Noise 102

Noise processes generate incoherent signals and, collectively, do so over a broad frequency range—from beyond your application's upper frequency limit to values approaching dc. Noise 101 introduced noise sources at the device level, presented their root mechanisms, and described their spectral characteristics (Reference 1). But low-noise design requires that you consider not only device issues, but also circuit-topology perspectives. Simple device combinations illuminate how the noise terms combine in practical circuits.

Setting bounds
Take, for example, the thermal noise voltage of a resistor, En:


which derives from


These expressions suggest that any nonzero resistance gives rise to a noise-voltage magnitude bounded only by the measurement bandwidth. In other words, as the measurement bandwidth approaches the infinite, so does the perception of a resistor's thermal noise. This result doesn't occur in practice, though why it doesn't might not be obvious at first glance.


Figure 1: A resistor-noise model includes the parasitic shunt capacitance.


A simple resistor model explains why (Figure 1). This model of a physical resistor comprises an ideal resistance, R, in series with its thermal noise voltage, ER, shunted by its parasitic capacitance, C. The shunt combination of the resistance and its parasitic capacitance limits the noise bandwidth. A doubling of the resistance increases the noise density by the square root of two but halves the noise bandwidth resulting in the same total noise. One implication of this observation is that, in the absence of another bandwidth limit, the resistor model's shunt capacitance sets the noise limit to


in mean squared volts (Reference 2). Another implication is that practical flatband noise measure-ments must account for the filter skirt that sets the measurement bandwidth due to the differences in definition between signal and noise bandwidth.

Making gains
Among the most common applications for low-noise circuits are analog-signal-input stages. Input signals may derive from sensors, antennas, or other low-level sources that require substantial gain before further processing or conversion to the digital domain. Other sources may provide signals with fairly large average amplitudes but require processing circuits with large dynamic range. Whatever your signal source may be, the source impedance sets the noise minimum at a given temperature—SNR only decays from there. The source impedance and the application's dynamic-range requirements provide the context for your consideration of input stages, which often dominate the system's noise performance.

Figure 2: The generalized gain-cell model includes the amplifier's input-referred voltage- and current-noise sources, and a noise voltage corresponding to the signal-source impedance.


Consider a generalized gain cell with an input impedance, ZI, and a voltage gain of AV (Figure 2). The noise-voltage and noise-current sources, respectively, En and In, model the amplifier noise referred to its input node. The signal source has a source impedance, RS, and noise voltage source, ER, which accounts for the source-impedance noise and any excess noise that the source presents to the amplifier's inputs. You can calculate the total input referred noise, Eni, as the root sum square of the input noise terms:


You can also calculate the total output referred noise by taking into account the effect of the amplifier's finite input impedance and the voltage gain. Invariably, you'll want to calculate—or at least estimate—both the input and the output referred noise early in your design. The two numbers are useful in evaluating different aspects of your paper design. The input-referred term allows you to compare amplifiers irrespective of their input impedances or the gain you set. The output-referred noise is the quantity the amplifier passes on to the subsequent signal-processing stage and must meet your application's noise criteria for that stage's input.

Making a difference

Figure 3: A difference amplifier demonstrates the noise calculations for feedback amplifiers.

A difference amplifier demonstrates the noise calculations for feedback amplifiers (Figure 3). If you set the R4/R3=R2/R1, then the circuit's transfer function is


you can slightly simplify the noise analysis by combining R3 and R4 and scaling V2 (Figure 4):


Figure 4: The complete noise model includes seven noise sources and two signal sources. Be careful when calculating the gains associated with the amplifier's input-noise source terms.

To facilitate comparisons among candidate amplifiers, you can perform the analysis in terms of noise-voltage and -current densities in keeping with the way IC vendors specify their devices.

Because noise sources are indistinguishable from any other floating signal source, the output noise calculation is straightforward with just a few caveats. Uncorrelated noise sources combine as root-sum-square terms. Some of the noise sources appear in unusual locations from a traditional signal-analysis perspective, which can lead to erroneous results if you don't carefully assess the gain that each source sees. For example, despite the fact that noise source en1 connects to the summing junction, the fact that it lies between that point and the noninverting input gives it a gain of G+1, where G=R2/R1. Similarly, the non-inverting input's noise current does not divide but flows entirely in R2.

