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Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation - Northrop.pdf
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342

Analysis and Application of Analog Electronic Circuits

Rs @ T

 

ena

Vi

Vo

 

 

 

 

 

 

+

Vs

R1 @ T

ina

 

Vi H(s)

 

 

rmsA/√Hz

 

 

FIGURE 9.5

The two-noise source model for a noisy amplifier. ena and ina are root power density spectra. Rout is neglected.

30

 

 

 

 

 

 

 

30

ena

 

 

 

 

 

 

 

ina

nVrms/√Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fArms/√Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ina

20

 

 

 

 

 

 

 

20

 

ena

 

white region of ena

 

 

 

10

 

 

 

 

 

 

 

10

0

 

 

 

 

 

 

 

0

1

10

102

1

03

104

105

106

107

 

 

 

 

f

(Hz)

 

 

 

FIGURE 9.6

Plots of the ena and ina root power density spectra vs. f for a typical low-noise FET headstage amplifier. Note that ena has a low frequency 1/ f component and ina does not. ina and ena increase at very high frequencies.

In addition to the equivalent short-circuited input noise voltage, the modeling of the net noise characteristics of amplifiers requires the inclusion of an equivalent input noise current source, ina, as shown in Figure 9.5. ina(f ) is

the root PDS of the input equivalent noise current; its units are RMS amps per Hertz. Note that ena(f ) and ina(f ) have flat mid-frequency portions that

invite approximation by white noise sources. At high frequencies, both equivalent noise root PDSs slope upward. For discrete JFETs and BJTs, and IC amplifiers, ena(f ) shows a distinct 1/f region at low frequencies.

9.2.4.3Noise in JFETs

Certain selected discrete JFETs are sometimes used in the design of low-noise amplifier headstages for biomedical signal conditioning systems. Some JFETs give good low-noise performance in the audio and subaudio frequency regions of the spectrum; others excel in the video and radio-frequency end of the spectrum, giving them applications for RF oscillators, mixers, and

© 2004 by CRC Press LLC

Noise and the Design of Low-Noise Amplifiers for Biomedical Applications

343

tuned amplifiers used in ultrasound applications. Amplifiers using JFET headstages have relatively low input dc bias currents (IB 10 pA) and high input resistances (>1010 Ω), both of which are desirable.

Van der Ziel (1974) showed that the theoretical thermal noise generated in the conducting channel of a JFET can be approximated by a white, equivalent short-circuited input noise with PDS given by:

e 2

= 4kT gm = 4kT g

m0

 

IDSS

MSV Hz

(9.31)

 

nn

 

 

IDQ

 

 

 

 

 

 

 

 

where gm0 is the FET’s small-signal transconductance measured when VGS =

0 and ID = IDSS; IDSS = the dc drain current measured for VGS = 0 and VDS > VP; and IDQ = the quiescent dc drain current at the FET’s operating point

where VGS = VGSQ. In reality, due to the presence of 1/f noise, the theoretical short-circuited input voltage PDS can be better modeled by:

ena2 (f ) = (4kT gm )(1+ fc f n ) MSV Hz

(9.32)

The exponent n has the range 1 < n < 1.5 and is determined by device and lot. For algebraic simplicity, n is usually set equal to one. The origins of the 1/fn effect in JFETs is poorly understood. Note that ena given by Equation 9.32 is temperature dependent; heat sinking or actively cooling the JFET will reduce ena. The parameter fc used in Equation 9.32 is the corner frequency of the 1/f noise spectum. Depending on the device, it can range from below 10 Hz to above 1 kHz. fc is generally quite high in RF and video frequency JFETs because, in this type of transistor, ena(f ) dips to around 2 nV/ Hz in the 105- to 107-Hz region, which is desirable.

