A) Parameters used in this paper

Table 1 shows the definition of the variables used herein. The following equations are used to express the relations among these variables:

(1)$$P_{input} = DP_{RF}$$

(2)$$P_{output} = \displaystyle{1 \over T}\int_0^T {\displaystyle{{V_{out}{(t)}^2} \over {R_{load}}}dt} $$

(3)$$P_{DC} = \displaystyle{{V_{DC}^2} \over {R_{load}}}$$

(4)$$V_{DC} = \displaystyle{1 \over T}\int_0^T {V_{out}\lpar t \rpar dt} $$

(5)$$\eta _{DC} = \displaystyle{{P_{DC}} \over {P_{input}}}$$

(6)$$\eta _{out} = \displaystyle{{P_{output}} \over {P_{input}}}$$

(7)$$M = \displaystyle{{{\rm max}\,{\rm (}V_{out}{\rm (}t{\rm ))}-{\rm min}\,{\rm (}V_{out}{\rm (}t{\rm ))}} \over {V_{DC}}}$$

Table 1.

Definition of variables.

B) Discussion on the fluctuation of V _out

The fluctuation of V _out is caused by the charging and discharging of the smoothing capacitor. The amount of electric charge stored during charging is designated as Q. In the steady state, the amount of electrical charge stored during the charging period is same as that flowing out from C _out during the discharging period. Hence, equation (8) is obtained:

(8)$${\eqalign{Q & = C_{out}(V_{out}(t_0 + DT)-V_{out}(t_0))\quad \cr & = \int_{t_0 + DT}^{t_0 + T} {\displaystyle{{V_{out}(t)} \over {R_L}}} dt.}} $$

In contrast, equation (9) is obtained:

(9)$$\int_{t_0 + DT}^{t_0 + T} {\displaystyle{{V_{out}(t)} \over {R_L}}dt \le \displaystyle{{{\rm max\;} (V_{out})} \over {R_L}}(1-D)T.} $$

Therefore, the maximum value of the voltage change is expressed using equation (10):

(10)$$\max (V_{out})-\min {\rm (}V_{out}{\rm )} \le \displaystyle{{{\rm max}(V_{out})} \over {C_{out}R_L}}(1-D)T.$$

Equation (11) is derived from equation (10):

(11)$$\displaystyle{{{\rm min(}V_{out}{\rm )}} \over {{\rm max}(V_{out})}} \ge 1-\displaystyle{{(1-D)T} \over {C_{out}R_L}}.$$

Consequently, equation (12) is derived as a condition wherein a sufficiently small fluctuation occurs:

(12)$$C_{out}\gg \displaystyle{{1-D} \over {R_Lf_r}}.$$

The operation of rectifiers with the condition satisfying equation (12) is discussed in the next subsection. With the condition equation (12) unsatisfied, the output voltage wave is in the steady state, as shown in Fig. 5. The waveform consists of two functions, f _rising and f _falling. Figure 6 shows the forms of their functions. E means the DC voltage in the steady state with CW input, which has P _RF input power. E _max and E _min respectively indicate the maximum and minimum voltages of an output wave under pulse-modulated input. Herein, the change in V _out during charging is denoted as f _rising(t) and that during discharging as f _falling(t). f _rising(t) and f _falling(t) satisfy equations (13) and (14), respectively:

(13)$$\eqalign{& f_{rising}(DT + t_{r0})-f_{rising}(t_{r0}) \cr & \quad = f_{falling}((1-D)T + t_{f0})-f_{falling}(t_{f0}),} $$

(14)$$f_{falling}(t_{f0}) = f_{rising}(DT + t_{r0}).$$

These are the boundary conditions of a steady state. When we know the transient operations of rectifiers during charging and discharging, the operations of rectifiers under the pulse-modulated waves can be calculated by equations (13) and (14). Figures 7 and 8 show the equivalent circuits at the rising and falling edges. f _rising(t) and f _falling(t) can be determined by the easy calculations shown below:

(15)$$f_{rising}(t) = E\left( {1-\exp {\rm \;} \left( {-\displaystyle{t \over {\tau_{rising}}}} \right)} \right),$$

(16)$$f_{falling}(t) = E\exp {\rm \;} \left( {-\displaystyle{t \over {\tau_{falling}}}} \right).$$

Fig. 5.

V _out with pulse-modulated waves in the steady state.

Fig. 6.

Definition of  f _rising(t) and f _falling(t).

Fig. 7.

Equivalent circuit at the rising edge.

Fig. 8.

Equivalent circuit at the falling edge.

