Oscillator 67
Besides a wide FTR, inductive tuning can also pr ovide the benefit of isolated
DC noise from the tuning element.
The loads on transformer for inductive tuning can be categorized into
three types: resistor [131], capacitor [147], and inductor [145]. Wide FTR is
then achieved by controlling the value of the load. However, traditional loaded
transformer topologies suffer from various limitations. For example, resistor-
loaded transformer has a nonlinear tuning-curve with larg e effective K
V CO
,
which can make PLL difficult to lock [13 1]. Capacitor-loaded tr ansformer s uf-
fers from a narrow FTR due to the limited tuning range and poor quality factor
of the varactor at high frequency region [147]. Inductor-loaded transformer re-
quires the use of multiple number o f transformers, which constrain the effective
number of sub-bands due to layout size and design complexity [145].
4.2 Frequency Tuning by Loaded Transformer
4.2.1 Inductive Tuning Analysis
The mechanism of lo aded transformers applied for inductive tuning can be
explained by Figure 4.1. The loaded transformer is utilized to tune the effective
inductance (c
eff
) in a LC-tank, while C
t
consists of the total capacitance in
the LC-tank. Note the 3 types of loaded transformer s can all be approximately
equalized to a RC tank and analyzed with the same equivalent circuit as shown
in Figure 4 .1.
The transformer is as sumed to be ideal with coupling factor k, and with
L
1
and L
2
as the primary inductance and secondary inductance, re spectively.
The equivalent circuit w ith l
eq
and R
eq
can then be calculated as
L
eq
= L
1
×
R
2
[1 − ω
2
CL
2
(1 − k
2
)]
2
+ ω
2
L
2
2
(1 − k
2
)
2
R
2
(1 − ω
2
CL
2
) [1 − ω
2
CL
2
(1 − k
2
)] + ω
2
L
2
2
(1 − k
2
)
R
eq
=
R
2
L
1
[1 − ω
2
CL
2
(1 − k
2
)]
2
+ ω
2
L
1
L
2
2
(1 − k
2
)
2
Rk
2
L
2
.
(4.1)
Thus the oscillation frequency becomes
Figure 4.1: Equivalent circuit model f or inductive tuning of loade d
transformer.
68 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
ω =
1
p
L
eq
C
t
. (4.2)
For a resistor or inductor-loaded transformer, the FTR of the equivalent
circuit can be estimated by considering the two extreme conditions of R in
Figure 4.1:
(
L
eq max
= L
eq
(R → ∞) = L
1
×
1−ω
2
CL
2
(1−k
2
)
1−ω
2
CL
2
L
eq min
= L
eq
(R → 0) = L
1
1 − k
2
.
(4.3)
By substituting (4.3) into (4.2), the FTR for LC-tank oscillation frequency
can be obtained:
ω
min
= ω (R → ∞) =
r
ω
2
1
+ω
2
2
−
√
(ω
2
1
+ω
2
2
)
2
−4ω
2
1
ω
2
2
(1−k
2
)
2(1−k
2
)
ω
max
= ω (R → 0) =
ω
1
√
1−k
2
, (4.4)
where ω
1
=
1
√
L
1
C
t
and ω
2
=
1
√
L
2
C
. As s hown in Figure 4.1, ω
1
and ω
2
represent the resonant frequencies at the primary side and the secondary side
of the transformer, respectively.
Note that ω
1
is pre-determined by pa rameters o f the transformer and the
LC-tank, while ω
2
would be affected by the load. By defining ω
2
= αω
1
, whe re
α>0 is the ratio between two re sonant frequencie s, we can further analyze the
value based on different α va lue s. Since
∂ω(R→∞)
∂α
stays positive for all α values,
by taking the extreme conditions for α, the range for can be estimated as
ω (R → ∞) ≈
ω
2
, 0 < α ≪ 1
ω
1
, α ≫ 1.
(4.5)
According to (4.5), when ω
2
is much higher than ω
1
or equals ω
1
, indicat-
ing negligible dep endence between value of ω (R → ∞) and the load. However,
as ω
2
drops below ω
1
, ω (R → ∞) is dec reased, approaching the value of ω
2
in-
stead. This is actually tie mechanism for frequency-tuning of capacitor-loaded
transformer.
The effect of ω
2
value on the quality factor for the effective LC-tank must
be considered, which can be easily derived from (4.1) as
Q
eq
=
R
eq
ωL
eq
=
R
ωL
2
×
1 −
ω
2
ω
2
2
[1 −
1 − k
2
ω
2
ω
2
2
]
k
2
+
ωL
2
R
×
1 − k
2
k
2
.
(4.6)
Note that here the loss fro m the transformer and the LC-tank is not in-
cluded in the calculation and Q
eq
quantifies the additional loss coupled from
transformer load into the LC-tank. As (4.6) shows, this coupled loss is con-
tributed by two portions. When R >> ωL
2
, the first item on the right-side
Oscillator 69
of the equation dominates. When R << ωL
2
, the second item dominates.
Clearly, in the case of ω (R → ∞), the first item should be considered. Un-
fortunately, as ω
2
drops, it approaches the value of ω
2
, forcing Q
eq
degrade
toward 0, which indicates a high degradation on the phase noise performance.
As a result, the approach of lowering ω
2
value for larger FTR, suffers sig-
nificant phase noise degradation. In fact, this is also one limitation for the
capacitor-loaded transformer.
