I would like to express my gratitude to those who helped me acquire this skill. I would also like to thank my mother, Tahereh, for her enormous impact on my life.
Contributions
Organization
Soliton
A soliton is a self-amplifying solitary wave caused by a delicate balance between nonlinear and dispersive effects in the medium. Ten years later, Zabusky and Kruskal [22] showed that this was correct and that it could be explained by solitons.
Wave Propagation in Periodic Structures
Fermi-Pasta-Ulam Experiment: Birth of Soliton
Ten years later, Zabusky and Kruskal [22] showed that for the FPU problem with cubic nonlinearity, the Korteweg-de Vries (KdV) equation [30] (discovered in 1895 by D. J. Korteweg and G. de Vries for modeling water waves) . in a shallow channel) represents a good approximation to the actual equations of motion of the system. In 1967, Gardner [31] showed that if the initial condition is localized, an analytical solution of the KdV equation can be found using the inverse scattering method.
Nonlinear Waves in Electronics
- Motivation
- Electrical Wave Propagation Medium
- Sources of Nonlinearity
- Historical Remarks
- Discreteness Generates Dispersion
- Traveling Wave Solutions
- Reduction to KdV
- Remark 1: Zero-Dispersion Case
- Remark 2: Linear Case
- Frequency Response
It is established that at t → ∞ the solution of the KdV equation consists of a system of interacting solitons. In this work, we try to analytically solve the frequency response problem only in the linear regime.
Nonuniform Linear 1D
- Linear case
- Physical Scenario
- Non-Dimensionalization
- Exponential Tapering
- Perturbative Solution
- Discussion
- General Case
From the setup of the problem, it is clear that the solution will consist of a wave train moving to the right at a certain finite speed. Finally, discreteness causes a decrease in the maximum speed of the wave train; it immediately follows from ω+/k < ω0/k.
Nonuniform Nonlinear Case
In practice, given particular values of all required constants, the solution can easily be found using Matlab. The now classic result[64] is that for λ small but positive, the usual soliton waveform of the KdV equation is modified by a shelf of height that follows the solitary wave.
Numerics
- Scheme
- Remark
- Results
- Large Lattice
- Lens/Funnel
- A Physical Scenario
- Exact Solutions
- Properties
- Discussion
In this case we apply the reductive perturbative method and show that a modified Kadomtsev-Petviashvili (KP) equation describes weakly nonlinear wave propagation in the lattice. Since (4.5) is linear, we expect that for small h the solution can be expanded into the form. Note that this is the precise impedance surface used in the following 2D numerical simulations.
Nonuniform Nonlinear Case
A general funnel-shaped inhomogeneity causes rays to bend; the solution will be of the form. Of course, (4.27) reveals a wealth of phenomena beyond the KdV equation, of which soliton resonance is perhaps the most intriguing from an application point of view.
Numerics
Linear case
As shown in Figure 4.3, the voltage increases as the wavefront moves to the right. Based on Figure 4.3, we see that for each fixed i, the voltage is constant over all j. However, the impedance is greater at the vertical sides (see Figure 4.2), so the current is smaller there.
Nonlinear Case
Remember that at t = 0 the power is distributed evenly, which means that the power fraction in the central element ati= 0 is exactly 1/M.). For the same simulation, we plot the current in figure 4.8 and observe the focusing of the current along the center line j =M/2. Focusing is not only sharp, but it happens faster than in the linear case.
One-Dimensional Transmission Line
- Continuum Model
- Schr¨ odinger Equation
- Exponentially Tapered Line
- Scattering
- Reflection Coefficient for Matched Case
Because the solution of (5.8) is known when L and C are constant, the anadiabatic approximation can be tried when Land Care slowly changes the position functions z. Here a1(ω) is the amplitude of the incident wave, b1(ω) is the amplitude of the reflected wave and c1(ω) is the amplitude of the transmitted wave. Note that the maximum value of the reflection coefficient is less than 10−13, so, for all practical purposes, we have 100% transmission of the input signal.
Two-Dimensional Transmission Lattice
- Continuum Model
- Electrical Funnel
- Electrical Lens
- Intuitive Explanation
- Pulse Degeneration Problem
Line dispersion, on the other hand, causes the waveform to spread out, as shown in the lower half of Figure 6.2. We note that the characteristic pulse width of the line is controlled by the node spacing, h, and the propagation velocity, ν, which in turn is controlled by L and C. A design consideration is that the characteristic impedance of each segment matches those of adjacent segments to avoid reflections.
Edge Sharpening Line
Intuitive Explanation
Initially, the voltage is low, which corresponds to a smaller capacitance in Figure 6.6 and thus a faster instantaneous propagation speed for the lower end of the pulse. This pushes the lower part of the transition forward in time and causes the rising edge to sharpen. The upper part of the transition (voltages above V2) will be accelerated due to the decrease in capacitance and will create an advancing front, as symbolically shown in the middle waveform of Figure 6.7.
