7.4 Implementation
7.4.2 Comparison and Conclusion
A comparison of presented power amplifier with previous work on mm-wave power amplifiers (mostly in silicon) is summarized in table 7.1. Among other amplifiers on silicon substrate this work demonstrates the highest achieved center frequency of operation (85GHz) and the highest achieved power output (120mW) at this frequency.
It is noteworthy that the relatively low efficiency of the amplifier is due to the low efficiency of the drivers as we used class A distributed amplifiers. The peak efficiency of the power combiner is around 70% and we could use different classes of drivers to increase the power efficiency by sacrificing some bandwidth.
Power-meter
Agilent E4418B
Signal Generator
20 GHz
Frequency Quadrupler (Spacek Labs
AE-4XW)
WR-12
waveguide WR-12
waveguide Attenuator
HP 83650B
Chip
Picoprobe
Power Sensor HP W8486A Power Supply
Vcc Vee
Figure 7.11: Measurement setup
F req. Device Pout(dBm) P AEmax(%) Gain(dB) Ref.
85GHz 0.12µm SiGe HBT 20.8 4 8 This work
77GHz 0.12µm SiGe HBT 17.5 12.8 17 [80]
77GHz 0.12µm SiGe HBT 13 3.5 6.1 [76]
60GHz 0.12µm SiGe HBT 16 4.3 10.8 [81]
90GHz 0.12µm GaAs pHEMPT 21 8 19 [82]
Table 7.1: A comparison between this work and other designs
Power supply wire-bonds
Ground supply wire-bonds RF probe
Chip under the test Brass substrate
Printed circuit board (PCB)
CVD diamond
Figure 7.12: The power amplifier chip under test
72 74 76 78 80 82 84 86 88 90 6
6.5 7 7.5
8 8.5
9
Figure 7.13: Measured small signal gain of the amplifier
70 75 80 85 90 95 100
16.5 17 17.5 18 18.5 19 19.5 20 20.5 21
Frequency (GHz)
Power (dBm)
Low Gain Pout, measured with BWO (dBm)
Pout, measured with Multiplier (dBm)
Figure 7.14: Measured peak output power of the amplifier
-10 -5 0 5 10 15 -5
0 5 10 15 20 25
0 1 2 3 4 5 6
Figure 7.15: Large signal behavior of the amplifier at 85GHz
Chapter 8
Nonlinear Resonance in
Two-Dimensional Electrical Lattices
8.1 Introduction
Consider an electrical lattice comprised of inductors and capacitors, as shown in Fig- ure 8.1. Suppose that voltages are applied at the boundaries, producing two or more
Figure 8.1: 2-D Nonlinear Transmission Lattice
wave fronts that propagate towards the center of the lattice. In this chapter, we show that for certain configurations of inductors and voltage-dependent capacitors, the incoming waves combine nonlinearly, producing a single outgoing wave with peak amplitudegreater than the sum of the incoming waves’ amplitudes. Through numer- ical experiments, we shall examine how this resonant wave interaction is affected by varying the incoming waves’ amplitudes, phases, and frequency content. In all cases, we are concerned with the regime in which (1) the ratio of amplitude to wavelength is large and (2) the ratio of lattice spacing to wavelength is non-negligible.
The lattice in Figure 8.1 is the natural generalization to two spatial dimensions of the classical one-dimensional transmission line shown in Figure 8.2. It has been
Figure 8.2: 1-D Nonlinear Transmission Line
known [38][39][40][4] since the 1960s that the presence of voltage-dependent capacitors in these one-dimensional structures leads to nonlinear wave phenomena, including the formation of solitons. As a result, nonlinear transmission lines (NLTLs) have been studied by various groups [45][46][43][47][48] with a focus on solitonic generation of ultrashort, high-power, stable electrical pulses. Recent developments [56; 55] have demonstrated that NLTLs are suitable for a variety of ultra-wideband pulse-shaping applications, and that they can be built inexpensively on silicon chips.
