- •Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation
- •Dedication
- •Preface
- •Reader Background
- •Rationale
- •Description of the Chapters
- •Features
- •The Author
- •Table of Contents
- •1.1 Introduction
- •1.2 Sources of Endogenous Bioelectric Signals
- •1.3 Nerve Action Potentials
- •1.4 Muscle Action Potentials
- •1.4.1 Introduction
- •1.4.2 The Origin of EMGs
- •1.5 The Electrocardiogram
- •1.5.1 Introduction
- •1.6 Other Biopotentials
- •1.6.1 Introduction
- •1.6.2 EEGs
- •1.6.3 Other Body Surface Potentials
- •1.7 Discussion
- •1.8 Electrical Properties of Bioelectrodes
- •1.9 Exogenous Bioelectric Signals
- •1.10 Chapter Summary
- •2.1 Introduction
- •2.2.1 Introduction
- •2.2.4 Schottky Diodes
- •2.3.1 Introduction
- •2.4.1 Introduction
- •2.5.1 Introduction
- •2.5.5 Broadbanding Strategies
- •2.6 Photons, Photodiodes, Photoconductors, LEDs, and Laser Diodes
- •2.6.1 Introduction
- •2.6.2 PIN Photodiodes
- •2.6.3 Avalanche Photodiodes
- •2.6.4 Signal Conditioning Circuits for Photodiodes
- •2.6.5 Photoconductors
- •2.6.6 LEDs
- •2.6.7 Laser Diodes
- •2.7 Chapter Summary
- •Home Problems
- •3.1 Introduction
- •3.2 DA Circuit Architecture
- •3.4 CM and DM Gain of Simple DA Stages at High Frequencies
- •3.4.1 Introduction
- •3.5 Input Resistance of Simple Transistor DAs
- •3.7 How Op Amps Can Be Used To Make DAs for Medical Applications
- •3.7.1 Introduction
- •3.8 Chapter Summary
- •Home Problems
- •4.1 Introduction
- •4.3 Some Effects of Negative Voltage Feedback
- •4.3.1 Reduction of Output Resistance
- •4.3.2 Reduction of Total Harmonic Distortion
- •4.3.4 Decrease in Gain Sensitivity
- •4.4 Effects of Negative Current Feedback
- •4.5 Positive Voltage Feedback
- •4.5.1 Introduction
- •4.6 Chapter Summary
- •Home Problems
- •5.1 Introduction
- •5.2.1 Introduction
- •5.2.2 Bode Plots
- •5.5.1 Introduction
- •5.5.3 The Wien Bridge Oscillator
- •5.6 Chapter Summary
- •Home Problems
- •6.1 Ideal Op Amps
- •6.1.1 Introduction
- •6.1.2 Properties of Ideal OP Amps
- •6.1.3 Some Examples of OP Amp Circuits Analyzed Using IOAs
- •6.2 Practical Op Amps
- •6.2.1 Introduction
- •6.2.2 Functional Categories of Real Op Amps
- •6.3.1 The GBWP of an Inverting Summer
- •6.4.3 Limitations of CFOAs
- •6.5 Voltage Comparators
- •6.5.1 Introduction
- •6.5.2. Applications of Voltage Comparators
- •6.5.3 Discussion
- •6.6 Some Applications of Op Amps in Biomedicine
- •6.6.1 Introduction
- •6.6.2 Analog Integrators and Differentiators
- •6.7 Chapter Summary
- •Home Problems
- •7.1 Introduction
- •7.2 Types of Analog Active Filters
- •7.2.1 Introduction
- •7.2.3 Biquad Active Filters
- •7.2.4 Generalized Impedance Converter AFs
- •7.3 Electronically Tunable AFs
- •7.3.1 Introduction
- •7.3.3 Use of Digitally Controlled Potentiometers To Tune a Sallen and Key LPF
- •7.5 Chapter Summary
- •7.5.1 Active Filters
- •7.5.2 Choice of AF Components
- •Home Problems
- •8.1 Introduction
- •8.2 Instrumentation Amps
- •8.3 Medical Isolation Amps
- •8.3.1 Introduction
- •8.3.3 A Prototype Magnetic IsoA
- •8.4.1 Introduction
- •8.6 Chapter Summary
- •9.1 Introduction
- •9.2 Descriptors of Random Noise in Biomedical Measurement Systems