With these issues in mind, you can determine the output referred noise, eno, by inspection:


Each noise voltage source sees a gain—G+1 for the amplifier's input noise and for RP, G for the noise contribution of R1, and unity for the noise contribution of R2. Each noise current flows through a resistance and the resultant voltage appears at the output through a gain. In1 flows through R2 and directly adds to the output noise. In2 flows through RP and contributes to the output through the noninverting gain, G+1.

Amplifier data sheets generally do not give independent input-noise-voltage densities for each input but rather publish a total figure. With few exceptions, IC amplifiers use balanced input structures, which tend to produce equal input-noise magnitudes on the two inputs. Divide the amplifier's total input referred voltage noise by and ascribe the resulting quantity to en1 and en2. Other models include a single input-noise-voltage source and connect it to one of the amplifier's inputs. This input is often the noninverting one, a choice that can simplify the drawing and make clear the forward gain that applies to the noise source.

An amp for every app
In the past, ultra-low-noise op amps came with slow slew rates, modest gain-bandwidth products, and high quiescent currents. However, long-standing industry trends have prompted op-amp manufacturers to develop higher performing designs. The cost and capability of DSPs, micro-processors, and data converters have radically improved over the years. Faster and higher resolution signal processing in the digital domain demands faster low-noise signal processing in the analog front end. This relationship is evident in areas as disparate as medical imaging, ATE, and wireless communication. Additionally, both channel bandwidths and channel densities have pushed higher, pressuring amplifier designers to ever more tightly squeeze their power budgets.

The recent results have been impressive: A spate of amplifiers has entered the market during the last few quarters. Among them is the AD8099 from Analog Devices—one of the ICs nominated for this year's EDN Innovation Awards—which the company developed for precision applications, such as radar collision-avoidance systems, medical ultrasound signal processing, and precision instrumentation.

The 8099 exploits a novel front-end design that beats the traditional trade-off between low noise and input-stage linearity. For decades, op-amp designers have used resistors in the input pair's emitter path to degenerate the first-stage transconductance and improve the amplifier's linearity in the presence of large signal swings, which in op-amp parlance can mean anything larger than a large fraction of a thermal voltage—about 26mV. Unfortunately, the resistors add noise, and, therefore, most designs would begin a balancing act among noise, linearity, and quiescent current. The AD8099 improves on the traditional approach by degenerating the input stage in a way that puts the noise source in the common-mode instead of the differential-mode path.

The result is an amplifier that offers 0.95nV/ voltage noise and -90dB distortion at 10MHz, 2V p-p, at a gain of 2. Under the same operating conditions, the $1.98 (1000) AD8099 can slew 600V/µsec, increasing to 1600V/ µsec at a gain of 10 where its gain bandwidth is 5GHz.
The 8099 is the first amplifier to offer a new pinout that the manufacturer is proposing to reduce distortion due to mutual inductance between the non-inverting input and the negative supply pins. The new pinout also provides two output pins: one for the signal path to the following signal-processing stage and the other for the amplifier's feedback network. The second output pin simplifies your pc-board layout and reduces feedback parasitics, enhancing the amplifier's stability. The new op amp is available in either an eight-pin LFCSP (lead-frame chip-scale package) that offers low inductance and excellent thermal characteristics or in a more traditional eight-pin SOIC.

The VCA8613 from Texas Instruments exemplifies low-noise amplifiers in a more application-specific and more highly integrated form. The eight-channel 8613 variable-gain amplifier targets imaging applications that demand high channel count, compact size, low power, and low noise.

Each of the VCA8613's eight channels comprises an LNA (low-noise amplifier) with internal clamp diodes; a voltage-controlled attenuator; a programmable-gain amplifier; and a two-pole, 14MHz output filter. In addition to this differential time-gain-control path, the eight LNAs also feed an 8310 single-ended crosspoint switch, which is programmable through a serial-interface port and provides continuous-wave outputs. The eight LNAs offer 70MHz gain bandwidth and 1.2nV/ at 5MHz in time-gain-control mode and 1.6nV/ in continuous-wave mode. You can program the voltage-controlled attenuator for 0- to 29-, 33-, 36.5-, or 40dB ranges and the programmable-gain amplifier for 21- or 26dB gain.