When the C _in and junction capacitance of the diode are small, τ _falling is equal to C _outR _load. In addition, equation (16) has been discussed and used in some papers [Reference Claessens, Schreurs and Pollin4, Reference Claessens, Pan, Rajabi, Schreurs and Pollin5]. The rise operation is complex for the microwave propagation; therefore, τ _rise cannot be obtained from a theoretical analysis.

C) Analysis of the rectifier with a sufficiently large smoothing capacitor

This subsection discusses the rectifier operation under the conditions of equation (12). The current is defined as shown in Fig. 9. We let the average of I and V _out be represented as $\bar{I}$ and $\overline {V_{out}} $. In addition, the average current flowing through the capacitor during charging is a times as large as $\bar{I}$. When the fluctuation of V _out is sufficiently small, the fluctuation of I is also very small. Equation (17) is obtained by considering the conservation of charge during the charging and discharging of a steady state capacitor:

(17)$$a\bar{I}D = (1-D)\bar{I}.$$

Fig. 9.

Rectifier load.

With a sufficiently small voltage fluctuation, the apparent impedance also has a sufficiently small change. Let R be the resistance observed from the rectifier circuit during charging as expressed in equation (18):

(18)$$R = \displaystyle{{\overline {V_{out}}} \over {(a + 1)\bar{I}}}.$$

R _load is represented by equation (19):

(19)$$R_{load} = \displaystyle{{\overline {V_{out}}} \over {\bar{I}}}.$$

Equation (20) is derived using equations (17)–(19):

(20)$$R = DR_{load}.$$

Equation (20) shows that the rectifier operates under the conditions of input power P _input and load resistance DR _L during charging. Using η _RF with P _input of input power and DR _L of load resistance, we obtain equation (21):

(21)$$\overline {V_{out}} = \sqrt {\displaystyle{{\eta _{RF}P_{input}R} \over D}} = \sqrt {\eta _{RF}P_{input}R_{load}}. $$

Equation (22) is derived from equation (21):

(22)$$P_{DC} = \displaystyle{{{\overline {V_{out}}} ^2} \over {R_{load}}} = \eta _{RF}P_{input}.$$

Equation (23) follows from the above equations:

(23)$$\eta _{DC} = \eta _{RF}.$$

Equation (23) shows that the RF-DC conversion efficiency with CW having P _input/D can be obtained. In addition, equation (20) states that the optimum load of this condition becomes 1/D times that under the CW drive conditions. These equations are obtained with satisfied conditions of equation (12) and our theory is not related to the matching network and the value of R _load. Therefore, these equations are generally obtained when using pulse-modulated input and equation (12) is satisfied. Therefore, the optimum matching network is same as CW input rectifiers with DR _load.

A) Estimation of the rectifier operation

The pulse-modulated waves have two basic frequencies: repetition and carrier frequencies. This study exploits the carrier frequency of 2.45 GHz and the repetition frequency is below 1 MHz. Thus, the pulse-modulated waves possess excessive frequency components. In this case, the rectifier operation cannot be analyzed using the harmonic balance method. Therefore, only the transient simulation is useful for analyzing the operation of rectifiers with the pulse-modulated input. However, for the transient analysis, the transient simulation requires the time step to be sufficiently small compared to the period of the carrier wave. Hence, analyzing the operation of the rectifiers by taking into account only one condition is time consuming. For this reason, the output voltage waveform is estimated using equations (13) and (14) using f _rising(t) and f _falling(t). The exponential function approximates the output voltage wave. It is estimated that f _rising(t) and f _falling(t) can be expressed by equations (15) and (16), respectively. Then, the simulation result should confirm consistency of this assumption. The output voltage of the rectifiers with a single-shot CW input is simulated. The chosen simulation parameters are the input duration, the average power, and the frequency being equal to 250 µs, 1 mW, and 2.45 GHz, respectively; R _load and C _out are 2 kΩ and 47 nF, respectively. The green line in Fig. 10 shows the simulation result of the transient. Equations (15) and (16) fit the green line using the Levenberg–Marquardt method [Reference Moré18]. The red and blue lines in Fig. 10 show the fitting result of equations (15) and (16), respectively. Figure 10 exhibits a complete match of the green, red, and blue lines. Therefore, the former estimation is quite correct. The application of the fitting procedure allows us to deduce the value of τ _rising as 3.576 × 10⁻⁵ and that of τ _falling as 9.408 × 10⁻⁵. In this case, the rise and fall times become 78.6  and 206.7 µs, respectively.

