B is the Hertz noise bandwidth of the DA acting on the white PDSs. Now the MS output SNR of the DA for a pure DM input is easily found to be:

Note that CMRR−2 1 and is negligible. The 2 in the numerator is real, however, and illustrates the advantage of using a DA with pure DM signals.

9.7Effect of Feedback on Noise

9.7.1Introduction

Chapter 4 and Chapter 5 showed that negative voltage feedback (NVFB), correctly applied, has many important beneficial effects in signal conditioning. These include but are not limited to reduction of harmonic distortion, extension of bandwidth, and reduction of output impedance. A common misconception is that negative voltage feedback also acts to improve output SNR and reduce the NF. It will be shown next that NVFB has quite the opposite effect: it reduces the output SNR and increases the NF.

9.7.2Calculation of SNR0 of an Amplifier with NVFB

Figure 9.15 illustrates a noisy differential amplifier with NVFB applied through a voltage divider to the inverting input. To simplify calculations, assume that the DA has a finite difference-mode gain, AD, and an infinite CMRR, i.e., AC 0. Also, neglect the input voltages produced by the current noise sources. The DA’s output is given by:

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vi

vs

vo

ena’

vi’

FIGURE 9.15

Circuit model for a DA with negative voltage feedback. The ina s are neglected.

When the expressions for vi and vi′ are substituted into Equation 9.75, vo is found to be:

The mean-squared output signal voltage is thus:

When considering the noises, vs is set to zero and, for the time signals:

Note that β ∫ R1/(R1 + RF) and ena = ena′ statistically but not necessarily in the time domain; ens, en1, and enf = the thermal noise voltages from Rs, R1, and RF, respectively — all at Kelvin temperature T. One can now write the expression for the mean squared vo over the noise Hertz bandwidth B:

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The MS signal-to-noise ratio of the feedback amplifier is found by taking the ratio of Equation 9.78 to Equation 9.81:

It is left as an exercise for the reader to show that, in the absence of feedback (no feedback resistors at all), the output SNR is:

Note that the presence of the feedback voltage divider resistors adds noise to the output and gives a lower SNRo. Contemplate the effect of applying feedback through a purely capacitive voltage divider. Capacitors do not generate thermal noise.

9.8Examples of Noise-Limited Resolution of Certain Signal Conditioning Systems

9.8.1Introduction

In following examples, assume that the stationary sources of noise (resistor thermal noise, amplifier ena and ina) are statistically independent, are uncorrelated, have white spectra in the range of frequencies of interest, and are added in the MS sense as PDSs are. Aside from their short-circuit input voltage noise, ena, the op amps are assumed to be ideal.

9.8.2Calculation of the Minimum Resolvable AC Input Voltage to a Noisy Op Amp

Figure 9.16 illustrates a simple inverting op amp circuit with a sinusoidal input. Assume that ina RF ena and thus ina produces negligible noise at the amplifier output and is deleted from the model. Only amplifier ena and resistor Johnson noise contribute to the amplifier’s noise output. The meansquared signal output is given by:

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OA

VS

vi

FIGURE 9.16

A simple inverting op amp circuit with a sinusoidal voltage input. White thermal noise is assumed to come from the two resistors and also from the op amp’s ena.

The noise voltages in this circuit are all conditioned by different gains: en1 is the thermal noise from R1; it is conditioned by the same gain as vs, i.e., (–RF/R1). Because the summing junction is at 0 V due to the ideal op amp assumption, the thermal noise in RF, enf, is seen at the output as simply enf (a gain of unity). Again, the gain for ena is found by assuming vi′ = 0. The voltage at the R1–RF node thus must be ena; therefore, the voltage divider ratio gives vo = ena (1 + RF/R1). Note that these three gains are derived assuming ena, enf, and en1 are voltages varying in time. When the MS noise at the output is examined, the gains must be squared.

Using the principles described previously, the total MS noise at the op amp output is:

von2 = {4kTR1(−RF R1)2 + 4kTRF + ena2 (1+ RF R1)2}B

Note that the equivalent noise bandwidth, B, must be used to effectively integrate the white output PDS, giving MSV. The MS SNRo of the amplifier is thus:

¬

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To maximize Equation 9.89 for SNRo, it is clear that R1/RF must be small or, equivalently, that the amplifier’s signal gain magnitude, RF/R1, must be large. This is an unusual result because it says the SNRo is gain dependent. Most SNRos to be calculated are independent of gain.

Equation 9.89 can be solved for the minimum VS to give a specified SNRo. For example, let the required MS SNRo be set to 3; 4kT ∫ 1.656 ∞ 10−20; R1 = 1k; RF = 10k; ena = 10 nVRMS/ Hz; and B = 1000 Hz. The source frequency is 10 kHz and lies in the center of B. The minimum peak sinusoidal voltage, VS, is found to be 0.914 μV.

