14
Josh EE40
Lecture Hug
7/26/2010
EE40 Summer 2010
Hug
1
Logisticals
Midterm Wednesday
Study guide online
Study room on Monday
Cory 531, 2:00
Cooper, Tony, and I will be there 3:00-5:10
Study room on Tuesday
Cory 521, 2:30 and on
Completed homeworks that have not been
picked up have been moved into the lab
cabinet
If you have custom Project 2 parts, Ive
emailed you with details about how to pick
them up
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Lab
Lab will be open on Tuesday if you want to
work on Project 2 or the Booster Lab or
something else
Not required to start Project 2 tomorrow
No lab on Wednesday (wont be open)
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Power in AC Circuits
One last thing to discuss for Unit 2 is
power in AC circuits
Lets start by considering the power
dissipated in a resistor:
10(50)
+
-
5
10
= () = 10 cos 50
cos 50)
(
5
= 20 cos2 50
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Or graphically
10(50)
+
-
5
10
= () = 10 cos 50
cos 50)
(
5
= 20 cos 2 50
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Average Power
10(50)
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+
-
5
Peak Power: 20W
Min Power: 0W
Avg Power: 10W
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Capacitor example
Find p(t)
10(50)
+
-
1
= 103 500 sin 50
0.5 sin 50
= 5 sin 50 cos 50)
(
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=
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7
Graphically
10(50)
+
-
1
Peak Power: 5/2
Min Power: 5/2
Avg Power: 0W
= 5 sin 50 cos 50)
(
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Is there some easier way of calculating power?
Like maybe with phasors?
+
-
10(50)
1 = 0.5 cos 50 +
2
= 5cos 50) cos 50 + )
(
(
2
Phasors are:
= 100
=
0.5
2
How about = ?
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Is there some easier way of measuring power?
Phasors are:
= 5cos 50) cos 50 +
(
(
= 100
= 0.5
2
Does = ?
=
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5
2
A.
B.
C.
D.
)
2
Yes, matches p(t)
No, wrong magnitude
No, wrong phase
No, wrong frequency
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It gets worse
For the resistor, there is no phasor which
represents the power (never goes
negative)
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Average Power
Tracking the time function of power with
some sort of phasor-like quantity is annoying
Frequency changes
Sometimes have an offset (e.g. with resistor)
Often, the thing we care about is the average
power, useful for e.g.
Battery drain
Heat dissipation
Useful to define a measure of average other
than the handwavy thing we did before
Average power given periodic power is:
=
0
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1
T is time for 1 period
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Power in terms of phasors
Weve seen that we cannot use phasors to
find an expression for p(t)
Average power given periodic power is:
=
0
1
T is time for 1 period
Well use this definition of average power
to derive an expression for average power
in terms of phasors
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Average Power
=
Note:
e.g. = 5 cos , = 4 cos
Average of each cosine is zero
Average of their product is 10
Our goal will be to get the average power
from phasors and
Well utilize =
1
[ ]
2
* denotes complex conjugate
See extra slides for proof of this identity
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Power from Phasors
=
1
2
=
=
=
+
+
=
=
=
=
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+
+
1
+ +
2
1
[ ]
2
1
[ ]
2
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15
Power from Phasors
Thus, given a voltage phasor and a
current phasor , the average power
absorbed is
1
=
2
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Capacitor Example
1 = 0.5 cos 50 +
2
= 5cos 50) cos 50 + )
(
(
+
-
10(50)
2
Phasors are:
= 100
=
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0.5
2
1
=
2
1
=
2
100 0.5
5
2
2
=0
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Resistor Example
10(50)
= 100
= 20
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=
1
2
1
=
2
+
-
5
Find avg power across resistor
20 = 10
A. 0 Watts
B. 10 Watts
C. 20 Watts
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Resistor Example
1
10(50)
= 5 20
= 100
+
-
5
Find avg power from source
= /
1
1
100
1
=
= 2
=
2
2
5 + 20
1
1
100
= 1.17 4.7 = 0.58
=
2
20.61551.3258
2
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Reactive Power
So if power dissipated is
what is
1
2
?
