a coupled resonators-circuit, and a load. In the WPT system, the variable IM network is used to match the transmitter’s impedance to the source impedance through feedback control. The source operating frequency is adjusted to change the receiver’s impedance instead of using another variable IM network, which can simplify the structure of the receiver circuit.Figure 3
Proposed frequency-tracking and impedance-matching combined WPT system.The circuit model shown in Figure3 can be looked at as a cascade connection of the power source, the variable IM network, the coupled resonators-circuit, and the load. For simplicity, the characteristics of them are analyzed separately.


### 3.1.1. Coupled Resonators-Circuit

Based on the two-port network theory [31–33], the transmission matrix equation can be used to describe the input and output characteristics of the resonators-circuit, as (2)
U
1
I
1
=
A
B
C
D
U
2
I
2
,where U
1 and I
1 are the input voltage and current at port 1, respectively. U
2 and I
2 are the input voltage and current at port 2, respectively. A, B, C, and D are the transmission parameters of the coupled resonators-circuit. For a series-compensated wireless power transfer system, A, B, C, and D can be calculated as follows: A
=
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
/
j
ω
M, B
=
[
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
+
ω
2
M
2
]
/
j
ω
M, C
=
1
/
j
ω
M, and D
=
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
/
j
ω
M.From the circuit model shown in Figure3, the input impedance at port 1 can be found:(3)
Z
i
n
=
U
1
I
1
=
A
Z
L
+
B
C
Z
L
+
D
=
R
i
n
+
j
X
i
n
,where R
i
n and X
i
n represent the real part and the imaginary part of Z
i
n, respectively. Substituting A, B, C, and D parameters into (3) yields(4)
R
i
n
=
R
T
R
R
+
R
L
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
+
R
R
ω
2
M
2
+
R
L
ω
2
M
2
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
,
X
i
n
=
ω
L
T
-
1
/
ω
C
T
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
-
ω
2
M
2
X
L
+
ω
L
R
-
1
/
ω
C
R
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the input active power at port 1 can be calculated by(5)
P
i
n
=
I
1
2
R
i
n
.Because the load in a practical system may not be a pure resistance [34, 35], the load impedance Z
L is assumed to be R
L
+
j
X
L, where R
L and X
L represent the resistance and reactance of the load, respectively. Then, the output active power at port 2 can be calculated by(6)
P
o
u
t
=
I
2
2
R
L
.In previous studies, two kinds of power transfer efficiencies are defined to analyze the performance of the WPT system [36–38]. The system efficiency η
S refers to the ratio of the output power P
o
u
t at port 2 and total input power P
S from the wall outlet. It considers both the power losses in the AC power source and the resonators-circuit. Therefore, the system efficiency η
S cannot exceed 50% if the conjugate impedance-matching theorem is adopted because the source internal resistance will consume a half of the total input power P
S. The power transmission efficiency η
T is the ratio of the output power P
o
u
t at port 2 and the input power at port 1 P
i
n. It only includes the power losses in the resonators-circuit. In this paper, because the goal is not to improve the system power efficiency, the power transmission efficiency η
T is used to characterize the WPT system. η
T can be calculated using (7)
η
T
=
P
o
u
t
P
i
n
.By substituting (5) and (6) into (7), the power transmission efficiency of the resonators-circuit can be found:(8)
η
T
=
ω
2
M
2
R
L
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.


### 3.1.2. IM Network

As shown in Figure3, the IM network includes two compensating components. X
1 and X
2 are the values of them. The transmission matrix equation of the IM network can be expressed as(9)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
U
2
′
I
2
′
,where U
1
′ and I
1
′ are the input voltage and current at port 1′, respectively. U
2
′ and I
2
′ are the input voltage and current at port 2′, respectively. A
′, B
′, C
′, and D
′ are the transmission parameters of the IM network. Based on the structure of the IM network, A
′, B
′, C
′, and D
′ can be calculated as follows: A
′
=
1, B
′
=
j
X
2, C
′
=
1
/
j
X
1, and D
′
=
X
1
+
X
2
/
X
1.Then, the input impedance at port 1′ can be found:(10)
Z
i
n
′
=
U
1
′
I
1
′
=
A
′
Z
i
n
+
B
′
C
′
Z
i
n
+
D
′
=
R
i
n
′
+
j
X
i
n
′
,where R
i
n
′ and X
i
n
′ represent the real part and the imaginary part of Z
i
n
′, respectively. Substituting A
′, B
′, C
′, and D
′ parameters into (10) yields(11)
R
i
n
′
=
-
X
1
X
2
R
i
n
-
X
1
X
i
n
R
i
n
+
X
1
+
X
2
+
X
i
n
X
1
R
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
,
X
i
n
′
=
X
1
R
i
n
2
+
X
1
+
X
2
+
X
i
n
X
1
X
2
+
X
1
+
X
2
+
X
i
n
X
1
X
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
.In this paper, the ratio of the output active powerP
o
u
t
′ at port 2′ and input active power P
i
n
′ at port 1′ is defined as the power transmission efficiency of the IM network. It can be calculated by(12)
η
I
M
=
P
o
u
t
′
P
i
n
′
,where P
i
n
′
=
I
1
′
2
R
i
n
′ and P
o
u
t
′
=
I
2
′
2
R
i
n.Substituting (11) into (12) yields η
I
M
=
100
%. This is because the compensating components are assumed to be ideal energy-storage components without any energy losses. But, in fact, X
1 and X
2 also consume part of the total energy in the forms of heat and radiation [39, 40]. Therefore, the power transmission efficiency of the IM network cannot reach 100% in a practical system. To simplify the analysis, η
I
M is assumed to be 100
% in the theoretical analysis.


