#### 4.1. Simulation Analysis of the Energy Harvesting Device

In this part, the optimal parameters of the power harvesting device are determined with threshold constraints to demonstrate the direct influences of those parameters on output power. Simulation is carried out to visually display the calculations and the discussions in

Section 3. Though the specific working environment is determined in advance, any major conclusions drawn from the analysis should be suitable for other circumstances too. An example of the practical design of a power harvesting device is presented here.

The current in the HVPL is assumed to be 500 A. Cold rolled silicon steel sheet is selected for the core according to 3.2.1. Further analysis of parameter optimization factors such as core inner radius a, core external radius b, radial thickness (b-a), height h, gap δ and coil turns N_{s} is introduced to find out the general rules for achieving maximum output power.

Since multiple parameters interact with each other, the dimensions of the analysis are reduced step by step to simplify the analysis and ensure the accuracy at the same time. The analytical method adopted in the paper is fractional reduction of dimensions, that is, the influences of a parameter with less impact on the output power is analyzed first, and then a certain law can be obtained. If this complies with the principle of monotony, this parameter can be set as a reference point when we analyze how other parameters affect output power. If not, then this parameter may interact with others, so a second parameter should be chosen.

After a series of computational analysis and dimension reduction analyses, core inner radius

a, radial thickness (

b-

a) are analyzed first. The results are shown in

Figure 5.

It is apparent from

Figure 5 that output power increases as the radial thickness increases (

b-

a). Besides, a larger core inner radius contributes to more output power when other parameters remain the same, which corresponds with the theoretical analysis. It is surprising to find that 40 W is obtained without any other optimization. Since both the core inner radius and radial thickness have a monotonic increasing relationship with output power, which meets the criteria of dimension reduction analysis, parameters

a and (

b-

a) are determined first. Additionally, too large a core inner radius

a and radial thickness (

b-

a) have much smaller influences on the increase of output power and may increase the risk of wire deformation. Thus, suitable evaluation parameters should be chosen in advance to do further optimization.

To verify the specific influence of core cross-sectional area

S_{eq} on output power illustrated in Equation (15), radial thickness (

b-

a) and core height

h are chosen to assess their impacts on determining the output power. Similarly, the changing trend while other parameters remain the same is shown in

Figure 6.

As shown in

Figure 6, output power increases with the increase of

S_{eq}, and the rise of the radial thickness (

b-

a) and core height also leads to the increase of output power. An important distinction between these two dimensions, namely, (

b-

a) and

h is that same increase leads to different output power changes. Take

$\{\begin{array}{l}[b-a]=h=8\text{cm}\\ {P}_{o}=39.96\mathrm{W}\end{array}$ as a reference point, Δ

Z_{1} = Δ

P_{o1} = 0.96 W when Δ

x = Δ

h = 2 cm but

Z_{2} = Δ

P_{o2} = 6.94 W when Δ

y = Δ

h = 2 cm. Apparently, Δ

Z_{2} is 5.98 W more than Δ

Z_{1} which is in accordance with theoretical analysis.

In sum, increasing the core inner radius and cross-sectional areas both lead to larger output power. However, the increase rate in

Figure 5 and

Figure 6 is pretty small, so these optimization methods commonly serve as helping measures. As a result, optimizations of the core gap

δ and secondary coil turns

N_{s} are discussed to increase the output power by a large margin. A part of parameters are determined as follows: core inner radius

a = 10 cm, core external radius

b = 18 cm, core height

h = 8 cm. The relationship between output power and core gap as well as secondary coil turns is displayed in

Figure 7.

It is obvious from

Figure 7 that there is an optimal secondary coil turn where output power is maximized. Meanwhile, output power increases prominently with the decrease of core gap

δ. For instance, when

δ decreases from 2 mm to 1.8 mm, the maximum output power increases by 6.3 W. When

δ decreases from 1.2 mm to 1 mm, the maximum output power increases by 28.4 W. In conclusion, for a core of fixed parameters, maximum output power can only be achieved with a certain secondary coil turn. When the primary current is 500 A, the optimum core dimension is suggested as follows: core inner radius

a = 10 cm, core external radius

b = 18 cm, core height

h = 8 cm, core gap

δ = 1 mm and secondary coil turns

N_{s} = 60–80.

Though a smaller core gap contributes to larger output power, it is impossible to minimize the core gap as much as possible since the core gap plays an important role in broadening the linear working regions of the core. In a situation when the gap is small and the current is large, the core is likely to work in deep saturation mode which will result in a drastic rise in temperature. Therefore it is very necessary to keep the energy extracting device working in the critical range between linear and saturated regions. In order to increase the output power when the current is small and discharge output power when current is large, a stability control strategy of extracting energy is proposed to adjust to the fluctuation of current. The basic principle is to control the core gap and the secondary coil turn as shown in

Figure 7. An overall scheme of the control strategy is specifically illustrated in

Figure 8.