1
2
, then
Imaginary part is called reactive power
Physical intuition is that its power that you
put into an element with memory, but
which the element eventually gives back
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Capacitor Reactive Power Example
1 = 0.5 cos 50 +
2
= 5cos 50) cos 50 + )
(
(
+
-
10(50)
2
Phasors are:
= 100
=
0.5
2
=
1
2
=
1
2
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100 0.5
5
2
=
2
5
2
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Graphically
10(50)
+
-
1
Peak Power: 5/2
Min Power: 5/2
Avg Power: 0W
Avg Reactive Power: -5/2W
Like a frictionless car with perfect regenerative brakes,
starting and stopping again and again and again
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Note on Reactive Power
Providing reactive power and
consuming reactive power are physically
the same thing
Usually we say capacitors provide
reactive power, which comes from our
definition, whereas inductors consume
reactive power
preactive =
1
2
?
As youll see on HW7, capacitors and
inductors can be chosen to get rid of
reactive power
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And that rounds out Unit 2
Weve covered all that needs to be
covered on capacitors and inductors, so
its time to (continue) moving on to the
next big thing
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Back to Unit 3 Integrated Circuits
Last Friday, we started talking about
integrated circuits
Analog integrated circuits
Behave mostly like our discrete circuits in lab,
can reuse old analysis
Digital integrated circuits
We havent discussed discrete digital circuits,
so in order to understand digital ICs, we will
first have to do a bunch of new definitions
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Digital Representations of Logical Functions
Digital signals offer an easy way to
perform logical functions, using Boolean
algebra
Example: Hot tub controller with the
following algorithm
Turn on heating element if
A: Temperature is less than desired (T < Tset)
and B: The motor is on
and C: The hot tub key is turned to on
OR
T: Test heater button is pressed
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Hot Tub Controller Example
Example: Hot tub controller with the
following algorithm
Turn on heating element if
A: Temperature is less than desired (T < Tset)
and B: The motor is on
and C: The hot tub key is turned to on
OR
T: Test heater button is pressed
C
110V
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B
T
A
Heater
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Hot Tub Controller Example
Example: Hot tub controller with the
following algorithm
A: Temperature is less than desired (T < Tset)
B: The motor is on
C: The hot tub key is turned to on
T: Test heater button is pressed
Or more briefly: ON=(A and B and C) or T
C
110V
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B
T
A
Heater
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28
Boolean Algebra and Truth Tables
Well next formalize some useful
mathematical expressions for dealing with
logical functions
These will be useful in understanding the
function of digital circuits
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Boolean Logic Functions
Example: ON=(A and B and C) or T
Boolean logic functions are like algebraic
equations
Domain of variables is 0 and 1
Operations are AND, OR, and NOT
In contrast to our usual algebra on real
numbers
Domain of variables is the real numbers
Operations are addition, multiplication,
exponentiation, etc
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Examples
In normal algebra, we can have
3+5=8
A+B=C
In Boolean algebra, well have
1 and 0=0
A and B=C
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Have you seen boolean algebra before?
A. Yes
B. No
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Formal Definitions
not is a unary operator (takes 1 argument)
Returns 1 if its argument is 0, and 0 if its
argument is 1, e.g.
not 0=1
There exist many shorthand ways of writing
the not operation e.g.
0=1
0 = 1
0 = 1
I will use bar notation for consistency with the
book.