### 3.1.3. Overall WPT System

By substitutingU
2
′
=
U
1 and I
2
′
=
-
I
1 into (2), the transmission matrix equation of the overall WPT system can be found:(13)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
A
B
C
D
U
2
I
2
.Thus, the power transmission efficiency of the overall WPT system can be calculated as(14)
η
t
o
t
a
l
=
P
o
u
t
P
i
n
′
=
P
o
u
t
′
P
i
n
′
×
P
o
u
t
P
i
n
=
η
I
M
η
T
.Substituting (7) and (12) into (14) yields(15)
η
t
o
t
a
l
=
ω
2
M
2
R
L
η
I
M
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the output active powerP
o
u
t of the overall WPT system can be expressed as(16)
P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l
.From here it can be seen that the output powerP
o
u
t is the function of the input power P
i
n
′ and the power transmission efficiency η
t
o
t
a
l. To maximize the output power of the WPT system, both P
i
n
′ and η
t
o
t
a
l should be improved.


### 3.1.4. Efficiency and Output Power Optimization

According to (15), the operating frequency can be adjusted to increase the power transmission efficiency η
t
o
t
a
l. By solving “d
η
t
o
t
a
l
/
d
f
=
0,” the optimal frequencies to maximize η
t
o
t
a
l can be found:(17)
f
o
p
t
=
C
R
+
C
L
π
4
L
R
C
R
C
L
2
+
4
L
R
C
R
2
C
L
-
2
C
R
2
C
L
2
R
R
+
R
L
2
,
for Capacitive Load,
1
π
4
L
R
C
R
+
4
L
L
C
R
-
2
C
R
2
R
R
+
R
L
2
,
for Inductive Load,where C
L
=
-
1
/
ω
X
L represents the equivalent capacitance of a capacitive load and L
L
=
X
L
/
ω represents the equivalent inductance of inductive load. The operating frequency is f
=
ω
/
2
π.To maximize the input active powerP
i
n
′, the IM network is used to conjugately match the input impedance Z
i
n
′ to the source impedance Z
S. Then the output power of the overall WPT system can be maximized because P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l. By introducing (10) into Z
i
n
′
=
Z
S
∗, the optimal values of X
1 and X
2 can be obtained:(18)
X
o
p
t
,
1
=
±
R
S
R
i
n
R
S
R
i
n
-
R
i
n
2
,
X
o
p
t
,
2
=
-
X
i
n
±
R
S
R
i
n
-
R
i
n
2
.The compensating componentsX
1 and X
2 can be realized using variable capacitors/inductances. The optimal values of the capacitors/inductances can be calculated by(19)
L
o
p
t
,
i
=
X
o
p
t
,
i
2
π
f
o
p
t
if
X
o
p
t
,
i
>
0
,
C
o
p
t
,
i
=
-
1
2
π
f
o
p
t
X
o
p
t
,
i
if
X
o
p
t
,
i
<
0
,
i
=
1,2
.According to the analysis, when the load impedance changes, the power transmission efficiency and the output power of the WPT system can be optimized following these steps: (1) adjusting the source operating frequencyf to the optimal frequency f
o
p
t at first and (2) adjusting the variable components, X
1 and X
2, to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


## 3.1.1. Coupled Resonators-Circuit

Based on the two-port network theory [31–33], the transmission matrix equation can be used to describe the input and output characteristics of the resonators-circuit, as (2)
U
1
I
1
=
A
B
C
D
U
2
I
2
,where U
1 and I
1 are the input voltage and current at port 1, respectively. U
2 and I
2 are the input voltage and current at port 2, respectively. A, B, C, and D are the transmission parameters of the coupled resonators-circuit. For a series-compensated wireless power transfer system, A, B, C, and D can be calculated as follows: A
=
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
/
j
ω
M, B
=
[
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
(
R
T
+
j
ω
L
T
-
j
/
ω
C
T
)
+
ω
2
M
2
]
/
j
ω
M, C
=
1
/
j
ω
M, and D
=
(
R
R
+
j
ω
L
R
-
j
/
ω
C
R
)
/
j
ω
M.From the circuit model shown in Figure3, the input impedance at port 1 can be found:(3)
Z
i
n
=
U
1
I
1
=
A
Z
L
+
B
C
Z
L
+
D
=
R
i
n
+
j
X
i
n
,where R
i
n and X
i
n represent the real part and the imaginary part of Z
i
n, respectively. Substituting A, B, C, and D parameters into (3) yields(4)
R
i
n
=
R
T
R
R
+
R
L
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
+
R
R
ω
2
M
2
+
R
L
ω
2
M
2
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
,
X
i
n
=
ω
L
T
-
1
/
ω
C
T
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
-
ω
2
M
2
X
L
+
ω
L
R
-
1
/
ω
C
R
R
R
+
R
L
2
+
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the input active power at port 1 can be calculated by(5)
P
i
n
=
I
1
2
R
i
n
.Because the load in a practical system may not be a pure resistance [34, 35], the load impedance Z
L is assumed to be R
L
+
j
X
L, where R
L and X
L represent the resistance and reactance of the load, respectively. Then, the output active power at port 2 can be calculated by(6)
P
o
u
t
=
I
2
2
R
L
.In previous studies, two kinds of power transfer efficiencies are defined to analyze the performance of the WPT system [36–38]. The system efficiency η
S refers to the ratio of the output power P
o
u
t at port 2 and total input power P
S from the wall outlet. It considers both the power losses in the AC power source and the resonators-circuit. Therefore, the system efficiency η
S cannot exceed 50% if the conjugate impedance-matching theorem is adopted because the source internal resistance will consume a half of the total input power P
S. The power transmission efficiency η
T is the ratio of the output power P
o
u
t at port 2 and the input power at port 1 P
i
n. It only includes the power losses in the resonators-circuit. In this paper, because the goal is not to improve the system power efficiency, the power transmission efficiency η
T is used to characterize the WPT system. η
T can be calculated using (7)
η
T
=
P
o
u
t
P
i
n
.By substituting (5) and (6) into (7), the power transmission efficiency of the resonators-circuit can be found:(8)
η
T
=
ω
2
M
2
R
L
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.


