Texas Instruments | OA-12 Noise Analysis for Comlinear Amplifiers (Rev. B) | Application notes | Texas Instruments OA-12 Noise Analysis for Comlinear Amplifiers (Rev. B) Application notes

Texas Instruments OA-12 Noise Analysis for Comlinear Amplifiers (Rev. B) Application notes
Application Report
SNOA375B – April 1996 – Revised April 2013
OA-12 Noise Analysis for Comlinear Amplifiers
.....................................................................................................................................................
ABSTRACT
This application report covers the noise model for all current-feedback op amps, simple design techniques
and useful approximations. This is a frequency-domain model to simplify circuit analysis and design. This
information simplifies the selection of a low-noise current-feedback op amp.
This revision obsoletes the previous revision of this document, and covers additional material.
Contents
1
Contents ......................................................................................................................
2
Scope of Noise Analysis ...................................................................................................
3
Noise Model .................................................................................................................
4
Integrated Noise .............................................................................................................
5
Dynamic Range .............................................................................................................
6
Improving Output Noise ....................................................................................................
7
1/f Noise ......................................................................................................................
8
SPICE Models ...............................................................................................................
9
Design Example .............................................................................................................
10
Conclusions ..................................................................................................................
11
References ...................................................................................................................
Appendix A
Derivation of Noise Power Bandwidth Formula ................................................................
2
2
2
3
4
4
5
5
6
8
8
9
List of Figures
1
2
................................................................................................................................
..............................................................................................
Non-Inverting Gain Amplifier
3
6
List of Tables
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Contents
1
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Contents
The subjects covered are:
• The noise model for current-feedback op amps
• Converting noise densities to integrated noise
• Interpreting integrated noise as SNR
• Output noise improvement
• 1/f noise calculations
• SPICE models
• A design example
• A derivation of the noise power bandwidth (NPBW) approximation (see Appendix A)
• A reference section
2
Scope of Noise Analysis
The noise analysis in this application report deals with random noise generated by the devices and
components in a circuit. Noise analysis gives the greatest benefit when:
• The signal level is low
• The signal to noise ratio (SNR) is high
• The signal sees a substantial gain
Noise analysis will not help:
• Identify and eliminate oscillation or instability problems
• Reduce EMI (Electro-Magnetic Interference)
• Reduce cross talk
3
Noise Model
Three input-referred noise density (spot noise) sources model the noise generated by current-feedback
(CFB) op amps. Noise power density (en2 or in2 ) is the power measured in a narrow bandwidth,
normalized to the load resistance, in units of V2/Hz or A2/Hz. Voltage noise density (en) and current noise
density (in) are the square-root of noise power density in units of V/√Hz or A/√Hz. Notice that these noise
densities are functions of frequency.
Figure 1 shows the three input noise density sources, eni2 , ibn2 and ibi2 , in a standard amplifier circuit. The
specifications give densities that are constant over frequency (white noise). Ground RT for inverting gain
circuits, and ground Rg for non-inverting gain circuits
2
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Integrated Noise
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Figure 1.
The equation for the output voltage noise density is:
(1)
The load resistor (RL ) has a negligible contribution to the noise because the output resistance of the op
amp is very small.
The system transfer function will shape the output noise. See References [1] and [2] for information on
how to generate noise transfer functions. The 1/f Noise section covers excess noise (noise that exceeds
the white noise specifications).
4
Integrated Noise
Convert the output voltage noise density to the integrated output voltage noise by integrating over
frequency:
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Dynamic Range
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(2)
where:
• Heno (jf) is the noise transfer function for eno
• f1 is the lower -3dB corner frequency for AC-coupled systems, or the lowest frequency that affects your
system’s performance
• f2 is the upper -3dB corner frequency
• The NPBW approximation holds when:
– There is ≤ 3dB of gain peaking
– f1 << f2
– If the NPBW approximation does not hold, use numerical integration instead
The integrated output noise, Eno , is the standard deviation of the output noise in units of Vrms . It is also a
measure of the lower end of the useful dynamic range. Because integrated output noise depends on the
circuit architecture, component values and the op amp, it is best to compare op amps based on the
input noise densities
To see how each noise source contributes to Eno , integrate each term separately:
(3)
This information is useful for improving the amplifier’s SNR.
