Analog Signals and Electronics
• An analog signal is a continuous time and
continuous amplitude signal representing a
physical phenomenon
• It is converted to a corresponding electrical
signal in the form of a voltage or current for
processing in analog electronic systems
• E.G. A microphone converts sound to current
• The voltage or current is still an analog signal
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Analog Signals and Electronics
• One of the easiest forms of analog electronics
to explain is an amplifier or pre-amplifier
• An analog signal may be too weak to interpret
or use
• A phonograph needle produces a weak voltage
from the grooves on the record
• An amplifier magnifies the amplitude of the
voltage or current to power the loudspeakers
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Analog Signals and Electronics
• Vacuum tubes or more recently transistors can
be used to amplify analog signals across a range
of frequencies based on the amplifier design
• Note: Although most of the time current is really
electrons flowing from minus to plus, electrical
engineers use the convention of arrows from
plus to minus for current on schematic diagrams
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Analog Signals and Electronics
• Vacuum tubes allow a small voltage to be
amplified and control a larger current to a “load”
• A typical Triode
Load
(Loud Speaker)
Plate
Analog
Signal
Input
Control
Current
Load Current =
beta * Control Current
+
Power
Source
Grid
-
Heat
Cathode
Ground
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Analog Signals and Electronics
• Solid state transistors allow a small current to be
amplified and control a larger current to a “load”
• Typical NPN Transistor Load
(Loud Speaker)
Collector
Analog
Signal
Input
Base
Control
Current
N material
Load Current =
beta * Control Current
+
Power
Source
P material
N material
-
Emitter
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Analog Signals and Electronics
• A high quality stereo system amplifier amplifies
signals over a bandwidth of 20 Hz to 20K Hz –
the normal range of human hearing
• That introduces another concept – a filter
• A filter passes signals of different frequencies
with different output amplitudes/magnification
• All analog signals can be represented as a sum
of multiple signals at different frequencies
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Analog Signals and Electronics
• A filter is an amplifier that magnifies each signal
or each portion of an aggregate signal more or
less depending on its frequency
• If a signal or a portion of an aggregate signal is
outside the range of the filter, it is not amplified
and may be difficult to detect at all
• Within the “pass band”, a filter may have some
fluctuations based its design, but usually a flat
response in the pass band is the goal
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Analog Signals and Electronics
• A filter may be low pass:
Frequency 0
• High pass
• Band pass
• Band reject (or notch)
Frequency 0
Frequency 0
Frequency 0
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Analog Signals and Electronics
• We use the LaPlace transform and complex math
to analyze the behavior of signals and filters
• We will analyze a couple filters after covering the
math behind the analysis
• Note: Electrical engineers use i for current so we
use j as the imaginary square root of - 1
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LaPlace Transform
• Using the LaPlace transform “L()”, problems defined by
differential equations can be solved using algebra
• Either a signal or a system that processes signals can be
represented by a differential equation
• Any signal’s or system’s differential equation can be
represented by its LaPlace transform as a function of a
complex frequency variable s
• If signal A is input to system B, L(A) can be multiplied
by L(B) to get the LaPlace transform of the output L(C)
• The inverse transform L -1(L(C)) is the signal itself
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LaPlace Transform
• The Laplace transform of a function f(t), defined
for all real numbers t ≥ 0, is the function F(s):
• The parameter s is a complex number with real
part σ and imaginary part ω: s = σ + j ω
• The imaginary component ω = 2 x Pi x frequency
• We usually look up LaPlace transforms and
inverse LaPlace transforms in tables
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LaPlace Transform
• For the unit impulse function (a value of one at t
= 0 and zero value everywhere else), the LaPlace
transform is 1
• For the unit step function (a value of zero for all
t < 0 and a value of 1 for all t >= 0) is 1/s
• For the function sin(at), the LaPlace transform is
a/(s2 + a2)
• For the function cos(at), the LaPlace transform
is s/(s2 + a2)
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LaPlace Transform
• To evaluate the stability of a system, plot the poles