## 3.1.2. IM Network

As shown in Figure3, the IM network includes two compensating components. X
1 and X
2 are the values of them. The transmission matrix equation of the IM network can be expressed as(9)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
U
2
′
I
2
′
,where U
1
′ and I
1
′ are the input voltage and current at port 1′, respectively. U
2
′ and I
2
′ are the input voltage and current at port 2′, respectively. A
′, B
′, C
′, and D
′ are the transmission parameters of the IM network. Based on the structure of the IM network, A
′, B
′, C
′, and D
′ can be calculated as follows: A
′
=
1, B
′
=
j
X
2, C
′
=
1
/
j
X
1, and D
′
=
X
1
+
X
2
/
X
1.Then, the input impedance at port 1′ can be found:(10)
Z
i
n
′
=
U
1
′
I
1
′
=
A
′
Z
i
n
+
B
′
C
′
Z
i
n
+
D
′
=
R
i
n
′
+
j
X
i
n
′
,where R
i
n
′ and X
i
n
′ represent the real part and the imaginary part of Z
i
n
′, respectively. Substituting A
′, B
′, C
′, and D
′ parameters into (10) yields(11)
R
i
n
′
=
-
X
1
X
2
R
i
n
-
X
1
X
i
n
R
i
n
+
X
1
+
X
2
+
X
i
n
X
1
R
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
,
X
i
n
′
=
X
1
R
i
n
2
+
X
1
+
X
2
+
X
i
n
X
1
X
2
+
X
1
+
X
2
+
X
i
n
X
1
X
i
n
R
i
n
2
+
X
1
+
X
2
+
X
i
n
2
.In this paper, the ratio of the output active powerP
o
u
t
′ at port 2′ and input active power P
i
n
′ at port 1′ is defined as the power transmission efficiency of the IM network. It can be calculated by(12)
η
I
M
=
P
o
u
t
′
P
i
n
′
,where P
i
n
′
=
I
1
′
2
R
i
n
′ and P
o
u
t
′
=
I
2
′
2
R
i
n.Substituting (11) into (12) yields η
I
M
=
100
%. This is because the compensating components are assumed to be ideal energy-storage components without any energy losses. But, in fact, X
1 and X
2 also consume part of the total energy in the forms of heat and radiation [39, 40]. Therefore, the power transmission efficiency of the IM network cannot reach 100% in a practical system. To simplify the analysis, η
I
M is assumed to be 100
% in the theoretical analysis.


## 3.1.3. Overall WPT System

By substitutingU
2
′
=
U
1 and I
2
′
=
-
I
1 into (2), the transmission matrix equation of the overall WPT system can be found:(13)
U
1
′
I
1
′
=
A
′
B
′
C
′
D
′
A
B
C
D
U
2
I
2
.Thus, the power transmission efficiency of the overall WPT system can be calculated as(14)
η
t
o
t
a
l
=
P
o
u
t
P
i
n
′
=
P
o
u
t
′
P
i
n
′
×
P
o
u
t
P
i
n
=
η
I
M
η
T
.Substituting (7) and (12) into (14) yields(15)
η
t
o
t
a
l
=
ω
2
M
2
R
L
η
I
M
R
T
R
R
+
R
L
2
+
R
R
+
R
L
ω
2
M
2
+
R
T
X
L
+
ω
L
R
-
1
/
ω
C
R
2
.Then, the output active powerP
o
u
t of the overall WPT system can be expressed as(16)
P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l
.From here it can be seen that the output powerP
o
u
t is the function of the input power P
i
n
′ and the power transmission efficiency η
t
o
t
a
l. To maximize the output power of the WPT system, both P
i
n
′ and η
t
o
t
a
l should be improved.


## 3.1.4. Efficiency and Output Power Optimization

According to (15), the operating frequency can be adjusted to increase the power transmission efficiency η
t
o
t
a
l. By solving “d
η
t
o
t
a
l
/
d
f
=
0,” the optimal frequencies to maximize η
t
o
t
a
l can be found:(17)
f
o
p
t
=
C
R
+
C
L
π
4
L
R
C
R
C
L
2
+
4
L
R
C
R
2
C
L
-
2
C
R
2
C
L
2
R
R
+
R
L
2
,
for Capacitive Load,
1
π
4
L
R
C
R
+
4
L
L
C
R
-
2
C
R
2
R
R
+
R
L
2
,
for Inductive Load,where C
L
=
-
1
/
ω
X
L represents the equivalent capacitance of a capacitive load and L
L
=
X
L
/
ω represents the equivalent inductance of inductive load. The operating frequency is f
=
ω
/
2
π.To maximize the input active powerP
i
n
′, the IM network is used to conjugately match the input impedance Z
i
n
′ to the source impedance Z
S. Then the output power of the overall WPT system can be maximized because P
o
u
t
=
P
i
n
′
×
η
t
o
t
a
l. By introducing (10) into Z
i
n
′
=
Z
S
∗, the optimal values of X
1 and X
2 can be obtained:(18)
X
o
p
t
,
1
=
±
R
S
R
i
n
R
S
R
i
n
-
R
i
n
2
,
X
o
p
t
,
2
=
-
X
i
n
±
R
S
R
i
n
-
R
i
n
2
.The compensating componentsX
1 and X
2 can be realized using variable capacitors/inductances. The optimal values of the capacitors/inductances can be calculated by(19)
L
o
p
t
,
i
=
X
o
p
t
,
i
2
π
f
o
p
t
if
X
o
p
t
,
i
>
0
,
C
o
p
t
,
i
=
-
1
2
π
f
o
p
t
X
o
p
t
,
i
if
X
o
p
t
,
i
<
0
,
i
=
1,2
.According to the analysis, when the load impedance changes, the power transmission efficiency and the output power of the WPT system can be optimized following these steps: (1) adjusting the source operating frequencyf to the optimal frequency f
o
p
t at first and (2) adjusting the variable components, X
1 and X
2, to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