Therefore, for a resistor or inductor-loade d transformer, the condition
ω
1
≪ ω
2
is required in design optimization. The FTR can then be estimated
by
FTR =
ω
1
√
1−k
2
− ω
1
ω
1
√
1−k
2
+ ω
1
× 2 =
1 −
√
1 − k
2
1 +
√
1 − k
2
× 2. (4.7)
According to (4.7), to achieve a large FTR, a large k is required. Moreover,
according to (4.6), Q
eq
approaches infinity when R approaches 0 or infinity,
but drops when R moves from the two boundaries. This explains the per-
formance degradation for resistor-loaded transformer since its major tuning
region locates away from these two boundaries.
4.2.1.1 Model of I nductor-Loaded Transformer with Switches
The three types of loaded transformer (r esistor, capac itor and inductor) are
shown in Figure 4.2. In Section 4.2.2 .1, they a re analyzed by the same equiv-
alent circuit shown in Figur e 4.1. Since both resistor and capacitor-loaded
transformers have the loading at one fixed lo cation on the secondary coil of
the transformer, their cir cuit behavior can be fully emulated by the same
equivalent circuit shown in Figure 4.1. The inductor-loaded transformer, on
the other hand, has switches located on several different locations on the sec-
ondary coil. When one part of the secondary c oil is turned on and plays the
majo r role in determining the effective inductance on the primary coil, the
remaining part of the secondary coil can still affect the performance due to
parasitic effect, which is ig nored by the semilar equivalent circuit in Figure
4.1. As a result, one more comprehensive circuit model is developed in Fig-
ure 4.3 that can provide a more comprehensive model for an inductor-lo aded
transformer.
Figure 4.3 s hows the circuit models for an traditional inductor-loaded
transformer. Three inductors (L
1
, L
2
, and L
3
) are used to form the simplest
top ology, and are coupled with each other by the mutual inductances M
12
,
M
13
, and M
23
. The terminal voltage and loop current for each inductor are
represented by (V
1
, I
1
), (V
2
, I
2
), and (d
3
, I
3
), respectively. The loaded in-
ductance is varied by switching on different combinations of L
2
and L
3
, with
their switches represented by (R
2
, C
2
, i
2
n2
) and (R
3
, C
3
, i
2
n3
), respectively.
The resulting L
eq
and R
eq
can then be calculated by solving the following
70 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 4.2: Traditional resistor-, capacitor-, and inductor-loaded
transformers.
Figure 4.3: Equivalent circuit model for conventional inductor-loaded
transformer.
equations:
I
1
= V
1
×
1
R
eq
+
1
sL
eq
+
i
neq
V
1
= sL
1
I
1
+ sM
12
I
2
+ sM
13
I
3
V
2
= sM
12
I
1
+ sL
2
I
2
+ sM
23
I
3
V
3
= sM
13
I
1
+ sM
23
I
2
+ sL
3
I
3
V
2
+
R
2
I
2
1+sR
2
C
2
=
R
2
i
n2
1+sR
2
C
2
V
3
+
R
3
I
3
1+sR
3
C
3
=
R
3
i
n3
1+sR
3
C
3
(4.8)
where an e quivalent noise curre nt (i
2
eq
) pa rallel to the LC tank can b e trans-
ferred from the noise components of the switches.
4.2.1.2 Switch Design Parameters
To have the performance analysis in (4.8), we need to extract and optimize
switch parameter s for performance.
Oscillator 71
Figure 4.4: Equivalent circuit model for switches. R
on
, C
on
and i
2
n
on
form the equivalent circuit of switch when it is turned on. R
off
, C
off
and i
2
n
off
form the equivale nt circuit of switch when it i s turned off.
Table 4.1: Switch Parameters Extracted from CMOS 65 nm Technol-
ogy
W
s
(µm) 10 20 50 100 200 400
R
on
(Ω) 41.4 20.5 8.16 4.07 2.04 1.02
R
of f
(Ω) 2.72K 1.36K 543 272 136 67.9
C
on
(fF) 7.78 15.6 38.9 77.8 156 311
C
of f
(fF) 6.10 12.2 30.5 61.0 122 244
As shown in Figure 4.4, the switch state parameter in (4.8) is approxi-
mated by a RC-tank, where R
on
and R
off
are used to represent its effective
resistance for the on and off states. Further more, the effective capacitances are
represented by C
on
and C
off
. One can build an FC-library model for switches
by extracting RC-values base d on the 65-nm CMOS technology by sweeping
as listed in Table 4.1. No te that since the minimum length of the switch is
used to minimize parasitic c apacitance, the size of the switch is determined
by its width W
s
.
For the switch noise parameter in (4.8), the gate noise (
v
2
gn
) and channel
noise (
i
2
chn
) of the switch are transformed to equivalent current noise so urces
(
i
2
n
on
and i
2
n
off
) that a re added to the RC tank in para llel. Since the trans-
former filters out the low-frequency noise, flicker noise is not considered in the
model for simplicity. As a result, the noise sources (
i
2
n
on
and i
2
n
off
) of the
switch are estimated at thermal noise in equivalent model.
4.2.2 Inductor-Loaded Transformer by Switching
Return-Path
According to Figures 4.1 and (4.1), there are 4 variables: R, C, L
2
, and k on
the secondary coil of the tr ansformer, which control the oscillation frequency.
Resistor-lo aded transformer tunes R , capacitor-loaded transformer tunes C,
while inductor-loaded transformer tunes both L
2
and k. Conventionally, L
2
and k are tuned by switching o n different combinations of transformers [145].
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