ETC AB-076

®
NOISE ANALYSIS OF FET TRANSIMPEDANCE AMPLIFIERS
The availability of detailed noise spectral density characteristics for the OPA111 amplifier allows an accurate noise
error analysis in a variety of different circuit configurations.
The fact that the spectral characteristics are guaranteed
maximums allows absolute noise errors to be truly bounded.
Other FET amplifiers normally use simpler specifications of
rms noise in a given bandwidth (typically 10Hz to 10kHz)
and peak-to-peak noise (typically specified in the band
0.1Hz to 10Hz). These specifications do not contain enough
information to allow accurate analysis of noise behavior in
any but the simplest of circuit configurations.
where N rms is the rms value of some random variable n(t).
In the case of amplifier noise, n(t) represents either en(t) or
in(t).
R2
–
V
eO(t) = en(t) [1 + R2/R1] + In–(t) R2 + In+(t) R3 [1 + R2/R1]
FIGURE 2. Circuit With Error Sources.
The internal noise sources in operational amplifiers are
normally uncorrelated. That is, they are randomly related to
each other in time and there is no systematic phase relationship. Uncorrelated noise quantities are combined as rootsum-squares. Thus, if n1(t), n2(t), and n3(t) are uncorrelated
then their combined value is
Noiseless Amplifier
In– (t)
V
N TOTAL rms = N12 rms + N 2 2 rms + N 3 2 rms
Amplifier With Noise
TYPICAL APPLICATION
The circuit in Figure 3 is a common application of a low
noise FET amplifier. It will be used to demonstrate the
above noise calculation method.
FIGURE 1. Noise Model of OPA111.
If the instantaneous terms represent DC errors (i.e., offset
voltage and bias currents) the equation is a useful tool to
compute actual errors. It is not, however, useful in the same
direct way to computer noise errors. The basic problem is
that noise cannot be predicted as a function of time. It is a
random variable and must be described in probabilistic
terms. It is normally described by some type of average—
most commonly the rms value.
∆
N rms = 1 / T ∫ QT n 2 (t) dt
(2)
The basic approach in noise error calculations then is to
identify the noise sources, segment them into conveniently
handled groups (in terms of the shape of their noise spectral
densities), compute the rms value of each group, and then
combine them by root-sum-squares to get the total noise.
In+ (t)
1994 Burr-Brown Corporation
en (t)
In+ (t)
R3
+
en (t)
In– (t)
Noise in the OPA111 can be modeled as shown in Figure 1.
This model is the same form as the DC model for offset
voltage (EOS) and bias currents (IB). In fact, if the voltage
en(t) and currents in(t) are thought of as general instantaneous
error sources, then they could represent either noise or DC
offsets. The error equations for the general instantaneous
model are shown in Figure 2.
en (t)
en (t)
R1
R2
IIN
λ
CR1
eO
eO = IINR2
(1)
AB-076
1
FIGURE 3. Pin Photo Diode Application.
Printed in U.S.A. February, 1994
CR1 is a PIN photodiode connected in the photovoltaic
mode (no bias voltage) which produces an output current iIN
when exposed to the light, λ.
109
Transimpedance (V/A)
A more complete circuit is shown in Figure 4. The values
shown for C1 and R1 are typical for small geometry PIN
diodes with sensitivities in the range of 0.5 A/W. The value
of C2 is what would be expected from stray capacitance with
moderately careful layout (0.5pF to 2pF). A larger value of
C2 would normally be used to limit the bandwidth and
reduce the voltage noise at higher frequencies.
C2 = 1pF
R2 =
f2 = 15.9k
106
10
100
1k
10k
100k
1M
10M
FIGURE 6. Transimpedance.
C1
25pF
eO
VOLTAGE NOISE
Figure 7 shows the noise voltage gain for the circuit in
Figure 4. It is derived from the equation
In–


 A 
1 1 
eO = en 

 = en β 
1 + 1 
 1 + Aβ 
Aβ 

NOTE: In+ shorted in this configuration.
FIGURE 4. Noise Model of Photodiode Application.
(3)
where:
In Figure 4, en and in represent the amplifier’s voltage and
current spectral densities, en(ω) and in(ω), respectively. These
are shown in Figure 5.
A
β
= A(ω) is the open-loop gain.
= β(ω) is the feedback factor. It is the amount of output
voltage feedback to the input of the op amp.
Aβ = A(ω) β(ω) is the loop gain. It is the amount of the
output voltage feedback to the input and then amplified and returned to the output.
Figure 6 shows the desired “gain” of the circuit
(transimpedance of eO/iIN = Z2(s)). It has a single-pole rolloff
at f2 = 1/(2πR2C2) = ω2/2π. Output noise is minimized if f2
is made smaller. Normally R2 is chosen for the desired DC
transimpedance based on the full scale input current (iIN full
scale) and maximum output (eO max). Then C2 is chosen to
make f2 as small as possible consistent with the necessary
signal frequency response.
Note that for large loop gain (Aβ >> 1)
eO ≅ en
1
β
(4)
1,000
10,000
1,000
Current Noise (fA/√Hz)
Voltage Noise (nV/√Hz)
1
Frequency (Hz)
V
K1 = 80nV/√Hz
100
10
K2 = 6nV/√Hz
1
0
0.01
0.1
107Ω
en
R1
108Ω
107
105
0.01
Equivalent Circuit for CR1
IIN
108
0.1
1
10
100
1k
10k
100k
1M
100
10
K4 = 0.4fA/√Hz
1
0.1
0.01
0.01
10M
0.1
1
10
100
1k
Frequency (Hz)
Frequency (Hz)
(a) Voltage Noise
(b) Current Noise
FIGURE 5. Noise Voltage and Current Spectral Density.
2
10k
100k
1M
10M
For the circuit in Figure 4 it can be shown that
f3 = 80kHz
(1)
This may be rearranged to
1 R 2 + R1
=
R1 '
β
 τ As + 1 


