digital compensation of dynamic acquisition errors at the front-end of ...

30 Chapter 2: Nonlinearity at the ADC’s Front-End
a linear function with memory which is still compatible with our initial model
(equation (2.8)).
Vin ( k)
= fnonlin
( Vout
( k))
× flin
( Vout
( k),
Vout
( k −1),
Vout
( k + 1),...) (2.10)
where f nonlin is the result of RC product and f lin is generated from different derivative
orders.
Depending on the initial voltages on the capacitors the response of this system to
a sinusoidal input voltage can be shown as
t
−
t
−
τ1 τ 2
Vout 2
= k1e
+ k2e
+ Asin(
ωt
+ ϕ)
(2.11)
where τ 1 and τ 2 are time constants of the circuit and k 1 and k 2 are defined based on
initial voltages and the input sinewave. This shows that the output voltage has a
transient response first, before reaching the steady-state sinewave response. If this
transient has not settled completely by the time of the sampling, it causes dependency
to V out (n-1) in the nonlinear part of the output voltage.
Now we can also add package parasitics to our model. The simplified circuit
model including wire-bond inductance and pad capacitances is shown in Figure 2.23.
The relation between V in and V out in this circuit can be modeled with a 3 rd order
differential equation as shown below
V
V
in
out 2
= V
out 2
+ ( R C
2
( t = 0) = V
20
2
+ R C
, V
1
out1
2
dV
+ R1C
1)
dt
( t = 0) = V
10
out 2
, I
l1
+ ( R R C C
1
2
( t = 0) = 0
1
2
2
d V
+ L1C
1
+ L1C
2)
dt
3
d V
+ ( L1C 1R2C2
)
dt
out 2
2
out 2
3
(2.12)
Again, in this equation memory comes from different orders of signal derivative and is
a separate factor from static nonlinearity. This can be used for simplifying the
distortion function into equation (2.10).