Takashi Ohira
The circuit diagram is shown in Figure 1. Recall that the current waveform can be written as \[{i}_{s}{\left({t}\right)} = \frac{\left[{V}_{P}{V}_{Q}\right]}{{\omega}{L}}{\left[\begin{array}{c}{1}{-}{\cos}\,{\omega}{t} \\ {\sin}\,{\omega}{t}{-}{\omega}{t}\end{array}\right]} \tag{1} \]
Figure 1. The rectifier circuit diagram. When f = 50 MHz, L = 50 nH, and Ro = 50 Ω, the diode’s on-duty cycle becomes approximately 50%.
along with the voltage relations \begin{align*}{V}_{Q} & = {V}_{o} \tag{2} \\ {V}_{P} & = \frac{\pi}{2}{V}_{Q} \tag{3} \end{align*} from the previous two puzzles [1], [2].
We now focus on the output port. Thanks to the smoothing capacitor C∞, the diode current is(t) is fully averaged for one cycle and yields the dc output current Vo/Ro. From (1), the time average of is(t) results in \begin{align*}{\frac{{V}_{o}}{{R}_{o}}} & = \frac{1}{T} \mathop{\int}\nolimits_{0}\nolimits^{\frac{1}{2}T}{i}_{s}{\left({t}\right)}{dt} \\ & = \frac{\left[{V}_{P}{V}_{Q}\right]}{{4}{\pi}{\omega}{L}}{\left[\begin{array}{c}{{2}{\pi}} \\ {4}{-}{\pi}^{2}\end{array}\right]}{.} \tag{4} \end{align*}
Note that the integral is not carried out for the entire period T but only for its first half-cycle when D stays on.
For the three equations (2), (3), and (4) to simultaneously have a nonzero solution, we reach a quite explicit formula \[{R}_{o} = {\pi}{\omega}{L} = {2}{\pi}^{2}{fL}{.} \tag{5} \]
Therefore, the correct answer to last month’s quiz is (c). If we approximate π 2 ≈ 10, we can quickly calculate Ro ≈ 50 Ω when f = 50 MHz and L = 50 nH. This makes an instructive numeric instance of (5) easy to remember.
Looking at the simple circuit diagram, one may have supposed that the diode always works at a 50% duty cycle. However, this is not true. As derived in the aforementioned solution, the duty cycle becomes 50% only when (5) is satisfied. The resistance Ro specified by (5) is called the nominal load for half-wave rectification.
[1] T. Ohira, “Rectifier output voltage,” IEEE Microw. Mag., vol. 24, no. 3, p. 99, Mar. 2023, doi: 10.1109/MMM.2022.3226632.
[2] T. Ohira, “Half-wave rectification [Enigmas, etc.] ,” IEEE Microw. Mag., vol. 24, no. 5, p. 149, May 2023, doi: 10.1109/MMM.2023.3242520.
Digital Object Identifier 10.1109/MMM.2023.3277407