Decibel (dB) – 3dB point P dB 10 log o Pi
V dB 20 log o Vi • Resonance • Bode Plots • Wires – theory vs reality • Amplitude Modulation • Diodes Acknowledgements: Lecture material adapted from Prof Qing Hu & Prof Jae Lim, 6.003 Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky
6.101 Spring 2016
Lecture 2
100 dB = 100,000 = 105
log10(2)=.301
1
80 dB = 10,000 = 104
3 dB point = ?
60 dB =
1,000 = 103
half power point
40 dB =
100 = 102
6.101 Spring 2016
Bode Plot ‐ Review
Lecture 2
MATLAB
• A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. • Magnitude plot on log‐log scale – Slope: 20dB/decade, same as 6dB/octave • Bode plot provides insight into impact of RLC in frequency response. • Stable networks must always have poles and zeroes in the left‐half plane.
• Matlab windows: current folder, command, work space (workspace), command history (commandhistory) • Set folder to your favorite folder • Built in help in command window • docking/undocking
Command window Current folder
6.101 Spring 2016
Lecture 2
2
3
Workspace Command history
MATLAB
MATLAB commands • •
• • • • • •
% comment delimiter MATLAB arrays starts with index=1 – a = [4,5,6] is a row vector a(2)=5 – b = [7;8;9] is a column vector “;” don’t print values Variables are case sensitive A a Variables must start with a letter who/whos: list the current variables in short/long form shg – show recent graph, pop to the front use apostrophes for FILENAME
pi i,j
3.14159265 sqrt(-1) imaginary unit
zeros(n,m) ones(n,m)
an n x m matrix of zeros an n x m matrix of ones
+*/ ^
addition, subtraction multiplication, division, power
sqrt
square root
MATLAB Flow Control
MATLAB Matrix Operation >> a=[2,3,4] a= 2 3
4
>> b=[1,0,0] b= 1 0
0
>> c=a+b c= 3 3
4
>> d=a*b % dot product operation ??? Error using ==> mtimes Inner matrix dimensions must agree. >> d=a.*b d= 2 0
0
>> e=a*b' % ' transpose e= 2
• if else statement
• for loop • while loop
if a == 0 b = a; else b = 1/a; end n = 100 for m = 1:n a(m) = a(m) + 1; end
n = 10 while n > 0 n = n – 1 end
Bode vs Freqs Plots
MATLAB example sin(x) >> t=[0:1/100:1-1/100]; % create t from 0 to .99, 100 values >> x=sin(2*pi*t); >> plot(t, x); >> stem(t,x); >> shg
R=1 L=47uh C=1.8nf f=540khz
not same scale
6.101 Spring 2016
R
L
Z R sL C
1 1 R j (L ) C sC
2 1
• BW =
• Q* (quality factor) • Applies to more complex RLC circuits Q
• At resonance: power is maximum • At resonace: phase angle zero, i.e. capacitive reactance = inductive reactance, or impedance is real
6.101 Spring 2016
Lecture 2
bode
freqs Lecture 2
10
Bandwidth and Q (Series RLC)
Resonance (Series RLC) – Key points + V -
num=[1/L 0] denom=[1 R/L 1/(L*C)] f=1/(2*pi*sqrt(L*C)) w=2*pi*f w_range = [.8*w:20:1.2*w] h=bode(num,denom,w_range) magh=abs(h) plot(w_range,magh) shg freqs(num,denom,w_range)
1 s 1 I ( s) L R V ( s) Z ( s) s2 s 1 L LC
R L resonant frequency bandwidth
1 L R C
• Higher Q implies more selectivity *Agarwal/Lang
11
6.101 Spring 2016
Foundation of Analog Digital Elect Circuits equation 14.47, p 794
Lecture 2
12
Summary – Parallel Series RLC
Series Parallel Duality
Series
Parallel
L
R V
1 V I R jwL jwC
C I
O
Q
Q w RC o
I
6.101 Spring 2016
R
+ C V -
L
1 1 I V jwC jwL R
Series
Parallel
V
I
R
1/R
L
C
C
L
BW
1 RC
O
BW ( w2 w1 )
R L
1 2f LC 1 f 2 LC
13
6.101 Spring 2016
Lecture 2
14
Selectivity and Q
L=47uh C=1.8nf f=540khz
Lecture 2
wL R
0
Selectivity and Q
6.101 Spring 2016
1 LC
wO
1 LC
w
L=47uh C=1.8nf f=540khz
R
Q
R
Q
1
160
1
160
5
32
5
32
10
16
10
16
15
6.101 Spring 2016
Lecture 2
16
Lab 1 Topics
Lab 1
• Resonance, Q, bandwidth • Transformers and impact on load and bandwidth • Diode detector, demodulation • Simple AM transmitter and receiver
12-100 pF trimmer
C
Rsource
L [Antenna]
VS
10
Pri-Sec turns ratio = 4:1 PRI [3 pins]
RSERIES C
VS [Function Generator]
SEC [2 pins]
RLOAD
Metal shield ["can"]
6.101 Spring 2016
17
6.101 Spring 2016
Proper External Grounding for Lab 1 IF Transformer
• IC power supply connections generally not drawn. All integrated circuits need power! • Use standard color coded wires to avoid confusion. – red: positive – black: ground or common reference point – Other colors: signals • Circuit flow, signal flow left to right • Higher voltage on top, ground negative voltage on bottom • Neat wiring helps in debugging!
