3.4 Numerics
3.4.3 Results
First we simulate a linear exponentially tapered line. As predicted by the perturbative theory, we see two modes propagating inside an exponentially shaped envelope. As shown in Figure 3.2, the amplitude of the wave increases slowly as a function of element number.
Next we simulate both uniform and nonuniform NLTLs. In the nonuniform case,
we use the exponential tapering described above. We observe in Figure 3.3(a) that sinusoids are now converted to soliton-like waveforms. If we switch on nonuniformity, multiple pulse generation is suppressed, as shown in Figure 3.3(b). That is, fewer solitonic pulses are generated from the same incoming sinusoidal signal.
The nonuniformity also allows us to narrow the width of pulses considerably, as demonstrated in 3.4. Note that Figure 3.4 also shows that the resulting pulses are not symmetric, as predicted by theory. The asymmetry appears on the left (trailing) side of the pulse.
To summarize,
• The nonuniform linear transmission line can be used for pulse compression / voltage amplification. However, the frequency and speed of outgoing waves cannot be significantly altered using a linear circuit.
• The nonuniform nonlinear transmission line can increase both the voltage am- plitude and the frequency content of incoming waves.
In the next chapter, we generalize 1D transmission lines to 2D transmission lattices.
The extra dimension allows us to create a circuit that can simultaneously upconvert and combine incoming signals.
(a) Uniform
(b) Nonuniform
Figure 3.3: Voltage Vi versus element number i at various times for the (a) uniform NLTL, with b = 0.5,λ = 0, and (b) nonuniform NLTL, with b = 0.25, λ = 0.02. All other parameters are the same as in the linear case. The input frequency is α = 5 GHz.
Figure 3.4: Voltage Vi versus element number i for the 1D nonuniform NLTL, with parameters identical to the previous figure. The outgoing pulse has a larger ampli- tude and much smaller wavelength than the sinusoidal signal that enters at the left boundary.
Chapter 4
Theory of Two-Dimensional Transmission Lattice
In this chapter, the extension of 1D transmission line to two dimensions is considered.
For the description of long waves in a 2D lattice consisting of 1D lines coupled together by capacitors, one obtains a modified Zakharov-Kuznetsov (ZK) equation [57]. It should be mentioned that in §2.9 of Scott’s treatise[4], precisely this sort of lattice is considered, and a coupled mode theory is introduced. These lattices consist of weakly coupled 1D transmission lines, in which wave propagation in one direction is strongly and inherently favored.
When a small transverse perturbation is added to the KdV equation, one obtains a Kadomtsev-Petviashvili (KP) model equation. Dinkel et al. [50] carry out this procedure for a uniform nonlinear 2D lattice, and mention that the circuit may be useful for ”mixing” purposes; however, no physical applications are described beyond this brief mention in the paper’s concluding remarks.
In this work, for a linear nonuniform lattice, we write the continuum model and derive a family of exact solutions. A continuum model is also derived for the nonlinear nonuniform lattice. 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.
Furthermore, we present a variety of numerical results. We choose the inductance and capacitance of lattice elements in a particular way, which we call an electric lens
Figure 4.1: 2D transmission lattice
or funnel configuration. We solve the semi-discrete model of the lattice numerically, and show that the resulting solutions have physically useful properties. For example, our numerical study predicts that a linear nonuniform lattice can focus up to 70%
of the power of input signals with frequency content in the range 0-100 GHz. We present numerical studies of nonlinear lattices as well. In this case, power focusing is present alongside frequency upconversion, or the ability of the lattice to increase the frequency content of input signals. The numerical studies show that nonlinear nonuniform lattices can be used for wideband signal shaping applications.
4.1 Nonuniform Linear Case
Consider a section of the two-dimensional transmission lattice shown in Figure 4.1.
Using only regular polygons, there are three possible lattice blocks that can be used to tile the two-dimensional plane: triangular, rectangular, and hexagonal. Though the governing equations in each case will be different, at the continuum limit, they will have the same physical properties. Therefore, for mathematical simplicity, we analyze only the rectangular case. As in the previous section, we suppose that the
lattice is nonuniform, meaning
∇L(x, y)6=0, ∇C(x, y)6=0.
For now, we assume the lattice is linear:
∂C
∂V = 0, ∂L
∂I = 0.
Then Kirchoff’s laws yield the semi-discrete system:
Ii,j−1/2+Ii−1/2,j−Ii+1/2,j−Ii,j+1/2 =cijdVij
dt (4.1a)
Vij −Vi,j−1 =−`i,j−1/2
d
dtIi,j−1/2 (4.1b) Vij −Vi+1,j =`i+1/2,j d
dtIi+1/2,j (4.1c)
Differentiating (4.1a) with respect to time, we substitute (4.1b-4.1c), yielding Vij−Vi,j−1
`i,j−1/2
+Vij −Vi−1,j
`i−1/2,j
+ Vij −Vi+1,j
`i+1/2,j + Vij −Vi,j+1
`i,j+1/2 =−cijd2Vij
dt2 . (4.2) Taking the continuum limit in the usual way, we obtain the O(h0) lattice model
∇2V −LCVtt = ∇V · ∇L
L , (4.3)
where
∇2V =Vxx+Vyy. Or, if we keep all terms at order h2, we obtain
∇2V −LCVtt = ∇V · ∇L L
−h2 1
12(Vxxxx+Vyyyy)−1 6
LxVxxx+LyVyyy
L − 1
4
L2xVyy+L2yVxx L2
(4.4)
In the practical examples we consider, L will be a slowly varying function of both x and y, rendering negligible the terms involving squared derivatives of L, i.e., L2x/L2 and L2y/L2. Our O(h2) lattice model is
∇2V −LCVtt = ∇V · ∇L L − h2
12(Vxxxx+Vyyyy) + h2 6
LxVxxx +LyVyyy
L (4.5)
In this equation, the left-hand side is the normal wave propagation equation. On the right, the first term corresponds to the inhomogeneity, the second term is due to the discreteness and the last term represents both inhomogeneity and discreteness.