 τ 2s + 1 
(5a)
where τa = (R1 || R2) (C1 || C2)
and τ2 = R2C2.
1
1
and f 2 =
2π τ a
2π τ 2
K2 [1 + (C1/C2)]
1000
100
10
156
88
K1 [1 + (R2/R1)]
6.6
K2 [1 + (R2/R1)]
.0098
0.01
0.01 0.1 1
K3 • K2
10 100 1k 10k 100k 1M 10M
Frequency (Hz)
FIGURE 8. Output Voltage Noise Spectral Density.
(5d)
The total rms noise at the amplifier’s output due to the
amplifier’s internal voltage noise is derived from the eO(ω)
function in Figure 8 with the following expression:
For very low frequencies (f << fa), s approaches zero and
equation 5 becomes
R
1
= 1+ 2 .
(6)
β
R1
E O rms = ∫ +– ∞∞ e O 2 (ω) dω
For very high frequencies (f >> f2), s approaches infinity and
equation 5 becomes
C
1
= 1+ 1 .
C2
β
(3)
0.1
(5b)
(5c)
Then, f a =
(2)
Regions
1
1  R1 R 2 
=
(C1 + C 2 )
β  R1 + R 2 
f2 = 15.9kHz
(5)
eO (ω) Voltage (nV/√Hz)
R ( R C + 1)
1
= 1 + 2 1 1S
R1 ( R 2 C 2S + 1)
β
fa = 673Hz
(8)
It is both convenient and informative to calculate the rms
noise using a piecewise approach (region-by-region) for
each of the three regions indicated in Figure 8.
(7)
Region 1; f1 = 0.01Hz to fC = 100Hz
 R 
f 
E n1 rms = K1 1 + 2  ln  C 
R
 f1 
1

10M
(9)
1M
 10 7 
100 
= 80nV / Hz 1 + 8  ln 

0.
01 
10


100k
A(ω)
Gain (V/V)
10k
f3 = 80kHz
1k
1/β Aβ
100
= 0. 267µV
1 + (C1/C2) = 26
fa = 673Hz
10
1
(9a)
This region has the characteristic of 1/f or “pink” noise
(slope of –10dB per decade on the log-log plot of enω). The
selection of 0.01Hz is somewhat arbitrary but it can be
shown that for this example there would be only negligible
Note that K1 (l + R2/R1) is the value of eO at f = 1Hz.
1 + (R2/R1)
f2 = 15.9kHz
0.1
0.01
K3 = 1.63 x 10–3 = (1Hz/fa) [1 + (R2/R1)]
0.001
0.01 0.1
1
10 100 1k 10k 100k 1M 10M
Frequency (Hz)
Region 2; fa = 673Hz to f2 = 15.9kHz
FIGURE 7. Noise Voltage Gain.
E n 2 rms = K 2 • K 3
The noise voltage spectral density at the output is obtained
by multiplying the amplifier’s noise voltage spectral density
(Figure 5a) times the circuits noise gain (Figure 7). Since
both curves are plotted on log-log scales, the multiplication
can be performed by the addition of the two curves. The
result is shown in Figure 8.
(
)
= 6nV/ Hz (1.63 × 10 –3 )
= 11.3µV
3
f 23 f a 3
–
3
3
(10)
(15. 9kHz )3 (673)3
3
–
3
(10a)
(9)
This is the region of increasing noise gain (slope of +20dB/
formed by the resistance R1 || R2 and the capacitance (C1 +
C2). Note that K3 • K2 is the value of the eO (ω) function for
this segment projected back to 1Hz.
(2)
(1)

