Decibel (dB) – 3dB point
P 
dB  10 log o 
 Pi 
V 
dB  20 log o 
 Vi 
• Bode Plots
• Resonance
• 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 2014
Lecture 2
1
reference
60 dB =
1,000 = 103
half power point
40 dB =
100 = 102
6.101 Spring 2014
Lecture 2
2
Sound Levels*
application
dbV
1 Volt rms
dBm
1 mW into 50 [0.224V] or radio-frequency [50] or audio [600] power
measurements [in England, the dBu is used to
600 [0.775V]
mean 0.775V reference without regard to
impedance or power]
routine voltage measurements [comparisons!]
dB mV
1 millivolt rms
signal levels in cable systems
dbW
1 Watt
audio power amplifier output [usually into 8, 4,
or 2 impedances]
dBf
1 femtowatt [10-15 watt]
communications and stereo receiver
sensitivity [usually 50, 75unbalanced, or
300balanced antenna input impedances]
dB (SPL) 0.0002bar,
=
[=Pascals] [1 bar
2
dynes/cm ~1AT]
80 dB = 10,000 = 104
3 dB point = ?
Common Decibel Units
dB UNIT
100 dB = 100,000 = 105
log10(2)=.301
noise induced hearing loss (NIHL)
20Pa Sound Pressure Level measurements: the
= 106 reference is the “threshold of hearing”.
mosquito at 3 yards
* www.osha.gov
6.101 Spring 2014
Lecture 2
3
6.101 Spring 2014
4
Bode Plot ‐ Review
Magnitude & Phase
• 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.
Lecture 2
6.101 Spring 2014
1
1
s  j

s  a j  a
1
H ( j ) 
2
  a2
recall z  z e jz
H (s) 
H (s) 
 
 
H ( j )  m tan 1    n tan 1  
b
 
a
 
H ( j )   tan 1  
a
5
N ( j ) ( j  b ) m

D ( j ) ( j  a ) n
Lecture 2
6.101 Spring 2014
Low Pass Filter LPF
6
High Pass Filter HPF
AV (dB)
log scale
AV (dB)
C
R
V1
C
V2
V1
0
-3dB
R
V2
0
-3dB
slope = 6 dB / octave
slope = 20 dB / decade
slope = -6 dB / octave
slope = -20 dB / decade
log f
1
Av 
Av 
 j XC
V2


V1 R   j X C
1
sRC  1
j C
1

1
j RC  1
j C
log f
Av 
fHI or f-3dB
R
V2
R
j CR
s CR



V1 R  1
j CR  1 s CR  1
j C
fLO or f-3dB
Degrees
Degrees
PHASE LEAD
90o
PHASE LAG
0o
45o
-45o
0o
-90o
-45o
log f
fHI or f-3dB
6.101 Spring 2014
Lecture 2
fLO or f-3dB
log f
7
6.101 Spring 2014
Lecture 2
8
Proper External Grounding for
Lab 1 IF Transformer
Transformer – Step Up
V1
V1
V2
R2
V1
V
 V2 2
R1
R2
2
PC Board
2
V1
V
 2
R1
R2
1:n
R1 
V1 I 1  V2 I 2
pri
sec
NC
2
1
V R2
V22
Can
but V2  nV1
R1 
R1 is equivalent resistance
of R2 reflected back to
primary.
V12 R2
nV1 
2

V12 R2
n 2V12
so
R1 
R2
n2
Lecture 2
6.101 Spring 2014
9
Lecture 2
6.101 Spring 2014
Resonance
Series Resonance
L
R
+
V
-
I
C
6.101 Spring 2014
V

