Lecture Notes

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ECE 206L

• Lecture Notes
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DC Current vs. AC Current 

• Direct current (DC) flows in one direction the
  circuit. 
• Alternating current (AC) flows first in one
  direction then in the opposite direction.
• The same definitions apply to alternating voltage
  (AC voltage) 
• DC voltage has a fixed polarity.
• AC voltage switches polarity back and forth.
• There are numerous sources of DC and AC current
  and voltage. However  Sources of DC are commonly
  shown as a cell or battery, and for the AC
  current Generators 

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The Sinusoidal AC Waveform

• The most common AC waveform is a sine (or
  sinusoidal) waveform.   
• The vertical axis represents the amplitude of the
  AC current or voltage, in amperes or volts. 
• The horizontal axis represents the angular
  displacement of the waveform. The units can be
  degrees or radians. 
• The sine waveform is accurately represented by
  the sine function of plane trigonometry  y
  rsinq
• where  y the instantaneous amplitude
• r the maximum amplitude
• q the horizontal
  displacement

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DefinitionPeak and Peak-to-Peak Voltage 

•     
• Peak and peak-to-peak values are most often used
  when measuring the amplitude of ac waveforms
  directly from an oscilloscope display.
• Peak voltage is the voltage measured from the
  baseline of an ac waveform to its maximum, or
  peak, level.  Unit Volts peak (Vp)  Symbol Vp
• For a typical sinusoidal waveform, the positive
  peak voltage is equal to the negative peak
  voltage.   Peak voltages are expressed without a
  or - sign.
•  
• Peak-to-peak voltage is the voltage measured from
  the maximum positive level to the maximum
  negative level.  Unit Volts peak-to-peak 
  (Vp-p)  Symbol Vp-p
• For a typical sinusoidal waveform, the
  peak-to-peak voltage is equal to 2 times the peak
  voltage.  Peak-to-peak voltages are expressed
  without a or - sign .

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Conversion

• Convert Vp to Vp-p  Vp-p 2 Vp
• Convert Vp-p to Vp  Vp 0.5Vp-p
• What is the peak-to-peak value of a sinusoidal
  waveform that has a peak value of 10 V? 
• What is the peak value of a sine wave that has a
  peak-to-peak value of 240 V?   

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Instantaneous Current and Voltage 

• i Ipsin?
• Where  
• i instantaneous current in amperes  Ip
  the maximum, or peak, current in amperes  ?
  the angular displacement in degrees or radians
• v Vpsin?
• Where 
• v instantaneous voltage in volts  Vp
  the maximum, or peak, voltage in volts  ? the
  angular displacement in degrees or radians

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Average Voltage

• Average voltage is the average value of all the
  values for one half-cycle of the waveform.  Unit
  Volts average (Vave)  Symbol Vave 
• The average voltage of a sinusoidal waveform is
  equal to 0.637 times its peak value. 
• Vave 0.637Vp
• The average voltage is determined from just one
  half-cycle of the waveform because the average
  value of a full cycle is zero.  Average voltages
  are expressed without a or - sign 

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Average Voltage

• Convert Vp to Vave  Vave 0.637Vp 
• Convert  Vave to Vp Vp 1.57Vave
• Determine the average value of a waveform that
  measured 16 Vp.  Ans 10.2 Vave
• What is the peak value of a waveform that has an
  average value of 22.4 V?  Ans 35.1 Vp

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Root-Mean-Square (RMS) Voltage 

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• AC levels are assumed to be expressed as RMS
  values unless clearly specified otherwise.
• RMS voltage is the amount of dc voltage that is
  required for producing the same amount of power
  as the ac waveform.  Unit Volts (V)  Symbol
  Vrms 
• The RMS voltage of a sinusoidal waveform is equal
  to 0.707 times its peak value. 
• Vrms 0.707Vp
• In a dc circuit, applying 2 V to a 1 W resistance
  produces 4 W of power.   In an ac circuit,
  applying 2 Vrms to a 1 W  resistance produces 4 W
  of power. 
• RMS voltages are expressed without a or -
  sign. 

