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Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits with Metamaterials by Yang Shang, Hao Yu

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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
×
1ω
2
CL
2
(1k
2
)
1ω
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
(1k
2
)
2(1k
2
)
ω
max
= ω (R 0) =
ω
1
1k
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

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