- •9.2.1 Introduction
- •9.2.2 The Probability Density Function
- •9.2.3 The Power Density Spectrum
- •9.2.4 Sources of Random Noise in Signal Conditioning Systems
- •9.2.4.1 Noise from Resistors
- •9.2.4.3 Noise in JFETs
- •9.2.4.4 Noise in BJTs
- •9.3 Propagation of Noise through LTI Filters
- •9.4.2 Spot Noise Factor and Figure
- •9.5.1 Introduction
- •9.6.1 Introduction
- •9.7 Effect of Feedback on Noise
- •9.7.1 Introduction
- •9.8.1 Introduction
- •9.8.2 Calculation of the Minimum Resolvable AC Input Voltage to a Noisy Op Amp
- •9.8.5.1 Introduction
- •9.8.5.2 Bridge Sensitivity Calculations
- •9.8.7.1 Introduction
- •9.8.7.2 Analysis of SNR Improvement by Averaging
- •9.8.7.3 Discussion
- •9.10.1 Introduction
- •9.11 Chapter Summary
- •Home Problems
- •10.1 Introduction
- •10.2 Aliasing and the Sampling Theorem
- •10.2.1 Introduction
- •10.2.2 The Sampling Theorem
- •10.3 Digital-to-Analog Converters (DACs)
- •10.3.1 Introduction
- •10.3.2 DAC Designs
- •10.3.3 Static and Dynamic Characteristics of DACs
- •10.4 Hold Circuits
- •10.5 Analog-to-Digital Converters (ADCs)
- •10.5.1 Introduction
- •10.5.2 The Tracking (Servo) ADC
- •10.5.3 The Successive Approximation ADC
- •10.5.4 Integrating Converters
- •10.5.5 Flash Converters
- •10.6 Quantization Noise
- •10.7 Chapter Summary
- •Home Problems
- •11.1 Introduction
- •11.2 Modulation of a Sinusoidal Carrier Viewed in the Frequency Domain
- •11.3 Implementation of AM
- •11.3.1 Introduction
- •11.3.2 Some Amplitude Modulation Circuits
- •11.4 Generation of Phase and Frequency Modulation
- •11.4.1 Introduction
- •11.4.3 Integral Pulse Frequency Modulation as a Means of Frequency Modulation
- •11.5 Demodulation of Modulated Sinusoidal Carriers
- •11.5.1 Introduction
- •11.5.2 Detection of AM
- •11.5.3 Detection of FM Signals
- •11.5.4 Demodulation of DSBSCM Signals
- •11.6 Modulation and Demodulation of Digital Carriers
- •11.6.1 Introduction
- •11.6.2 Delta Modulation
- •11.7 Chapter Summary
- •Home Problems
- •12.1 Introduction
- •12.2.1 Introduction
- •12.2.2 The Analog Multiplier/LPF PSR
- •12.2.3 The Switched Op Amp PSR
- •12.2.4 The Chopper PSR
- •12.2.5 The Balanced Diode Bridge PSR
- •12.3 Phase Detectors
- •12.3.1 Introduction
- •12.3.2 The Analog Multiplier Phase Detector
- •12.3.3 Digital Phase Detectors
- •12.4 Voltage and Current-Controlled Oscillators
- •12.4.1 Introduction
- •12.4.2 An Analog VCO
- •12.4.3 Switched Integrating Capacitor VCOs
- •12.4.6 Summary
- •12.5 Phase-Locked Loops
- •12.5.1 Introduction
- •12.5.2 PLL Components
- •12.5.3 PLL Applications in Biomedicine
- •12.5.4 Discussion
- •12.6 True RMS Converters
- •12.6.1 Introduction
- •12.6.2 True RMS Circuits
- •12.7 IC Thermometers
- •12.7.1 Introduction
- •12.7.2 IC Temperature Transducers
- •12.8 Instrumentation Systems
- •12.8.1 Introduction
- •12.8.5 Respiratory Acoustic Impedance Measurement System
- •12.9 Chapter Summary
- •References
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Analysis and Application of Analog Electronic Circuits |
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FIGURE 5.23
The Wien bridge oscillator’s root-locus diagram. Oscillation frequency is near 1/RC r/s.