As early as the 1940’s, L´eon Brillouin analyzed wave propagation in two-dimensional linear lattices [33]. In contrast, two-dimensional nonlinear electrical lattices have not received as much attention, and we are aware of only three works on this subject other than our own. These papers were concerned with establishing that 2-D soliton for- mation was possible, either through experiments [83][84] or through weakly nonlinear asymptotics [50]. For the remainder of this chapter, we label inductors and capacitors in the 2-D lattice as shown in Figure 8.3. In [50], the authors assume a uniform, non-
Figure 8.3: 2-D Nonlinear Lattice Block
linear 2-D lattice with Lij = L and Cij(V) = C(V) everywhere; they show that the Kadomtsev-Petviashvili (KP) equation describes weakly nonlinear wave propagation in such lattices. In [68], it is shown how to chooseLij and Cij nonuniformly in space, for both linear and nonlinear lattices, in order to design circuits that focus power from different input signals. Let us quickly review the mathematical modeling of these 2-D lattices; a detailed description of what follows may be found in [68].
Using Kirchoff’s laws, one writes equations for the current I and voltage V in the lattice:
Ii,j−1/2+Ii−1/2,j−Ii+1/2,j−Ii,j+1/2 = d
dt(Cij(Vij)Vij) (8.1a) Vij −Vi,j−1 =−Li,j−1/2 d
dtIi,j−1/2 (8.1b) Vij −Vi+1,j =Li+1/2,j d
dtIi+1/2,j (8.1c)
HereLij andCij are prescribed, while Iij and Vij are unknown functions of time only.
System (8.1), with appropriate boundary conditions that we will make precise later, is what we solve numerically in order to study wave interactions in 2-D electrical lattices. In order to obtain analytical insight into the problem, we may examine various continuum limits of (8.1).
From the lattice equations to the KP equation. Starting from (8.1), we may outline one possible path to the KP equation. For now we takeLij =LandCij(V) = C0(1− bV) everywhere, and consider a continuum model of (8.1) for the case of an equispaced lattice, where h is the distance between any two adjacent nodes. We switch fromL andC to, respectively, the inductance and capacitance per unit length L0 =L/h and C00 = C0/h. Next, we decouple (8.1) and obtain a single second-order equation for the unknown Vij:
1
h2 (Vi,j−1−2Vij +Vi,j+1+Vi−1,j−2Vij +Vi+1,j) =L0C00
"
(1−bVij)d2Vij
dt2 −b dVij
dt 2#
. (8.2) Standard Taylor series∗ then allows us to approximate the discrete functionVij(t) by a smooth functionV(x, y, t), and then rewrite the left-hand side of (8.2) as follows:
∇2V +h2
12(Vxxxx+Vyyyy) = L0C00
(1−bV)Vtt−b(Vt)2
. (8.3)
Here and in what follows, subscripts denote partial derivatives. Starting from the continuum model (8.3), we now seek an equation that describes, asymptotically, small- amplitude two-dimensional long waves propagating through the lattice. To derive such an equation, we set ν0 = (L0C00)−1/2 and introduce scaled variables
ξ=ε1/2(x−ν0t), η=εy, T = 1
2ν0ε3/2t. (8.4)
∗Inherent in this application of Taylor series is the assumption that h/λ is sufficiently small, whereλis a characteristic wavelength.
We insert (8.4) into (8.3), expand V = εV1 +ε2V2 +· · ·, and drop all terms except those of lowest order in ε. The scaling (8.4) selects certain waves; specifically those that propagate mainly in the x-direction with the y-direction treated as a perturba- tion. The expansion/truncation of V is valid only if A/λ, the ratio of amplitude to wavelength, is sufficiently small. The result of this procedure will be the Kadomtsev- Petviashvili (KP) equation †:
h
(V1)t+V1(V1)x+ (V1)xxxi
x
+ (V1)yy = 0. (8.5) Equation (8.5) is a weakly nonlinear limit of the continuum lattice model (8.3).For more details, please consult [68].
Remark. Starting from (8.1), suppose we choose Lij and Cij such that they are not constant functions of i and j. Following the same procedure as outlined above, the final equation will be KP plus two extra terms that involve the spatial derivativesLx and Ly; again, see [68] for full details.