Super-Regenerative Detection 233
Recently, a metamateria l-based resonator has been explored in [254] to
improve the Q with compact area at the mm-wave frequency region. A split-
ring-res onator (SRR) can be des igned in the CMOS proces s top-metal layer
to provide nega tive permeability (µ) for mm-wave propagation. When loading
SRR to a host transmission-line (TL-SRR), the integrated structure becomes
a non-transmission medium with s ingle-negative property (µ · ε < 0) in the
vicinity of re sonance frequency. A sharp stop-band is thereby formed such
that the incident mm-wave can be perfectly reflected at SRR load with a
stable sta nding-wave esta blished in the host T-line [254]. Compared to the
traditional L C -tank-based resonator, TL-SRR has stable EM-energy storage
within a compact area, which results in a much higher Q factor. As such,
it becomes relevant to study the CMOS on-chip SRR for the compact and
high-sensitivity SRX design of the THz imager.
In this chapter, firstly, the design of on-chip metamaterial resonators is
explored based on differ ential T-line loaded with SRR and CSRR beyond 70
GHz. As a demonstration of the idea, two oscillators based on SRR and CSRR
resonators are implemented with 65nm CMOS process at 76 GHz and 96 GHz,
respectively. The s tate-of-the-art performance shows that the phase noise and
FOM of SRR achieves -108.8 dBc/Hz and -182.1 dBc/Hz (@10 MHz offset),
respectively. The power is reduced dramatically to 2.7 mW compared to the
existing designs on SWOs [246, 238, 239]. And the CSRR oscillator shows
a state-of-the-art pha se noise of -111.5 dBc/Hz (@10 -MHz offset) and FOM
of -182.4 dBc/Hz. Secondly, two super-regenerative receivers with quench-
controlled metamaterial high-Q oscillators by TL-C SRR and TL-SRR are
demonstrated with improved sensitivity over traditional LC-tank resona tor-
based designs at 96 GHz and 135 GHz, respectively. With a sharp stop-ba nd
introduced by the metamater ial resonators, high-Q oscillato ry a mplifications
are a chieved. The 96-GHz DTL-CSRR-based SRX has a compact core chip
area of 0.014 mm
2
, and it is measured with power consumption of 2.8 mW,
sensitivity of -79 dBm, noise figure (NF) of 8.5 dB, and noise equivalent power
(NEP) of 0.67 fW/
√
Hz. T he 135-GHz DTL-SRR-based SRX has a compact
core chip area of 0.0085 mm
2
, and it is measured with power consumption of
6.2 mW, sensitivity of -76.8 dBm, NF of 9.7 dB, and NEP of 0.9 fW/
√
Hz.
The proposed SRXs have 2 .8-4 dB sensitivity improvement a nd 6 0% smaller
core chip are a when compared to the conventional SRX with LC-ta nk-based
resonator at similar fre quencies.
10.2 Fundamentals of Super-Regenerat ive
Amplification
Generally, a SRX consists of a quench-controlled oscillator injected by an
external sig nal and an envelope detector. The process of injecting an external
signal into a quench-controlled oscillator is firstly reviewed to under stand the
operation of SRX, called super-regener ative amplification (SRA).
234 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 10.3: Simplified equivalent circuit model of super-regenerative
amplifier.
10.2.1 Equivalent Circuit of SRA
A simplified circuit model of SRA is shown in Figure 10.3. The resonator is
modeled by RLC block, and its oscillation is quench-controlled by a time-
depe ndent negative resistance −1/G
m
(t), where G
m
is the equivalent con-
ductance determined by the associa ted active dev ices. The external signal
injected is modeled as a time-dep endent current source I
i
(t). V
o
(t) is the out-
put voltage. The resonance frequency is ω
0
= 1/
√
LC; the quality factor is
Q
0
= R/Z
0
= 0 .5ζ
−1
0
; Z
0
and ζ
0
are the characteristic impedance and quies-
cent damping factor, r espectively.