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ConversionRoot-Mean-Square (RMS) Voltage 

• Convert Vp to Vrms  Vrms 0.707Vp 
• Convert Vrms to Vp   Vp 1.414Vrms 
• Determine the RMS value of a waveform that
  measures 15 Vp.  Ans 10.6 V
• Determine the peak value of 120 V.
  .
• (Assume 120 V is in RMS)  Ans 170 Vp

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Resistors in Series
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Resistors in Parallel
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Measuring Voltage
Total Voltage VR1VR2
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Voltage Dividers
The voltage is divided up in such that it is
proportional to the resistances of the resistors
in a series circuit.
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Statistical Evaluation of Measurement Data and
Errors

• Average or mean value of a set of measurement
• Deviation from the average value
• Average value of the deviation
• Standard deviation(from the concept of RMS)
• Probability of error size in one observation

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The Decibel (dB)

• The decibel, or dB, is a means of expressing
  the gain of an active device (such as an
  amplifier) or the loss in a passive device (such
  as an attenuator or length of cable). It is
  simply the ratio of output to input expressed in
  logarithmic form. The decibel was developed by
  the telephone company(Bel, to express the gain or
  loss in telephone transmission systems.

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Calculating the Decibel (dB)

• Now, imagine for a moment what it would be like
  to calculate the total gain of a string of
  amplifiers. It would be a cumbersome task at
  best, and especially so if there were portions of
  the cascade which were lossy and reduced the
  total gain, thereby requiring division as well as
  multiplication.
• log (A x B) log A log B
• log (A/B) log A - log B
• Using the Decibel
•                     
• G 10 log (Po/Pi)     ,
• Where
• G Gain in dB
• Po Power output from the device
• Pi Power input to the device
• Ex. A length of coaxial transmission line is
  being fed with 150 watts from a transmitter, but
  the power measured at the output end of the line
  is only 112 watts. What is the line loss in dB?
• G 10 log (112/150)
• G 10 log 0.747
• G 10 (-0.127)
• G -1.27 dB
•                          

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Capacitors

• Capacitors consist of two plates with a
  dielectric material in-between. When a potential
  difference is placed across the plates, a charge
  builds up until it is large enough to cause a
  discharge across the plates through the material.

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Reading Capacitors
Larger capacitors have the number of microfarads
written on them directly. Smaller capacitors use
a code based on the number of picofarads. We
generally use microfarads, so XYZ
XY 10Z 10-6 mF
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Capacitors in Series
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Capacitors in Parallel
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Impedance vs. Resistance

• Resistance is a property of a material that
  causes a reduction in the rate of flow of
  electrons.
• Impedance is the reduction in the rate of flow of
  electrons caused by the material (resistance) AND
  other the properties of the component involved
  (reactance).
• Resistors have no reactance. So the impedance of
  a resistor is equal to its resistance only.
• Reactance varies with the frequency of the input.
  Resistance remains the same at all frequencies.
• Both impedance and resistance are measured in
  ohms.

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ImpedanceDefinition

• A general measure of how a component or group of
  components pushes against the current flowing
  through it.
• Impedance resistance reactance
• Impedance is used to refer to the behavior of
  circuits with resistors, capacitors and other
  components.
• When we consider components in a theoretical
  circuit diagram, the impedance of inductors and
  capacitors is their reactance only. Any
  resistance is modeled separately as a resistor.
  So theoretical capacitors and inductors have
  impedance, but no resistance.

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Capacitor Impedance
Real capacitors have effectively no resistance,
so impedance is reactance for all capacitors.
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What is Reactance

• Reactance is the property of resisting or
  impeding the flow of ac current or ac voltage in
  inductors and capacitors. Note particularly we
  speak of alternating current only ac, which
  expression includes audio af and radio
  frequencies rf. NOT direct current dc.
• Inductive Reactance
• When ac current flows through an inductance a
  back emf or voltage develops opposing any change
  in the initial current. This opposition or
  impedance to a change in current flow is measured
  in terms of inductive reactance. 2 pi f L
• where 2 pi 6.2832 f frequency in hertz
  and L inductance in Henries
• Capacitive Reactance
• When ac voltage flows through a capacitance an
  opposing change in the initial voltage occurs,
  this opposition or impedance to a change in
  voltage is measured in terms of capacitive
  reactance. 1 / (2 pi f C)
• where 2 pi 6.2832 f frequency in hertz
  and C capacitance in Farads

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Some examples of Reactance

• What reactance does a 6.8 uH inductor present at
  7 Mhz? Using the formula above we get
• 2 pi f L
• where 2 pi 6.2832 f 7 X 106 Hz and L
  6.8 X -6 Henries
• Answer 299 ohms
• What reactance does a 33 pF capacitor present at
  7 Mhz? Using the formula above we get
• 1 / (2 pi f C)
• where 2 pi 6.2832 f 7 X 106 Hz and C
  33 X -12 Farads
• Answer 689 ohms

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Inductors
An inductor is a coil of wire through which a
current is passed. The current can be either AC
or DC.
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Inductors
This generates a magnetic field, which induces a
voltage proportional to the rate of change of the
current.
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Combining Inductors

• Inductances add like resistances
• Series
• Parallel

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Inductor Impedance
Real inductors always have a small resistance
(that is not shown in these circuits). The
impedance of the theoretical inductor shown is
only its reactance.
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Comparison of Components
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ImpedanceDefinition

• A general measure of how a component or group of
  components pushes against the current flowing
  through it.
• Impedance resistance reactance
• Impedance is used to refer to the behavior of
  circuits with resistors, capacitors and other
  components.
• When we consider components in a theoretical
  circuit diagram, the impedance of inductors and
  capacitors is their reactance only. Any
  resistance is modeled separately as a resistor.
  So theoretical capacitors and inductors have
  impedance, but no resistance.

