This section introduces the power circuit configuration, control approach and reliability model that are considered in the following discussion of this paper.

#### 3.1. Control and Reliability of Power Circuits

This paper considers two representative power processing circuit configurations.

Figure 3 shows the configuration and control approach of a dc-dc buck converter. Thanks to its ability to step down the output voltage, a buck converter is commonly used for energy harvesting in low power applications, such as wireless sensor networks [

29] and small portable devices [

30]. The buck converter can be modeled as follows [

31]:

where

${i}_{L}$ is the inductor current,

$E$ is the input (source) voltage,

${v}_{o}$ is the capacitor (load) voltage,

L is the inductance,

C is the capacitance,

R is the load resistance, and

q(

t) is the switching function of the power switch

Q. The switching function

q(

t) is defined as:

A current-fed push-pull interface can be considered when a boost operation through isolation at higher power levels is required for safety.

Figure 4 shows the circuit configuration of this boost power processing unit, and the circuit can be modeled as [

32]:

where

i_{i} is the current of inductor

L_{i};

v_{i} is the input voltage;

v_{p} is the primary voltage;

v_{o} is the capacitor (output) voltage;

C_{o} is the output capacitance;

i_{s} is the secondary current of the transformer;

i_{o} is the output current;

$\varphi $ is the transformer magnetization flux;

v_{s} is the transformer secondary voltage;

N_{s} is the number of turns in the transformer secondary side;

N_{p} is the number of turns in the transformer primary side; and

q_{i}(

t) is the switching function of the power switch

Q_{i} (

i = 1, 2). The switching function

q_{i}(

t) is defined in the same manner as Equation (3).

To regulate the output voltage,

v_{o}, a cascaded control approach [

33] (i.e.,

Figure 3b) was considered. As shown in

Figure 3b, the outer voltage controller generates the reference command for the inner current controller, and the current controller generates the control command for the power switch,

Q. To regulate the load voltage, the voltage controller requires the feedback of the load voltage value, and the current controller uses the feedback of the inductor current through sensors [

34,

35,

36]. Therefore, failure to provide the feedback of accurate information on the inductor current to the current controller might cause an unsatisfactory performance or failure in the voltage regulation of the power circuit. This paper uses the inductor current for voltage regulation purposes, whereas the current information has been used in other applications of power circuits for various objectives. A representative example of energy harvesting purposes is the maximum power point tracking (MPPT) method of PV modules [

7].

While the control performance is determined by the controller configuration, controller gains, and the circuit parameter values, the reliability performance is determined by the failure rate of the interface hardware. In this paper, it is assumed that the reliability of each hardware component can be modeled to have an exponential distribution with a constant failure rate (

$\mathsf{\lambda}$), as in previous studies [

9,

10,

12,

25]. That is, the reliability function of a component (i.e.,

R(

t)) is expressed as [

9,

10,

22]:

where

$\mathsf{\lambda}$ is the failure rate (failures/hours), and

t is time (hours).

#### 3.2. Reliability Analysis Considering Sensors

Although each hardware device of the power circuit (e.g., capacitor, inductor, and power switch) can fail during the system operation, this study simplifies the reliability analysis by assuming that only the current sensors are subject to failures. Reliability studies that consider the failures of other components can be found in other studies [

22,

24]. By assuming that only the inductor current sensor can fail, the probability that the power circuit is operational (i.e.,

R_{PC}(

t)) can be calculated by using the failure rate data of the current sensor as follows:

where

R_{CS}(

t) is the reliability function of the current sensor, and

λ_{CS} is the failure rate of the current sensor. Examples of current sensor failure rate values can be found in previous reliability analyses [

14,

25] or in datasheets provided by the manufacturers [

37].

While the reliability can be calculated using Equation (11) based on the assumption that the inductor current sensor is the only method for sensing the inductor current value, the inductor current can still be known without requiring explicit feedback information of the sensor output. For example, the inductor current can be estimated using the inductor voltage value as follows [

16,

17]:

where

v_{L} is the inductor voltage. This estimation approach is possible based on the voltage-current relationship of the inductors. When the inductor current feedback is performed based on an alternative method instead of using the current sensor, the failure rate to be considered in Equation (11) should be modified to represent the actual sensing configuration. For example, the failure rate in Equation (11) should represent the properties of the voltage sensor instead of the current sensor if the inductor voltage based estimation approach (i.e., Equation (12)) is used. A different approach for estimating the inductor current is by reconstructing the inductor current value through other current information of the interface. In particular, the capacitor current and load current can be used for such purposes. Furthermore, such alternative sensing approaches can be used as a backup mechanism with an objective to achieving fault tolerant capability against the faults of the inductor current sensor. The system can be designed such that the controller receives feedback from the inductor current sensor as primary, and the source of the inductor current value feedback can be switched from the current sensor to an alternative approach (e.g., Equation (12)) when a fault in the inductor current sensor is detected.

When such alternative approaches are available, a more comprehensive analysis can be performed to quantify the level of reliability improvement.

Table 1 lists the possible operation conditions of the three current sensors in the buck converter.

