# Lecture Overview

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```					           Announcements
• Transistor lab continues this week – if you
finish early, start on Op-amps 1. (2 labs
over 3 weeks)
• Assignment 4 posted, due Thursday
• Assignment 2 solutions were corrected on
the webpage on Saturday
• mid-term: Thursday, October 21st
Lecture 11 Overview
•   The operational amplifier
•   Ideal op-amp
•   Negative feedback
•   Applications
– Amplifiers
– Summing/ subtracting circuits
Reminder: Impedances
• Attach an input - a source voltage VS plus source impedance RS

RS                            ROUT

VIN     RIN         AVIN             VOUT
VS

• Note the voltage divider RS + RIN.
• VIN=VS(RIN/(RIN+RS))
• We want VIN = VS regardless of source impedance
• So want RIN to be large.
• The ideal amplifier has an infinite input impedance
Reminder: Impedances
• Attach a load - an output circuit with a resistance RL

RS                              ROUT

RIN            AVIN              RL
VS                 VIN                             VOUT

• Note the voltage divider ROUT + RL.
• VOUT=AVIN(RL/(RL+ROUT))
• Want VOUT=AVIN regardless of load
• We want ROUT to be small.
• The ideal amplifier has zero output impedance
Operational Amplifier
• Integrated circuit containing ~20 transistors
Operational Amplifier
• An op amp is a high voltage gain, DC amplifier with high input
impedance, low output impedance, and differential inputs.
• Positive input at the non-inverting input produces positive output,
positive input at the inverting input produces negative output.
• Can model any amplifier as a "black-box" with a parallel input
impedance Rin, and a voltage source with gain Av in series with an
output impedance Rout.
Ideal op-amp
• Place a source and a load on the model

RS

+
RL
vout
-

So the equivalent circuit of an
• Infinite internal resistance Rin (so vin=vs).   ideal op-amp looks like this:
• Zero output resistance Rout (so vout=Avvin).
• "A" very large
• iin=0; no current flow into op-amp
Many Applications e.g.

•   Amplifiers
•   Integrators and differentiators
•   Clock generators
•   Active Filters
•   Digital-to-analog converters
Applications
Originally developed for use in analog computers:

Applications
Originally developed for use in analog computers:

Using op-amps

• Power the op-amp and apply a voltage
• Works as an amplifier, but:
• No flexibility (A~105-6)
• Exact gain is unreliable (depends on chip, frequency and temp)
• Saturates at very low input voltages (Max vout=power supply voltage)
• To operate as an amp, v+-v-<VS/A=12/105 so v+≈v-
• In the ideal case, when an op-amp is functioning properly in the
active region, the voltage difference between the inverting and non-
inverting inputs≈0
Noninverting Amplifier

vO  A(v   v  )
            R2 
vO  A vIN  vO
                  
          R1  R2 


     AR2 
1 
vO               AvIN
    R1  R2 


AvIN
vO 
1 AR2
R1  R2
When A is very large:
Take A=106, R1=9R, R2=R

106 vIN
vO 
1  106 R
9R  R
106 vIN
vO 
AvIN                       vO 
6 1
>>1
1  AR2                              1  10 
R1  R2                               10
vO 
AvIN                         vO  vIN 10
R2
A
R1  R2        • Gain now determined only by resistance ratio
• Doesn’t depend on A, (or temperature,
R1  R2     frequency, variations in fabrication)
vO  vIN
R2
Negative feedback:
• How did we get to stable operation in the linear
amplification region???
• Feed a portion of the output signal back into the input
(feeding it back into the inverting input = negative feedback)
• This cancels most of the input
• Maintains (very) small differential signal at input
• Reduces the gain, but if the open loop gain is ~, who
cares?
• Good discussion of negative feedback here:
Why use Negative feedback?:
• Helps to overcome distortion and non-linearity

• Improves the frequency response

• Makes properties predictable - independent of
temperature, manufacturing differences or other
properties of the opamp

• Circuit properties only depend upon the
external feedback network and so can be easily
controlled

• Simplifies circuit design - can concentrate on
circuit function (as opposed to details of
operating points, biasing etc.)
More insight
• Under negative feedback:

 R1  R2 
vO 

 R vIN  
v  v            1    0
A          A
v  v

• We also know
• i+ ≈ 0
• i- ≈ 0
• Helpful for analysis (under negative feedback)
• Two "Golden Rules"
1) No current flows into the op-amp
2) v+ ≈ v-
More insight
• Allows us to label almost every point in circuit terms of vIN!

1) No current flows into the op-amp
2) v+ ≈ v-
Op amp circuit 1: Voltage follower
• So vO=vIN
•or, using equations
R1  R2
vO  vIN
R2
R1  0
R2  

• What's the gain of this circuit?
Op amp circuit 1: Voltage follower
• So vO=vIN
•or, using equations
R1  R2
vO  vIN
R2
R1  0
R2  
• What's the application of this circuit?
•Buffer
voltage gain = 1                   Useful interface between different circuits:
input impedance=∞                  Has minimum effect on previous and next
output impedance=0                 circuit in signal chain
RS                               ROUT

RIN              AVIN                   RL
VS                   VIN                               VOUT
Op amp circuit 2: Inverting Amplifier
• Signal and feedback resistor,
connected to inverting (-) input.

