Report

ECE 271 Electronic Circuits I Topic 1 Introduction to Electronics NJIT ECE-271 Dr. S. Levkov Chap 1 - 1 Topic Goals • Explore the history of electronics. • Describe classification of electronic signals. • Introduce tolerance impacts and analysis. NJIT ECE-271 Dr. S. Levkov Chap 1 - 2 1. The Subject of the Course • The subject of the course is modern electronics, or microelectronics. • Microelectronics refers to the integrated-circuit (IC) technology • IC – can contains hundreds of millions of components on a IC chip with the area of the order 100 sq. mm. • Subject of study: - electronic components/devices that can be used singly (discrete circuits) - electronic components/devices that can be used as components of the IC NJIT ECE-271 Dr. S. Levkov Chap 1 - 3 2. Brief History The Start of the Modern Electronics Era It can be said that the invention of the transistor and the subsequent development of the microelectronics have done more to shape the modern era than any other invention. Bardeen, Shockley, and Brattain at Bell Labs - Brattain and Bardeen invented the bipolar transistor in 1947. NJIT ECE-271 Dr. S. Levkov The first germanium bipolar transistor. Roughly 50 years later, electronics account for 10% (4 trillion dollars) of the world GDP. Chap 1 - 4 Electronics Milestones 1874 Braun invents the solid-state rectifier (using point contact based on lead sulphide) 1906 DeForest invents triode vacuum tube. 1907-1927 First radio circuits developed from diodes and triodes. 1925 Lilienfeld field-effect device patent filed. 1947 Bardeen and Brattain at Bell Laboratories invent bipolar transistors. 1952 Commercial bipolar transistor production at Texas Instruments. 1956 Bardeen, Brattain, and Shockley receive Nobel prize. NJIT ECE-271 Dr. S. Levkov 1958 1961 1963 1968 1970 1971 1978 1974 1984 1995 Integrated circuits developed by Kilby (TI) and Noyce and Moore (Fairchild Semiconductor) First commercial IC from Fairchild Semiconductor IEEE formed from merger of IRE and AIEE First commercial IC opamp One transistor DRAM cell invented by Dennard at IBM. 4004 Intel microprocessor introduced. First commercial 1-kilobit memory. 8080 microprocessor introduced. Megabit memory chip introduced. Gigabite memory chip presented. Chap 1 - 5 Evolution of Electronic Devices Vacuum Tubes Discrete Transistors SSI and MSI Integrated Circuits VLSI Surface-Mount Circuits NJIT ECE-271 Dr. S. Levkov Chap 1 - 6 Evolution of Electronic Devices A work of art from the Museum of Modern Art, Paris NJIT ECE-271 Dr. S. Levkov Chap 1 - 7 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. NJIT ECE-271 Dr. S. Levkov Chap 1 - 8 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. NJIT ECE-271 Dr. S. Levkov Chap 1 - 9 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - Chap 1 - 10 NJIT ECE-271 Dr. S. Levkov Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - 1014 Chap 1 - 11 NJIT ECE-271 Dr. S. Levkov Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - 1014 – Number of seconds elapsed since Big Bang – NJIT ECE-271 Dr. S. Levkov Chap 1 - 12 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - 1014 – Number of seconds elapsed since Big Bang – 1017 NJIT ECE-271 Dr. S. Levkov Chap 1 - 13 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - 1014 – Number of seconds elapsed since Big Bang – 1017 – Number of ants in the world - NJIT ECE-271 Dr. S. Levkov Chap 1 - 14 Microelectronics Proliferation • The integrated circuit was invented in 1958. • World transistor production has more than doubled every year for the past twenty years. • Every year, more transistors are produced than in all previous years combined. • Approximately 