IEEE Solid-State Circuits Magazine - Fall 2016 - 88

A second important tradeoff in VCOs is the one
between phase noise and the tuning range,
which is the ratio between the frequency range
covered by the VCO and the average value of
this range.
Fourier coefficients). It is interesting
to note that the technique inherently
implies (see Figure 13) 1) the folding
of out-of-band noise, since a noise
components at n ~ 0 + ~ m will be
folded at ~ m by multiplication with
the C coefficients at n ~ 0; and 2) an
integration effect (also evident from
the impulse-to-step function link)
that, in the case of white noise, produces a 1/~ 2m shaped S U (~ m). From
the latter spectrum, the phase noise
L (~ m) will immediately follow.

The evaluation of the C for each
noise source is not straightforward;
a rigorous derivation for the two VCO
topologies discussed here was provided in [8] and [9]. For instance, consider again the impact of the channel
thermal noise. As a first approximation, we can imagine that the noise is
injected only at the zero crossing for
the same Tw used in Figure 9 (see Figure 14). Under this assumption, the
evaluation of C 2rms, thus of F, is easy.
For the differential pair, it is again

SnI

∞
a0
+ ∑ an cos(nω0t + θn )
Γ (t ) =
2 n=1

∆φ (t ) =

qMAX

SnI 0

∆q

t

1

ωm

0

Γ (ω0 τ ) ⋅ in(τ )dτ

-∞

a0

S
SΦ (ωm ) =

SnI 0
2
4qMAX

ω0
a1

2ω0
a2

3ω0
a3

2

2
SnI 0
a
. ∑ n =
. Γrms
2
2
2
ωm
2qMAX ωm

ω

ωm

Figure 13: The folding and integration of a generic white noise obtained from the ISF.

V (t )

A0
tt

2Vov
in /2

in /2
in

1/2

ΓMOS

t
T0

TW
2

F = 2⋅

Γrms, MOS
1/2

.

SnI, MOS
4kT/R

≅ 2.

(TW /2T0)
1/2

-1/2

.

4kTγ gm
4kT/R

=γ

Figure 14: The contribution of MOS thermal noise to F, evaluated by the ISF.

88

fa l l 2 0 16

IEEE SOLID-STATE CIRCUITS MAGAZINE

ω

F = c, which is the value obtained by
a rigorous analysis [8], [9]. Of course,
the same technique can be applied
to all the other noise sources of the
VCO, e.g., the noise from the tail.

Minimum F
We will see that the noise contribution from the tail can be, at least
in principle, eliminated. Instead, it
seems that it is not possible to reduce
F below c. One may wonder whether
the CMOS oscillator performs intrinsically better. From the analysis
in [9], it seems that the answer is
no. The minimum F is the average
between c PMOS and c NMOS (i.e., F = c,
assuming c PMOS = c NMOS = c) .
This can also be intuitively recalled
using the simplified model in Figure 14,
comparing the two VCOs featuring the
same tank, at the optimum bias point.
Also assume that the nMOS transistors'
(W/L)s are the same (the pMOS is scaled
proportionally to the ratio between
mobilities). If the single pair drains a
current IT, the CMOS VCO dissipates
I T /4 (half amplitude and doubled current efficiency), thus it features half
overdrive and half g m with respect to
its nMOS-only counterpart. It also follows that the injection time Tw will be
the same (in the CMOS, both the amplitude and the overdrive are halved
with respect to the single pair VCO).
Therefore, in this example, the noise
power injected by a single transistor is
halved in the CMOS VCO, but the number of noise sources is doubled.
As a matter of fact, an important general result has been demonstrated in [6] and [10], i.e., that under
some conditions commonly achieved
(e.g., the active element does not add
losses to the tank, the transistors
noise is proportional to gm, etc.), for
this topology of oscillators the minimum value for F is c. This result is
valid not only for MOS. For instance,
the equivalent c in bipolar transistors is 1/2, assuming the only noise
source is the collector shot noise of
the device, 2qIc . What is interesting
to note is that, at the first order, the
noise added by the active element
does not depend on transistors' size



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