236 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
where I
x
is the equivalent induced AC in SRA in response to the AC input
I
i
, and σ
2
x
is the varianc e of I
x
.
As dis cussed in [257], for a typical ra mp- damping function with normalized
ramping slope of k, we have
I
x
=
I
0
ω
0
σ
s
2
, σ
2
x
=
Nω
2
0
E
g
2
(10.8)
where σ
s
=
q
2Q
0
ω
0
k
is the SRA time constant with a unit of s/
√
rad, E
g
=
σ
s
√
π is the energy of density function, and N is the noise power density with
N = 4K · T · F/ R. Note that K and F denote the Boltzmann constant and
noise factor of SRA contributed by active devices, respectively.
As such, the sensitivity of SRX can be found by substituting (10.8) into
(10.7):
S = 2KT F
s
kω
0
πQ
0
. (10.9)
Note that the receiver noise figur e (NF) can be approximated as [89]:
NF =
S
K · T · B
. (10.10)
Note that the NF of an SRX is independent of quench signal. For a typical
3-dB bandwidth of the SRX (B = 1.177Ω
0
), the NF becomes 0.958 F. In
addition, the noise equivalent power (NEP) can be calculated by S/
√
B:
NEP = 1 .38KT F
4
s
kω
0
πQ
0
. (10.11)
Note that k is usually determined by the freq ue nc y of the quench signal and
the sampling rate of one SRA. Therefore, it can be observed from (10.9) and
(10.11) that, for a given ω
0
and k, the s ensitivity and NEP are inversely
proportional to the square-root and the fourth-root of Q
0
, respectively. So
the resona tor with higher Q will significantly improve the sensitivity within
the interested bandwidth for imaging application.
10.3 Super-Regenerative Receiver by
SRR/CSRR Resonator
Two SRXs working at 96 GHz and 135 GHz are implemented in the CMOS
process to demonstrate the advantages of applying quench-controlled oscilla-
tors with metamater ial resonators in super-regenerative rec eivers (SRX). The
fundamentals o f quench-controlled oscillator design is introduced first.
Super-Regenerative Detection 237
Figure 10.4: Reflection coefficient of T-line loaded with CSRR unit-
cells.
10.3.1 Quench-Controlled Oscillation
10.3.1.1 High -Q Resonance with Standing Wave
In the practical on-chip resonator design with finite Q of SRR or CSRR,
the reflection c oefficient (|Γ|) depends on the number of cascading TL-SRR
or TL -CSRR unit-cells. Figure 10.4 shows the circuit-level simulation of TL-
CSRR at 96-GHz resonance frequency with following observations. First, the
reflection coefficient |Γ| is more sensitive to the cells number when Q is below
200. Second, |Γ| can be improved by cascading more unit-c ells.
10.3.1.2 Voltage Controlled Negative Resistance
The oscillation can be sustaine d by compensating the reflection loss (|Γ| < 1)
with a negative resistance. Similarly, a quench-contro lled oscillation can be
achieved by controlling voltage controlled negative resistance (VCNR), which
determines the instantaneous damping factor (ζ(t)) of (10.1) as discussed in
Section II. The sensitivity of SRX is also a function of ζ(t) that is de termined
by VCNR. Usually a cross-coupled NMOS pair is applied for the differential
negative resis tance design as depicted in Figure 10 .5, where the tail current
of the cross-coupled NMOS pair (I
D
) can b e quench-controlled by another
NMOS biased in the satura tion re gion. The equivalent differential negative
conductance between nodes “a” and “b” can be expressed as below by ne-
238 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 10.5: Reflection loss compensation by cross-coupled NMOS
pair with controlled tail current.
glecting the channel-length modulation
G
m
=
gm
2
2
=
I
D
2V
od2
(10.12)
where gm
2
and V
od2
are the transc onductance and overdrive voltage of cro ss-
coupled NMOS FETs, respectively. Note that I
D
can be obtained by
I
D
=
W
1
2L
1
µ
n
C
ox
V
2
od1
=
W
2
L
2
µ
n
C
ox
V
2
od2
(10.13)
where W
1
, L
1
and V
od1
= V
Q
−V
T
are the channel width, length and overdrive
voltage of tail NMOS; W
2
and L
2
and V
od2
are the channel width, length and
overdrive voltage of the cross-coupled NMOS pair; µ
n
C
ox
and V
T
are the
process related parameters. As such, (10.12) can be written as a function of
V
Q
by
G
m
=
µ
n
C
ox
(V
Q
− V
T
)
4
r
2W
1
W
2
L
1
L
2
. (10.14)
One can obser ve from (10.14) that G
m
is linearly controlled by V
Q
, of
which the slope is determined by the product of W
1
/L
1
and W
2
/L
2
. Note
that the oscillation starts when 1/G
m
< R and stops when 1/G
m
> R. As
such, (W
1
/L
1
)(W
2
/L
2
) must be large enough to satisfy the oscillation start
conduction (1/G
m
< R ). However, large W
2
/L
2
will introduce additional
parasitic capacitance, which will be counted into the res onator rank and reduce
the oscillation frequency. Moreover, in order to provide sufficient head room
for the cross-coupled NMOS pair, W 1/L1 is selected several times larger than
W 2/L2.