Fig. 10.

Simulated V _out using transient analysis.

B) Estimated and simulation characteristics of the rectifiers

In this subsection, the output voltage is estimated in a steady state using the values of τ _rising and τ _falling in subsection A). Using this estimation, the V _out values in various conditions derived using numerical calculations are known as long as the load components are fixed. Since numerical calculations are quick, the time for analyzing the operation of rectifiers under the pulse-modulated input is reduced. If the value of E is known, the performances of rectifiers can be calculated by equations (13) and (14) using numerical analysis. E is equal to the output voltage of the rectifiers with a CW input black, which has the same power as a career wave (P _RF). Thus, E is calculated using the harmonic balance or transient simulations. Table 3 shows the simulated DC voltage in a steady state with the condition that the values of R _load and C _out are 2 kΩ and 47 nF, respectively. Using the data in Table 3, η _DC, η _out, and M are calculated at P _input of 1 mW. Figures 11, 12, and 13 show η _out, η _DC, and M, respectively; they demonstrate a good agreement between the estimation and simulation results. But with D = 0.1, Figs 11 and 12 show differences and Fig. 13 shows a complete match. These results indicate that our proposed method is not good for estimating η _out and η _DC under a condition in which the output voltage has little fluctuation and D is also little. When D is very small, the error of the rising waveform will affect the calculation results. Figures 11 and 12 show an error of 20%. This large error can occur from only a few percent errors in the rising waveform. In addition, the transient simulation can easily generate a numerical error with small D. Therefore, the simulation and our proposed estimation method are not good for analyzing the steady state with too small D. Hence, when D is not so small and f _r is less than or equal to 100 kHz, the estimation method mentioned in subsection A) is consistent with the simulation of rectifiers under a pulse-modulated wave input.

Fig. 11.

Simulated and estimated η _out depending on f _r and D.

Fig. 12.

Simulated and estimated η _DC depending on f _r and D.

Fig. 13.

Simulated and estimated M depending on f _r and D.

Table 3.

Simulated DC voltage in the steady state.

C) Experimental characteristics of rectifiers

Figure 14 presents a rectifier fabricated using the parameters in Fig. 4. Figure 15 shows the change of η _DC with CW depending on P _input and R _load. In the experiments and simulations, C _in and C _out are set as 100 pF and 47 nF, respectively. The results of the experiments and simulations show similar tendencies. The rectifier experiments used the conditions shown in Table 4. Figures 16, 17, and 18 exemplify the experimental and estimation results indicating a good agreement. But, with D = 0.1, Figs 16 and 17 show around 10% errors. Also the experimented and simulated values (Figs. 11 and 12) have around 10% errors. These errors occur by the reasons mentioned in the previous section. Figure 16 reveals that η _out increases with the decrease of f _r at low D. When f _r is high, equation (12) is satisfied. In this case, the observed load resistance of rectifiers becomes 1/D times R _load because of equation (20). Letting the apparent resistance R _eq = R/D, the rectifiers operate with P _RF of input power and R _eq of load resistance. Figure 15 shows that a 20 kΩ load resistance is much higher than the optimum resistance, and the RF-DC conversion efficiency is very low under such conditions. In contrast, if equation (12) is not satisfied, and the output voltage wave becomes a pulse wave with a low f _r, the rectifier operates with P _input/D and 2 kΩ of input power and load resistance, respectively. A 2 kΩ load impedance is around the optimum value with from 1 to 10 mW input. Therefore, the lower D is, the more η _output become.

Fig. 14.

An experimented rectifier.

Fig. 15.

The change of η _DC with CW input depending on P _input and R _load.

Fig. 16.

Experimental and estimated η _out depending on f _r and D.

Fig. 17.

Experimental and estimated η _DC depending on f _r and D.

Fig. 18.

Experimental and estimated M depending on f _r and D.

Table 4.

Parameters in the experiments in Section III subsection C).

Figure 17 shows the increase in the experimental η _DC and the simulated η _DC depending on f _r. η _DC is proportional to D with a low f _r, when equation (12) is not satisfied, and the output voltage becomes a pulse wave. Equation (24) in this case provides the relation between η _DC and η _out:

(24)$$\eta _{DC} = D\eta _{out}.$$

Thus, η _DC is proportional to D, consistent with the results in Fig. 17. On the other hand, Fig. 17 shows the peak around D = 0.4 with f _r is 10 or 100 kHz. In these cases, f _r is sufficiently high and the output voltage has little fluctuation. The apparent power and impedance are P _input/D and DR _L, respectively. Figure 15 shows the optimum impedance of our experimented rectifier was more than ~1 kΩ when the input power was lower than or equal to 10 mW. In this paper, D is larger than or equal to 0.1. Therefore, as D becomes smaller, the apparent input power further closer to the optimum power and the apparent resistance becomes further away from the optimum impedance. For this trade-off relationship between the apparent power and impedance, our experimented rectifier had the optimum value of D around 0.4.

