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
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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
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Chap 1 - 20
Analog and Digital Signals
Analog signal
• Analog signals usually are
continuous in time and in
values.
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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  2n 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  2n 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  2n VFS , f .i. {0000001}
• The most significant bit (MSB) - VMSB  21VFS , 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  2n VFS , f .i. {0000001}
• The most significant bit (MSB) - VMSB  21VFS , f .i. {1000000}
• Then for an n-bit D/A converter, the output voltage is expressed as:
VO  (b1 21  b2 22  ...  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  2n VFS , f .i. {0000001}
• The most significant bit (MSB) - VMSB  21VFS , f .i. {1000000}
• Then for an n-bit D/A converter, the output voltage is expressed as:
VO  (b1 21  b2 22  ...  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  2n VFS , f .i. {0000001}
• The most significant bit (MSB) - VMSB  21VFS , f .i. {1000000}
• Then for an n-bit D/A converter, the output voltage is expressed as:
VO  (b1 21  b2 22  ...  bn 2 n )VFS
 (b1 2n1  b2 2n 2  ...  bn 20 )2 n VFS
 (b1 2n1  b2 2n2  ...  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 21  b2 22  ...  bn 2 n )VFS
 (b1 2n1  b2 2n2  ...  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 21  b2 22  ...  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 
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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  11 k

vi 
vi
   1 RS  R1
 50  11 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 k0t  bk sin k0t
k 0
1
a0 
T
t0  T

t0
2
v(t )dt , ak 
T
t0 T

t0
2
v(t ) cos k0 dt , bk 
T
t0 T

v(t ) sin k0 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(k0 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

sin0 t  sin30 t  sin50 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

 322A
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

 322A
R1  R2min 18k(0.95)  36k(0.95)
IImin
VImin
15V (0.9)
 max
 238A
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

 322A
R1  R2min 18k(0.95)  36k(0.95)
IImin
VImin
15V (0.9)
 max
 238A
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