We report the experimental observation of the generation of continuous electrical waves in a
switch line, which is a transmission line periodically loaded with electronic switches. The oscillating
motions of a wave front have been experimentally demonstrated in a line with discrete Esaki diodes
employed as switches, when a rising step-pulse signal was passed through the line.

A switch line [1, 2], defined as a lumped transmission line
containing a series resistor, a shunt capacitor, and a shunt switch (switch
open for voltages greater than a fixed threshold; switch closed otherwise) in
each section, is first discussed by Richer in 1966. We investigated a switch
line for use in high-speed electronics, and found an interesting propagation
property of the wave front observed when a step pulse is input: the wave front
travels forward at the beginning, and then returns to the input port [3]. The returned wave is
reflected at the input port; therefore this oscillating behavior continues
indefinitely. By extracting a part of these waves from a switch line, we can generate an
oscillatory electromagnetic wave. The oscillation frequency is determined by
the round-trip time of the traveling wave; thus, it can be increased by
decreasing the traveling distance or increasing the wave velocity, as long as
it is not limited by the cut-off value of the operation frequency of the loaded
switches.

Figure 1(a)
shows two sections of a switch line, where L and R are the series inductor and series resistor of
the unit section, respectively. A shunt switch is modeled by a capacitor C and a bias-dependent conductance is placed in
parallel. The ideal current-voltage relationship is shown by the solid curve in
Figure 1(b). We can then write the current through the switch as GVθ(Vth−V),
where θ(V) shows the Heaviside function (θ(x)=1 for x>0; θ(x)=0 otherwise). The current-voltage relationship
of the typical tunneling diodes is shown by the dotted curve in Figure 1(b). If
the voltage range that exhibits the negative differential resistance becomes
narrow and if the thermal currents at the greater voltages than the valley are
negligible, the tunneling diodes such as the Esaki diodes and the resonant
tunneling diodes (RTDs) can be good candidates for the switches. We define two
dimensionless variables α and β for convenience:α=R2CL,β=G2LC. Hereafter, we
call the voltage range greater (less) than Vth region I (II). Then, the voltage wave is
influenced by the finite shunt conductance in region II and not by the
conductance in region I. We consider the situation when the input rising
step-pulse crosses Vth as shown in Figure 1(c). When the wave front
travels forward, it is expected that an exponential wave develops in region II,
while the ordinary sinusoidal wave develops in region I. By examining the
dispersion relationship, the exponential mode in region II can travel forward,
when αβ>sinh2k2, where k represents its wave number normalized by the
inverse of the length of the unit cell. On the other hand, the sinusoidal mode
in region I has to satisfyα<2sink2.By combining (2) and (3), we
obtain the necessary condition of the development of the sinusoidal-exponential
wave:β>β0≡1αsinh2[sin−1(α2)].This exotic pulse is not stable;
therefore it becomes attenuated with transmission and finally disappears. At
this point, a stable dynamical-steady-state (dss) pulse develops at the point
where the forward pulse disappears, and starts to travel backward [4]. This process is illustrated
in Figure 2. The waveforms in Figure 2 result from a time-domain
finite-difference calculation of a switch line with ideal electronic switch.
The spatial position on the line is shown on the horizontal axis, and the
voltage is shown on the vertical axis. Figure 2(a) shows the behavior of the
forward pulse. The forward pulse combining a sinusoidal mode in region I and an
exponential mode in region II is not stable; therefore it becomes attenuated
and finally disappears, so that only the exponential pulse edge is left in
region II, as shown in Figure 2(b). This edge develops an exponential mode in
region I to form a dss pulse, and then it starts to travel backward as shown in
Figure 2(c). When the backward dss pulse reaches the input end, it is reflected
as shown in Figure 2(d), so that it again starts to travel forward as an
unstable sinusoidal-exponential wave. The cycle continues indefinitely with
suitable boundary conditions at the input port (the input requires the
lower-impedance termination for time-invariant voltage supply), thus succeeding
in the generation of continuous electromagnetic waves.

Definitions.
(a) Equivalent representation of switch line. L: series inductance, R: series
resistance, C: shunt capacitance, G: shunt conductance simulating loaded
switch; (b) current-voltage relationship of shunt switch; (c) pulse input to
the line (the voltage range greater (less) than Vth is called region I (II)).

Oscillating behavior of a pulse traveling on a switch
line. (a) Forward-going quasisteady pulse; (b) pulse at the turning point; (c)
backward-going dss pulse; (d) reflection at input port.

Because the
increase in R contributes to the decrease in the round-trip
distance, the oscillation frequency becomes higher for larger R.
The increase in G has the same effect. Moreover, the larger part
of the pulse edge lies in region II for larger Vth.
Therefore, the increase in Vth may also decrease the round-trip distance. On
the other hand, when the amplitude of the input pulse is increased, the round
trip time increased; as a result, the oscillation frequency, defined as the
inverse of the period of the oscillating pulse edge, decreases. This simple
relationship allows the oscillation frequency to be widely tuned by varying the
voltage amplitude of the input pulse. In contrast to the conventional
oscillator, the device does not include any LC tanks for resonance that limit
the oscillation bandwidth. By proper design of loss elements such as electrode
loss and switch conductance and/or signal application, we can obtain an
oscillator having the required oscillation frequency with rather a wide tuning
range.

Although we
have to allow for the presence of finite conductance in regions I and II, the
shunt electronic switches can be designed using the devices that exhibit
negative differential resistance, such as Esaki diodes or RTDs [5, 6], the peak voltages of which
function as Vth of the switches. Although the above discussion
is based on ideal switch lines, the same properties have been established for
the switch lines implemented with these tunneling devices through SPICE- and
finite-difference-time-domain-based calculations [7].