5
Dynamic Range
Signal to noise ratio (SNR) describes how much dynamic range a signal has. It compares the lower end of
the useful dynamic range (Eno) to the signal magnitude (in units of Vrms). The input and output signal to
noise ratios are:
(4)
where:
• Vin(rms is the signal voltage at the input (VS1 or VS2), Vrms
• Enin is the integrated voltage noise at the input (at VS1 or VS2 ), Vrms
• Vo(rms) is the signal voltage at the output, Vrms
• Eno is the integrated voltage noise at the output, Vrms
6
Improving Output Noise
To
•
•
•
•
•
4
reduce output noise, do the following:
Band-limit the signal after the op amp to limit the final output noise
AC couple when possible
Use a low-pass filter, or a band-pass filter
Reduce gain peaking to lower the NPBW
Reduce resistor values to lower thermal noise, but keep in mind that:
– Rf values smaller than that recommended in the device-specific data sheet causes gain peaking
and increased bandwidth; the NPBW may increase faster than the intended noise reduction
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1/f Noise
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– Smaller loads at the op amp’s output increase distortion and power consumption
– Resistors connected to the input of current-sensing amplifiers act as current noise sources;
increase these resistor values to reduce thermal noise
For those op amps with an adjustable supply current, the input noise sources change with supply current.
As the supply current increases, the input voltage noise decreases, the input current noises increase, the
distortion improves and the bandwidth increases. For the best voltage noise performance, use the highest
supply current. For the best current noise performance, use the lowest supply current.
7
1/f Noise
At low frequencies, the three input noise density terms are larger than predicted by the specifications. The
dominant source of this excess noise is 1/f (or flicker) noise. Burst noise also contributes to excess noise,
but is not covered in this application report. The input noise sources, with both the 1/f noise and white
noise terms included, are:
(5)
where:
•
•
•
eni2(f) is the sum of the white noise term, eni2, and the 1/f noise term,
fc(eni) is the corner frequency of the 1/f noise for eni2(f); this is the point where eni2(f) doubles its white
noise value
the other input noise terms are defined similarly
Notice that flicker noise power density is proportional to 1/f; flicker voltage noise density and flicker current
noise densities are proportional to 1/√f.
To integrate both white noise and 1/f noise, evaluate individual noise terms separately. For each term
obtain:
(6)
The 1/f noise contribution is negligible when f2 >> fc . f1 is the largest frequency that does not affect your
system’s performance when the amplifier is DC-coupled.
Use metal-film resistors to minimize 1/f noise.
8
SPICE Models
SPICE models are available for most of Comlinear’s amplifiers. These models support AC noise
simulations at room temperature. We recommend simulating with Comlinear’s SPICE models to:
• Predict a better value for NPBW
• Support quicker design cycles
To verify your simulations, we recommend breadboarding your circuit. Evaluation boards are available for
building and testing Comlinear’s amplifiers.
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Design Example
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Design Example
This design example demonstrates the noise design of a simple circuit. The CFB op amp in this example
is not an actual product; the parameter values shown are arbitrary and are for illustration purposes
only.
This example uses the non-inverting gain amplifier in Figure 2. The components shown are:
• VS1 is the input voltage source (with very low output impedance). The signal at VS1 is 100mVrms , and
the voltage noise ens1 (at VS1) is 3.0nV/√Hz.
• A 50Ω coax cable is placed between the source and the amplifier
• RT1 = 50Ω to match the coax cable’s impedance and prevent reflections
• RT2 prevents gain peaking, and filters the input signal with CT
• CT filters the input signal (this reduces the signal’s slew rate)
• Rf and Rg set the gain; the recommended Rf is 250Ω for a gain of 10
• RL is 100Ω
• The op amp noise terms are:
• eni = 3.0nV/√Hz and fc(eni) = 1.0kHz
• ibn = 2.0pA/√Hz and fc(ibn) = 5.0kHz
• ibi = 12pA/√Hz and fc(ibi) = 10kHz
• Ambient temperature (T) is 25°C
• Power dissipation of the op amp causes a 15°C junction temperature rise
Figure 2. Non-Inverting Gain Amplifier
The design goals are:
• Provide a gain of 10 (= Gn for non-inverting gains)
• DC-couple the signal; the lowest frequency that affects system performance is 10Hz (f1)
• Set an upper 3dB corner frequency of 10MHz (f2)
• Achieve an output SNR of 74dB
The initial design choices that are made are:
• 20MHz pole at the input set by CT and RT2 (this will cause reflections in the coax cable for any signal
above this pole)
• 10MHz filter after this amplifier (not shown); this will set f 2 (NPBW)
6
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Design Example
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•
•
•
•
RT2 = 1.0kΩ
CT = 8pF
Rf = 250Ω, its recommended value, to avoid gain peaking
Rg = 27.8Ω to set the gain to Gn = 10
The resulting junction temperature of the op amp, input integrated noise and input SNR are:
(7)
RT1 does not contribute to the output noise; VS1is a nearly ideal voltage source.
