Takashi Ohira
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Duality is an elegant and convenient way to create new electronic systems. Based on an existing system, we can effectively find out its counterpart by simply applying the duality theorem. The duality theorem works in a wide scope: not only for linear passive systems, but also for active, nonlinear, and even nonreciprocal systems.
The class-E (CE) diode rectifier, proposed in the 1980s for high-frequency dc-to-dc conversion, is currently attracting attention as a promising key component of high-frequency/very high-frequency wireless power transfer systems because of its substantially high RF-to-dc power conversion efficiency.
This article formulates CE and its counterpart inverse-class-E (ICE) diode rectifiers from the viewpoint of duality. We model these rectifiers as simply as possible and characterize them in the words of linear algebra. Therefore, technical university students (even in undergraduate classes) can follow the fundamental theory using a pencil and paper.
The history of the two topologies, CE and ICE, dates back to 1988. Their basic schematics and ideal voltage and current waveforms were introduced as a new family of resonant rectifiers in [1]. Let us revisit the two topologies now from an educator’s view.
As shown in Figure 1 (upper), the CE rectifier employs a series ${L}_{1}{C}_{1}$ resonator, a shunt capacitor, a single diode, and a series choke. The RF input power is filtered by the ${L}_{1}{C}_{1}$ resonator. Only the fundamental wave from the RF source can pass through this resonator. In other words, the dc current, second-order, and higher-order harmonics generated by the diode are totally reflected back to the right-hand side of the resonator. The shunt capacitor ${C}_{2}$ helps the diode to perform a zero-current switching, which is the core of CE operation at the moment the diode turns off [2]. The diode, leading part of the rectifier, converts the RF input wave into dc along with nonlinear harmonics. The choke ${L}_{\infty}$ has a sufficiently high inductance to suppress the ripples, and thus admits only dc current to the output port.
Figure 1. Pair of counterparts: (upper) a CE rectifier, and (lower) an ICE rectifier.
We then divide the CE topology into four blocks. Each block is replaced by its counterpart. Table 1 shows the replacement rule. This replacement brings a new topology shown in Figure 1 (lower), which is called ICE diode rectifier. As one may presume by analogy with CE, the core of ICE is a zero-voltage switching operation at the moment the diode turns on [3]. Let us learn how the two topologies work in the following sections.
Table 1. Pair of counterparts in electronics.
Table 2. CE diode rectifier design lookup table (D = 50%, Q = 5).
Table 3. ICE diode rectifier design lookup table (D = 50%, Q = 5).
To understand how the CE and ICE circuits work, it is instructive to write down a simple formula for each LC element. For the first block, we make the series ${L}_{1}{C}_{1}$ of CE resonate at the RF input frequency (angular frequency ${\omega} = {2}{\pi}{f}$), expressed as \begin{align*}{\omega}{L}_{1} & = {QR}_{\text{in}} \tag{1} \\ {\omega}{C}_{1} & = \frac{1}{{QR}_{\text{in}}} \tag{2} \end{align*} where Q stands for the loaded quality factor that specifies the resonance sharpness (empirically set at 5 to 10). The RF input resistance ${R}_{\text{in}}$ will be explained later. Juxtaposing CE and ICE, we now apply the duality theorem, resulting in the series ${L}_{3}{C}_{3}$ of ICE to meet \begin{align*}{\omega}{C}_{3} & = {QG}_{\text{in}} \tag{3} \\ {\omega}{L}_{3} & = \frac{1}{{QG}_{\text{in}}} \tag{4} \end{align*} where the RF input conductance ${G}_{\text{in}}$ will also be explained later.