9.8.3Calculation of the Minimum Resolvable AC Input Signal to Obtain a Specified SNR0 in a Transformer-Coupled Amplifier

Figure 9.17 illustrates an op amp circuit used to condition an extremely small AC signal from a low impedance source. An impedance-matching trans-

former is used because RS is much less than RSopt = ena/ina ohms. Assume that the op amp is ideal except for ena and ina. The summing junction of the op

amp, looking into the transformer’s secondary winding, can be shown to

Ideal Xfmr

FIGURE 9.17

Circuit showing the use of an ideal impedance-matching transformer to maximize the output SNR. Rs and RF are assumed to make thermal white noise; ena and ina are assumed to have white spectra. An ideal unity gain BPF is used to limit output noise msv.

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“see” an input Thevenin equivalent circuit with an open-circuit voltage of n vs and a Thevenin resistance of n2 RS (Northrop, 1990). Thus, the output voltage due to the signal is vos = n vs [−RF/n2 RS] and the MS signal is:

The op amp’s output is filtered by an (ideal) band-pass filter with peak gain = 1 and noise bandwidth, B, Hertz centered at fs, the signal frequency. The filter is used to restrict the noise power at the output while passing the signal.

Finding the output noise is somewhat more complicated. The gain for ens, the thermal noise from RS, is the same as for vs, i.e., [RF/n2RS ]. The gain for inf, the thermal noise from RF, is 1. The gain (transresistance) for ina is RF. The gain for ena can be shown to be [1 + RF/(n2RS)] (Northrop, 1990). By using the superposition of mean-squared voltages, the total noise MS output voltage can be written:

The denominator of Equation 9.92 has a minimum for some non-negative transformer turns ratio, no; thus, SNRo is maximum for n = no. To find no, differentiate the denominator with respect to n2 and set the derivative equal to zero, then solve for no.

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Now calculate the turns ratio, no, that will maximize SNRo. Define the parameters: ena = 3 nVRMS/ Hz (white); ina = 0.4 pARMS/ Hz (white); RF = 104 ohms; RS = 10 ohms; 4kT = 1.656 ∞ 10−20; and B = 103 Hz. Using Equation 9.95, no = 14.73 (the secondary-to-primary ratio does not need to be an integer). Using no = 14.73, the desirable MS SNRo = 3; substitute the preceding numerical values into Equation 9.92 and solve for the vs RMS required: vs = 28.3 nVRMS. If no transformer is used (equivalent to n = 1 in Equation 9.92), vs = 166 nVRMS is needed for the MS SNRo = 3.

9.8.4The Effect of Capacitance Neutralization on the SNR0

of an Electrometer Amplifier Used for Glass Micropipette Intracellular Recording

Measurement of the transmembrane potential of neurons, muscle cells, and other cells is generally done with hollow glass micropipettes filled with a conductive electrolyte solution such as 3M KCl (Lavallée et al., 1969). In order for the glass micropipette electrode to penetrate the cell membrane, the tips are drawn down to diameters of the order of 0.5 μm; the small diameter tips give micropipettes their high series resistance, Rμ. Figure 9.18 illustrates a simplified lumped-parameter model for a glass microelectrode with its tip in a cell. Rμ is on the order of 50 to 500 megohms. CT is the equivalent lumped shunt capacitance across the electrode’s tip glass; it is on the order of single picofarads.

The electrometer amplifier (EA) used to condition signals recorded intracellularly with a glass micropipette microelectrode is direct-coupled, has a low noninverting gain between 2 and 5, and is characterized by its extremely high input impedance (approximately 1015 Ω) and very low dc bias current (approximately 100 fA). The latter two properties are necessary because the

FIGURE 9.18

Circuit used to model noise in a capacity-neutralized electrometer amplifier supplied by a glass micropipette electrode. Only white thermal noise from the microelectrode’s internal resistance is considered along with the white noises, ena and ina.

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Thevenin source resistance of the microelectrode is so high and, for practical reasons, negligible dc bias current should flow through it to prevent ion drift at the tip. For all practical purposes, the EA can be treated as a noisy but otherwise ideal voltage amplifier. The so-called capacitive neutralization is accomplished by positive feedback applied though the variable capacitor, CN.

The transfer functions for the biosignal, Vb, and the amplifier noises, ena and ina, will now be found. Assume the EA’s gain is +3. To do this, first write the node equation for the v1 node in terms of the Laplace variable, s:

Equation 9.96 can be solved for V1:

Note that, in this simple case, when CN is made CT/2, the amplifier’s time constant 0, thus the break frequency fb = 1/2πτ • and Vo = 3Vb.

Now consider the transfer function for the EA’s current noise, ina:

Now find the transfer function for ena. Again, write a node equation:

Note that when V1′ = V1 − ena is substituted into Equation 9.101, it is possible to solve for the transfer function:

In summary, when CN = CT/2, the amplifier is neutralized and the noise gains are:

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