The input source produces an output noise of: The individual white noise contributions of the op amp to
the output noise are:
(8)
The individual white noise contributes of the op amp to output noise are:
(9)
The individual 1/f noise contributions of the op amp to the output noise are:
(10)
The contributions of the other components to the output noise are:
(11)
The resulting output integrated noise, output signal and output SNR are:
Eno ≈ 227μVrms
Vo(rms) = GnVin(rms) ≈ 1.00Vrms
SNRo ≈ 72.9dB
(12)
(13)
(14)
Reduce RT2 to improve SNR; this has little impact on other performance parameters. Changing RT2 to
200Ω gives:
CT = 40pF
Eno ≈ 169μVrms
SNRo ≈ 75.4dB
(15)
(16)
(17)
In an actual design, the next step would be SPICE simulations, then breadboarding the circuit.
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Conclusions
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Conclusions
The important points to remember when designing low noise circuits are:
• Employ noise analysis where small signals are present
• Select correct resistor values to reduce thermal noise
• Select op amps based on their input noise densities (integrated noise is circuit-dependent)
• Reduce NPBW and gain peaking to minimize integrated output noise
• Estimate your signal’s dynamic range using SNR
• Simulate with Comlinear’s SPICE models to estimate noise performance
• Build and measure your circuit to verify the design
• Refer to Section 11 for additional background information
11
References
1. C. D. Motchenbacher and J. A. Connelly, Low-Noise Electronic System Design, New York, John Wiley
& Sons, 1993.
2. P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, 2nd Ed. New York:
John Wiley & Sons, 1984.
3. J. D. Gibson, Principles of Digital and Analog Communications, New York: Macmillan, 1989.
4. A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical
Communication, 3rd Ed. New York: McGraw-Hill, 1986.
5. P. Antognetti and G. Massobrio (Editors), Semiconductor Device Modeling with SPICE, New York:
McGraw-Hill, 1988.
6. CLC to LMH Conversion Table (SNOA428)
•
NOTE: The circuits included in this application report have been tested with Texas Instruments parts
that may have been obsoleted and/or replaced with newer products. To find the appropriate
replacement part for the obsolete device, see the CLC to LMH Conversion Table
(SNOA428).
8
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Appendix A Derivation of Noise Power Bandwidth Formula
The goal is to estimate NPBW using common, easy to measure parameters: the -3dB bandwidth and gain
peaking. Assume a second-order transfer function for the op amp circuit’s high-frequency behavior:
(18)
where, ωo = 2πfo is the natural frequency of this transfer function.
Integrating the magnitude squared of the transfer function gives:
(19)
Solving for the upper -3dB corner frequency (f2), and substituting the result in Equation 19, gives:
(20)
Gain peaking is easy to measure, and is a strong function of Q for large Q. It is easy to show that:
(21)
where, Hmax is the peak gain magnitude.
These results support the following approximations:
(22)
with a 20% maximum error. This translates to a 0.8dB maximum error in the estimated SNR.
If the amplifier transfer function has a single pole response, it is easy to show that:
NPBW = (π/2) · f2, Single pole transfer function
(23)
High-order filters will have:
NPBW ≈ f2, high-order filters
(24)
The approximation formula includes both of these cases.
The above results hold for the lower corner -3dB frequency (f1) with minor modifications. When the corner
-3dB frequencies do not interact (f1 << f2 ), we obtain:
(25)
It is easy to extend this result when there is more than 3.0dB of peaking, but it is better to reduce the
peaking, or to numerically integrate the output noise.
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