Coupled Oscillator Network 107
5.2 In-Ph ase Signal Generation by MPW
In this work, a magnetic plasmon waveguide (MPW) with zer o phase propa-
gation is introduced in the coupling network design with 2k/N = 0 to largely
improve the output power within compact are a. A zero phase propagation is
not only important for power combination but also the phase noise reduction.
The noise coupling network becomes r eciproc al in zero-phase mode, and the
phase noise at CON output becomes 1/N of a single free-running oscillator
[77].
The MPW unit-cell can be implemented on-chip by coupled T-line-based
resonator with C contributed by the parasitic capacitances of transistors as
shown in Figure 5.1(a). The two-port Y-parameters for a c onventional coupled
T-lines structure can be expressed as [171]:
Y 11 Y 12
Y 21 Y 22
=
"
−j(Y
0o
+Y
0e
) cot θ
2
−j(Y
0o
−Y
0e
) cot θ
2
−j(Y
0o
−Y
0e
) cot θ
2
−j(Y
0o
+Y
0e
) cot θ
2
#
(5.1)
where Y
0o
, Y
0e
denotes the odd-mode and even-mode admittance, respectively;
θ = βl is the electrical length of the coupler.
When two identical capacitors (C) are introduced on both sides of the
coupler, the two-port Y-parameters becomes:
Y 11
′
Y 12
′
Y 21
′
Y 22
′
=
"
−j(Y
0o
+Y
0e
) cot θ
2
+ jωC
−j(Y
0o
−Y
0e
) cot θ
2
−j(Y
0o
−Y
0e
) cot θ
2
−j(Y
0o
+Y
0e
) cot θ
2
+ jωC
#
.
(5.2)
Equation (5.2) can be converted into S-parameters according to the
Figure 5.1: (a) Equivalent circuit of differential ZPC loaded with par-
asitic capacitance; (b) on-chip realization of inter-digital coupled T-
lines.
108 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
metho d introduced in [172];
S11 S12
S21 S22
=
"
1+Z
2
0
(Y
0o
cot θ−ωC)(Y
0e
cot θ−ωC)
∆Y
jZ
0
(Y
0o
−Y
0e
) cot θ
∆Y
jZ
0
(Y
0o
−Y
0e
) cot θ
∆Y
1+Z
2
0
(Y
0o
cot θ−ωC)(Y
0e
cot θ−ωC)
∆Y
#
(5.3)
with
∆Y = Z
2
0
[ωC(Y
0o
+ Y
0e
) cot θ − ω
2
C
2
− Y
0o
Y
0e
cot
2
θ]
+ jZ
0
[2ωC − (Y
0o
+ Y
0e
) cot θ] + 1. (5.4)
As such, the coupling phase (φ) can be expressed as:
φ =
π
2
−tan
−1
2ωZ
0
C − Z
0
(Y
0o
+ Y
0e
)cotθ
1 + Z
2
0
[ωC(Y
0o
+ Y
0e
) cot θ − ω
2
C
2
− Y
0o
Y
0e
cot
2
θ]
. (5.5)
When the impedance of both ends are perfectly matched (Z
2
0
Y
0o
Y
0e
= 1), a
zero coupling phase condition (φ = 0) is s atisfied in (5.5) with
cot(βl) −
ωC
Y
0e
cot(βl) −
ωC
Y
0o
= 1. (5.6)
The required physical length l can be derived a s
l =
1
ω
√
µε
cot
−1
ωC(Y
0o
+ Y
0e
)
2Y
0o
Y
0e
+
s
1 +
ωC(Y
0o
− Y
0e
)
2Y
0o
Y
0e
2
. (5.7)
Note that (5.7) is obtained as the minimum p ositive solution of (5.6), which
provides the smallest feature size of zero-phase-coupler in the practical IC
layout.
With an inter-digital configur ation layout, a s shown in Figure 5.1(b), lower
coupling loss can be achieved [173]. The coupling coefficient in zero-phase
mode |S21
ZP
| can be derived from (5.3 ) and (5.6):
|S21
ZP
| =
(Y
0e
cot θ − ωC) − (Y
0o
cot θ − ωC)
(Y
0e
cot θ − ωC) + (Y
0o
cot θ − ωC)
, (5.8 )
which can be optimized for the start-up condition by a higher odd-mode ad-
mittance (Y
0o
) or a lower even-mode admittance (Y
0o
).
Clearly, a low loss can be obta ine d by a much smaller physical length l and
a lower Y
e
for multiple ZPC-based oscillator unit-cells under the zero-phase
condition. Compared to the conventional coupler design with T-line by single
strip on e ach side, the proposed ZPC structure can simultaneously increase
Y
0o
and reduce Y
0e
, which can be further optimized based on the relation of
coupler length l vs. loaded capacita nc es and even-mode admittance as shown
in Figure 5.2. The coupling loss needs to be compensated to start os cillation,
which means |S21
G
m
| · |S21
ZP
| > 1, where S21
G
m
is the equivale nt gain of
shunt negative conductance from active devices.
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