In

Figure 8, the control strategy is divided into the following steps:

- (1)
According to

Figure 7, both the optimal secondary coil turn

N_{s1} to achieve maximum output power and secondary coil turn

N_{s2} to ensure the minimum demanded power can be determined, which should comply with

N_{s2} >

N_{s1}.

- (2)
Use a Rogowski coil to sample the current of HVPL in the form of induced voltage. A digital integrator is applied to transform the transient voltage signal e(t) into a stable signal E. The sampled voltage when the current equal to 500 A (less than or equal to the saturation current of core) is set as the standard value.

- (3)
The digital comparator is in charge of comparing the sampled E with the standard value. It outputs high-level signals (“1”) when d1 < d2 and outputs low-level signals (“0”) when d1 > d2.

- (4)
The diode is shut off when the comparator outputs high-level signals and the secondary coil turn is set to be N_{s1} so that maximum output power is achieved. On the contrary, the diode is turned on and the secondary coil turn is set to be N_{s2} so that minimum demanded power is acquired. The control strategy greatly reduces the impact of power increase on the load and realizes the goal to obtain relatively stable output power, regardless of current fluctuation.

If the comparator shown in

Figure 8 is equipped with multiple channels, several preset values can be set accordingly, so that the turns of secondary coil will be divided into several segments by controllable diodes. This can bring about two benefits: (1) the receiving power can be regulated to be more stable when the HVPL current is in fluctuation; (2) in addition, the core saturation can be limited by changing the turns of secondary coil to increase secondary current when HVPL current is too large. What we need to do is just to determine the most suitable number of turns of the secondary coil.

#### 4.2. Experimental verification

To verify the accuracy of theoretical and simulated analysis, multiple paired experiments are carried out by using the designed devices as shown in

Figure 9. A single wire is enwound into several turns to generate large primary current so that a real working status of HVPL can be simulated. An iodine-tungsten lamp (220 V, 1000 W) is connected to the source (220 V, 50 Hz) with a wire to generate the primary current in the experiments. When the current in single wire is 4.1 A and wire turns is 25, then the primary current of power harvesting device is 102.5 A.

With regard to the parameters optimization design with threshold constraints, nine paired experiments are designed to implement comprehensive validations, when the condition of

δ/h ≤ 0.2 and

δ/(

b-a)

≤ 0.2 is considered. Specific experimental parameters are shown in

Table 1.

Test 1: the influence of 0.5 cm’s increase in core inner radius on output power is studied by comparing 1 and 2 in

Table 1.

Test 2: 1, 3 and 4 are analyzed comparatively to investigate the influence of core radial thickness (b-a) and core height h on output power when the two parameters are increased by 0.1 cm, respectively.

Test 3: the influence of core gap δ on output power is demonstrated through a comparison among 2, 8 and 9.

Test 4: 2, 5, 6 and 7 are tested to verify the influence of secondary turns N_{s} on output power.

The corresponding results of Test 1–Test 4 are displayed in

Figure 10a–d. Several conclusions can be drawn from the experimental results above:

- (1)
Output power increases with the increase of core inner radius when other parameters remain certain;

- (2)
The increase of core radial thickness (b-a) and core height h will contribute to the increase of output power but the former one has a more pronounced effect;

- (3)
The smaller core gap is, the more output power the device can obtain once it holds true for Equation (16) and other parameters remain unchanged;

- (4)
Maximum output power can be acquired with the optimized secondary coil turns when other parameters remain certain. In this experiment specifically, P_{o} (N_{s} = 80) > P_{o} (N_{s} = 100).

In order to further validate the proposed optimization theory, the power calculations are performed when the parameters remain the same as that of experiments, which are shown in

Table 1. The theoretical calculated results are compared with those of the experiments, as shown in

Figure 11.

The primary calculated current is the same as that seen in the experiments, that is, 102.5 A. From

Figure 11, it is obvious that the calculated load power and experimental results have the same trend, which proves the accuracy of the theoretical analysis. There are some misalignments between calculations and experiments that are mainly caused by the different formations of the primary current. In the experiments, the primary current is generated by 25 turns of wires, and the current in a single wire is 4.1 A. For the small radius of every wire, the wires in the middle of the group would have little contribution to the main flux of the core, so the effect of the equivalent primary current in the experiments is actually less than that of the single thick wire with the same current value, which is the main reason why the extracted power in the experiments is less than that in the calculations.