The $25.40 (1000) VCA8613 dissipates an average of 75mW per amplifier and operates on 3V supplies. Texas Instruments packages the eight-channel analog front end in a TQFP-64.
Linear Technology offers 0.95nV/ rail-to-rail-input and -output op amps for medical diagnostics, communications, and optoelectronics in the form of the LT6200-10 and -5. The LT6200-10 version provides a 1.6GHz gain bandwidth and a slew rate of 450V/µsec. It is stable at gains of 10 or more. The LT6200-5 offers half the gain bandwidth of its faster sibling and a slew rate of 250V/µsec but is stable at gains as small as 5. Two other members of the family, the LT6200 singleton and LT6201 dual amplifier are unity-gain-stable and provide 165-MHz gain bandwidth.

The TL6200 family of amplifiers produces -80dB distortion at 1MHz. On the dc side, the amplifiers' offset is limited to 1mV. You can drive these op amps with unipolar or bipolar supplies totaling 2.5 to 12.6V. They are specified at 3, 5, and ±5V. The singletons are available in SOT-23-6 and SO-8 packages at prices starting at $1.50 (1000).

The Elantec division of Intersil, long known as providers of high-speed amplifiers, makes
a 0.9nV/ amplifier in the form of the EL5132. Applications include instrumentation, communications, and imaging. The 5132 has a 6.7GHz gain-bandwidth product and is stable at a gain of 10. The amplifier slews 1kV/µsec.

The $1.05 (1000) EL5132 draws 11mA and operates from either unipolar or bipolar supplies totaling 5 to 12V. It features an enable pin and is available in SO-8 packages. A sibling part, the EL5133, drops the enable pin to fit into a SOT-23-5 package.

National Semiconductor's LMH6624 and LMH6626 single and dual ultra-low-noise op amps typically measure 0.92nV/. They offer 1.5- and 1.3GHz gain bandwidth and slew 360 and 340V/µsec, single and dual, respectively, at a gain of 10. Second- and third-harmonic-distortion figures are -63 and -80- dBc, both at 10MHz driving 100Ω.

The 6624 and 6626, $1.67 and $1.99 (1000), respectively, operate on either unipolar or bipolar supplies totaling 5 to 12V. They draw a maximum of 16mA per amplifier at room temperature and 18mA over the -40 to +125°C operating-temperature range. National houses the LMH6624 in SOT-23-5 and SO-8 packages and the LMH6626 in MSOP-8 packages.

These selections represent just a tiny sample of the many ultra-low-noise amplifiers available from these vendors. Visit the vendors' Web sites for information on other devices in their respective product lines and continue to watch this space for more information about future developments in low-noise analog ICs.

References
1.Israelsohn, Joshua, "Noise 101," EDN, Jan 8, 2004, pg 41, 2. Motchenbacher, CD and JA Connelly, Low-Noise Electronic System Design, Wiley, 1993.

Going off on a tangent gets to the point
In most engineering discussions, the word "bandwidth' carries an implied "-3dB" stipulation. By convention, the half-power points define signal bandwidth in most applications.

Noise bandwidth, ∆f, by contrast, is the frequency span of a rectangular spectrum equal in area to the actual power curve. In most cases, voltage gain rather than power gain characterizes these circuits, so, for convenience, you can recast the noise bandwidth expression in terms of the voltage gain:

Equation A

where Av(f) is the voltage gain, and Avo is the peak gain. Noise bandwidth is always greater than the signal bandwidth because the integral extends beyond the -3dB frequency.