The Effect of Loss
This way we can get a symmetric C(V) curve, but the capacity of each node would be twice as large, limiting the cutoff frequency of the line by a factor of √. Another limitation of this method is the additional parasitic capacitance to the substrate that can reduce the effective nonlinearity factor, b, of the capacitors. In both models, the numerical solution of the governing equations shows that loss has an effect similar to the spreading, meaning that loss causes the waveform to spread, thus to have a soliton pulse in a nonlinear transmission line with loss, non-linearity must be strong enough to cancel out both spread and loss.
Simulations
Pulse Narrowing Line
The inductances and capacitances within each segment are lower than those of the previous segment. The lines are designed so that the characteristic pulse width of each segment (given by equation 6.4) is half that of the previous segment, so that the line can compress at least the input pulse by a factor of sixteen without degenerate. in multiple pulses. The simulated line output waveform at a 65 ps wide input pulse is shown in Figure 6.12.
Edge Sharpening Line
The rise and fall times of the output pulses are different due to the asymmetric behavior of the nonlinear element for two different edges. These parasitic capacitors are voltage independent, therefore linear, and will result in an effective reduction of the non-linearity factor, b. There appears to be some data-dependent delay due to the non-linear behavior of the lines in the simulations (see Fig. 13).
Experimental Results
The amplitude response of the entire measurement setup is the product of Figure 6.15 and Figure 6.16. Based on the response of the measurement setup (Figure 6.15 and Figure 6.16), the response of the measurement setup to a 2.5ps pulse is 21.5ps wide. Most of the traditional power combining methods use either resonant circuits and are therefore narrowband or.
A variation of Electrical Funnel
We call this anelectric funnel because of the way it combines and channels the power in the center to the output. Efficiency is determined by the ratio of the power at the output node to the sum of the input powers. For other frequencies, the phase shift from input to output is different, resulting in a different focal length.
Power Amplifier Architecture
Driver Design
To obtain a wideband response, we use degenerate cascode distributed amplifiers with emitter degeneracy as input drivers, shown in Figure 7.6. The main advantage of cascode stage over single transistor is the higher maximum stable power gain. As Figure 7.7 shows, a non-degenerate cascode gain stage in this process has a maximum stable power gain of 15dB at 80GHz, as opposed to 7dB for a standard common emitter.
Implementation
- Measurement Results
- Comparison and Conclusion
- KP Resonance
- Resonance in Electrical Lattices
- Main Results
It is worth noting that the relatively low efficiency of the amplifier is due to the low efficiency of the drivers, since we used class A distributed amplifiers. The grid in Figure 8.1 is the natural generalization to two spatial dimensions of the classical one-dimensional transmission line shown in Figure 8.2. Since we are concerned with resonant wave interactions, which are impossible for KP-I (see Fig. 7.3 in [8]), we will only consider KP-II and refer to it as simply "KP" for the rest of this work.
Numerical Setup
Occasionally we consider different types of forcing applied to the same or different boundaries of the grid; when we do, we must mention it explicitly. In each numerical experiment, waves introduced into the lattice through boundary force interact and produce outgoing waves. We focus on the amplitudes of the outgoing waves as a function of the amplitudes/angles of the incoming waves.
Results of Numerical Experiments
Equal amplitude and in-phase
By repeating the experiment with different values of A, we study the dependence of the resonance amplitude AR on the input amplitude A. The ratio AR/Measures the efficiency of the combination; for linear combination, output amplitude AR = 2A always. Increasing the input voltage to A = 0.5 has no effect on the output voltage - at this point, the nonlinearity of the capacitors is saturated.
Unequal Amplitude and In-Phase
Recall from (8.6) that for KP soliton interactions, input signals with these wave vectors would combine to form an output signal with the wave vector k3 = (1,1), which is exactly what we observed in all the experiments we performed to generate Figure 8.5. Other functions in the odd-amplitude case are the same as the even-amplitude case. The net effect of the circuit is to generate concentrated soliton-like pulses from sinusoidal inputs.
Equal amplitude but out-of-phase
Practical Considerations
Double resonance
Non-sinusoidal inputs
We find that the network combines and converts the two input square waves into a single nonlinear pulse. Comparison with Figure 8.5 shows that the amplitude of the AR resonance output is smaller for square wave inputs than for sinusoidal inputs. The reason is that square wave inputs, unlike the sinusoidal inputs we considered earlier, have their energy spread across the entire Fourier spectrum.
Discussion
The quasi-continuum models consist of the continuum models plus higher-order dispersive corrections designed to account for the discreteness of the lattice. For clarity, in this work we use (1,1) and (N, M) to denote the lower left and upper right corners of the grid, respectively. Here α2 is a suitable scaling factor, yj is the jth component of the output vector y, and.