In the
experiments, we used a switch line whose unit section is represented by Figure 1(a). Thirty-six sections were devised. The circuit was built on a standard
bread board. The shunt electronic switches were constructed with NEC 1S1763
Esaki diodes. The peak current and voltage, the latter of which corresponds to Vth in the ideal switch model, are typically 6.0 mA and 60 mV, respectively. Moreover, the typical parasitic capacitance is 30.0 pF. Series inductances, resistances, and shunt capacitances were implemented
using 1.0μH inductors (TDK SP0508), 1.0Ω resistors (Tyco Electronics CFR25J), and 470 pF capacitors (TDK FK18C0G1), respectively. The size of the unit cell is about
10 mm. The design parameters using these components are estimated as L=1.0μH, C=500 pF, R=1Ω (α=0.02), and G=0.1 S (β=4.47). Because the condition β>β0 is satisfied for the present parameters, the
oscillating motion of the wave front is expected. The test switch line was fed
with the signal generated by an Agilent 81150A function generator. The signals
along the test switch line were detected and monitored in the time domain using
an Agilent DSO90254A oscilloscope. In order to monitor the waveforms using the
oscilloscope, we input a pulse having finite duration instead of a step pulse.
The input pulse had rise and fall times of 1.0μs with a duration of 30.0μs. The output impedance of the function
generator was set to 50Ω.

Figure 3 shows
voltage waveforms monitored at the first cell. Figure 3(a) shows a measured
waveform resulting from a single sweep with the oscilloscope. Although the
input signal was a simple pulse, we can observe a cycle of short-period pulses.
Because of the weakness of the coherence between the input step pulse and the
short-period pulses, the temporal positions of short-period pulses varied for
different single sweeps. We numerically solved the following transmission
equation of the test switch line by the standard finite difference method
[8]:LdIndt=−RIn+Vn−Vn+1,CdVndt=−Idio(Vn)+In−1−In,where In and Vn are the current and voltage at the nth cell. The Esaki diodes are modeled by the
voltage-controlled current source Idio,
whose current-voltage relationship is given by the data sheet of 1S1763. The
variation of parameters of each device is neglected. The resulting waveform at
the first cell is shown in Figure 3(b). Although the calculation does not
simulate the amplitude and period exactly, it qualitatively characterizes the
measured short-period pulses. By examining the numerically obtained waveforms
at various cells, we see that the generation of short-period pulses results
from the oscillating propagation of the edge of the input step pulse. The
measured waveform of the short-period pulse is typically shown in the inset of
Figure 3(a). It exhibits a trapezoid-like shape with oscillation. When the
pulse edge passes a point, the voltage at the point remains constant until the
edge returns. Moreover, the forward edge is carried by the
sinusoidal-exponential hybrid mode. These observations are consistent with the
measured pulse shape.

Measured waveforms monitored at first cell. (a)
Waveform measured using a single sweep with the oscilloscope (the inset shows
the typical waveform of the short-period pulses); (b) the waveform calculated
by numerically solving the transmission equations of the test switch line.

Figure 4(a)
shows the measured waveform monitored at the first cell, which was averaged
1024 times with the oscilloscope. As mentioned above, the short-period pulses
have weak coherence with the step input. Through the waveform averaging, we can
extract the coherent part. It is interesting to note that the averaged
amplitude of the short-period pulse becomes smaller, that is, the degree of
decoherence becomes greater, with time after the rising edge of the input step
pulse. By using the coherent parts, it is possible to compare the phase of the
voltage waveform at a cell with that at the other cells. The spatiotemporal
distribution of the averaged voltage wave is shown in Figure 4(b). The
waveforms at the first twenty cells having a 7.5μs duration are plotted. The edge first travels
forward. Then, at the turning point Pi (i=1,2,3,4) in Figure 4(b), the wave edge
starts to travel backward. The backward wave front is reflected at the input
port, starts to travel forward again, and reaches the turning point Pi+1.
This spatiotemporal voltage distribution clearly shows that the oscillating
motion of the edge is established in the test switch line.

Coherent part of measured waveforms. (a) The waveform
measured by averaging 1024 times with the oscilloscope; (b) spatiotemporal
voltage distribution. Waveforms measured at the first twenty cells with a
duration of 7.5μs are plotted.

In conclusion,
we successfully demonstrated the generation of electrical continuous waves
using switch lines. We believe that further investigations result in the
quantified characterization of the oscillating pulse edge, such that we can
manage this oscillating phenomena as we need. Our approach could be also scaled
from its current MHz form into microwave, millimeter-wave, and terahertz form,
when implemented with the state-of-the-art tunneling devices such as InP RTDs.

RicherI.The switch-line: a simple lumped transmission line that can support unattenuated propagationScottA.NaraharaK.narahara@yz.yamagata-u.ac.jpElectromagnetic continuous-wave generation using switch linesNaraharaK.narahara@yz.yamagata-u.ac.jpYamakiT.TakahashiT.NakamichiT.Characterization of voltage-controlled oscillator using RTD transmission lineSugiyamaH.hiroki.sugiyama@ntt-at.co.jpMatsuzakiH.OdaY.YokoyamaH.EnokiT.KobayashiT.Metal-organic vapor-phase epitaxy growth of InP-based resonant tunneling diodes with a strained In0.8Ga0.2As well and AlAs barriersOrihashiN.HattoriS.SuzukiS.AsadaM.Experimental and theoretical characteristics of sub-terahertz and terahertz oscillations of resonant tunneling diodes integrated with slot antennasNaraharaK.narahara@yz.yamagata-u.ac.jpYokotaA.Full-wave analysis of quasi-steady propagation along transmission lines periodically loaded with resonant tunneling diodesPaulC. R.