The behavior of diode rectifiers can generally be characterized in terms of the diode’s duty cycle or flow angle [4]. Especially for CE and ICE rectifiers, it is quite convenient to introduce the radian angle \[{\phi} = {\pi}\left({{1}{-}{D}}\right) \tag{5} \] where D denotes the diode’s on-duty ratio. When D ranges from 0% to 100%, ${\phi}$ linearly decreases from $\begin{gathered}{\pi}\end{gathered}$ down to zero. Employing (5), we obtain the circuit’s characteristic equation in an elegant fashion as \begin{align*}{\left({{1}{-}{\phi}\cot{\phi}}\right)}^{2} = \begin{cases}{\begin{array}{l}{{\pi}{\omega}{C}_{2}{R}_{o}\,{\text{for}}\,{\text{CE}}}\\{{\pi}{\omega}{L}_{4}{G}_{o}\,{\text{for}}\,{\text{ICE}}}\end{array}}\end{cases} \tag{6} \end{align*} where ${R}_{o}$ stands for the dc load resistance we suppose to connect at the rectifier’s output port, and ${G}_{o}$ makes its inverse, i.e., ${G}_{o} = {1}{/}{R}_{o}$. Equation (6) exactly dominates the diode’s on–off operation. Given the time constant ${C}_{2}{R}_{o}$ or ${L}_{4}{G}_{o}$ of the circuit, ${\phi}$ and D are uniquely determined by (5) and (6).
As mentioned previously, the rectifier operates at any duty cycle D between 0% and 100%. However, we now specify ${D} = {50}{\%}$, which is called nominal operation or half-wave rectification. The 50% duty means ${\phi} = {\pi}{/}{2}$ from (5), thereby eliminating $\cot{\phi}$, and thus further simplifying (6) into \begin{align*}{\omega}{C}_{2} & = \frac{1}{{\pi}{R}_{o}} \tag{7} \\ {\omega}{L}_{4} & = \frac{1}{{\pi}{G}_{o}}{.} \tag{8} \end{align*}
Under this nominal condition, we no longer need to solve the transcendental (6) for ${\phi}$. For technical and educational convenience, we choose ${D} = {50}{\%}$ in this article. If interested in further exploring (5) and (6) with D other than 50%, visit [3].
To proceed formulation on CE and ICE, we assume sinusoidal waveforms for their RF input voltage and current as \begin{align*}\left[{\begin{array}{c}{{v}_{s}\left({t}\right)}\\{{i}_{s}\left({t}\right)}\end{array}}\right] = \left[{\begin{array}{c}{\begin{array}{cc}{{V}_{P}}&{{V}_{Q}}\end{array}}\\{\begin{array}{cc}{{I}_{P}}&{{I}_{Q}}\end{array}}\end{array}}\right]\left[{\begin{array}{c}{\sin{\omega}{t}}\\{\cos{\omega}{t}}\end{array}}\right] \tag{9} \end{align*} where subscripts P and Q stand for the in-phase and quadrature components, respectively. Note that we take the time origin ${t} = {0}$ exactly at the moment the diode turns off for CE, and turns on for ICE. Also note that the diode is a nonlinear device, which generates the second- and higher-order harmonics. However, thanks to the front-end LC resonant filter, the harmonics are barely observed at the RF input port. This is why we can employ (9) for the stimulus of both CE and ICE. The two-by-two voltage/current matrix that appears in (9) is the starting point of our linear algebra on the diode rectifiers [2].
Once employing the nominal condition (7) or (8) together with sinusoidal stimulus (9), we find that the RF input quantities are linearly related with the dc output quantities as \begin{align*}\left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right] & = \frac{1}{2}\left[{\begin{array}{c}{\pi}\\{2}\end{array}}\right]{I}_{o} \tag{10} \\ \left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right] & = \frac{1}{2}\left[{\begin{array}{cc}{4}&{{-}{\pi}}\\{{-}{\pi}}&{4}\end{array}}\right]\left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right]{R}_{o} \\ & = \frac{1}{4}\left[{\begin{array}{c}{{2}{\pi}}\\{{8}{-}{\pi}^{2}}\end{array}}\right]{V}_{o} \tag{11} \end{align*} for CE, and \begin{align*}\left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right] & = \frac{1}{2}\left[{\begin{array}{c}{\pi}\\{2}\end{array}}\right]{V}_{o} \tag{12} \\ \left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right] & = \frac{1}{2}\left[{\begin{array}{cc}{4}&{{-}{\pi}}\\{{-}{\pi}}&{4}\end{array}}\right]\left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right]{G}_{o} \\ & = \frac{1}{4}\left[{\begin{array}{c}{{2}{\pi}}\\{{8}{-}{\pi}^{2}}\end{array}}\right]{I}_{o} \tag{13} \end{align*} for ICE. The subscript o implies a dc output quantity. Not to mention, the dc output voltage ${V}_{\mathrm{o}}$ and current ${I}_{\mathrm{o}}$ comply with the Ohm’s law: ${V}_{\mathrm{o}} = {R}_{\mathrm{o}}{I}_{\mathrm{o}}$ or ${I}_{\mathrm{o}} = {G}_{\mathrm{o}}{V}_{\mathrm{o}}$.