Thus, when Avcrit = 3, the WB oscillator’s poles are on the jω axis at s =
±j/(RC), the condition that sustains steady-state output oscillations of ampli-
tude Vo = 3V1Q.
The Wien bridge and the phase-shift oscillators described have low total harmonic distortion (THD) in their outputs. The low distortion is the result of using the tungsten lamp in the automatic gain control nonlinearity. Other oscillators that use diode or zener diode clipping to regulate the oscillator’s loop gain have outputs with greater THD.
5.6Chapter Summary
This chapter began with a review of the frequency response as a descriptor for linear amplifier performance. The basics of Bode plotting an amplifier’s frequency response were examined, including the use of asymptotes to expedite pencil-and-paper Bode plots, given an amplifier’s transfer function or frequency response function. How to obtain the frequency response phase function from the frequency response function was also covered.
The concept of amplifier stability was set forth and stability was considered from the viewpoint of the location of a system’s poles in the s-plane. Various stability tests were mentioned, and the root-locus (RL) technique was stressed as a useful method for characterizing a linear feedback system’s closed-loop poles as a function of gain. Root-locus plotting rules were given with examples of RL plots. Application of the RL method was given in the
© 2004 by CRC Press LLC
Feedback, Frequency Response, and Amplifier Stability |
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design of feedback amplifiers with useful pole locations. The design of the phase-shift oscillator and the Wien bridge oscillator were shown to be made easier with the RL method. The venerable Barkhausen technique was introduced as an alternate to RL analysis.
Home Problems
5.1A power op amp is used to drive a CRT deflection yoke coil as shown in
Figure P5.1. Assume that the op amp is ideal except for finite differential gain, Vo = Kv (Vi − Vi′). Negative current feedback is used to increase the frequency
response of the coil current, IL, thus the coil’s B field. Assume the current from the VF node into RF is negligible, L = 0.01 Hy; RL = 0.5 Ω; R1 = 104 Ω; RF = 104 Ω; Kv = 104; RM = 1 Ω; and RF, R1 RM.
a.Derive an algebraic expression for the amplifier’s transfer admit-
tance, YL(jω) = IL/V1, in time V1 constant form. Show what happens to YL(jω) as Kv. gets very large.
b.Evaluate YL(jω) numerically. Give the time constant of the coil with and without feedback.
RF
R1
Vi’ |
Vo |
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V1
L
IL
RL
↑
VF
0
RM
1Ω
FIGURE P5.1
5.2Negative voltage feedback is used to extend the bandwidth of a differential amplifier (not an op amp), as shown in Figure P5.2. The DA gain is given by the transfer function:
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© 2004 by CRC Press LLC
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Analysis and Application of Analog Electronic Circuits |
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FIGURE P5.2
a.Draw the system’s root-locus diagram to scale. Let β ∫ R1/(R1 + RF).
b.Find the numerical value of β required to give the closed-loop poles a damping factor ξ = 0.707.
c.What is the system’s dc gain with the β value found in part B?
Give the system’s fT with this β. Compare this fT to the fT of the amplifier without feedback (β = 0).
d.The same amplifier is compensated with a zero in the feedback path. Thus, β = βo (3.333 ∞ 10−5 s + 1). Repeat parts b through d.