## 3.2. Numerical Simulations

In this section, the impacts of load impedance change on the power transmission efficiencyη
t
o
t
a
l and output power P
o
u
t are investigated through numerical simulations. Two different situations, the load reactance change and the load resistance change, are considered separately. The parameters used for calculations are given in Table 1.Table 1
Parameters used for calculations.
Parameters
Values
Source resistanceR
S
50 Ω
Lumped capacitorsC
T, C
R
216 pF
Coil resistanceR
T, R
R
2 Ω
Coil inductanceL
T, L
R
29 uH
Coil frequencyf
T, f
R
2 MHz
Coupling coefficientk
0.13
Load resistanceR
L
10~500 Ω
Load reactanceX
L
−500~500 Ω


### 3.2.1. Power Transmission Efficiency

When the load reactanceX
L increases from −500 Ω to 500 Ω, the power transmission efficiency of the WPT system can be obtained using (15). Figure 4(a) shows the calculated results of power transmission efficiency η
t
o
t
a
l versus load reactance X
L and source operating frequency f. When f is fixed at 2 MHz, it can be seen from Figure 4(a) that η
t
o
t
a
l increases firstly and then decreases with the increase of X
L, and the maximum power transmission efficiency can only be achieved at a single load reactance X
L
=
0
Ω.Figure 4
(a) Calculatedη
t
o
t
a
l versus X
L and f. (b) Calculated η
t
o
t
a
l versus R
L and f.
(a)
(b)Figure4(b) shows the calculated results of η
t
o
t
a
l versus load resistance R
L and source operating frequency f, where the load resistance R
L increases from 10 Ω to 500 Ω. From Figure 4(b), the similar trend can be found: when f is fixed at 2 MHz, η
t
o
t
a
l increases firstly and then decreases with the increase of R
L, and the maximum power transmission efficiency can only be achieved at an optimal load resistance R
L
=
50
Ω.Based on the theoretical analysis, the source operating frequency is adjusted to improve the power transmission efficiency firstly when the load impedance changes. The optimal frequenciesf
o
p
t for maximizing η
t
o
t
a
l can be calculated using (17). They are marked with blue dashed line in Figure 4. It can be observed that the WPT system can maintain high efficiency when the source frequency is adjusted along the optimal frequency line.


### 3.2.2. Output Power

SettingX
L
=
-
500
,
-
250
,
250
,
500
Ω and R
L
=
100
,
200,300,400
Ω, respectively, yields corresponding optimal frequencies f
o
p
t
≈
3.11, 2.64, 1.57, and 1.32 MHz and f
o
p
t
≈
2.05, 2.18, 2.47, and 3.19 MHz. To increase the output power of the WPT system, the variable components X
1 and X
2 are used to match the impedance of the WPT system at the optimal frequencies.Figures5 and 6 show the calculated results of output power P
o
u
t versus X
1 and X
2. The best matching points (X
o
p
t
,
1, X
o
p
t
,
2) for maximizing P
o
u
t can be found using (18). They are marked with black points in Figures 5 and 6. It can be seen that P
o
u
t will reach the maximum value if X
1 and X
2 are adjusted to the best matching points.Figure 5
CalculatedP
o
u
t versus X
1 and X
2. The load resistance R
L
=
100
Ω.
(a)
X
L
=
-
500
Ω and f
o
p
t
=
3.11
M
H
z
(b)
X
L
=
-
250
Ω and f
o
p
t
=
2.64
M
H
z
(c)
X
L
=
250
Ω and f
o
p
t
=
1.57
M
H
z
(d)
X
L
=
500
Ω and f
o
p
t
=
1.32
M
H
zFigure 6
CalculatedP
L versus X
1 and X
2. The load reactance X
L
=
0
Ω.
(a)
R
L
=
100
Ω and f
o
p
t
=
2.05
M
H
z
(b)
R
L
=
200
Ω and f
o
p
t
=
2.18
M
H
z
(c)
R
L
=
300
Ω and f
o
p
t
=
2.47
M
H
z
(d)
R
L
=
400
Ω and f
o
p
t
=
3.19
M
H
zFrom Figures5 and 6, it can be seen that the numerical simulation results agree with the theoretical analysis well. The results suggest that the power transmission efficiency of the WPT system can be significantly improved by changing the source operating frequency f to the optimal frequency f
o
p
t, and the output power can achieve high value by adjusting the IM network to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


## 3.2.1. Power Transmission Efficiency

When the load reactanceX
L increases from −500 Ω to 500 Ω, the power transmission efficiency of the WPT system can be obtained using (15). Figure 4(a) shows the calculated results of power transmission efficiency η
t
o
t
a
l versus load reactance X
L and source operating frequency f. When f is fixed at 2 MHz, it can be seen from Figure 4(a) that η
t
o
t
a
l increases firstly and then decreases with the increase of X
L, and the maximum power transmission efficiency can only be achieved at a single load reactance X
L
=
0
Ω.Figure 4
(a) Calculatedη
t
o
t
a
l versus X
L and f. (b) Calculated η
t
o
t
a
l versus R
L and f.
(a)
(b)Figure4(b) shows the calculated results of η
t
o
t
a
l versus load resistance R
L and source operating frequency f, where the load resistance R
L increases from 10 Ω to 500 Ω. From Figure 4(b), the similar trend can be found: when f is fixed at 2 MHz, η
t
o
t
a
l increases firstly and then decreases with the increase of R
L, and the maximum power transmission efficiency can only be achieved at an optimal load resistance R
L
=
50
Ω.Based on the theoretical analysis, the source operating frequency is adjusted to improve the power transmission efficiency firstly when the load impedance changes. The optimal frequenciesf
o
p
t for maximizing η
t
o
t
a
l can be calculated using (17). They are marked with blue dashed line in Figure 4. It can be observed that the WPT system can maintain high efficiency when the source frequency is adjusted along the optimal frequency line.