The RF input impedance of diode rectifiers is, in general, a complicated function of the dc load resistance because the diode’s duty cycle changes with the loading condition [4]. To make it simple for education purpose, we assume 50% duty here again. If we use a standard complex notation \[{Z}_{\text{in}} = {R}_{\text{in}} + {jX}_{\text{in}} \tag{14} \] of the RF input impedance, the resistance ${R}_{\text{in}}$ and reactance ${X}_{\text{in}}$ are derived as \begin{align*}\left[{\begin{array}{c}{{R}_{\text{in}}}\\{{X}_{\text{in}}}\end{array}}\right] & = {\left[{\begin{array}{cc}{{I}_{P}}&{{-}{I}_{Q}}\\{{I}_{Q}}&{{I}_{P}}\end{array}}\right]}^{{-}{1}}\left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right] \\ & = \frac{1}{{2}\left({{4} + {\pi}^{2}}\right)}\left[{\begin{array}{c}{16}\\{{\pi}\left({{4}{-}{\pi}^{2}}\right)}\end{array}}\right]{R}_{o} \\ & \approx\left[{\begin{array}{c}{0.577}\\{{-}{0}{.}{665}}\end{array}}\right]{R}_{o} \tag{15} \end{align*} for CE from linear algebra [2], [5]. Thanks to duality, we do not need to repeat the circuit analysis for ICE, but just apply the rule of Table 1 to (14) and (15), resulting in the RF input admittance as \begin{align*}{Y}_{\text{in}} & = {G}_{\text{in}} + {jB}_{\text{in}} \tag{16} \\ \left[{\begin{array}{c}{{G}_{\text{in}}}\\{{B}_{\text{in}}}\end{array}}\right] & = {\left[{\begin{array}{cc}{{V}_{P}}&{{-}{V}_{Q}}\\{{V}_{Q}}&{{V}_{P}}\end{array}}\right]}^{{-}{1}}\left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right] \\ & = \frac{1}{{2}\left({{4} + {\pi}^{2}}\right)}\left[{\begin{array}{c}{16}\\{{\pi}\left({{4}{-}{\pi}^{2}}\right)}\end{array}}\right]{G}_{o} \\ & \approx\left[{\begin{array}{c}{0.577}\\{{-}{0}{.}{665}}\end{array}}\right]{G}_{o}{.} \tag{17} \end{align*}
We define the rectifier’s voltage gain ${G}_{v}$ as the output-to-input voltage ratio. Looking back at (11), we find \begin{align*}{G}_{v} & = \frac{{V}_{o}}{\left|{{V}_{P} + {jV}_{Q}}\right|} \\ & = \frac{4}{\sqrt{{(}{2}{\pi}{)}^{2} + {\left({{8}{-}{\pi}^{2}}\right)}^{2}}} \\ & \approx{0}{.}{610}{.} \tag{18} \end{align*}
In a similar manner, the relation (10) reveals the current gain \begin{align*}{G}_{i} & = \frac{{I}_{o}}{\left|{{I}_{P} + {jI}_{Q}}\right|} \\ & = \frac{2}{\sqrt{{\pi}^{2} + {4}}} \\ & \approx{0}{.}{537} \tag{19} \end{align*} for CE. The duality theorem indicates that we can just interchange the results of ${G}_{v}$ and ${G}_{i}$ as \begin{align*}{G}_{v} & \approx {0}{.}{537} \tag{20} \\ {G}_{i} & \approx {0}{.}{610} \tag{21} \end{align*} for ICE.