C 
E n3 rms = K 2  1 + 1 
C2 

 π f – f
2
 2 3
Voltage (nV/√Hz)
Region 3; f > 15.9kHz
(11)
)
25
= 6nV/ Hz  1 + 

1
10–7
f2 = 15.9kHz
10–8
6.36
10–9
10–10
0.01 0.1
(11a)
(
(3)
10–6
1
10 100 1k 10k 100k 1M 10M
 π  80k – 15. 9k
( )
 2
Frequency (Hz)
FIGURE 9. Output Voltage Due to Noise Current.
= 51. 7µV
RESISTOR NOISE
For a complete noise analysis of the circuit in Figure 4, the
noise of the feedback resistor, R2, must also be included. The
thermal noise of the resistor is given by:
This is a region of white noise with a single order rolloff at
f3 = 80kHz caused by the intersection of the 1/β curve and
the open-loop gain curve. The value of 80kHz is obtained
from observing the intersection point of Figure 7. The π/2
applied to f3 is to convert from a 3dB corner frequency to an
effective noise bandwidth.
E R rms = 4kTRB
(14)
K = Boltzmann’s constant = 1.38 x 10
Joules/°Kelvin
T = Absolute temperature (°K)
R = Resistance (Ω)
B = Effective noise bandwidth (Hz) (ideal filter assumed)
–23
CURRENT NOISE
The output voltage component due to current noise is equal
to:
(12)
E = i x Z (s)
ni
n
2
(12a)
where Z2(s) = R2 || XC2
At 25°C this becomes
E R rms ≅ 0.13 RB
ER rms in µV
R in MΩ
B in Hz
This voltage may be obtained by combining the information
from Figures 5 (b) and 6 together with the open loop gain
curve of Figure 7. The result is shown in Figure 9.
Using the same techniques that were used for the voltage
noise:
For the circuit in Figure 4
Region l; 0.1Hz to 10kHz
E ni1 = 4 × 10
R 2 = 10 7 Ω = 10MΩ
–9
10k – 0.1
(13)
B=
= 0. 4µV
Then
Region 2; 10kHz to 15.9kHz
E ni2 = 4 × 10 –13
(
= ( 411nV/
3
–
3
= 0. 4µV
(13a)
= 64. 9µVrms
Region 3; f > 15.9kHz
π
(80kHz ) – 15. 9kHz
2
= 2.1µV
(13b)
E ni TOTAL = 10 –6 (0. 4)2 + (0. 4)2 + (2.1)2
= 2. 2µVrms
)
Hz )
E R rms = 411nV/ Hz
(15. 9kHz )3 (10kHz )3
E ni3 = 6.36 × 10 –9
π
π
( f 2 ) = 2 15. 9kHz
2
(13c)
4
B
π
15. 9kHz
2
TOTAL NOISE
It should be noted that increasing C2 will also lower the
value of K2 (1 + C1/C2), and the value of f2 (see equation 5b).
This reduces signal bandwidth and the final value of C2 is
normally a compromise between noise gain and necessary
signal bandwidth.
The total noise may now be computed from
E n TOTAL = E n12 + E n 2 2 + E n3 2 + E nR 2 + E ni 2
(15)
It is interesting to note that the current noise of the amplifier
accounted for only 0.1% of the total En. This is different than
would be expected when comparing the current and voltage
spectral densities with the size of the feedback resistor. For
example, if we define a characteristic value of resistance as
(15a)
= 10
–6
( 0. 293) + (11.3) + (51. 7) + (64. 9) + ( 2. 2 )
2
2
2
2
2
= 83.8µVrms
CONCLUSIONS
Examination of the results in equation (16b) together with
the curves in Figure 8 leads to some interesting conclusions.
R CHARACTERISTIC =
The largest component is the resistor noise EnR (60% of the
total noise). A lower resistor value decreases resistor noise
as a function of √R, but it also lowers the desired signal gain
as a direct function of R. Thus, lowering R reduces the
signal-to-noise ratio at the output which shows that the
feedback resistor should be as large as possible. The noise
contribution due to R2 can be decreased by raising the value
of C2 (lowering f2) but this reduces signal bandwidth.
=
e n (ω)
at f = 10kHz
i n (ω)
6nV/ Hz
0. 4fA Hz
= 15MΩ
Thus, in simple transimpedance circuits with feedback resistors greater than the characteristic value, the amplifier’s
current noise would cause more output noise than the
amplifier’s voltage noise. Based on this and the 10MΩ
feedback resistor in the example, the amplifier noise current
would be expected to have a higher contribution than the
noise voltage. The reason it does not in the example of
Figure 4 is that the noise voltage has high gain at higher
frequencies (Figure 7) and the noise current does not (Figure
6).
The second largest component of total noise comes from En3
(38%). Decreasing C1 will also lower the term K2 (1 + C1/C2).
In this case, f2 will stay fixed and fa will move to the right
(i.e., the +20dB/decade slope segment will move to the
right). This can have a significant reduction on noise without
lowering the signal bandwidth. This points out the importance of maintaining low capacitance at the amplifier’s input
in low noise applications.
The information provided herein is believed to be reliable; however, BURR-BROWN assumes no responsibility for inaccuracies or omissions. BURR-BROWN assumes
no responsibility for the use of this information, and all use of such information shall be entirely at the user’s own risk. Prices and specifications are subject to change
without notice. No patent rights or licenses to any of the circuits described herein are implied or granted to any third party. BURR-BROWN does not authorize or warrant
any BURR-BROWN product for use in life support devices and/or systems.
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