## 3.2.2. Output Power

SettingX
L
=
-
500
,
-
250
,
250
,
500
Ω and R
L
=
100
,
200,300,400
Ω, respectively, yields corresponding optimal frequencies f
o
p
t
≈
3.11, 2.64, 1.57, and 1.32 MHz and f
o
p
t
≈
2.05, 2.18, 2.47, and 3.19 MHz. To increase the output power of the WPT system, the variable components X
1 and X
2 are used to match the impedance of the WPT system at the optimal frequencies.Figures5 and 6 show the calculated results of output power P
o
u
t versus X
1 and X
2. The best matching points (X
o
p
t
,
1, X
o
p
t
,
2) for maximizing P
o
u
t can be found using (18). They are marked with black points in Figures 5 and 6. It can be seen that P
o
u
t will reach the maximum value if X
1 and X
2 are adjusted to the best matching points.Figure 5
CalculatedP
o
u
t versus X
1 and X
2. The load resistance R
L
=
100
Ω.
(a)
X
L
=
-
500
Ω and f
o
p
t
=
3.11
M
H
z
(b)
X
L
=
-
250
Ω and f
o
p
t
=
2.64
M
H
z
(c)
X
L
=
250
Ω and f
o
p
t
=
1.57
M
H
z
(d)
X
L
=
500
Ω and f
o
p
t
=
1.32
M
H
zFigure 6
CalculatedP
L versus X
1 and X
2. The load reactance X
L
=
0
Ω.
(a)
R
L
=
100
Ω and f
o
p
t
=
2.05
M
H
z
(b)
R
L
=
200
Ω and f
o
p
t
=
2.18
M
H
z
(c)
R
L
=
300
Ω and f
o
p
t
=
2.47
M
H
z
(d)
R
L
=
400
Ω and f
o
p
t
=
3.19
M
H
zFrom Figures5 and 6, it can be seen that the numerical simulation results agree with the theoretical analysis well. The results suggest that the power transmission efficiency of the WPT system can be significantly improved by changing the source operating frequency f to the optimal frequency f
o
p
t, and the output power can achieve high value by adjusting the IM network to the best matching point (X
o
p
t
,
1, X
o
p
t
,
2).


## 4. Experiments and Discussion

### 4.1. Hardware Implementation

To demonstrate the validity of the proposed frequency-tracking and impedance-matching combined approach for the WPT system, an experimental system is designed and implemented. Two circular air-core coils, which are fabricated by Litz wires, are used as the power transmitter and receiver. The radii of the transmitter coil and the receiver coil are 13 cm and 11 cm, and the number of turns of the two coils is 8. In order to compensate the reactive power, two lumped capacitors are connected to the transmitter coil and receiver coil, respectively. A signal generator, which has an adjustable frequency range of 1–10 MHz, is used to change the operating frequency of the power amplifier. An L-section impedance-matching network consisting of variable-capacitor and fixed-inductance arrays (as shown in Figure7) is inserted in the transmitter circuit to match the input impedance Z
i
n
′ to the source impedance Z
S. Each variable air capacitor has a tunable range of 360 pF to 20 pF, for the rotation angle of θ
=
0
° to θ
=
180
°, respectively. Small steering engines are used to change the rotation angle of the variable air capacitors through feedback control.Figure 7
L-section IM network.A directional power meter, which is connected between the power amplifier and the IM network, is used to measure the input powerP
i
n
′. A load, which includes a resistance R
L and a reactance X
L, is connected to the output port of the receiver coil. Another power meter is connected between the receiver coil and the load, providing output power P
o
u
t feedback to the transmitter-side through wireless communication. A controller is used to analyze the power transmission efficiency and the output power of the overall WPT system and change the output frequency of the signal generator and the rotation angle of the steering engines. The actual experimental system and the adaptive IM circuit are shown in Figures 8 and 9, respectively. An Agilent 4294A impedance analyzer is used to measure the precise electrical parameters of the experimental system, which are listed in Table 2.Table 2
Measured parameters of the experimental WPT system.
Parameters
Values
Tx coil inductanceL
T
13.6μH
Tx capacitorC
T
203.9 pF
Tx coil frequencyf
T
3.0 MHz
Rx coil inductanceL
R
10.6μH
Rx capacitorC
R
260.5 pF
Rx coil frequencyf
R
3.0 MHz
Source resistanceR
S
50 Ω
Load resistanceR
L
50, 100, 150, 200, 250 Ω
Load reactanceX
L
-
400, −200, 0, 200, 400 ΩFigure 8
Experimental frequency-tracking and impedance-matching combined WPT system.Figure 9
Adaptive impedance-matching circuit.When the load impedance changes, the WPT system can be automatically adjusted following these steps: (1) first step is adjusting the source operating frequencyf until the measured power transmission efficiency η
t
o
t
a
l is maximized. The source operating frequency to achieve the maximum η
t
o
t
a
l is marked as the optimal frequency f
o
p
t; (2) second step is adjusting the variable capacitors in the IM network at the optimal frequency until the measured output active power P
o
u
t
′ is maximized.