of its LaPlace transform (values of s for which the
transform value is infinite) on the complex s plane
• A system with all poles in the left half plane is stable
but if any are in the right half plane it is unstable
• A system with poles on the j axis is marginally stable
• For example, the LaPlace transform of sin(at) has
poles at s = +ja and –ja, so a system represented by
that function is marginally stable – it oscillates
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LaPlace Transform
• Each electronic circuit component has a voltage
to current relationship defined by a differential
equation so it has a LaPlace Transform
– A resistor is just a constant – R (V = iR)
– A capacitor has an integral transform – 1/sC
– An inductor has a derivative transform – sL
• We add the LaPlace transforms for components
connected in series to solve for the current that
will be created by an applied voltage waveform
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Analog Signals and Electronics
• A passive low pass filter (resistor – capacitor):
LaPlace Transform
Vout/Vin = (1/sC)/(R + 1/sC)
Vout/Vin = 1 / (1 + sRC)
R
Vin
C
Vout
Cutoff Frequency = 1 / (2 Pi RC)
• An active low pass filter (with an op amp):
R
R
Vin
C
-
+
Vout
The voltage and current at the
input to the op amp must be 0
so Vin/R – Vout (1/R + sC ) = 0
giving the same LaPlace transform
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Analog Signals and Electronics
• The LaPlace Transform for the Low Pass filter is
1 / (1 + sRC) which has one pole at s = - 1/RC
+j ω
• S-plane Plot:
Left Half Plane
Right Half Plane
Filter bandwidth
-σ
+σ
- 1/RC
-j ω
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Analog Signals and Electronics
• A passive high pass filter (capacitor - resistor):
C
Vin
R
Vout
LaPlace Transform
Vout/Vin = R / (R + 1/sC)
Vout/Vin = sRC / (1 + sRC)
Cutoff Frequency = 1 / (2 Pi RC)
• An active high pass filter (with an op amp):
R
C
R
-
Vin
+
Vout
The voltage and current at the
input to the op amp must be 0
so Vin/(R + 1/sC) – Vout (1/R) = 0
giving the same LaPlace transform
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Analog Signals and Electronics
• The LaPlace Transform for the High Pass filter is
sRC / (1 + sRC) which has one pole at s = - 1/RC
and one zero at the origin
+j ω
Filter bandwidth
• S-plane Plot:
Left Half Plane
Right Half Plane
-σ
+σ
- 1/RC
-j ω
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Analog Computers
• Rather than use math analysis, we can simulate
the behavior of analog systems with a computer
• Analog computers process continuous time and
amplitude signals to solve differential equations
• Although some analog computers have been
based on mechanics, hydraulics, pneumatics, etc.
the predominant technology has been electrical
• The limitations of designing precision analog
electronic circuits on a large scale can introduce
artifacts into the solution
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Analog Computers
• A small scale analog computer
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Analog Computers
• Each input is a continuous time analog signal
• An analog computer input may be generated by
a potentiometer controlled by the operator or
attached to a part of the device being controlled
• Each output is a continuous time analog signal
• An analog computer output may be displayed
on an oscilloscope or traced on a strip chart
recorder (like an ECG) or attached to a part of
the device being controlled
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Analog Computers
• Although mostly made obsolete by digital
computers today, analog computers were the
brains inside artillery fire control systems in the
WWII and Korean War timeframe
• Guns on warships were mounted on a moving
platform and an analog fire control system would
be used to aim at either fixed or moving targets
• Anti-aircraft gun fire was directed by analog
computers using signals from radar systems
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Analog Computers
• In the 1960’s, RPI still had analog computers
• I did some work on them for my master’s thesis
to compare an analog computer output with a
digital simulation output with good agreement
• They are programmed with patch cords that are
used to connect summer, integrator, multiplier,
inverter, and divider devices
• Scaling voltages and time are required to keep
all signals within the range of supported voltage
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Analog Computers
• For this electronic circuit and differential equation:
http://www.edn.com/design/analog/4376400/A-virtual-analog-computer-for-your-desktop
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Analog Computers
• The analog computer circuit would look like this:
http://www.edn.com/design/analog/4376400/A-virtual-analog-computer-for-your-desktop
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