Assuming G
m
(t) varies much slower than ω
0
such that a quasi-static con-
dition holds in the sys tem to have a time-varying transfer function in s-domain
by
V
o
(s, t)
I
i
(s)
=
Z
0
ω
0
s
s
2
+ 2ζ(t)ω
0
s + ω
2
0
, (10.1)
where ζ(t) = ζ
0
[1 − G
m
(t)R] is the instantaneous damping factor.
A second-order linear time variant system can be o bserved from (10.1 ).
By varying ζ(t), the pole can be shifted between left and right sides of the
s-plane per iodically. In other words, the oscillation starts in SRA w he n ζ(t)
is negative, and stops when ζ(t) is positive. Note that (10.1) is only valid
when SRA works in linear mode, such that V
o
(s, t) is small enough to prevent
significant distortion in each quench cycle. Generally, SRA working in linear
mode is pr eferred in the application of millimeter-wave imaging since it has a
better sensitivity than that in the logarithmic mode [255, 256].
After a Laplace trans form, (10 .1) can be written as a second-order differ-
ential equation in the time domain:
v
′′
o
(t) + 2ζ(t)ω
0
v
′
o
(t) + ω
2
0
v
o
(t) = Z
0
ω
0
I
′
i
(t). (10.2)
Assuming the oscillation is fully q ue nched in each cycle, such that v
o
(t) is
independent of the previous ones. For a particular quench cycle t ∈ (t
a
, t
b
]
with t
a
< 0 < t
b
, if ζ(t) is positive fo r t ∈ (t
a
, 0] and negative for t ∈ (0, t
b
],
Super-Regenerative Detection 235
(10.2) can be written as [256]:
v
o
(t) =
Z
0
s(t)
Z
t
t
a
I
′
i
(τ)s(τ)sin[ω
0
(t − τ)]dτ (10.3)
where s(t) = e
ω
0
R
t
0
ζ(λ)dλ
is called the sensitivity function, and it rea ches
maximum when t=0; and decays rapidly with t. As a res ult, the SRA is only
sensitive to the input I
′
i
(t) in the time window centered at t=0 when ζ(t)
turns from positive to negative.
10.2.2 Freq ue ncy Response of SRA
The frequency response of SRA c an be analyzed with a convolution model
[257]. For an AC input with I
i
(t) = I
0
sin(ω
i
t + φ
i
), the output waveform can
be approximated by
v
o
(t) ≈
Z
0
ω
i
I
0
2s(t)
|S(∆ω)|sin(ω
0
t + φ
i
) (10.4)
where ∆ω = ω
0
− ω
i
and S(ω) is the Fourier transform of s(t). In the appli-
cation of millimeter-wave imaging, we are more interested in the envelope of
v
o
, which is
Env[v
o
(t)] =
Z
0
ω
i
I
0
2s(t)
|S(∆ω)|. (10.5)
Assuming ω
i
is very close to ω
0
(∆ω << ω
i
), a quasi-static condition holds
in (10.5) that the frequency response of Env(v
o
(t)) is determined by |S(∆ω)|.
For a typical ra mping quench signal with time variant conductance G
m
=
1
R
(kt + 1), where k is the normalized ramping slope of G
m
with the unit of
1/s, the instantaneous damping facto r is ζ(t) = −
k
2Q
0
t. Thus the envelope of
v
o
(t) can be solved by
Env[v
o
(t)]
ramp
=
√
πZ
0
ω
i
I
0
Ω
0
e
Ω
2
0
4
t
2
e
−
∆ω
2
Ω
2
0
, (10.6)
where Ω
0
=
p
kω
0
/Q
0
is a constant that determines the frequency response
of SRA, e.g., the 3-dB bandwidth of SRA equals 1.177Ω
0
. As such, one can
observe that for the given k and ω
0
, the bandwidth is inversely proportional
to Q
0
.
10.2.3 Sensitivity of SR A
The sensitivity of SRA is defined as the minimum detected power, which
means the induced output signal power is the same as its variance:
S
SRA
= P
min
|
I
2
x
=σ
2
x
=
I
2
0
R
2
|
I
2
x
=σ
2
x
(10.7)
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