### 4.2. Experimental Results

In this section, the performances of three kinds of WPT systems are analyzed and compared, including the following:System 1: The Original WPT System, As Shown in Figure1. In order to achieve resonance, the source operating frequency is fixed at 3 MHz. The output power and power transmission efficiency of the system can be calculated using Equation (6) and Equation (8).
System 2: The Dual-Side Impedance-Matching WPT System, As Shown in Figure2. The operating frequency is fixed at 3 MHz. Based on the method proposed in [18], the system can be adjusted to satisfy dual-side conjugate matching equations “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗” using two IM networks. The output power and power transmission efficiency of the system can be calculated by introducing “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗” into (6) and (8).
System 3: The Proposed Frequency-Tracking and Impedance-Matching Combined WPT System, As Shown in Figure3. The optimal frequencies used and the values of the compensating components in the proposed system are listed in Tables 3 and 4.Table 3
Optimal frequencies and values of the compensating components. (The load resistanceR
L is fixed at 50 Ω.)
X
L (Ω)
f
o
p
t (MHz)
X
o
p
t
,
1 (Ω)
X
o
p
t
,
2 (Ω)
−400
5.3
−126.4
191.5
−200
4.3
−92.8
145.2
0
3.1
−126.0
11.1
200
2.2
−44.8
197.3
400
1.8
−33.1
307.5Table 4
Optimal frequencies and values of the compensating components. (The load reactanceX
L is fixed at 0 Ω.)
R
L (Ω)
f
o
p
t (MHz)
X
o
p
t
,
1 (Ω)
X
o
p
t
,
2 (Ω)
50
3.1
−126.0
11.1
100
3.3
−49.5
0.1
150
3.6
−39.2
−59.2
200
4.3
−35.7
−152.5
250
6.5
−35.4
−393.5The measured power transmission efficiencies and output powers are listed in Tables5 and 6. To simplify the analysis, the measured output powers are normalized to eliminate the influence of dimension. The normalized output power P
n is defined as P
n
=
P
o
u
t
/
P
m, where P
m is the maximum output power of the amplifier. In the experimental system, P
m
≈
1.5
W when the load is matched to 50 Ω.Table 5
Measured power transmission efficiencies and output powers. (The load resistanceR
L is fixed at 50 Ω.)
X
L (Ω)
P
o
u
t (W)
Efficiency (%)
S1
S3
S1
S3
−400
0.03
0.96
5.1
75.0
−200
0.26
1.15
25.0
81.8
0
1.21
1.27
85.4
83.6
200
0.26
1.08
22.3
73.9
400
0.02
0.83
3.3
59.1Table 6
Measured power transmission efficiencies and output powers. (The load reactanceX
L is fixed at 0 Ω.)
R
L (Ω)
P
o
u
t (W)
Efficiency (%)
S1
S3
S1
S3
50
1.20
1.28
84.2
85.3
100
1.06
1.23
77.8
82.0
150
0.90
1.23
70.6
80.4
200
0.75
1.20
66.7
78.2
250
0.69
1.16
64.3
77.5As shown in Figures10 and 11, the blue solid line represents the measured results of the original WPT system, the purple dashed line represents the calculated results of the original WPT system, the black dashed line represents the calculated results of the dual-side impedance-matching WPT system, the red solid line represents the measured results of the frequency-tracking and impedance-matching combined WPT system, and the green dashed line represents the calculated results of the frequency-tracking and impedance-matching combined WPT system.Figure 10
Comparison of three kinds of WPT systems. S1: the original WPT system; S2: the dual-side impedance-matching WPT system; S3: the frequency-tracking and impedance-matching combined WPT system. (a) Power transmission efficiency versus load reactance. (b) Output power versus load reactance.
(a)
(b)Figure 11
Comparison of three kinds of WPT systems. S1: the original WPT system; S2: the dual-side impedance-matching WPT system; S3: the frequency-tracking and impedance-matching combined WPT system. (a) Power transmission efficiency versus load resistance. (b) Output power versus load resistance.
(a)
(b)Figure10 shows that the power transmission efficiency and the output power of the original WPT system decrease dramatically when the load reactance deviates from X
L
=
0
Ω. By employing the frequency-tracking and impedance-matching combined approach, the power transmission efficiency and the output power of the WPT system can be significantly improved.In Figure11, the similar trend can be found: the power transmission efficiency and the output power of the original WPT system decrease when the load resistance deviates from R
L
=
50
Ω. Through the proposed approach, the WPT system can maintain high power transmission efficiency and output power even when the load resistance R
L changes. The experimental results suggest that the proposed frequency-tracking and impedance-matching combined system can significantly improve the robustness of the WPT system with load impedance variations.From Figures10 and 11, it can be observed that the measured power transmission efficiencies are lower than calculation results. It is because that the capacitors/inductances in the IM network are assumed to be ideal energy-storage components in calculations. But these components also consume part of the total energy in the form of heat and radiation in the practical WPT system. The more components are used, the more energy will be dissipated in the IM network. Therefore, adding too many impedance-matching components in the WPT circuit will decrease the total power transmission efficiency of the system.As shown in Figures10 and 11, the output powers of the proposed frequency-tracking and impedance-matching combined WPT system are lower than traditional dual-side impedance-matching WPT system. According to the maximum power transfer theorem [14, 30], the output power P
o
u
t will achieve the maximum value if the WPT system satisfies the dual-side conjugate matching equations “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗.” However, using the proposed frequency-tracking and impedance-matching combined approach, only the transmitter-side can satisfy “Z
i
n
=
Z
S
∗.” Because no variable IM networks are used in the receiver circuit, “Z
o
u
t
=
Z
L
∗” are not satisfied. Therefore, the output power of the WPT system does not reach the maximum value in fact. Although the output power of the frequency-tracking and impedance-matching combined WPT system is lower than traditional dual-side impedance-matching WPT system, it can still maintain high values without adding more components in the receiver circuit.In this study, the output power feedback at the receiver-side should be provided to the transmitter-side. There are two approaches to provide output power feedback to the transmitter-side. The mathematical approach requires only information of the input voltageU
1 and current I
1 at the transmitter-side, and then P
o
u
t can be calculated using (20), thereby eliminating the requirements of using wireless communication system.(20)
U
2
=
D
U
1
-
B
I
1
A
D
-
B
C
,
I
2
=
C
U
1
-
A
I
1
B
C
-
A
D
,
P
o
u
t
=
U
2
I
2
.However, in practice, the mathematical approach is very dependent on the accuracy of the measurements ofU
1, I
1 and A, B, C, and D parameters, which are complex numbers. The wireless communication approach is more accurate and efficient but requires using wireless communication system to give feedback about P
o
u
t measurements to the transmitter-side. For the embedded applications, the size of the wireless communication unit can be further reduced so as to integrate it into the receiver.Moreover, the selection of the operating frequency is important for resonant WPT systems. For many applications, the operating frequency of the WPT system is limited to a very narrow band (e.g., 6.765–6.795 MHz and 13.553–13.567 MHz according to the ISM band standard [15, 27]). In some previous studies [10–14], the feasibility and advantages of using frequency-tuning technique in a wider frequency range have also been demonstrated. Based on the frequency-tracking and impedance-matching combined approach proposed in this paper, the operating frequency of the WPT system should be adjusted in the range of a few megahertz to maximize the power transmission efficiency and output power, thereby eliminating the requirements of using receiver-side IM networks. The frequency range used in this study is commercially available and easy to implement in practical applications. Since fewer components are used in the receiver-side, the proposed approach is more suitable for some applications such as wireless power transfer for embedded devices.