1018 transistors were produced in a recent year. • To compare: – Number of cells in a human body - 1014 – Number of seconds elapsed since Big Bang – 1017 – Number of ants in the world - roughly 50 transistors for every ant in the world. *Source: Gordon Moore’s Plenary address at the 2003 International Solid State Circuits Conference. NJIT ECE-271 Dr. S. Levkov Chap 1 - 15 Rapid Increase in Density of Microelectronics Memory chip density versus time. NJIT ECE-271 Dr. S. Levkov Microprocessor complexity versus time. Chap 1 - 16 Device Feature Size • Feature size reductions enabled by process innovations. • Smaller features lead to more transistors per unit area and therefore higher density. • • • • • NJIT ECE-271 Dr. S. Levkov SSI – small scale integration (< 102) MSI – medium SI (102- 103) LSI – large SI (103- 104) VLSI – very large SI (104- 109) ULSI & GSI– ultra large SI & gigascale integration (> 109) Chap 1 - 17 3. Types of Signals • Analog signals take on continuous values - typically current or voltage. NJIT ECE-271 Dr. S. Levkov Chap 1 - 18 3. Types of Signals • Analog signals take on continuous values - typically current or voltage. • Digital signals appear at discrete levels (do not confuse with discrete times). NJIT ECE-271 Dr. S. Levkov Chap 1 - 19 3. Types of Signals • Analog signals take on continuous values - typically current or voltage. • Digital signals appear at discrete levels (do not confuse with discrete times). • Usually we use binary signals with only two levels - VL and VH • One level is referred to as logical 1 and logical 0 is assigned to the other level. • Typically: VL 0V , VH 5V - was standard for many years VL 0V , VH 3.3, 2.5,1.5V used now. • Bipolar levels also exist NJIT ECE-271 Dr. S. Levkov Chap 1 - 20 Analog and Digital Signals Analog signal • Analog signals usually are continuous in time and in values. NJIT ECE-271 Dr. S. Levkov Chap 1 - 21 Analog and Digital Signals Analog signal • Analog signals usually are continuous in time and in values. NJIT ECE-271 Dr. S. Levkov Discrete time signal • Sampled, discrete time signals are discrete in time (values are typically separated by fixed time intervals). • The values are continuous. • Needs digitization. Chap 1 - 22 Analog and Digital Signals • Sampled discrete time signal NJIT ECE-271 Dr. S. Levkov Chap 1 - 23 Analog and Digital Signals • Sampled discrete time signal NJIT ECE-271 Dr. S. Levkov • Digitized discrete time signal discrete time and digitized discrete values. • The values are not continuous – belong to a finite set. Chap 1 - 24 Analog and Digital Signals • Sampled discrete time signal NJIT ECE-271 Dr. S. Levkov • Digitized discrete time signal discrete time and digitized discrete values. • The values are not continuous – belong to a finite set. Chap 1 - 25 Analog and Digital Signals • Sampled discrete time signal NJIT ECE-271 Dr. S. Levkov • Digitized discrete time signal discrete time and digitized discrete values. • The values are not continuous – belong to a finite set. Chap 1 - 26 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define NJIT ECE-271 Dr. S. Levkov Chap 1 - 27 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} NJIT ECE-271 Dr. S. Levkov Chap 1 - 28 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} • The most significant bit (MSB) - VMSB NJIT ECE-271 Dr. S. Levkov ? Chap 1 - 29 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} • The most significant bit (MSB) - VMSB 21VFS , f .i. {1000000} NJIT ECE-271 Dr. S. Levkov Chap 1 - 30 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} • The most significant bit (MSB) - VMSB 21VFS , f .i. {1000000} • Then for an n-bit D/A