Figure 18 shows the monotonic decrease of M depending on the increase of f _r and D. V _DC increases depending on D and max (V _out) − min (V _out) decreases depending on f _r, which explains the decrease of M in Fig. 18.

A) Influences of C _out on the rectifier operation

In this subsection, the validity of the condition expressed in equation (12) is examined. The parameter settings and conditions from Table 5 are used in the experiments with the outcome presented in Figs. 19, 20, and 21. In this case, C _out ≫ (1 − D) × 10⁻⁶ is necessary for satisfying equation (12). Large changes occur at around C _out when it is equal to (1 − D) × 10⁻⁶ as shown in Figs. 19, 20, and 21. This indicates that the condition in equation (12) is consistent with the actual operation of the rectifiers under the pulse-modulated input.

Fig. 19.

Changes in η _out depending on C _out and D.

Fig. 20.

Changes in η _DC depending on C _out and D.

Fig. 21.

Changes in M depending on C _out and D.

Table 5.

Parameters setting in Section IV subsection A).

When C _out is sufficiently small, the output voltage becomes a pulse wave and the rectifier operates with the P _RF of input power and R _load of load resistance. Also, η _DC is expressed by equation (24) and η _out is same as that of CW input. From Fig. 15, η _out with D = 0.1 and D = 0.5 should be ~70 and 60% respectively, and η _DC with D = 0.1 and D = 0.5 should be ~7 and 30%, respectively. This assumption is consistent with Figs 19 and 20. Next, we consider the condition that C _out is sufficiently large. η _out and η _DC should be the same value and equal to the η _DC with P _RF of input power and R _eq of load resistance. This assumption is also consistent with Figs 19 and 20. These results indicate that the rectifier operation is changed by the value of C _out. This is the reason for the crossing in Fig. 19. If the application can use the pulse wave, the condition, where equation (12) is not satisfied, is better than satisfying equation (12). Therefore, we should change the conditions depending on applications for improving the RF-DC conversion efficiency. Figures 19 and 20 show that the C _out value is useful and important for managing conditions. Even if f _r is not so high, large C _out satisfies the conditions of equation (12). Therefore, pulse modulation is useful for all applications by managing the C _out value.

B) Improvement in η _DC with a sufficiently large smoothing capacitor

In this subsection, the η _DC is discussed depending on the R _load, which satisfies the condition of equation (12). The parameters shown in Table 6 are used in the experiments. A 100 µF C _out is sufficiently large for satisfying the condition of equation (12), wherein V _out is regarded as a constant. Figure 22 reveals the change of η _DC depending on R _load and D and the increasing maximum η _DC at the vanishing D, in accordance with equation (23). Figure 23 shows the comparison between the η _DC with the pulse-modulated-input (a duty ratio of 0.5 and an input power of 1 mW) and that with CW input power of 2 mW. The green line indicates the relation of the RF-DC conversion efficiency with the CW, which is 2 mW versus 1/D times R _load. Figure 24 shows the comparison between the η _DC under the pulse-modulated input (a duty ratio of 0.1 and an input power of 1 mW) and that with CW input of 10 mW input power. The green line indicates the relation of the RF-DC conversion efficiency with the CW, which is 10 mW versus 1/D times R _load. The red and green lines match as shown in Figs 23 and 24, respectively. This is consistent with equations (20) and (23) and proves that by using pulse modulation, the RF-DC conversion efficiency with CW, which has P _input/D and the optimum resistance shift 1/D times R _load, can be obtained. This indicates that the RF-DC conversion efficiency of the rectifiers having a lower input can be improved using pulse modulation.

Fig. 22.

The change in the η _DC depends on the R _load with sufficiently large C _out.

Fig. 23.

The comparison between the η _DC with pulse-modulated input, which has the duty ratio of 0.5 and the input power of 1 mW, and that with CW input, which has the input power of 2 mW.

Fig. 24.

The comparison between the η _DC with pulse-modulated input, which has the duty ratio of 0.1 and the input power of 1 mW, and that with CW input, which has the input power of 10 mW.

Table 6.

Parameters setting in Section IV subsection B).