## 4.1. Hardware Implementation

To demonstrate the validity of the proposed frequency-tracking and impedance-matching combined approach for the WPT system, an experimental system is designed and implemented. Two circular air-core coils, which are fabricated by Litz wires, are used as the power transmitter and receiver. The radii of the transmitter coil and the receiver coil are 13 cm and 11 cm, and the number of turns of the two coils is 8. In order to compensate the reactive power, two lumped capacitors are connected to the transmitter coil and receiver coil, respectively. A signal generator, which has an adjustable frequency range of 1–10 MHz, is used to change the operating frequency of the power amplifier. An L-section impedance-matching network consisting of variable-capacitor and fixed-inductance arrays (as shown in Figure7) is inserted in the transmitter circuit to match the input impedance Z
i
n
′ to the source impedance Z
S. Each variable air capacitor has a tunable range of 360 pF to 20 pF, for the rotation angle of θ
=
0
° to θ
=
180
°, respectively. Small steering engines are used to change the rotation angle of the variable air capacitors through feedback control.Figure 7
L-section IM network.A directional power meter, which is connected between the power amplifier and the IM network, is used to measure the input powerP
i
n
′. A load, which includes a resistance R
L and a reactance X
L, is connected to the output port of the receiver coil. Another power meter is connected between the receiver coil and the load, providing output power P
o
u
t feedback to the transmitter-side through wireless communication. A controller is used to analyze the power transmission efficiency and the output power of the overall WPT system and change the output frequency of the signal generator and the rotation angle of the steering engines. The actual experimental system and the adaptive IM circuit are shown in Figures 8 and 9, respectively. An Agilent 4294A impedance analyzer is used to measure the precise electrical parameters of the experimental system, which are listed in Table 2.Table 2
Measured parameters of the experimental WPT system.
Parameters
Values
Tx coil inductanceL
T
13.6μH
Tx capacitorC
T
203.9 pF
Tx coil frequencyf
T
3.0 MHz
Rx coil inductanceL
R
10.6μH
Rx capacitorC
R
260.5 pF
Rx coil frequencyf
R
3.0 MHz
Source resistanceR
S
50 Ω
Load resistanceR
L
50, 100, 150, 200, 250 Ω
Load reactanceX
L
-
400, −200, 0, 200, 400 ΩFigure 8
Experimental frequency-tracking and impedance-matching combined WPT system.Figure 9
Adaptive impedance-matching circuit.When the load impedance changes, the WPT system can be automatically adjusted following these steps: (1) first step is adjusting the source operating frequencyf until the measured power transmission efficiency η
t
o
t
a
l is maximized. The source operating frequency to achieve the maximum η
t
o
t
a
l is marked as the optimal frequency f
o
p
t; (2) second step is adjusting the variable capacitors in the IM network at the optimal frequency until the measured output active power P
o
u
t
′ is maximized.