Z
XC 
V
R 2  (L 
Lecture 2
V

Z
1 2
)
C
11
•
V
R 2  (L 
X L  j L
1
1
Z  R  sL 
 R  j (L 
)
C
sC
I
10
1
j C
6.101 Spring 2014
1 2
)
C
I is a maximum at resonance
L 
1
C
1
 2f
LC
1
f 
2 LC

inductive
reactance
capactive
reactance
Lecture 2
12
Bandwidth – 3dB Point
Resonance (Series RLC) – Key points
+
V
-
R
Z  R  sL 
L
C
1
1
 R  j (L 
)
C
sC
• Applies to more complex RLC circuits
• At resonance: power is maximum
• At resonace: phase angle zero, i.e. capacitive reactance = inductive reactance, or impedance is real 1
Lecture 2
6.101 Spring 2014
13
6.101 Spring 2014
Bandwith ‐ BW
2  1
•
BW =
•
Peak power is at resonance
•
Half power points
2 R  R 2  ( wL 
•
where
I
1 2
)
wc
2 , 1
V
2R
I
• Q* (quality factor)
V
R 2  (L 
1
C
R
1
 R 
   
2L
 2 L  LC
Q
2
2 
R
1
 R 
   
2L
 2 L  LC
  12
Note: resonant frequency
bandwidth
1 L
 
R C 
2
• Higher Q implies more selectivity *Agarwal/Lang
6.101 Spring 2014
R
L
)2
2
1  
2  1 
• BW =
V2
R
V

Z
14
Bandwidth and Q (Series RLC)
are the half power frequencies
P
2
Lecture 2
15
6.101 Spring 2014
Foundation of Analog Digital Elect Circuits equation 14.47, p 794
Lecture 2
16
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
R
+
C V
-
L
6.101 Spring 2014
1
1 
I  V   jwC 
jwL 
R
Series
Parallel
V
I
R
1/R
L
C
C
L
BW 
+
V
-
R
L
C
17
BW  ( w2  w1 ) 
R
L
Lecture 2
6.101 Spring 2014
18
MATLAB Functions bode, freqs
1
1
 R  j (L 
)
sC
C
1
s
I ( s)
1
L


V ( s) Z ( s) s2  R s  1
L
LC
•
BODE(SYS,W) uses the vector W of frequencies (in radians/TimeUnit) to evaluate the frequency response
•
[MAG,PHASE] = BODE(SYS,W) and [MAG,PHASE,W] = BODE(SYS) return the response magnitudes and phases in degrees (along with the frequency vector W if unspecified).
•
SYS is the transfer function expressed as numerator and denominator in the form
SYS 
• Magnitude of Z is minium at resonance
• Bode plots are graphs of the transfer function: magnitude and phase angle
Lecture 2
O
1
 2f
LC
1
f 
2 LC
• Pole at DC or zero frequency
6.101 Spring 2014
1
RC
wL
R
0 
Observations & Bode Plots
Z  R  sL 
1
LC
wO 
1
LC
w 
19
d  es
as  bs  c
2
•
bode(num,denom,range)
num=[d e], denom=[a b c], range= desired freqencies in radians
•
freqs(num,denom,range) plots frequency response and phase angle
6.101 Spring 2014
Lecture 2
20
Bode vs Freqs Plots
1
s
1
I ( s)
L


