TABLE OF CONTENTS
Analog Electronic Computation of the Finite-Time Fourier Transform
for Power Spectral Density Measurements
I. Introduction 1
II. Mathematical Modeling of Finite-Time Fourier Transform System 6
Theoretical Basis of the Fourier Transform 6
Hardware Implemented 7
Phase Relationship of the Noise and Reference Signals 7
Model Cosine Signal - Cosine Transform Result 8
MathCad Modeling of Flicker Noise and High-Pass Filtering
of Model 9
Computer Modeling of Flicker Noise and Filter of White Noise 14
A General Result for the Complete Fourier Transform 18
A Result for a Cosine Noise Source 19
III. Description of the Measurement System 25
Overview of the Gain Path of the MOSFET Under Test 27
Description of the Cosine Transform Circuit 33
Timing Considerations of the Cosine Transform Circuit 40
Multiplier Autozero Loop Hardware 42
Integrator Autozero Loop Hardware 46
Preamplifier Circuit 46
Test Device Circuit 49
Data Collection System- Digital Oscilloscope and IBM PC clone 51
Description of Lotus Macros 51
Sampling Considerations 52
IV. Accuracy Limitations 54
Analog Multiplier Considerations 54
Cross-Coupling between the Data Signal and Reference Channels 54
Offset Effects / Autozero Circuits 61
Data Signal and / or Reference Channel Offsets 61
Multiplier Offsets 62
Integrator Offsets 62
Signal Overload conditions 62
Low and High Frequency Response Limitations 63
A. Preamplifier Circuit 64
B. MOSFET Bias Circuit 66
Extraneous Effects 68
Noise Analysis 68
A. Preamplifier Circuit 68
B. MOSFET Bias Circuit 84
Pspice Gate Resistor Simulation 90
V. Overall gain determination and Reference Calibration 93
Theoretical Considerations 93
Noise Spectral Density 93
Transfer Function Concept and Noise Bandwidth 95
Practical Considerations 97
Reference Calibration Method 1: Series Gate Resistance 98
Reference Calibration Method 2: Twin-T Filters 98
Reference Calibration Method 3: Standard Noise Source 101
VI. MOSIS MOSFET measurements 103
VII. Summary 110
Bibliography 112
Appendices 114
A. MathCad Program Listing 115
B. Test Equipment Used 116
C. Lotus Macro Listing 117
D. Listing of OP AMP IC's used in Circuits 119
E. Software Packages Used 120
F. Pspice input file for Preamp. simulations 121
G. Pspice input file for Bias Circuit simulations 125
LIST OF TABLES
Table
3.1 Tektronix 2230 oscilloscope gain factors 24
4.1 Preamp. U1 stage midband noise calculation 69
4.2 Preamp. U2A stage midband noise calculation 70
4.3 Preamp. U3A stage midband noise calculation 70
4.4 Preamp. U3B stage midband noise calculation 71
4.5 Preamp. U2B stage midband noise calculation 71
4.6 Transfer fucntions at three low frequencies 72
4.7 RC filter capacitor impedance at three low frequencies 73
4.8 Op. Amp. flicker noise contributions at three low frequencies 73
4.9 Preamp. U1 stage 100 Hz. noise calculation 74
4.10 Preamp. U2A stage 100 Hz. noise calculation 74
4.11 Preamp. U3A stage 100 Hz. noise calculation 75
4.12 Preamp. U3B stage 100 Hz. noise calculation 75
4.13 Preamp. U2B stage 100 Hz. noise calculation 75
4.14 Preamp. U1 stage 10 Hz. noise calculation 76
4.15 Preamp. U2A stage 10 Hz. noise calculation 77
4.16 Preamp. U3A stage 10 Hz. noise calculation 77
4.17 Preamp. U3B stage 10 Hz. noise calculation 77
4.18 Preamp. U2B stage 10 Hz. noise calculation 78
4.19 Preamp. U1 stage 0.3162 Hz. noise calculation 78
4.20 Preamp. U2A stage 0.3162 Hz. noise calculation 79
4.21 Preamp. U3A stage 0.3162 Hz. noise calculation 79
4.22 Preamp. U3B stage 0.3162 Hz. noise calculation 80
4.23 Preamp. U2B stage 0.3162 Hz. noise calculation 80
4.24 Preamp. noise hand calculation and Pspice comparison 81
4.25 Bias Circuit noise hand calculation 84
4.26 Bias Circuit noise hand calculation vs. Pspice 85
4.27 Bias Circuit OP-27 noise current at four frequencies 86
4.28 Bias Circuit noise terms with flicker noise at four frequencies 86
5.1 data table 90
LIST OF FIGURES
Figure
2.1 Fourier Transform using two cosine transform circuits 6
2.2 Plot of I2 / T as a function of frT for flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 for high-pass
filter frequency xf = 0.1 11
2.3 Plot of I2 / T as a function of frT for flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 for high-pass
filter frequency xf = 0.020 12
2.4 Plot of I2 / T as a function of frT for flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 for no high-pass
filter 13
2.5 Plot of Power Spectral Density of white noise with flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 for high-pass filter frequency
xf = 0.1 15
2.6 Plot of Power Spectral Density of white noise with flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 for high-pass filter frequency
xf = 0.020 16
2.7 Plot of Power Spectral Density of white noise with flicker noise corner
frequencies of xc = 0, 0.1, 0.2, and 0.5 with no filter 17
3.1 Block diagram of the system 26
3.2 Overall gain path diagram 28
3.3 Schematic of the cosine transform circuit 34
3.4 Noise Data Input 35
3.5 U1 output - amplified gated noise 36
3.6 U2 output - amplified reference signal 37
3.7 U3 output - amplified multiplier output 38
3.8 U7 output - amplified integrated output 39
3.9 Timing circuit and timing diagram for the cosine transform
circuit 41
3.10 U6 output - multiplier autozero output 43
3.11 Nulled output 44
3.12 Non-Nulled output 45
3.13 U5 output - integrator autozero output 47
3.14 Schematic of the preamplifier circuit 48
3.15 Schematic of the test transistor bias circuit 50
4.1a X feedthrough effect 55
4.1b Y feedthrough effect 55
4.2 100Hz Output 57
4.3 100Hz Sampled Output 58
4.4 1kHz Output 59
4.5 1kHz Sampled Output 60
4.6a X DC offset effect 61
4.6b Y DC offset effect 61
4.7 RC High Pass Filter 63
4.8 Bode Plot Frequency Response of Preamplifier 65
4.9 Bode Plot Frequency Response of MOSFET Bias Circuit 67
4.10 Preamplifier Pspice output noise plot 82
4.11 Preamplifier Pspice equivalent input noise plot 83
4.12 Bias Circuit Pspice output noise plot 88
4.13 Bias Circuit Pspice equivalent input noise plot 89
4.14 Bias Circuit Pspice output noise with 20k½ 91
4.15 Bias Circuit Pspice input noise with 20k½ 92
5.1 Twin-T Filter Setup 99
5.2 Standard Noise Source System 102
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