sec NC
Can
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18
Schematics & Wiring
PC Board pri
Lecture 2
Lecture 2
19
6.101 Spring 2016
20
Wires Theory vs Reality ‐ Lab 1
Wire Gauge • Wire gauge: diameter is inversely proportional to the wire gauge number. Diameter increases as the wire gauge decreases. 2, 1, 0, 00, 000(3/0) up to 7/0. • Resistance – 22 gauge .0254 in 16 ohm/1000 feet – 12 gauge .08 in 1.5 ohm/1000 feet – High voltage AC used to reduce loss
30-50mv voltage drop in chip Wires have inductance and resistance
• 1cm cube of copper has a resistance of 1.68 micro ohm (resistance of copper wire scales linearly : length/area)
ܮ
ௗ ௗ௧
noise during transitions
This image cannot currently be display ed.
power supply noise
Voltage drop across wires LC ringing after transitions 6.101 Spring 2016
21
Bypass (Decoupling) Capacitors Electrolytic Capacitor 10uf
Lecture 2
Lecture 2
22
The Concept of Modulation (modulating a carrier)
Bypass capacitor 0.1uf typical
Through hole PCB (ancient) shown for clarity.
6.101 Spring 2016
6.101 Spring 2016
Transmitted Signal
x(t) • Provides additional filtering from main power supply • Used as local energy source – provides peak current during transitions • Provided decoupling of noise spikes during transitions • Placed as close to the IC as possible. • Use small capacitors for high frequency response. • Use large capacitors to localize bulk energy storage 23
Carrier Signal
Why? • • • •
More efficient to transmit E&M signals at higher frequencies. Transmitting multiple signals through the same medium using different carriers. Increase signal/noise ratio in lock‐in measurements. others...
How? • Many methods
6.101 Spring 2016
Lecture 2
24
Two of Many Methods of Modulation
Fourier Series ? T= 1/f1
? V=|Asin(ω0t)| ω0 = 2π f1
Focus is on Amplitude Modulation (AM)
?
6.101 Spring 2016
25
Time Domain Analysis
6.101 Spring 2016
26
Amplitude Modulation (AM) of a Complex Exponential Carrier
v Ac cos c t * KAm cos mt
ct e j c t , c — carrier frequency
KAm v Ac cos c t [cos(c m )t cos(c m )t ] 2
y(t) x(t) e
j ct
1 X j C j 2 1 X j 2 c 2
Y j
X j c 6.101 Spring 2016
Lecture 2
27
6.101 Spring 2016
Lecture 2
28
AM with Carrier (for different Amplitudes of A)
Asynchronous Demodulation
xm (t ) A c(t )
Assume c >> M, so signal envelope looks like x(t) Add same carrier
• •
xm (t )
t c(t )
t
Time Domain
y (t ) xm (t ) c(t )
A + x(t) > 0 why?
Frequency Domain
t 6.101 Spring 2016
D1
Lecture 2
29
6.101 Spring 2016
30
Units
Asynchronous Demodulation (continued) Envelop Detector
In order for it to function properly, the envelop function must be positive definite, i.e. A + x(t) > 0. Simple envelop detection for asynchronous demodulation. D1: 1N914 or 1N4148
6.101 Spring 2016
Lecture 2
31
6.101 Spring 2016
Lecture 2
32
Standard Values
Diodes
I D I s (e
qv D kT
1)
kT/q is also known as the thermal voltage, VT. VT = 25.9 mV when T = 300K, room temperature.
