IEEE Solid-States Circuits Magazine - Summer 2019 - 27

noise and metastability generates a
PMF very close to the convolution of
the noise-only and metastability-only
PMFs. This tells us that random errors
due to noise and metastability effectively combine in an additive way, just
as in synchronous architecture. For
the practitioner, it's safe to approximate the combined PMF by convolving
the metastability-only PMF with the
noise-only PMF. We described how to
calculate the metastability-only PMF

errors greater than X to be smaller
than 10 -Y , what value of N do I need?"
Figure 12 answers this question for a
7-b ADC with ENOB = 6. As with the
synchronous architecture, the probability of errors greater than some
value has a thermal noise asymptote
at large N. Starting slow and then
speeding up the ADC (i.e., decreasing
N ), we eventually hit a corner where
metastability begins to cause errors
more significant than the thermal

previously, and the noise-only PMF
can be approximated by referring
comparator noise to the input of the
SAR ADC and finding the probability
mass that falls into each code (i.e.,
quantizing a Gaussian). With these
approximations, we can compute the
code error PMF of an asynchronous
SAR ADC with practical resolution in
a few minutes.
Lastly, let's return to the question "If I want the probability of code

P

X

10-15

2

10-15

2

4

6

8

Probability Code Error ≥ X

10-10

10

N
TFIX/t = 12

100
10-5
10-10
10-15

2

4

6
N

8

10

6,

7,

8,

9

10

3

X=5

1

4

5

N
(a)

6

7

10-5
10-10
10-15

2

4

6

8

10

N
TFIX/t = 16

100
10-5
10-10
10-15
10-20

9

TFIX/t = 4

100

10-20

8

Probability Code Error ≥ X

1

X=4
X=5
X=6
X=7
X=8

10-5

=

X=4

=

,1

Probability Code Error ≥ X

Probability Code Error ≥ X
Probability Code Error ≥ X

X

TFIX/t = 0

100

10-20

ta

10-10

10-20

10-20

me

10-5

Probability Code Error ≥ X

Probability Code Error ≥ X

100

2

4

6

8

10

10

TFIX/t = 8

100
10-5
10-10
10-15
10-20

2

4

2

4

100

6
N
TFIX/t = 20

8

10

8

10

10-5
10-10
10-15
10-20

N

6
N

(b)
FIGURE 12: A set of plots showing the probability of code errors exceeding a certain amount (X) as a function of N for a 7-b ADC with
ENOB = 6. Given a desired code error ceiling and the tolerable probability of crossing it, these plots allow the designer to read off the required
regeneration time in units of comparator time constants. (a) TFIX /x = 8, computed using the combined model for metastability and noise. The
overall rate of metastable events (Pmeta) versus N is superimposed in dashed red. (b) TFIX /x = 0, 4, 8, 12, 16, and 20, approximated by convolving the metastability-only PMF with the noise-only PMF.

IEEE SOLID-STATE CIRCUITS MAGAZINE

SU M M E R 2 0 19

27



IEEE Solid-States Circuits Magazine - Summer 2019

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Summer 2019

Contents
IEEE Solid-States Circuits Magazine - Summer 2019 - Cover1
IEEE Solid-States Circuits Magazine - Summer 2019 - Cover2
IEEE Solid-States Circuits Magazine - Summer 2019 - Contents
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