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No credit card required 254 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Substituting Z
RLC
(s) into equation (11.6), the transfer function of the
proposed SRX can be expressed as
Z
NT V
(s, t) =
Z
0
ω
0
s
s
2
+ 2ζ
n
(t)ω
0
s + ω
2
0
(11.7)
where the new damping function ζ
n
(t) becomes
ζ
n
(t) = ζ
0
[1 G
m1
(t) R
1 + e
jϕ
]. (11.8)
Note tha t the absolute value G
m2
(t) is equal to G
m1
(t), and a phase
diﬀerence ϕ is introduced due to the phase diﬀerence from the injected signals.
Therefore, when the damping signal is a ramping signal with slope β, the
damping function becomes
ζ
n
(t) = 1 (1 + e
jϕ
)βt.
As a result, the gain function µ
n
(t) and the sensitivity function g
n
(t)
become
µ
n
(t) = κe
1
2
ω
0
β(1+e
)t
2
(11.9)
g
n
(t) = κe
1
2
ω
0
β(1+e
jϕ
)t
2
. (11.10)
One can observe that the gain and sensitivity functions are both inﬂuenced
by the phase diﬀerence of the injected signal between two oscillators. When
the phase diﬀerence becomes zero, both the gain and the sens itivity functions
can be optimized.
We further compare the gain function and sensitivity function of the con-
ventional SRX w ith that of the proposed ZPS-coupled SRX by
U
C
=
µ
n
(t)
µ (t)
= e
1
2
ω
0
βt
2
(11.11)
G
C
=
g
n
(t)
g (t)
= e
1
2
ω
0
βt
2
. (11.12)
One can observe tha t the gain of the SRX enhancement is exponential
with ω
0
. When a signal frequency around ω
0
is injected into LC- tank-I, it is
ampliﬁed and injected into LC-tank-II in phase. Then, it is further ampliﬁed
by L C -tank-II and re -injected into LC-tank-I. Thus, a po sitive feedback loop is
established when in-phase coupling is realized by the ZPS, where the oscillator
ampliﬁcation gain is increased with the improved detection sensitivity.
11.3 Circuit Prototyping and Measureme nt
11.3.1 SRX Circuit Design
The schematic of the proposed SRX is shown in Figure 11.3. It consists of
two ZPS-coupled LC-tank resonators, one common source input buﬀer and In-Phase Detection 255
VDD
M
1
M
1
M
2
V
TUNE1
V
quench
M
2
L
Z
L
Z
C
Z
L
T
L
T
ZPS
LNA with
input matching
network
LC Tank-I
LC Tank-II
V
B
Output
R
D
R
B
M
5
Detector
V
B
M
5
L
1
L
2
V
G
M
3
M
3
M
4
M
4
M
3
M
3
R
B
Signal input
C
A
C
A
CON
M
dummy
M
6
V
TUNE2
Figure 11.3: Circuit diagram of propos ed SRX with ZPS-coupled os-
cillators.
one output e nvelope detector. Relatively small sized (2µm/100nm) NMOS
transistors (M3) are c onnected in both oscillator tanks, working as var actors
for freq ue nc y tuning. By tuning control voltages VTUNE1 and VTUNE2,
the process mismatch in two LC tanks is well cancelled to make sure that
the free-running fr equencies of two tanks are the same. As a result, CON
synchronization ma inly dep ends on the coupling network, which is ensured
by the ZPS given in this paper. The quench-controlled transco nductances are
implemented by cross-coupled transistor pairs (M
1
and M
2
), of which the tail
current is controlled by M
4
. Note that M
1
and M
2
have an identical size of
60nm length and 12µm width, and M4 has a size of 60nm length and 6 0µm
width. The input of LC-tank-I is connected to a common source buﬀer (M
6
),
of which the input is matched to 50 Ω by L
1
and L
2
. A dummy transistor
(M
dummy
) is introduced to compensate para sitic capac itor unbalance. The
output of LC-tank-II is connected to a diﬀerential envelope detector (M
5
).
The design of a pa ssive part such as ZPS  is shown in Figure 11.4. The
inductor is implemented in the top metal layer. The radius of the inductor

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