JFET gates have a dc leakage or bias current, IGL = IB, which produces broadband shot noise that is superimposed on the leakage current. This noise component in IGL is primarily due to the random occurrence of charge carriers that have enough energy to cross the reverse-biased gate–channel diode junction. The PDF of the gate current shot noise is generally assumed to be Gaussian and its PDS is approximated by:

i

2

= 2 q I

GL

MSA/Hz

(9.33)

 

na

 

 

 

where q = 1.602 10–19 C (electron charge) and IGL is the dc gate leakage current in amperes. IGL is typically about 2 pA, so ina is about 1.8 fA RMS/ Hz in the flat mid-range of ina(f ). Like ena(f ), ina(f ) shows a 1/f characteristic at low frequencies, which can be modeled by:

ina2 (f ) = 2qIGL (1+ f fic ) MSA Hz

(9.34)

where fic is the current noise corner frequency.

© 2004 by CRC Press LLC

344

Analysis and Application of Analog Electronic Circuits

 

 

VCC

 

 

 

RL

 

 

 

RB

 

 

 

C

Vo

 

 

 

 

RS

C

 

 

 

 

 

 

 

VS

 

 

 

 

 

A

 

RS B

rx

B’

 

 

 

gmvbe

 

VS

 

inb

inc RL Vo

 

rπ

B

FIGURE 9.7

(A) A simple, grounded emitter BJT amplifier relevant to noise calculations. (B) The noise equivalent circuit for the BJT amplifier.

For some transistors, the measured ena(f ) and ina(f ) have been found to be greater than the predicted, theoretical values; in other cases, they have been found to be less. No doubt the causes for these discrepancies lie in the oversimplifications used in their derivations. Note that MOSFETs have no gate–drain diode and thus do not have a 1/f component in their ina s.

9.2.4.4Noise in BJTs

The values of ena and ina associated with bipolar junction transistor amplifiers depend strongly on the device’s quiescent dc operating (Q) point because there are shot noise components superimposed on the quiescent base and collector currents. A mid-frequency, small-signal model of a simple grounded-emitter BJT amplifier is shown in Figure 9.7(B). In this circuit, negligible noise from RL, the voltage-controlled current source, gm vbe, and the small-signal base input resistance, rπ, is assumed. The shot noise PDSs are:

i

2

= 2 q I

BQ

MSA/Hz

(9.35)

 

nb

 

 

 

© 2004 by CRC Press LLC

Noise and the Design of Low-Noise Amplifiers for Biomedical Applications

345

inc2 = 2 q β IBQ MSA/Hz

(9.36)

where β = hfe is the BJT’s small-signal forward current gain evaluated at the BJT’s quiescent operating (Q) point. In this example, it is algebraically simpler not to find the equivalent input ena and ina, but to work directly with the two white shot noise sources in the mid-frequency, hybrid pi small-signal model. It can be shown (Northrop, 1990) that the total output noise voltage PDS is given by:

SNO (f ) = 4kTRL + 2q(IBQ β)(βRL )

2

+

4kTRs(βRL )2 + 2q IBQ Rs2 (βRL )2

MSV Hz

 

(VT IBQ + rx )2

 

 

 

 

(9.37)

where it is clear that rπ is approximated by VT/IBQ, Rs= rx + Rs, and the Johnson noise from RL is neglected because it is numerically small compared to the other terms.

It is easy to show (Northrop, 1990) that the mean squared output signal can be written as

 

=

 

(βRL )2 B (VT IBQ + rx )2

 

 

vos2

vs2

MSV

(9.38)

Thus, the MS signal-to-noise ratio at the amplifier output can be written:

SNRO =

v 2

B

(9.39)

s

 

4kTRs′ + 2qIBQRs2 + 2q(IBQ β)(VT IBQ + Rs)2

where B is the specified equivalent (Hz) noise bandwidth for the system. The SNRO given by the preceding equation has a maximum for some non-

negative IBQMAX. The IBQMAX that will give this maximum can be found by differentiating the denominator of Equation 9.39 and setting the derivative

equal to zero. This gives:

IBQMAX = VT (Rs′ β + 1) DC amperes

(9.40)

What should be remembered from the preceding exercise is that the best noise performance for BJT amplifiers is a function of quiescent biasing conditions (Q-point). Often these conditions must be found experimentally when working at high frequencies. Although individual BJT amplifiers may best be modeled for noise analysis with the two shot-noise current sources, it is more customary when describing complex BJT IC amplifier noise performance to use the more general and more easily used ena and ina two-source model parameterized for a given BJT Q-point.

© 2004 by CRC Press LLC