converter, the output voltage is expressed as: VO (b1 21 b2 22 ... bn 2 n )VFS NJIT ECE-271 Dr. S. Levkov Chap 1 - 31 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} • The most significant bit (MSB) - VMSB 21VFS , f .i. {1000000} • Then for an n-bit D/A converter, the output voltage is expressed as: VO (b1 21 b2 22 ... bn 2 n )VFS (b1 2n 1 b2 2n 2 ... bn 20 )2 n VFS NJIT ECE-271 Dr. S. Levkov Chap 1 - 32 Digital-to-Analog (D/A) Conversion • The input is a binary number b {b1b2 ...bn }, f .i. {1001011} • Let’s introduce VFS = Full-Scale Voltage and then define • The least significant bit (LSB) - the smallest possible binary number (smallest voltage change) is known as resolution of the converter. VLSB 2n VFS , f .i. {0000001} • The most significant bit (MSB) - VMSB 21VFS , f .i. {1000000} • Then for an n-bit D/A converter, the output voltage is expressed as: VO (b1 21 b2 22 ... bn 2 n )VFS (b1 2n1 b2 2n 2 ... bn 20 )2 n VFS (b1 2n1 b2 2n2 ... bn 20 )VLSB NJIT ECE-271 Dr. S. Levkov Chap 1 - 33 Analog-to-Digital (A/D) Conversion • Analog input voltage Vx is converted to the nearest n-bit number that represent VO - the closest (WRT to the accuracy = VLSB /2) value to the Vx VO (b1 21 b2 22 ... bn 2 n )VFS (b1 2n1 b2 2n2 ... bn 20 )VLSB • Output is approximation of input due to the limited resolution of the n-bit output. Error is expressed as: V Vx (b1 21 b2 22 ... bn 2 n )VFS NJIT ECE-271 Dr. S. Levkov Chap 1 - 34 A/D Converter Transfer Characteristic (input-output) NJIT ECE-271 Dr. S. Levkov Chap 1 - 35 A/D Converter Transfer Characteristic (input-output) NJIT ECE-271 Dr. S. Levkov Chap 1 - 36 A/D Converter Transfer Characteristic (input-output) NJIT ECE-271 Dr. S. Levkov Chap 1 - 37 A/D Converter Transfer Characteristic (input-output) NJIT ECE-271 Dr. S. Levkov Chap 1 - 38 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 39 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 40 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 41 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 42 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 43 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 44 A/D Converter Transfer Characteristic (input-output) VLSB / 2 NJIT ECE-271 Dr. S. Levkov VFS 16 Chap 1 - 45 A/D Converter Transfer Characteristic (input-output) NJIT ECE-271 Dr. S. Levkov Chap 1 - 46 4. Notational Conventions • In many circuits the signal will be a combination of the dc and time varying values. • Total signal = DC bias + time varying signal vT VDC vsig iT I DC isig • Resistance and conductance - R and G with same subscripts will denote reciprocal quantities. Most convenient form will be used within expressions. 1 Gx Rx NJIT ECE-271 Dr. S. Levkov and 1 g r Chap 1 - 47 5. Circuit Theory Review: Thévenin and Norton Equivalent Circuits Thévenin Norton NJIT ECE-271 Dr. S. Levkov Chap 1 - 48 5. Circuit Theory Review: Thévenin and Norton Equivalent Circuits Thévenin Norton NJIT ECE-271 Dr. S. Levkov Chap 1 - 49 5. Circuit Theory Review: Thévenin and Norton Equivalent Circuits Thévenin Norton NJIT ECE-271 Dr. S. Levkov Chap 1 - 50 Circuit Theory Review: Find the Thévenin Equivalent Voltage Problem: Find the Thévenin equivalent voltage at the output. Solution Approach: Voltage source vth is defined as the output voltage with no load. NJIT ECE-271 Dr. S. Levkov Chap 1 - 51 Circuit Theory Review: Find the Thévenin Equivalent Voltage NJIT ECE-271 Dr. S. Levkov Chap 1 - 52 Circuit Theory Review: Find the Thévenin Equivalent Voltage Applying KCL at the output node, vo vi vo i1 0 R1 RS Current i1 can be written as: i1 vo vi Substituting into previous expression: R1 1 1 1 vo , vi RS R1 R1 NJIT