R
V ( s) Z ( s) s2  s  1
L
LC
R=1 L=47uh C=1.8nf f=540khz
6.101 Spring 2014
not same scale
bode
Selectivity and Q
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)
L=47uh
C=1.8nf
f=540khz
Q
1
160
5
32
10
16
freqs
Lecture 2
21
6.101 Spring 2014
22
• Resonance, Q, bandwidth
• Transformers and impact on load and bandwidth
• Diode detector, demodulation
• Simple AM transmitter and receiver
L=47uh
C=1.8nf
f=540khz
Lecture 2
Lecture 2
Lab 1 Topics
Selectivity and Q
6.101 Spring 2014
R
R
Q
1
160
5
32
10
16
23
6.101 Spring 2014
24
Lab 1
Schematics & Wiring
12-100 pF trimmer
• 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!
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 2014
Lecture 2
25
6.101 Spring 2014
26
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
power
supply
noise
Voltage drop across wires
LC ringing after transitions
6.101 Spring 2014
27
6.101 Spring 2014
Lecture 2
28
Bypass (Decoupling) Capacitors
Electrolytic
Capacitor 10uf
The Concept of Modulation (modulating a carrier)
Bypass capacitor
0.1uf typical
Through hole PCB (ancient) shown for clarity.
6.101 Spring 2014
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
Lecture 2
29
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 2014
Two of Many Methods of Modulation
Lecture 2
30
Fourier Series ?
T= 1/f1
?
V=|Asin(ω0t)| ω0 = 2π f1
Focus is on Amplitude Modulation (AM)
6.101 Spring 2014
?
31
6.101 Spring 2014
32
Time Domain Analysis
Amplitude Modulation (AM) of a Complex Exponential Carrier
v  Ac cos c t * KAm cos mt
ct   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 
Lecture 2
6.101 Spring 2014
33
34
AM with Carrier (for different Amplitudes of A)
Asynchronous Demodulation
•
•
Lecture 2
6.101 Spring 2014
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 2014
Lecture 2
35
6.101 Spring 2014
36
D1
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 2014
Lecture 2
37
6.101 Spring 2014
Standard Values
Lecture 2
38
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.
6.101 Spring 2014
39
6.101 Spring 2014
Lecture 2
40
Finger Tips Facts
Diode V‐I Characteristic • Current thru pn junction doubles for every 26mv (at room temperature) or 10x for every 60mv
• Temperature coefficent 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
qv D
kt
qv D
kT
 26mv
I s  10 pa
6.101 Spring 2014
41
Lecture 2
6.101 Spring 2014
Reverse Breakdown Voltage
42
Zener Diode
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 2014
Lecture 2
43
6.101 Spring 2014
Lecture 2
44
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 2014
45
6.101 Spring 2014
1N4001‐1N4007
46
Transient Respopnse
Fast reverese recovery diode needed for switching power supplies
6.101 Spring 2014
47
6.101 Spring 2014
Lecture 2
48
Diodes
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 2014
Lecture 2
49
Lecture 2
6.101 Spring 2014
50
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
51
6.117 2104 IAP Lecture 2
52
RC Equation
Diode Circuits
Vs = 5 V
Vs = 5 V
Switch is closed t<0
Switch opens t>0
R
Vs = VR + VC
Vs = iR R+ Vc
+
t
Vc V  51  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 2014
Lecture 2
53
6.101 Spring 2014
More Diode Circuits
Lecture 2
54
Clamping Circuit
RC time constant limitation
6.101 Spring 2014
Lecture 2
55
6.101 Spring 2014
Lecture 2
56
Power Supply Ripple Voltage Calculation
Rectifier Circuits
+
1N4001
12.6 VCT RMS
120 V 60 Hz
CF
+
RL
Vout =
v OUT
-
Pri
Sec
3a) Half-wave rectifier circuit diagram
1N4001
+
12.6 VCT RMS
120 V 60 Hz
CF
+
RL
Vout =
v OUT
-
Pri
Sec
1N4001
3b) Full-wave rectifier circuit diagram
4x 1N4001
12.6 VCT RMS
120 V 60 Hz
+
CF
+
RL
Vout =
v OUT
-
Pri
Sec
3c) Bridge rectifier circuit diagram
RC >> 16.6ms why?
Lecture 2
6.101 Spring 2014
57
Lecture 2
6.101 Spring 2014
Voltage
58
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
v
i
120 2 sin 2 60t  169.7 sin 2 60t
+
v
-
340 V
6.101 Spring 2014
Lecture 2
59
6.101 Spring 2014
Lecture 2

r
0
2

1 2
v dt
r 0
vrms
60
Diode AC Resistance
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=
6.101 Spring 2014
2
vrms
Lecture 2
61
6.101 Spring 2014
Lecture 2
62