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Formal Definitions
and is a binary operator [takes 2
arguments] which returns 1 if both if its
arguments are 1, and 0 otherwise
Many ways to write A and B in
shorthand:
A
B
Z
0
0
0
As a table, if Z = , then:
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0
1
1
1
0
1
0
0
1
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Formal Definitions
or is a binary operator [takes 2
arguments] which returns 0 if either of its
arguments are 1, and 0 otherwise
Common ways to write A and B in
shorthand:
+
A
B
Z
As a table, if Z = + , then:
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0
0
1
1
0
1
0
1
0
1
1
1
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Boolean Algebra and Truth Tables
Just as in normal algebra, boolean algebra
operations can be applied recursively,
giving to rise complex
ABCZ
boolean functions
0000
Z=AB+C
0011
Any boolean function can
be represented by one of
these tables, called a
truth table
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0
0
1
1
1
1
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
1
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Boolean Algebra
Originally developed by George Boole as
a way to write logical propositions as
equations
Now, a very handy tool for specification
and simplification of logical systems
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Simplification Example
: Shine the bat signal
: Crime in progress
:Want to meet Batman
: Test bat signal
= + +
Simpler expression:
= + +
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C
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
T
0
1
0
1
0
1
0
1
Z
0
1
1
1
1
1
1
1
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Logic Simplification
In CS61C and optionally CS150, you will
learn a more thorough systematic way to
simplify logic expression
All digital arithmetic can be expressed in
terms of logical functions
Logic simplification is crucial to making
such functions efficient
You will also learn how to make logical
adders, multipliers, and all the other good
stuff inside of CPUs
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Quick Arithmetic-as-Logic Example
Assuming we have boolean input variables
1 , 2 , 2 , 2 and boolean output variables
1 , 2
Lets say that each variable represent one
digit of a binary number, we have 16
possibilities
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Logic Gates
Logic gates are the schematic equivalent
of our boolean logic functions
Example, the AND gate:
A
B
F
F = AB
AB
00
01
10
11
F
0
0
0
1
If were thinking about real circuits, this is
a device where the output voltage is high if
and only if both of the input voltages are
high
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Logic Functions, Symbols, & Notation
NAME
NOT
OR
AND
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SYMBOL
A
A
B
A
B
NOTATION
F
F
F
F=A
F = A+B
F = AB
TRUTH
TABLE
AF
01
10
AB
00
01
10
11
F
0
1
1
1
AB
00
01
10
11
F
0
0
0
1
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Multi Input Gates
AND and OR gates can also have many
inputs, e.g.
F
A
B
C
F = ABC
Can also define new gates which are
composites of basic boolean operations,
for example NAND:
A
B
C
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F
=
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Logic Gates
Can think of logic gates as a technology
independent way of representing logical
circuits
The exact voltages that well get will
depend on what types of components we
use to implement our gates
Useful when designing logical systems
Better to think in terms of logical operations
instead of circuit elements and all the
accompanying messy math
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44
Hot Tub Controller Example
Example: Hot tub controller with the
following algorithm
A: Temperature is less than desired (T < Tset)
B: The motor is on
C: The hot tub key is turned to on
T: Test heater button is pressed
Or more briefly: ON=(A and B and C) or T
C
110V
EE40 Summer 2010
B
T
A
Heater
Hug
45
Hot Tub Controller Example
Example: Hot tub controller with the
following algorithm
A: Temperature is less than desired (T < Tset)
B: The motor is on
C: The hot tub key is turned to on
T: Test heater button is pressed
Or more briefly: ON=(A and B and C) or T
110V
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A
B
C
T
ON
Heater
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How does this all relate to circuits?