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33
Lecture 2
6.101 Spring 2016
Finger Tips Facts
34
Diode V‐I Characteristic
• Current thru pn junction doubles for every 26mv (at room temperature) or 10x for every 60mv • Temperature coefficient of silicon diode is ~2mv/degC at room temperature • Small signal resistance of pn junction is – 1 ohm @26 ma, 26 ohm @1ma
I D I se kT q
qvD kT
26mv
thermal voltage
I s 10 pa
6.101 Spring 2016
35
6.101 Spring 2016
Lecture 2
36
Zener Diode
Reverse Breakdown Voltage 4.7k
Low doped diodes have higher breakdown voltage
+ V _
• Zener diodes will maintain a fixed voltage by breaking down at a predefined voltage (zener voltage).
http://en.wikipedia.org/wiki/File:Diode_current_wiki.png
6.101 Spring 2016
Lecture 2
37
6.101 Spring 2016
39
6.101 Spring 2016
Lecture 2
38
Zener Breakdown •
Actually caused by two effects: avalanche effect and zener effect.
•
Avalanche effect: electron/holes entering depletion region is accelerating by the electric field, collides and creates additional electron/hole pairs – like a snow avalanche; occurs above 5.6V; has positive temperature coefficient
•
Zener effect: heavy doping of PN junction results in a thin depletion layer. Quantum tunneling results in current flow; occurs below 5.6V; has negative temperature coefficient
•
At 5.6V, two effects balance is near zero temperature coefficient.
6.101 Spring 2016
40
1N4001‐1N4007
Transient Respopnse
Fast reverese recovery diode needed for switching power supplies
6.101 Spring 2016
41
6.101 Spring 2016
Diodes
Lecture 2
42
Lecture 2
44
1N4001
Type
Max Vr
Max I Continous
Recovery time
Capaciitance
1N914
75V
10ma
4ns
1.3pf
1N4002
100V
1000ma
3500ns
15pf
1N5625
400
3000ma
1N1084
4000
30,000ma (peak)
400
50,000ma
1N914, 1N4148 1N7XX
40pf
Pulse Ox
Diode types
6.101 Spring 2016
Lecture 2
43
6.101 Spring 2016
Optical Isolators
Light Emitting Diode
• Optical Isolators are used to transmit information optically without physical contact.
• LED’s are pn junction devices which emit light. The frequency of the light is determined by a combination of gallium, arsenic and phosphorus. • Red, yellow and green LED’s are in the lab • Diodes have polarity • Typical forward current 10‐20ma
• Single package with LED and photosensor (BJT, thyristor, etc.) • Isolation up to 4000 Vrms • Used in pulse oximetry Nellcor DS-100 Pulse-ox
6.117 2104 IAP Lecture 2
45
6.117 2104 IAP Lecture 2
46
RC Equation
Diode Circuits
Vs = 5 V Vs = 5 V
Switch is closed t0
R
Vs = VR + VC Vs = iR R+ Vc
+ t Vc V 51 e RC c
Vs =
RC
dV
c iR = C dt
dVc Vc dt
t Vc Vs 1 e RC
Is RC in units of time? 6.101 Spring 2016
Lecture 2
47
6.101 Spring 2016
Lecture 2
48
More Diode Circuits
Clamping Circuit
RC time constant limitation Lecture 2
6.101 Spring 2016
49
Lecture 2
6.101 Spring 2016
Voltage
RMS Voltage v A sin t
• What is the equation describing the voltage from a 120VAC outlet? • 120 VAC is the RMS (Root Mean Square Voltage) • 60 is the frequency in hz • Peak to peak voltage for 120VAC is 340 volts!
•
The RMS voltage for a sinusoid is that value which will produce the same heating effect (energy) as an equivalent DC voltage.
•
Energy =
Pdt vidt
•
For DC,
2 rms
•
Equating and solving, A =
vrms
t=π
i
120 2 sin 2 60t 169.7 sin 2 60t
+ v -
340 V
6.101 Spring 2016
50
51
6.101 Spring 2016
v
r
0
1 2 v dt r 0
2
vrms
52
Agilent Function Generator
RMS Derivation
2 vrms 1 2 1 v dt A2 sin 2 tdt r r0 r0
t 1 2 sin dt sin 2 t [ ] 0 0 2 4
2 A2 t 1 vrms sin 2t ] [ 0 r r 2 4
A=
2
Turn on output!
vrms
6.101 Spring 2016
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6.101 Spring 2016
Agilent DMM
54
Oscilloscope Cursor controls
Menu driven soft key/buttons 6.101 Spring 2016
55
6.101 Spring 2016
56
Oscilloscope Controls • Auto Set, soft menu keys
•
– – – – –
• Trigger – channel, – slope, – Level
• Input – – – – –
Signal measurement
•
time, frequency, voltage cursors single sweep
Image capture
AC, DC coupling, 10x probe, 1khz calibration source, probe calibration, bandwidth filter
6.101 Spring 2016
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