ECE-271 Dr. S. Levkov vo 1 RS vi 1 RS R1 Chap 1 - 53 Circuit Theory Review: Find the Thévenin Equivalent Voltage (cont.) Using the given component values: vo 1 RS 50 11 k vi vi 1 RS R1 50 11 k 20 k 0.718vi and v th 0.718v i NJIT ECE-271 Dr. S. Levkov Chap 1 - 54 Circuit Theory Review: Find the Thévenin Equivalent Resistance NJIT ECE-271 Dr. S. Levkov Chap 1 - 55 Circuit Theory Review: Find the Thévenin Equivalent Resistance Problem: Find the Thévenin equivalent resistance. Solution Approach: Find Rth as the output equivalent resistance with independent sources set to zero. NJIT ECE-271 Dr. S. Levkov Test voltage vx has been added to the previous circuit. Applying vx and solving for ix allows us to find the Thévenin resistance as vx/ix. Chap 1 - 56 Circuit Theory Review: Find the Thévenin Equivalent Resistance (cont.) Applying KCL, i1 i1 vx ix 0 RS where i1 we get vx R1 v 1 ix x vx 0 RS R1 Rth NJIT ECE-271 Dr. S. Levkov , or ix ( 1) RS R1 vx 0 R1 RS vx RS R1 (1 k)(20 k) 282 ix ( 1) RS R1 (50 1)(1 k) 20 k Chap 1 - 57 Circuit Theory Review: Find the Norton Equivalent Circuit Problem: Find the Norton equivalent circuit. Solution approach: Evaluate current through output short circuit. NJIT ECE-271 Dr. S. Levkov Chap 1 - 58 Circuit Theory Review: Find the Norton Equivalent Circuit NJIT ECE-271 Dr. S. Levkov Chap 1 - 59 Circuit Theory Review: Find the Norton Equivalent Circuit Problem: Find the Norton equivalent circuit. Solution approach: Evaluate current through output short circuit. A short circuit has been applied across the output. The Norton current is the current flowing through the short circuit at the output. NJIT ECE-271 Dr. S. Levkov Chap 1 - 60 Circuit Theory Review: Find the Norton Equivalent Circuit (cont.) Applying KCL, i1 i1 in 0 Where i1 Thus in 0 vi vi R1 R1 1 R1 vi Short circuit at the output causes zero current to flow through RS. vi 50 1 in vi (2.55 mS)vi 20 k 392 Rth is equal to Rth found earlier. NJIT ECE-271 Dr. S. Levkov Chap 1 - 61 Final Thévenin and Norton Circuits vth = 0.718vi in = (2.55x10-3)vi Check of Results: Note that vth = inRth and this can be used to check the calculations: inRth=(2.55 mS)vi(282 ) = 0.719vi, accurate within round-off error. NJIT ECE-271 Dr. S. Levkov Chap 1 - 62 6. Signal spectrum Any periodic signal can be represented in the form of Fourier series: v(t ) a0 ak cos k0t bk sin k0t k 0 1 a0 T t0 T t0 2 v(t )dt , ak T t0 T t0 2 v(t ) cos k0 dt , bk T t0 T v(t ) sin k0 dt t0 T is the period of the function; ak , bk Fourier coefficients, 0=2/T (rad/s) is the fundamental radian frequency and f0=1/T (Hz) is the fundamental frequency of the signal. 2f0, 3f0, 4f0 , ….. are called the harmonic frequencies. v(t ) A0 Ak cos(k0 t k ) Alternative representation: k 0 bk A0 a0 , Ak ak bk , k tan ak 2 NJIT ECE-271 Dr. S. Levkov 2 1 Chap 1 - 63 Fourier Series example For example, a square wave is represented by the following Fourier series: v(t) VDC 2VO 1 1 sin0 t sin30 t sin50 t ... 3 5 Signal Spectrum The spectrum of the periodic signal is the graph of the Fourier coefficients vs the harmonic frequencies. Periodic signals have discrete spectra. Non periodic signals have continuous spectra often occupying a broad range of frequencies. NJIT ECE-271 Dr. S. Levkov Chap 1 - 64 Frequencies of Some Common Signals • • • • • • • • • Audible sounds Baseband TV FM Radio Television (Channels 2-6) Television (Channels 7-13) Maritime and Govt. Comm. Cell phones and other wireless Satellite TV Wireless Devices 20 Hz - 20 0 - 4.5 88 - 108 54 - 88 174 - 216 216 - 450 1710 - 2690 3.7 - 4.2 5.0 - 5.5 KHz MHz MHz MHz MHz MHz MHz GHz GHz Show the Fourier applet here NJIT ECE-271 Dr. S. Levkov