A digital circuit is simply any circuit where
every voltage in the circuit is one of two
values
(typically ground) will represent boolean 0
(in modern CPUs, approximately 1V,
though you can set this on your computer) will
represent boolean 1
In truth, of course, values will vary
continuously, but entire design is
conceptualized as simply 1s and 0s
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The Static Discipline
We can think of the whole circuit as obeying
a contract to always provide output voltages
and at all outputs as long as the
inputs follow these same rules
Up to the circuit designer to ensure this
specification is met
In truth, voltages may be a little lower or
higher than these contractual values
However, as long as the output values are
close enough, the deviations are
unimportant
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Many Possible Ways to Realize Logic Gates
There are many ways to build logic gates,
for example, we can build gates with op1
1
amps
1
-5V
1
A
= (, )
5V
-5V
5V
Z
B
Far from optimal
5 resistors
Dozens of transistors
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Is this a(n):
A. AND gate
B. OR gate
C. NOT gate
D. Something else
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Switches as Gates
Example: Hot tub controller
ON=(A and B and C) or T
Switches are the most natural
implementation for logic gates
C
110V
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A
110V
B
C
B
T
T
A
ON
Heater
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Relays, Tubes, and Transistors as Switches
Electromechnical relays are ways to make
a controllable switch:
Zuses Z3 computer (1941) was entirely
electromechnical
Later vacuum tubes adopted:
Colossus (1943) 1500 tubes
ENIAC (1946) 17,468 tubes
Then transistors:
IBM 608 was first commercially available
(1957), 3000 transistors
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Electromechanical Relay
Inductor generates a magnetic field that
physically pulls a switch down
When current stops flowing through
inductor, a spring resets the switch to the
off position
Three
+
+
Terminals:
C
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C
+ : Plus
: Minus
C : Control
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Electromechnical Relay Summary
Switchiness due to physically
manipulation of a metal connector using a
magnetic field
Very large
Moving parts
No longer widely used in computational
systems as logic gates
Occasional use in failsafe systems
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Vacuum Tube
Inside the glass, there is a
hard vacuum
Current cannot flow
If you apply a current to
the minus terminal
(filament), it gets hot
This creates a gas of
electrons that can travel to
the positively charged
plate from the hot filament
When control port is used,
grid becomes charged
Acts to increase or
decrease ability of current
to flow from to +
+
C
(Wikipedia)
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Vacuum Tube Demo
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Vacuum Tube Summary
Switchiness is due to a charged cage which
can block the flow of free electrons from a
central electron emitter and a receiving plate
No moving parts
Inherently power inefficient due to
requirement for hot filament to release
electrons
No longer used in computational systems
Still used in:
CRTs
Very high power applications
Audio amplification (due to nicer saturation
behavior relative to transistors)
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Field Effect Transistor
+
-
(Drain)
+
(Gate)
C
(Source)
-------------
P is (effectively) a high resistance block of
material, so current can barely flow from + to
The n region is a reservoir of extra electrons (we
will discuss the role of the n region later)
When C is on, i.e. is relatively positive, then
electrons from inside the P region collect at bottom
of insulator, forming a channel
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57
Field Effect Transistor
+
-
(Drain)
+
(Gate)
C
(Source)
-------------
When the channel is present, then effective
resistance of P region dramatically decreases
Thus:
When C is off, switch is open
When C is on, switch is closed
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Field Effect Transistor
+
-
(Drain)
+
-
+
(Gate)
C
(Source)
------
If we apply a positive voltage to the plus side
Current begins to flow from + to
Channel on the + side is weakened
If we applied a different positive voltage to
both sides?
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Field Effect Transistor Summary
Switchiness is due to a controlling
voltage which induces a channel of free
electrons
Extremely easy to make in unbelievable
numbers
Ubiquitous in all computational technology
everywhere
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MOSFET Model
Schematically, we
represent the
MOSFET as a three
terminal device
Can represent all the
voltages and currents
between terminals as
shown to the right
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MOSFET Model
What do you expect
to be?
C
(Drain)
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+
(Gate)
(Source)
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S Model of the MOSFET
The simplest model
basically says that the
MOSFET is:
Open for <
Closed for >
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Building a NAND gate using MOSFETs
Consider the circuit
to the right where
On the board, well
show that =
Demonstration also
on page 294 of the
book
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Thats it for today
Next time, well discuss:
Building arbitrarily complex logic functions
Sequential logic
The resistive model of a MOSFET
Until then, study
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