Super-Regenerative Detection 239
Figure 10.6: Layout for CMOS on-chip implementation of DTL-CSRR
for 96 GHz SRX.
10.3.2 SRX Design by TL-CSRR
10.3.2.1 Folded Differential T-Line Loaded with CSRR
TL-CSRR structure cannot be directly employed for the SRX design. Firstly,
the single-ended approach will bring large common-mode noise in the oscilla-
tor; secondly, cascading more unit-cells will increase area overhead. A folde d
differential T-line loaded with CSRR (DTL-CSRR) structure is proposed to
reduce area by half while doubling the number of unit-cells [258]. As shown
in Figure 10.6, two c ascaded TL-CSRR unit-cells (with CSRR size of 60 ×60
µm
2
) are folded in the two topmost metal layers (M6 and M7 ).
The S-parameters of the proposed DTL-CSRR structure is verified by EM
simulation tool E MX with a parasitic capacitance of 40 fF from transistors.
Both ε and µ of DTL-CSRR are extracted from the simulation re sults accord-
ing to (9.3), which both become complex numbers due to the existence of loss
factor induced imaginary parts. The metamaterial property is illustrated by
the rea l par ts of ε and µ in Figur e 10.7. At the vicinity of 105-GHz resonance
frequency, an electric plasmonic medium is formed with ε < 0 and µ > 0. A
stop-band is thereby formed w ithin a narr ow bandwidth of 1.8 GHz, where the
Q factor is found to be 58 by Q = ω
0
/ω
3dB
from the differential impedance
(Z
diff
) between P1 and P2.
10.3.2.2 9 6-GHz DTL-CSRR-Based SRX
Figure 1 0.8 depicts the schematic of DTL-CSRR-based SRX. DTL-CSRR is
firstly connected to a differential negative resistance formed by cross-coupled
NMOS (M2 and M3). To further improve the detection efficiency, a virtual
ground at 96 GHz is fo rmed by two λ/4 stubs. The size of M4 is designed
as 4 times of that M2 and M3. Note that W
T otal
and W
Single
are the total
240 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 10.7: EM-simulation-based comparison of DTL-CSRR and LC-
tank resonator for CMOS 96 GHz SRX de sign.
and individual finger width of transistors in Figure 10.8, r espectively. And the
channel length of every active de vice is 60nm. The remaining circuit consists
of a common source input buffer (M1) for current injection and an envelope
detector formed by M5 and M6. The common source sta ge (M1) is designed
for input signal injection and a lso reverse isolation from the oscillator to the
input. The size o f M1 is optimized with consideration of minimized parasitic
capacitance as well as the input matching. Similarly, M5 and M6 als o need to
be minimized but doing so will reduce the detection efficiency. To solve this
problem, a capacitance coupling by C1 and C2 is introduced between the out-
puts of the o scillator tank and the e nvelope detection. Firstly, the capacitance
loading from M5 and M6 is reduced by series c onnection of the coupling capac-
itors; se condly, M5 and M6 are biased externally by large resistors (R1, R2)
to optimize the detection; thirdly, 1/f noise fro m M5 and M6 is also isola ted.
10.3.3 SRX Design by TL-SRR
10.3.3.1 Diff e rential T-Line Loaded with SRR
The TL-SRR structur e with horizontal placement of SRRs (Figure 2.13) is also
not suitable for the practical implementation for SRX, mainly due to the larg e
area overhead. Compared to TL-CSRR, TL-SRR inherently has better layout
flexibility because SRRs can be vertically stacke d within a compact area. One
differential T-line loaded with stacked SRRs (DTL-SRR) is proposed in this
work for the application of 135 GHz SRX design in the 65nm CMOS RF
process.
As shown in Figure 10.9(a), the DTL- SRR is designed by stacked SRRs
with the same dimensions of 24 × 24 µm
2
in 4 metal layers (M5 to M8). All
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