Chap 1 - 65 7. Circuit Element Variations • All electronic components have manufacturing tolerances. – Resistors can be purchased with 10%, 5%, and 1% tolerance. (IC resistors are often 10%.) – Capacitors can have asymmetrical tolerances such as +20%/-50%. – Power supply voltages typically vary from 1% to 10%. • Device parameters will also vary with temperature and age. • Circuits must be designed to accommodate these variations. • We will use worst-case and Monte Carlo (statistical) analysis to examine the effects of component parameter variations. NJIT ECE-271 Dr. S. Levkov Chap 1 - 66 Tolerance Modeling • For symmetrical parameter variations Pnom(1 - ) P Pnom(1 + ) Pnom - is the parameter specification - is the tolerance • For example, a 10K resistor with 5% percent tolerance could exhibit the resistance in the following range of values: 10k(1 - 0.05) R 10k(1 + 0.05) 9,500 R 10,500 Chap 1 - 67 NJIT ECE-271 Dr. S. Levkov Circuit Analysis with Tolerances • Worst-case analysis – Parameters are manipulated to produce the worst-case min and max values of desired quantities. – This can lead to over design since the worst-case combination of parameters is rare. – It may be less expensive to discard a rare failure than to design for 100% yield. NJIT ECE-271 Dr. S. Levkov Chap 1 - 68 Circuit Analysis with Tolerances • Worst-case analysis – Parameters are manipulated to produce the worst-case min and max values of desired quantities. – This can lead to over design since the worst-case combination of parameters is rare. – It may be less expensive to discard a rare failure than to design for 100% yield. • Monte-Carlo analysis – Parameters are randomly varied to generate a set of statistics for desired outputs. – Based on that we calculate the average values and optimize the design so that failures due to parameter variation are less frequent than failures due to other mechanisms. – In this way, the design difficulty is better managed than a worstcase approach. NJIT ECE-271 Dr. S. Levkov Chap 1 - 69 Worst Case Analysis Example Problem: Find the nominal and worst-case values for output voltage and source current. Solution: • Unknowns: VOnom, VOmin , VOmax, IInom, IImin, IImax . • Approach: Find nominal values and then select R1, R2, and VI values to generate extreme cases of the unknowns. Nominal Source current: I Inom VInom nom R1 R2nom 15V 278 A 18k 36k NJIT ECE-271 Dr. S. Levkov Nominal voltage solution: VOnom VInom 15V R1nom R1nom R2nom 18k 5V 18k 36k Chap 1 - 70 Worst-Case Analysis Example (cont.) Now we need to figure out how to find the min and max possible of the voltage and current in question. NJIT ECE-271 Dr. S. Levkov Chap 1 - 71 Worst-Case Analysis Example (cont.) Now we need to figure out how to find the min and max possible of the voltage and current in question. Rewrite VO to help us determine how to find the worst-case values. VO VI R1 V I R2 R1 R2 1 R1 NJIT ECE-271 Dr. S. Levkov Chap 1 - 72 Worst-Case Analysis Example (cont.) Now we need to figure out how to find the min and max possible of the voltage and current in question. Rewrite VO to help us determine how to find the worst-case values. R1 V VO VI I R2 R1 R2 1 R1 NJIT ECE-271 Dr. S. Levkov VO is maximized for max VI, R1 and min R2. VO is minimized for min VI, R1, and max R2. Chap 1 - 73 Worst-Case Analysis Example (cont.) Now we need to figure out how to find the min and max possible of the voltage and current in question. Rewrite VO to help us determine how to find the worst-case values. R1 V VO VI I R2 R1 R2 1 R1 max O V VO is maximized for max VI, R1 and min R2. VO is minimized for min VI, R1, and max R2. VImax 15V (1.1) 5.87V R2min 