Datasheet

AD8139
Rev. B | Page 19 of 24
APPLICATIONS
ESTIMATING NOISE, GAIN, AND BANDWIDTH
WITH MATCHED FEEDBACK NETWORKS
Estimating Output Noise Voltage
The total output noise is calculated as the root-sum-squared
total of several statistically independent sources. Because the
sources are statistically independent, the contributions of each
must be individually included in the root-sum-square calculation.
Table 6 lists recommended resistor values and estimates of
bandwidth and output differential voltage noise for various
closed-loop gains. For most applications, 1% resistors are
sufficient.
Table 6. Recommended Values of Gain-Setting Resistors and
Voltage Noise for Various Closed-Loop Gains
Gain R
G
(Ω) R
F
(Ω)
3 dB
Bandwidth (MHz)
Total Output
Noise (nV/√Hz)
1 200 200 400 5.8
2 200 400 160 9.3
5 200 1 k 53 19.7
10 200 2 k 26 37
The differential output voltage noise contains contributions
from the input voltage noise and input current noise of the
AD8139 as well as those from the external feedback networks.
The contribution from the input voltage noise spectral density
is computed as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
G
F
n
R
R
vVo_n1 1
, or equivalently, v
n
/β (7)
where v
n
is defined as the input-referred differential voltage
noise. This equation is the same as that of traditional op amps.
The contribution from the input current noise of each input is
computed as
Vo_n2 = i
n
(R
F
) (8)
where i
n
is defined as the input noise current of one input.
Each input needs to be treated separately because the two
input currents are statistically independent processes.
The contribution from each R
G
is computed as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
G
F
G
R
R
kTRVo_n3 4
(9)
This result can be intuitively viewed as the thermal noise of
each R
G
multiplied by the magnitude of the differential gain.
The contribution from each R
F
is computed as
Vo_n4 = √4kTR
F
(10)
Voltage Gain
The behavior of the node voltages of the single-ended-to-
differential output topology can be deduced from the previous
definitions. Referring to
Figure 59, (C
F
= 0) and setting V
IN
= 0,
one can write
F
ONAP
G
AP
IP
R
V
V
R
V
V
−
=
−
(11)
⎥
⎦
⎤
⎢
⎣
⎡
+
==
G
F
G
OPAPAN
RR
R
VVV
(12)
Solving the above two equations and setting V
IP
to V
i
gives the
gain relationship for V
O, dm
/V
i
.
i
G
F
dmO,
ONOP
V
R
R
VVV ==−
(13)
An inverting configuration with the same gain magnitude can
be implemented by simply applying the input signal to V
IN
and
setting V
IP
= 0. For a balanced differential input, the gain from
V
IN, dm
to V
O, dm
is also equal to R
F
/R
G
, where V
IN, dm
= V
IP
− V
IN
.
Feedback Factor Notation
When working with differential amplifiers, it is convenient to
introduce the feedback factor β, which is defined as
G
F
G
RR
R
+
=β
(14)
This notation is consistent with conventional feedback analysis
and is very useful, particularly when the two feedback loops are
not matched.
Input Common-Mode Voltage
The linear range of the V
AN
and V
AP
terminals extends to within
approximately 1 V of either supply rail. Because V
AN
and V
AP
are
essentially equal to each other, they are both equal to the input
common-mode voltage of the amplifier. Their range is indicated
in the
Specifications tables as input common-mode range. The
voltage at V
AN
and V
AP
for the connection diagram in Figure 59
can be expressed as
=
=
=
ACMAPAN
VVV
⎟
⎠
⎞
⎜
⎝
⎛
×
+
+
⎟
⎠
⎞
⎜
⎝
⎛
+
×
+
OCM
G
F
G
INIP
G
F
F
V
RR
RVV
RR
R
2
)(
(15)
where V
ACM
is the common-mode voltage present at the
amplifier input terminals.
Using the β notation, Equation 15 can be written as follows:
V
ACM
= βV
OCM
+ (1 − β)V
ICM
(16)
or equivalently,
V
ACM
= V
ICM
+ β(V
OCM
− V
ICM
) (17)
where V
ICM
is the common-mode voltage of the input signal,
that is, V
ICM
= V
IP
+ V
IN
/2.