5.3A nonideal op amp circuit is shown in Figure P5.3. The op amp transfer function is:
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Assume that R1 = 104 Ω; RF = 90 kΩ; Ro = 100 Ω; and β = R1/(R1 + RF).
RF
R1 βVo
Vi’
+
Ro 
Vo
Io
Vi
Hoc(s)(Vi − Vi’)
V1
FIGURE P5.3
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Feedback, Frequency Response, and Amplifier Stability |
233 |
a.Find an algebraic expression for the amplifier’s Thevenin output impedance, Zout (jω), at the Vo node, in time-constant form. (Neglect RF + R1 to ground at the output.)
b.Sketch and dimension the Bode plot asymptotes for Zout(jω) .
5.4Use the root-locus technique to find the range of feedback gain, β (positive
and negative values), over which the cubic system of Figure P5.4 is stable. What is the minimum value of β required for oscillation? At what Hertz frequency, fo, will the system first oscillate?
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FIGURE P5.4
5.5Positive voltage feedback is used around an op amp that is ideal except for its gain (see Figure P5.5):
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s + 1 |
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a.Sketch the system’s root-locus diagram to scale. Can this system oscillate?
b.Derive an expression for (Vo/V1)(s) in time-constant form. What conditions on R1 and R2 determine stability?
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V1 |
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FIGURE P5.5
5.6Two ideal op amps are used to make a “linear” oscillator, shown in Figure P5.6.
a.Give an expression for the oscillator’s loop gain, AL(s). Note that the oscillator uses positive voltage feedback.
b.Design the oscillator to oscillate at 3.0 kHz. Use a root-locus graphical approach. Specify the required numerical values for RF, R1, and C1.
c.Illustrate a nonlinear means to limit the peak amplitude of Vo to 5.0 V.
© 2004 by CRC Press LLC
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Analysis and Application of Analog Electronic Circuits |
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C2 |
10 kΩ R2
Vo C1
FIGURE P5.6
5.7A certain feedback amplifier has the loop gain:
A (s) = +K(a (s − 400) )
L s2 + s 1000 + 106
a. Assume Ka > 0. Draw the system’s root-locus diagram to scale as a function of increasing Ka. Find the numerical of Ka at which the system is on the verge of oscillation. Find the oscillation frequency, ωo r/s.
b. Now assume Ka < 0. Sketch the system’s root-locus to scale for increasingly negative Ka. Find the Ka value at which the system has two equal real, negative poles. (Use Matlab’s® RLocus program.)
5.8 A certain feedback amplifier has the loop gain:
AL (s) = |
−Kv |
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+ ω 2 |
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a. Plot and dimension the root-locus for this system for 0 < Kv ≤ •.
b. Plot and dimension the root-locus for this system for 0 > Kv ≥ –•. Give an expression for the Kv value at which the system becomes unstable.
5.9 A prototype “linear” oscillator is shown in Figure P5.9. Assume the op amps are ideal.
a. Find an expression for the transfer function (V2/V1)(s).
b. Find an expression for the circuit’s loop gain. Break the loop at the V1 – V1′ link. AL(jω) = V1′/V1.
c. Use Matlab’s RLocus program to draw the oscillator’s root-locus diagram as a function of β for positive β. 0 < β ≤ 1.
d. Use the Barkhausen criterion to find an expression for the oscillation frequency, ωo r/s, and the critical gain, β, required for oscillation.
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Feedback, Frequency Response, and Amplifier Stability |
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FIGURE P5.9
5.10 An otherwise ideal op amp has the gain:
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10−2 s + 1 |
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The op amp is connected as a simple inverting gain amplifier, as shown in Figure P5.10.
a.Find the maximum dc gain the amplifier can possess and have a −3-dB frequency of 50 kHz. That is, find the RF required.
b.For the RF of part a, find the inverting amplifier’s fT.