1 36 K (0.95) 1 max 18K (1.05) R1 NJIT ECE-271 Dr. S. Levkov Chap 1 - 74 Worst-Case Analysis Example (cont.) Now we need to figure out how to find the min and max possible of the voltage and current in question. Rewrite VO to help us determine how to find the worst-case values. R1 V VO VI I R2 R1 R2 1 R1 max O V VO is maximized for max VI, R1 and min R2. VO is minimized for min VI, R1, and max R2. VImax 15V (1.1) 5.87V R2min 1 36 K (0.95) 1 max 18K (1.05) R1 NJIT ECE-271 Dr. S. Levkov min O V VImin 15V (0.90) 4.20V R2max 1 36 K (1.05) 1 min 18K (0.95) R1 Chap 1 - 75 Worst-Case Analysis Example (cont.) Worst-case source currents: IImax VImax 15V (1.1) min 322A R1 R2min 18k(0.95) 36k(0.95) NJIT ECE-271 Dr. S. Levkov Chap 1 - 76 Worst-Case Analysis Example (cont.) Worst-case source currents: IImax VImax 15V (1.1) min 322A R1 R2min 18k(0.95) 36k(0.95) IImin VImin 15V (0.9) max 238A max R1 R2 18k(1.05) 36k(1.05) NJIT ECE-271 Dr. S. Levkov Chap 1 - 77 Worst-Case Analysis Example (cont.) Worst-case source currents: IImax VImax 15V (1.1) min 322A R1 R2min 18k(0.95) 36k(0.95) IImin VImin 15V (0.9) max 238A max R1 R2 18k(1.05) 36k(1.05) Check of Results: The worst-case values range from 14-17 percent above and below the nominal values. The sum of the three element tolerances is 20 percent, so our calculated values appear to be reasonable. NJIT ECE-271 Dr. S. Levkov Chap 1 - 78 Monte Carlo Analysis • All parameters are selected randomly from the possible distributions • Circuit is analysis is performed and solution is found • Many such solutions are performed and statistics are gathered. • The analysis can be done using programs like MATLAB, Mathcad, SPICE, or a spreadsheet to complete a statistically significant set of calculations. • For example, with Excel, a resistor with 5% tolerance can be expressed as: R Rnom (1 2(RAND() 0.5)) The RAND() function returns random numbers uniformly distributed between 0 and 1. NJIT ECE-271 Dr. S. Levkov Chap 1 - 79 Monte Carlo Analysis Result WC WC Histogram of output voltage from 1000 case Monte Carlo simulation. NJIT ECE-271 Dr. S. Levkov Chap 1 - 80 Monte Carlo Analysis Example Problem: Perform a Monte Carlo analysis and find the mean, standard deviation, min, and max for VO, IS, and power delivered from the source. Solution: • Unknowns: The mean, standard deviation, min, and max for VO, IS, and PS. • Approach: Use a spreadsheet to evaluate the circuit equations with random parameters. NJIT ECE-271 Dr. S. Levkov Chap 1 - 81 Monte Carlo Analysis Example (cont.) Monte Carlo parameter definitions: VI 15(1 0.2( RAND() 0.5)) R1 18, 000(1 0.1( RAND() 0.5)) R2 36, 000(1 0.1( RAND() 0.5)) Circuit equations based on Monte Carlo parameters: R1 VO VI R1 R2 VI II R1 R2 PI VI II Results: Vo (V) II (mA) P (mW) NJIT ECE-271 Dr. S. Levkov Avg Nom. 4.96 5.00 0.276 0.278 4.12 4.17 Stdev 0.30 0.0173 0.490 Max WC-max Min WC-Min 5.70 5.87 4.37 4.20 0.310 0.322 0.242 0.238 5.04 -3.29 -- Chap 1 - 82 Temperature Coefficients • Most circuit parameters are temperature sensitive. P = Pnom(1+1∆T+ 2∆T2) where ∆T = T-Tnom Pnom is defined at Tnom • Most versions of SPICE allow for the specification of TNOM, T, TC1(1), TC2(2). • SPICE temperature model for resistor: R(T) = R(TNOM)*[1+TC1*(T-TNOM)+TC2*(T-TNOM)2] • Many other components have similar models. NJIT ECE-271 Dr. S. Levkov Chap 1 - 83 Numeric Precision • Most circuit parameters vary from less than +- 1 % to greater than +- 50%. • As a consequence, more than three significant digits is meaningless. • Results in the text will be represented with three significant digits: 2.03 mA, 5.72 V, 0.0436 µA, and so on. NJIT ECE-271 Dr. S. Levkov Chap 1 - 84