The power coming to the rectifier at the input port is defined from the scalar product of voltage and current vectors. The input-to-output relations (10) and (11) find the incoming power \begin{align*}{P}_{\text{in}} & = \frac{1}{2}\left[{\begin{array}{cc}{{I}_{P}}&{{I}_{Q}}\end{array}}\right]\left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right] \\ & = \frac{1}{4}\left[{\begin{array}{cc}{{I}_{P}}&{{I}_{Q}}\end{array}}\right]\left[{\begin{array}{cc}{4}&{{-}{\pi}}\\{{-}{\pi}}&{4}\end{array}}\right]\left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right]{R}_{o} \\ & = {V}_{o}{I}_{o} \tag{22} \end{align*} for CE. In the same way, (12) and (13) yield \begin{align*}{P}_{\text{in}} & = \frac{1}{2}\left[{\begin{array}{cc}{{V}_{P}}&{{V}_{Q}}\end{array}}\right]\left[{\begin{array}{c}{{I}_{P}}\\{{I}_{Q}}\end{array}}\right] \\ & = \frac{1}{4}\left[{\begin{array}{cc}{{V}_{P}}&{{V}_{Q}}\end{array}}\right]\left[{\begin{array}{cc}{4}&{{-}{\pi}}\\{{-}{\pi}}&{4}\end{array}}\right]\left[{\begin{array}{c}{{V}_{P}}\\{{V}_{Q}}\end{array}}\right]{G}_{o} \\ & = {V}_{o}{I}_{o} \tag{23} \end{align*} for ICE. The final right-hand side of (22) or (23) signifies dc power delivered to the load at the output port. That is to say, the RF power is fully converted into dc power. We thus conclude that, in theory at least, both CE and ICE achieve 100% power conversion efficiency.
Finally, numerical design examples of CE and ICE rectifiers at three standard ISM frequencies (6.78, 13.56, and 27.12 MHz) are shown in Tables 2 and 3. Specifying the RF input frequency with required dc output load resistance, we can quickly look up the input impedance or admittance, and LC values we should use in the circuit. It is also worth noting that, since the reactance is a linear function of the operating frequency, the component values can easily be scaled for higher frequency applications. These tables familiarize us with the formulas presented in this article and are also useful for practical system design of RF power electronics featuring CE or ICE rectifiers.
[1] W. A. Nitz et al., “A new family of resonant rectifier circuits for high frequency DC-DC converter applications,” in Proc. IEEE Appl. Power Electron. Conf. Expo., New Orleans, LA, USA, Feb. 1988, pp. 12–22, doi: 10.1109/APEC.1988.10546.
[2] T. Ohira, “Linear algebra elucidates class-E diode rectifiers [Educator’s Corner] ,” IEEE Microw. Mag., vol. 23, no. 12, pp. 113–122, Dec. 2022, doi: 10.1109/MMM.2022.3203947.
[3] T. Ohira, “Inductor and diode [Enigmas, etc.] ,” IEEE Microw. Mag., vol. 24, no. 1, pp. 89–90, Jan. 2023, doi: 10.1109/MMM.2022.3211596.
[4] T. Ohira, “Power efficiency and optimum load formulas on RF rectifiers featuring flow-angle equations,” IEICE Electron. Exp., vol. 10, no. 11, pp. 1–9, Jun. 2013, doi: 10.1587/elex.10.20130230.
[5] T. Ohira, “RF input current [Enigmas, etc.] ,” IEEE Microw. Mag., vol. 24, no. 9, p. 80, Sep. 2023, doi: 10.1109/MMM.2023.3284796.
Digital Object Identifier 10.1109/MMM.2023.3321552