• v+=v- connected to ground
iS  iF  iin  0
i S  i F
vS  v     vout  v 
                 v+ grounded, so:
RS           RF
v  v  0
vS  0    v 0
  out
RS        RF
RF
vout       vS
RS
vout    RF
Gain       
vS      RS
Op amp circuit 3: Summing Amplifier
• Same as previous, but add more
voltage sources

i1  i2  ..... iN  iF
vS 1 vS 2          vSN    vout
      .....     
RS1 RS 2           RSN    RF
 RF      RF              RF    
vout        R vS1  R vS 2  ..... R vSN 
                                
 S1       S2              SN   

If RS1  RS 2  ...  RSN  RS
RF
vout       (vS 1  vS 2  ...  vSN )
RS
Summing Amplifier Applications
• Applications - audio mixer
• Adds signals from a number of waveforms
• http://wiredworld.tripod.com/tronics/mixer.html

• Can use unequal resistors to get a weighted sum
• For example - could make a 4 bit binary - decimal converter
• 4 inputs, each of which is +1V or zero
• Using input resistors of 10k (ones), 5k (twos), 2.5k (fours) and 1.25k (eights)
Op amp circuit 4: Another non-inverting amplifier
• Feedback resistor still to inverting input,
but no voltage source on inverting input
(note change of current flow)
• Input voltage to non-inverting input

v  v                     iS  i F
v   0 vout  v 
since iin  0                    
RS       RF
and v   v   vS
 RF  
vout      R v
 1    
    S 

 RF 
 R v S
vout  1      
      S 

vout      RF
Gain          1
vS        RS
Op amp circuit 5: Differential Amplifier (subtractor)

i1  i2  0
v1  v     vout  v 

R1          R2
v  v
R2
v            v2  v 
R1  R2
R2
vout       (v2  v1 )
R1

Useful terms:
• if both inputs change together, this is a common-mode input change
• if they change independently, this is a normal-mode change
• A good differential amp has a high common-mode rejection ratio (CMMR)
Differential Amplifier applications
• Very useful if you have two inputs corrupted with the same noise
• Subtract one from the other to remove noise, remainder is signal
• Many Applications : e.g. an electrocardiagram measures the
potential difference between two points on the body

http://www.picotech.com/applications/ecg.html

The AD624AD is an instrumentation amplifier - this is a high gain, dc
coupled differential amplifier with a high input impedance and high CMRR
(the chip actually contains a few opamps)
The 2d-Mesh of Trees (2d-MOT) combines the advantages of 2d-
meshes and binary trees. A 2d-mesh has large bisection width but
large diameter (√N). On the other hand a binary tree on N leaves
has small bisection width but small diameter. The 2d-MOT has
small diameter and large bisection width.

   A 3d-MOTcan be defined similarly.

22
Network Topologies - Fixed
Connection Networks (static)
   Hypercube
   The hypercube is the major representative of a class of networks that are called
hypercubic networks. Other such networks is the butterfly, the shuffle-exchange
graph, de-Bruijn graph, Cube-connected cycles etc.
   Each vertex of an n-dimensional hypercube is represented by a binary string of
length n. Therefore there are |V | = 2n = N vertices in such a hypercube. Two
vertices are connected by an edge if their strings differ in exactly one bit position.
   Let u = u1u2 . . .ui . . . un. An edge is a dimension i edge if it links two nodes that
differ in the i-th bit position.
   This way vertex u is connected to vertex ui= u1u2 . . . ūi . . .un with a dimension i
edge. Therefore |E| = N lg N/2 and d = lgN = n.
   The hypercube is the first network examined so far that has degree that is not a
constant but a very slowly growing function of N. The diameter of the hypercube
is D = lgN. A path from node u to node v can be determined by correcting the
bits of u to agree with those of v starting from dimension 1 in a “left-to-right”
fashion. The bisection width of the hypercube is bw = N. This is a result of the
following property of the hypercube. If all edges of dimension i are removed from
an n dimensional hypercube, we get two hypercubes each one of dimension n −
1.

23
Network Topologies:
Hypercubes and their Construction

Construction of hypercubes from hypercubes of lower dimension.   24
Network Topologies - Fixed
Connection Networks (static)
   Butterfly.
   The set of vertices of a butterfly are represented by (w, i),
where w is a binary string of length n and 0 ≤ i ≤ n.
   Therefore |V | = (n + 1)2n = (lgN + 1)N. Two vertices (w, i)
and (w‟, i‟ ) are connected by an edge if i„ = i + 1 and either
(a) w = w‟ or (b) w and w‟ differ in the i‟ bit. As a result |E|
= O(N lgN), d = 4, D = 2lgN = 2n, and bw = N.
   Nodes with i = j are called level-j nodes. If we remove the
nodes of level 0 we get two butterflies of size N/2. If we
collapse all levels of an n dimensional butterfly into one, we
get a hypercube.

25
Network Topologies - Fixed
Connection Networks (static)
   Other:
   Other hypercubic networks is cube-connected cycles (CCC), which
is a hypercube whose nodes are replaced by rings/cycles of length
n so that the resulting network has constant degree). This network
can also be viewed as a butterfly whose first and last levels
collapse into one. For such a network |V | = N lgN = n2n, |E| =
3n2n−1, d = 3, D = 2lgN, bw = N/2.

26
Evaluating
Static Interconnection Networks
Arc          Cost
Bisectio
Network                Diameter              Connectivi   (No. of
nWidth
Completely-connected
Star
Complete binary tree
Linear array
2-D mesh, no
wraparound
2-D wraparound mesh
Hypercube
Wraparound k-ary d-
cube
27
End

Thank you!

28

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