A normalized, single-pole, lowpass network makes a good example:

Equation B

where fc is the -3dB corner frequency. The voltage gain magnitude as a function of frequency is

Equation C


and the noise bandwidth is

Equation D


Integrating to infinity is the sort of thing that could take awhile, and most engineers would like an answer, say, within this lifetime. Various approximations exist, but an exact closed-form solution is always preferable—particularly absent a good assessment of the approximation's residual error. Instead, invoke a change of variable to gain an exact solution in just a few steps (Reference A):

Equation E


The integration limits are now 0 to π/2. Applying the change of variable to Equation D results in

Equation F


Recognizing the trigonometric identity sec²fө=1+tan2ө, Equation F collapses to

Equation G


So, the noise bandwidth of a single-pole system is greater than its signal bandwidth by a factor of π/2. You can extend the change-of-variable technique to other filter arrangements, as well. For example, you can connect a pair of identical poles with a buffer to prevent interaction (Figure A). This filter's gain function is

Equation H




Figure A: A pair of identical poles with a buffer to prevent loading demonstrates how to calculate noise bandwidth and the ratio of noise bandwidth to signal bandwidth.

Comparing this case with the single-pole case shows that the noise bandwidth of this filter is

Equation I


The same change of variable results in an exact solution for this structure.

Equation J


Remember, however, that fc is here the -3dB corner frequency of the individual poles, not of the filter overall. Two more steps allow you to calculate system bandwidth, fs, from the poles, and the noise bandwidth from the system bandwidth:

Equation K


Combining equations J and K,

Equation L


When performing noise measurements on the bench, you need to be aware of the roll-off characteristics of the system under test, your test equipment, and any interface circuits between the two. Understand which device sets the observed bandwidth and the shape of its roll-off so that you can properly calculate your measurement's noise bandwidth and properly attribute the results. Be sure that the other parts of your setup have roll-offs substantially higher in frequency than the one that sets the observation spectrum. Also as part of your setup, account for the noise floor that your test equipment and interface circuits establish.

Reference
A. Motchenbacher, CD and JA Connelly, Low-Noise Electronic System Design, Wiley, 1993.

Slow noise ain't noise
Often, when engineers talk about very-low-frequency noise phenomena, 1/f noise dominates the discussion. Though 1/f noise mechanisms exist in a range of observable phenomena, in semiconductor devices, 1/f noise results from crystal defects and surface states (references A and B). Circuit approaches to combating 1/f noise are essentially limited to adding a zero to the system-transfer function, which can be effective for nonbaseband applications. But for signal spectra that approach dc, a transfer-function zero can impose long start-up delays and play havoc with asymmetrical inputs.

Ironically perhaps, issues of drift are more readily addressed, but designers often ignore them during deliberations on noise (Reference C). Yet a careful system-transfer-function error analysis for many applications identifies offset drift as a low-frequency noise term.

For example, PTAT (proportional-to-absolute-temperature) current sources bias bipolar input stages that must operate at constant transconductance. As the die temperature varies, so do the input-bias currents. Designing your input circuit so that both noninverting and inverting nodes look into the same impedance cancels the bias current and its drift, which offset voltage induces.

Offsets can cause performance problems even if your favorite application includes no important signal energy near dc. For example, some imaging and audio-processing applications must accommodate signals with large dynamic ranges and use variable-gain structures to do so. Some of these circuits perform coarse scaling by implementing a number of discretely selectable gain steps. When these amplifiers switch gain in the presence of an input offset, their outputs must slew Vos∆G. The output contains in-band information during the slew interval, referred to as zipper noise, for the sound this type of noise makes in audio systems that switch rapidly through several discrete steps.

References

Lundberg, Kent H, "Noise sources in bulk CMOS,"http://web.mit.edu/klund/www/CMOSnoise.pdf.

Israelsohn, Joshua, "Noise 101," EDN, Jan 8, 2004, pg 41.

Williams, Jim, Linear Technology, interview, November 2003.

Author Information
You can reach Technical Editor Joshua Israelsohn at 1-617-558-4427, fax 1-617-558-4470, e-mail jisraelsohn@edn.com.

At a glance

• It's usually safe to assume that noise sources are uncorrelated and sum as root sum squares. Correlated sources, which would add directly, are far less common than uncorrelated ones.


• Noise bandwidth is always greater than signal bandwidth. You must account for this difference whenever you make noise measurements or calculate noise in a spectrum that includes the roll-off.


• Be careful when performing noise analysis of your circuits. Noise sources show up in odd places, and it's easy to ascribe an erroneous gain between a given source and the output.


Acknowledgments
Thanks to Scott Wurcer and Lew Counts of Analog Devices and Jim Williams of Linear Technology for their contributions to this article.