## 4.2. Experimental Results

In this section, the performances of three kinds of WPT systems are analyzed and compared, including the following:System 1: The Original WPT System, As Shown in Figure1. In order to achieve resonance, the source operating frequency is fixed at 3 MHz. The output power and power transmission efficiency of the system can be calculated using Equation (6) and Equation (8).
System 2: The Dual-Side Impedance-Matching WPT System, As Shown in Figure2. The operating frequency is fixed at 3 MHz. Based on the method proposed in [18], the system can be adjusted to satisfy dual-side conjugate matching equations “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗” using two IM networks. The output power and power transmission efficiency of the system can be calculated by introducing “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗” into (6) and (8).
System 3: The Proposed Frequency-Tracking and Impedance-Matching Combined WPT System, As Shown in Figure3. The optimal frequencies used and the values of the compensating components in the proposed system are listed in Tables 3 and 4.Table 3
Optimal frequencies and values of the compensating components. (The load resistanceR
L is fixed at 50 Ω.)
X
L (Ω)
f
o
p
t (MHz)
X
o
p
t
,
1 (Ω)
X
o
p
t
,
2 (Ω)
−400
5.3
−126.4
191.5
−200
4.3
−92.8
145.2
0
3.1
−126.0
11.1
200
2.2
−44.8
197.3
400
1.8
−33.1
307.5Table 4
Optimal frequencies and values of the compensating components. (The load reactanceX
L is fixed at 0 Ω.)
R
L (Ω)
f
o
p
t (MHz)
X
o
p
t
,
1 (Ω)
X
o
p
t
,
2 (Ω)
50
3.1
−126.0
11.1
100
3.3
−49.5
0.1
150
3.6
−39.2
−59.2
200
4.3
−35.7
−152.5
250
6.5
−35.4
−393.5The measured power transmission efficiencies and output powers are listed in Tables5 and 6. To simplify the analysis, the measured output powers are normalized to eliminate the influence of dimension. The normalized output power P
n is defined as P
n
=
P
o
u
t
/
P
m, where P
m is the maximum output power of the amplifier. In the experimental system, P
m
≈
1.5
W when the load is matched to 50 Ω.Table 5
Measured power transmission efficiencies and output powers. (The load resistanceR
L is fixed at 50 Ω.)
X
L (Ω)
P
o
u
t (W)
Efficiency (%)
S1
S3
S1
S3
−400
0.03
0.96
5.1
75.0
−200
0.26
1.15
25.0
81.8
0
1.21
1.27
85.4
83.6
200
0.26
1.08
22.3
73.9
400
0.02
0.83
3.3
59.1Table 6
Measured power transmission efficiencies and output powers. (The load reactanceX
L is fixed at 0 Ω.)
R
L (Ω)
P
o
u
t (W)
Efficiency (%)
S1
S3
S1
S3
50
1.20
1.28
84.2
85.3
100
1.06
1.23
77.8
82.0
150
0.90
1.23
70.6
80.4
200
0.75
1.20
66.7
78.2
250
0.69
1.16
64.3
77.5As shown in Figures10 and 11, the blue solid line represents the measured results of the original WPT system, the purple dashed line represents the calculated results of the original WPT system, the black dashed line represents the calculated results of the dual-side impedance-matching WPT system, the red solid line represents the measured results of the frequency-tracking and impedance-matching combined WPT system, and the green dashed line represents the calculated results of the frequency-tracking and impedance-matching combined WPT system.Figure 10
Comparison of three kinds of WPT systems. S1: the original WPT system; S2: the dual-side impedance-matching WPT system; S3: the frequency-tracking and impedance-matching combined WPT system. (a) Power transmission efficiency versus load reactance. (b) Output power versus load reactance.
(a)
(b)Figure 11
Comparison of three kinds of WPT systems. S1: the original WPT system; S2: the dual-side impedance-matching WPT system; S3: the frequency-tracking and impedance-matching combined WPT system. (a) Power transmission efficiency versus load resistance. (b) Output power versus load resistance.
(a)
(b)Figure10 shows that the power transmission efficiency and the output power of the original WPT system decrease dramatically when the load reactance deviates from X
L
=
0
Ω. By employing the frequency-tracking and impedance-matching combined approach, the power transmission efficiency and the output power of the WPT system can be significantly improved.In Figure11, the similar trend can be found: the power transmission efficiency and the output power of the original WPT system decrease when the load resistance deviates from R
L
=
50
Ω. Through the proposed approach, the WPT system can maintain high power transmission efficiency and output power even when the load resistance R
L changes. The experimental results suggest that the proposed frequency-tracking and impedance-matching combined system can significantly improve the robustness of the WPT system with load impedance variations.From Figures10 and 11, it can be observed that the measured power transmission efficiencies are lower than calculation results. It is because that the capacitors/inductances in the IM network are assumed to be ideal energy-storage components in calculations. But these components also consume part of the total energy in the form of heat and radiation in the practical WPT system. The more components are used, the more energy will be dissipated in the IM network. Therefore, adding too many impedance-matching components in the WPT circuit will decrease the total power transmission efficiency of the system.As shown in Figures10 and 11, the output powers of the proposed frequency-tracking and impedance-matching combined WPT system are lower than traditional dual-side impedance-matching WPT system. According to the maximum power transfer theorem [14, 30], the output power P
o
u
t will achieve the maximum value if the WPT system satisfies the dual-side conjugate matching equations “Z
i
n
=
Z
S
∗ and Z
o
u
t
=
Z
L
∗.” However, using the proposed frequency-tracking and impedance-matching combined approach, only the transmitter-side can satisfy “Z
i
n
=
Z
S
∗.” Because no variable IM networks are used in the receiver circuit, “Z
o
u
t
=
Z
L
∗” are not satisfied. Therefore, the output power of the WPT system does not reach the maximum value in fact. Although the output power of the frequency-tracking and impedance-matching combined WPT system is lower than traditional dual-side impedance-matching WPT system, it can still maintain high values without adding more components in the receiver circuit.In this study, the output power feedback at the receiver-side should be provided to the transmitter-side. There are two approaches to provide output power feedback to the transmitter-side. The mathematical approach requires only information of the input voltageU
1 and current I
1 at the transmitter-side, and then P
o
u
t can be calculated using (20), thereby eliminating the requirements of using wireless communication system.(20)
U
2
=
D
U
1
-
B
I
1
A
D
-
B
C
,
I
2
=
C
U
1
-
A
I
1
B
C
-
A
D
,
P
o
u
t
=
U
2
I
2
.However, in practice, the mathematical approach is very dependent on the accuracy of the measurements ofU
1, I
1 and A, B, C, and D parameters, which are complex numbers. The wireless communication approach is more accurate and efficient but requires using wireless communication system to give feedback about P
o
u
t measurements to the transmitter-side. For the embedded applications, the size of the wireless communication unit can be further reduced so as to integrate it into the receiver.Moreover, the selection of the operating frequency is important for resonant WPT systems. For many applications, the operating frequency of the WPT system is limited to a very narrow band (e.g., 6.765–6.795 MHz and 13.553–13.567 MHz according to the ISM band standard [15, 27]). In some previous studies [10–14], the feasibility and advantages of using frequency-tuning technique in a wider frequency range have also been demonstrated. Based on the frequency-tracking and impedance-matching combined approach proposed in this paper, the operating frequency of the WPT system should be adjusted in the range of a few megahertz to maximize the power transmission efficiency and output power, thereby eliminating the requirements of using receiver-side IM networks. The frequency range used in this study is commercially available and easy to implement in practical applications. Since fewer components are used in the receiver-side, the proposed approach is more suitable for some applications such as wireless power transfer for embedded devices.


## 5. Conclusions

To improve the robustness of wireless power transfer for embedded devices, a frequency-tracking and impedance-matching combined system is proposed in this paper. When the load impedance changes, the source operating frequency is optimized to increase the power transmission efficiency at first, and then the variable IM network in the transmitter circuit is automatically adjusted to increase the output power of the WPT system. An experimental system is implemented and measured, and a good agreement between the measurements and calculations is obtained. The results suggest that the proposed frequency-tracking and impedance-matching combined WPT system can quickly adapt to load impedance variations and maintain high power transmission efficiency and output power. Because no variable IM networks, relay coils, and other compensating components are used in the receiver circuit, the proposed approach is more beneficial to reduce the size, weight, and complexity of the receiver for embedded applications.



---

*Source: 5719835-2017-01-29.xml*