RF
1 kΩ
R1 Vi’
OA |
Vo |
V1
FIGURE P5.10
5.11Figure P5.11B illustrates a “regenerative magnetic pickup circuit” as described in NASA Tech Briefs #GSC-13309. The purpose of the circuit is to present a multiturn magnetic search coil with a negative terminating impedance that, when summed with the series impedance of the coil, forms a very low Thevenin impedance. This allows more current, Is, to flow from the
© 2004 by CRC Press LLC
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Analysis and Application of Analog Electronic Circuits |
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FIGURE P5.11
•
open-circuit induced voltage, Vs = NΦ, where N is the number of turns of the
•
search coil and Φ is the time rate of change of the magnetic flux linking the coil’s area. The op amp, less search coil, forms what is called a negative impedance converter circuit (NIC) (Ghaussi, 1971, Chapter 8). Assume the op amp is ideal in this analysis.
a.Refer to Figure P5.11A. Find an expression for the transfer func-
tion, Vo/Vs(s). Let Ls = 0.01 Hy; Rs = 100 Ω; and R3 = 104. Also evaluate the transfer function’s break frequency, fb = ωb/2π and its dc gain.
b.Refer to Figure P5.11B. Replace the search coil (Ls, Rs) with just
the voltage source, Vs. Find expressions for the input current, Is, and the input impedance looking into the Vi′ node. (Note that it emulates a negative inductance.)
c.Now consider the search coil in place with open-circuit, induced
EMF, Vs. Derive an expression for Vo/Vs(s) in time-constant form. Comment on the conditions on the circuit parameters (R1, R2, R3, and C) that will make the circuit unstable or the dc gain infinite.
5.12Negative voltage feedback and positive current feedback are used on a power op amp circuit, shown in Figure P5.12. The op amp is ideal except for the differential gain:
Vo |
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K V − V′ |
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τs + 1 |
Assume that (R1 + R2) (R3 + R4); α ∫ R2/(R1 + R2); β ∫ R1/(R1 + R2); and γ ∫ R4 /(R3 + R4).
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Feedback, Frequency Response, and Amplifier Stability |
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R2
R1
Vi’
POA |
Vo |
V1
Vi
R3
γVo
R4
FIGURE P5.12
a.Assume op amp Rout = 0. Find an expression for Vo/V1 in timeconstant form. Comment how the PVFB affects the dc gain, corner frequency, and gain-bandwidth product.
b.Now let Rout > 0. Replace the load resistor, R3, with a test voltage source, vt, to find an expression for the Thevenin output resistance that R3 “sees.”
5.13It is desired to measure the slew rate of a prototype op amp design physically.
The linear small-signal behavior of the op amp is known to be Vo = (Vi − Vi′)105/(10−3 s + 1). The op amp is connected as a unity-gain follower, as shown in Figure P5.13A.
a.Find Vo/Vs(s) in Laplace form. What is the follower’s time constant?
b.A step input of 100 mV is given, i.e., Vs(s) = 0.1/s. Sketch and dimension vo(t) at the follower output. What is the maximum slope of vo(t)?
c.The follower’s response to a 10-V input step, Vs(s) = 10/s, is shown in Figure P5.13B. Estimate the op amp’s slew rate, η, in V/μsec. Sketch and dimension vo(t) to the 10-V step when the slew rate is assumed to be infinite.
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100 ns |
FIGURE P5.13
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Analysis and Application of Analog Electronic Circuits |
5.14A certain op amp has open-loop poles at f1 = 100 Hz and at f2 = 100 kHz, and a negative real zero at f0 = 110 kHz. Its dc gain is 105. The op amp is connected as a noninverting amplifier (see text Figure 6.1(C)). Plot and dimension the amplifier’s root-locus diagram as a function of the closed-loop dc gain, Vo/Vs = (R1 + RF)/R1. Let Vo/Vs range from 1 to 103.
© 2004 by CRC Press LLC