Cases 1 to 4 represent the operation scenarios that the inductor current information is available through the inductor current sensor itself. Since the explicit feedback information of an inductor current value from the sensor is available, the system is regarded as being operational regardless of the operation status of the capacitor current sensor and the load current sensor. In other words, the system is operational as long as the inductor current sensor is healthy. Hence, the probability (or the system reliability function,

R_{PC1}) that the system is operational can be expressed as [

10]:

where

p_{IL} is the probability that the inductor current sensor is healthy.

When the inductor current sensor is not operational, as in the other cases (i.e., Cases 5 to 8), voltage regulation can still be performed using alternative approaches for current feedback information. A detailed explanation of these cases is as follows.

In the case that the capacitor current sensor and the load current sensor are operational (i.e., Case 5), the instantaneous inductor current value can be reconstructed from the output of the capacitor current sensor and the output of the load current sensor as follows:

where

i_{L_rec} is the reconstructed inductor current,

i_{C} is the measured capacitor current, and

i_{O} is the measured load current. Applying Kirchhoff’s current law at node A of

Figure 3a enables such a current reconstruction approach. Therefore, by installing a capacitor current sensor, a load current sensor, and a mechanism (e.g., algorithm) that can provide the result of Equation (14) to the current controller, the reliability of the system (i.e.,

R_{PC2}) can be increased from Equations (13) to (15) as follows:

where

q_{IL} is the probability that the inductor current sensor fails (

q_{IL} = 1 −

p_{IL}),

p_{IC} is the probability that the capacitor current sensor is operational, and

p_{IO} is the probability that the load current sensor is operational. An increase in reliability,

$\Delta {R}_{1}$, is achieved at the expense of installing additional current sensors (i.e., capacitor current sensor and load current sensor) and implementing the reconstruction approach, as expressed in Equation (14). Depending on the level of reliability required, some system design cases will require such a backup scheme at the cost of installing additional sensors.

Although only the capacitor current sensor is available as shown in Case 7, it is still possible to estimate the load current under the assumption that the load resistance is known. As the output voltage sensor would have already been installed to perform voltage regulation in the buck converter, the load current estimation can be performed using the sensed output voltage without installing extra sensors. The inductor current value can be estimated using the capacitor current and the output voltage value as follows:

where

i_{O_est} is the estimated load current. Accordingly, the reliability of the system (i.e.,

R_{PC3}) can be increased further from Equations (15) to (17) as

where

q_{IO} is the probability that the load current sensor fails (

q_{IO} = 1 −

p_{IO}), and

p_{VO} is the reliability function of the output voltage sensor. The primary objective of this study is not to develop fault tolerant control or sensing approaches for power circuits but to introduce and perform a quantitative reliability analysis considering the effects of sensors and various alternative feedback approaches. Based on this objective, this paper considered rather straightforward sensing approaches for the reliability analysis.

Figure 5 shows the reliability computation results from Equations (13), (15) and (17) using the sensor failure data [

14]:

$\mathsf{\lambda}$ = 2 failures/10

^{6} hours. The reliability value shows a remarkable increase with increasing number of cases that make the system operational by applying additional mechanisms for current sensing. In other words, the system reliability increases with increasing number of alternative approaches for acquiring the current value. For example, the reliability values at

t = 10,000 h for Equations (13), (15) and (17) are 0.9802, 0.9992 and 0.9996, respectively. The reliability analysis result of this study, such as the plot shown in

Figure 5, can be used to characterize the reliability of power circuits in various WoT devices. In addition, the analysis result can also provide estimation on system availability of networks that consist of such WoT devices. Using the quantitative reliability data (i.e.,

Figure 5) from the proposed reliability analysis process, a power circuit designer can also choose proper sensors or alternative sensing approaches for a reliable power circuit in a more systematic and comprehensive way. As reliability performance can be considered during the design process, reliability issues of power circuits for energy harvesting and power processing in WoT devices can be addressed by applying the analysis approach of this study.

#### 3.3. Further Considerations

As shown in

Figure 5, the system reliability can be improved by considering alternative approaches that perform identical tasks (e.g., control and sensing). In the case of buck converter control, voltage regulation can also be performed using the cascaded control configuration that relies on the feedback of the capacitor current instead of the inductor current [

38]. The capacitor current control approach can be activated when a fault in the original controller or in the inductor current sensor is detected so that the regulation performance is not interrupted by the resulting fault. With such back-up capability, Case 7 in

Table 1 can also be included as an operational case without requiring a load current estimation and can contribute to improving the system reliability. In a similar manner, the effects of alternative approaches for estimating the current on the system reliability can be studied further. Instead of relatively simple approaches that were considered in the previous analysis, it is possible to include the effects of different types of advanced current reconstructions or estimation approaches that can be used when the inductor current sensor fails [

39,

40]. An estimation of the capacitor current can be considered, and even all current values can be estimated when the direct feedback from all current sensors is unavailable, as in Case 8. As the system is equipped with some level of fault tolerant capability by introducing redundancy in the control or sensing function, it is possible to expect some improvement in reliability similar to previous studies [

14,

25,

26]. On the other hand, a reliability analysis that includes other alternative approaches was not performed in this paper because the development of alternatives is not in the scope of this paper. In addition, the actual applicability of these alternative approaches is decided not only by the reliability requirements, but also by the constraints of other factors, such as cost, complexity, and volume.