Signal growth in a pure time-modulated transmission line and the loss effect
Abstract
We present the first comprehensive study for signal growth in transmission lines (TL) with purely time-modulated characteristic impedance Zo (infinite superluminality). This study pioneers the investigation into the effects of varying the cell’s electrical length and the impact of loss on momentum band gaps and amplification levels. It also thoroughly examines how time-modulated transmission line truncation by a static load influences the sensitivity of amplification gain to the relative phase between the incoming signal and modulation, comparing these findings to the case of parametric amplification. Varying Zo is accomplished by loading TLs with a sinusoidally time-modulated capacitor (TMC). The study starts with a simple lumped model cell to facilitate understanding of the phenomena. Following this, transmission lines are introduced, and the effects of incorporating loss are examined. To accomplish this, three models are investigated: a lossless L-C TL lumped model loaded with a shunt lossless TMC and a TL loaded with a shunt lossless and lossy TMC. Dispersion diagrams are plotted and momentum bandgaps are identified at a modulation frequency double the signal frequency. Within the momentum bandgap, only imaginary frequencies are found and correlated to momentum bandgap width and signal growth level. Signal growth is confirmed using harmonic balance and transient simulations, and the results are consistent with the dispersion diagram outcomes.
The dataset encompasses various studies on the simulated transient output voltage and power for 9-unit cell transmission lines, each featuring a purely time-modulated characteristic impedance, ( Zo ). The modulation of ( Zo) is achieved by loading the transmission lines with a sinusoidally time-modulated capacitor (TMC). [Datasets: contain the Excel files generated from the transient time simulator and plotted in the manuscript (named according to the manuscript)].
Variables used in the dataset:
time: Time used in the transient simulations with an interval of 400 ns and time step 0.01 ns.
Po/Pout: Transient output power of the circuit.
Vo/TRAN.Vo: The circuit’s transient output voltage.
Pin: Transient input power input to the circuit.
Description of the data and file structure
Plot_ID: identifier for study plot
Fig.3(b): For 9 unit cells with 10 V input peak (1 Watt (rms)), time-modulated capacitor at MD = 0.66, L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency) loaded with a 50 Ω load
impedance. [ Output voltage (transient)].
Fig.4(a): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. [ Output voltage (transient) at time-modulated capacitor at MD = 0.5].
Fig.4(c): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. [Output voltage (TS) at time-modulated capacitor with MD = 0.33].
Fig.5(a): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. (a) Input instantaneous power (rms) in the absence of modulation (transient).
Fig.5(b): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. Transient simulation output power (rms) with the time-modulated capacitor at MD = 0.66.
Fig.5(c): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. Transient simulation output power (rms) with the time-modulated capacitor at (c) MD = 0.5.
Fig.5(d): For 9 unit cells with 10 V input peak (1 Watt(rms)), L = 12.5 nH, Co = 5 pF, Fs = 0.5 GHz (main signal frequency), Fm = 1 GHz (modulation frequency), loaded with a 50 Ω load impedance. Transient simulation output power (rms) with the time-modulated capacitor at MD = 0.33.
Fig.8(a): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and the TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For 0.1λm (m = 0.1) TL: output voltage (transient).
Fig.8(b): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and the TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For 0.15λm (m = 0.15) TL: output voltage (transient).
Fig.8(g): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and the TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For 0.1λm (m = 0.1) TL: output power (transient).
Fig.8(h): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and the TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For 0.15λm (m = 0.15) TL: output power (transient).
Fig.10(a): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and a 0.15λm (m = 0.15) TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For Rc=5 Ω: output voltage (transient).
Fig.10(b): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and a 0.15λm (m = 0.15) TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For Rc=15 Ω: output voltage (transient).
Fig.10(g): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and a 0.15λm (m = 0.15) TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For Rc=5 Ω: output Power (transient).
Fig.10(h): For 9 unit cells with 10 V input peak (1 Watt (rms)), Co = 4 pF Fs = 0.5 GHz, Fm = 1 GHz, MD = 0.58, and a 0.15λm (m = 0.15) TL with 90 Ω characteristic impedance (Zo) and different lengths loaded with a 50Ω load impedance. For Rc=15 Ω: output Power (transient).
Variable used to define the datasets:
MD: Modulation depth of capacitance time modulation.
Fs: Input signal frequency
Fm: Modulation frequency.
Co: Nominal capacitance.
λ: wavelength at input signal Frequency.
λm: wavelength at the modulation frequency.
Rc: Series resistance connected to the modulated capacitor.
Ω: Ohm.
rms: root mean square.
TL: transmission line.
Code/Software
This folder contains two main directories: one for MATLAB codes related to all proposed models, which plot the real and complex frequency dispersion as well as the loss dispersion, and another for the simulator file (Advanced Design System) that validates the results obtained from the dispersion plots.
Matlab files:
This folder contains all the MATLAB scripts for the proposed models, which are used to plot real and complex frequency dispersion, as well as loss dispersion. Each folder contains the eigenshuffle.m script, which arranges the eigenvalues resulting from the solution of the eigenvalue problem.
A.1. lumped model (Section 2(a))
L_C_Beta.m: Plots the real dispersion for the modulated TL lumped model.
L_C_imagfre.m: Plots the complex dispersion for the modulated TL lumped model.
A.2. Lossless and lossy TL (Section 2(b))
TL_RC_Beta.m: Plots the real dispersion for the modulated TL loaded with a shunt capacitor (lossy or lossless, depending on the value of Rc). TL_RC_imagfre.m: Plots the complex dispersion for the modulated TL loaded with a shunt capacitor (lossy or lossless, depending on the value of Rc).
TL_RC_ALFA.m: Plots the loss dispersion for the modulated TL loaded with a shunt capacitor (lossy or lossless, depending on the value of Rc).
Simulator File:
This folder includes the Advanced Design System (ADS) files that validate the results obtained from the dispersion plots.
The datasets are generated by simulating various types of 9-unit cell transmission lines, each featuring a purely time-modulated characteristic impedance ( Zo ). The modulation of ( Zo ) is achieved by loading the transmission lines with a sinusoidally time-modulated capacitor (TMC). These simulations are conducted using the Keysight Advanced Design System software.