Methodology and Merits
The latency is calculated by multiplying the characteristic delay of the grid, τ, by the number of nodes in the horizontal direction. Note that the latency is independent of the carrier frequency ω, and that it grows linearly in the size of the input vector M. One does not have to wait for an input signal to travel all the way from the left boundary to the right boundary of the grid before a new, different input signal is injected.
Historical Remarks
Next, we assume that the spatial part of the diffraction field is a solution of the Helmholtz equation. Knowing the radially symmetric solutions of (9.1), together with a choice of boundary conditions for the field and its normal derivative at the slit opening, enables us to pass from the integral I(P0) to a diffraction integral. The application of Keller's elegant methods in the context of 2-D LC gratings must wait until a future publication.
Lattice Equations and PDE Models
- Kirchhoff’s Laws
- Continuum Limit
- Range of Validity
- Dispersive Correction
- Effect of Boundaries
As long as one wavelength of the lattice wave covers more than 25 lattice distances, the continuum dispersion relation (9.6) and PDE (9.7) are a decent approximation of the completely discrete dispersion relation (9.4) and differential equation (9.3). As long as lattice waves occupy at least seven lattice distances, the dispersion relation (9.9) closely matches the true dispersion relation (9.4). Using the full dispersion relation (9.4), we determine that this condition holds for plane waves traveling in the x direction when ω < 198GHz, as before assuming a uniform lattice with inductance L = 30pH and C = 20fF.
Refraction
- Snell’s law
- Thick Parabolic Lens
- Paraxial Approximation
- Numerics
The angle ˆθT is the angle of incidence for the refraction problem at the right boundary of the lens. This is a simple consequence of the fact that the right border of the lens is vertical. For these simulations, we have one (or more) vertical interface separating two (or more) mesh sections.
Diffraction
Kirchhoff
If we now make the Kirchhoff assumptions, then both U and ∂U/∂n are zero everywhere on S1 except inside Σ. The Kirchhoff assumptions continue: assume that, within Σ, both U and ∂U/∂n are the same as if there were no screen. That is, assume that U(P1) is the field due to a radially symmetric point source located at P2, where P2 is a point on the left side of the screen, as in Figure 9.10.
Rayleigh-Sommerfeld
This integral could of course be specialized to the case where P1 is illuminated by a radially symmetric point source located at P2, an arbitrary point on the left side of the screen.
Huygens-Fresnel
The difference between the r01 and r01−1 approximations is that the term O(y−ξ)2 appears in r−101 with an additional factor of x−2. Since x is assumed large compared to the wavelength, we keep the term O(y−ξ)2 only when r01 appears in the numerator and drop it whenever r01 appears in the denominator. If we can design a lens that cancels the eikxξ2 phase shift, then we have designed a 2-D LC grating that takes the spatial Fourier transform of an input signal.
Applications
Comments on the Implementation
Fourier Transform
The output is not just a focused version of the input, but a focused and diffracted version of the input. Some of the phase shift from the original Huygens-Fresnel diffraction integral is not completely canceled out in the tails. Turing, "The Chemical Basis of Morphogenesis," Philosophical Transactions of the Royal Society of London, vol.
Reflection coefficient as a function of the ratio ω/β , for 2βl = 1. Note
Three normalized soliton shapes for different values of L and C (a)
Output waveforms of the normal and gradual soliton line
Schematic of the gradually scaled non-linear transmission line
Schematic of an accumulation mode MOS varactor
Capacitance versus voltage for a MOSVAR
How rise and fall time vary within the NLTL
A proposed NLTL for symmetrical edge sharpening
Simple model of a lossy non-linear transmission line with series resisitor 90
Simulated output waveform of the pulse narrowing line using ADS
Simulated input and scaled output waveforms of the edge sharpening
The frequency response of the oscilloscope
The frequency response of the cables, connectors, and probes
Input and output of pulse narrowing line
Response of the measurement setup to an ideal input
Input and output waveforms of the edge sharpening line
Output waveforms of the edge sharpening line with different amplitude
Basic idea of a funnel
Combiner structure
The architecture of power amplifier
Cascode architecture
Maximum stable power gain of a cascode stage vs. a single transistor . 108
Simulated gain of each distributed amplifier
Chip micro photograph
Measurement setup
The power amplifier chip under test
Measured small signal gain of the amplifier
Measured peak output power of the amplifier
Large signal behavior of the amplifier at 85GHz
Setup for deriving Kirchhoff diffraction integral
Illumination by point source in Kirchhoff diffraction integral
Setup for Sommerfeld Green’s function
Huygens-Fresnel picture showing illumination on a line several wave-
Architecture
Results for two different numerical simulations of the 2-D LC lattice
Numerical simulation of the 2-D LC lattice (in black) as compared with
Sinc input for the 2-D LC lattice, corresponding to Equation (9.35) with