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Equipment Impedances

• Each measuring device changes the circuit when
  you use it.
• The impedance of the device helps you understand
  how much.
• Device Impedances
• Function Generator 50 ohms
• Scope 1Meg ohms
• DMM (DC voltage) 10Meg ohms
• DMM (AC voltage) 1Meg ohms
• DMM (DC current) 5 ohms (negligible)

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Effect of Impedance on Circuit
Function generator thinks it is putting out the
same thing.
Output is clearly different.
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Effect of Impedance on Circuit
The function generator has an output impedance of
much less than 50O, so we can ignore it.
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Kirchoffs Laws
sum of currents entering a junction is the same
as the sum of the currents leaving a junction
sum of voltages in any loop is zero
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Circuit Analysis (Combination Method)
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SI Suffixes
pico p 10-12
nano n 10-9
micro ? (u) 10-6
milli m 10-3
Kilo k 103
Mega M (Meg) 106
Giga G 109
Tera T 1012
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Oscilloscope Tutorial

• The oscilloscope is basically a graph-displaying
  device
• It draws a graph of an electrical signal.
• In most applications the graph shows how signals
  change over time
• the vertical (Y) axis represents voltage
• the horizontal (X) axis represents time.

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Oscilloscopes
Horizontal sweeps at a constant rate. Vertical
plates are attached to an external voltage, the
signal you attach to the scope.
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Cathode Ray Tubes
Variation in potential difference (voltage)
placed on plates causes electron beam to bend
different amounts. Sweep refers to refreshing
repeatedly at a fixed rate.
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Scope (Cont)

• This simple graph can tell you many things about
  a signal
• You can determine the time and voltage values of
  a signal.
• You can calculate the frequency of an oscillating
  signal.
• You can see the "moving parts" of a circuit
  represented by the signal.
• You can tell if a malfunctioning component is
  distorting the signal.
• You can find out how much of a signal is direct
  current (DC) or alternating current (AC).
• You can tell how much of the signal is noise and
  whether the noise is changing with time.

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How does an Analog Scope work?
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How does a Digital Scope work?
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Triggering Stabilizes a Repeating Waveform
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Waveform shapes tell you a great deal about a
signal
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If a signal repeats, it has a frequency. The
frequency is measured in Hertz (Hz) and equals
the number of times the signal repeats itself in
one second
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Voltage, Current, Phase
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Performance Terms

• Bandwidth
• The bandwidth specification tells you the
  frequency range the oscilloscope accurately
  measures.
• Rise Time
• Rise time may be a more appropriate performance
  consideration when you expect to measure pulses
  and steps. An oscilloscope cannot accurately
  display pulses with rise times faster than the
  specified rise time of the oscilloscope.
• Vertical Sensitivity
• The vertical sensitivity indicates how much the
  vertical amplifier can amplify a weak signal.
  Vertical sensitivity is usually given in
  millivolts (mV) per division.
• Sweep Speed
• For analog oscilloscopes, this specification
  indicates how fast the trace can sweep across the
  screen, allowing you to see fine details. The
  fastest sweep speed of an oscilloscope is usually
  given in nanoseconds/div.
• Gain Accuracy
• The gain accuracy indicates how accurately the
  vertical system attenuates or amplifies a signal.
  
• Time Base or Horizontal Accuracy
• The time base or horizontal accuracy indicates
  how accurately the horizontal system displays the
  timing of a signal.
• Sample Rate
• On digital oscilloscopes, the sampling rate
  indicates how many samples per second the ADC can
  acquire. Maximum sample rates are usually given
  in megasamples per second (MS/s). The faster the
  oscilloscope can sample, the more accurately it
  can represent fine details in a fast signal..
• ADC Resolution (Or Vertical Resolution)
• The resolution, in bits, of the ADC indicates how
  precisely it can turn input voltages into digital
  values.
• Record Length
• The record length of a digital oscilloscope
  indicates how many waveform points the
  oscilloscope is able to acquire for one waveform
  record.

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Grounding

• Proper grounding is an important step when
  setting up to take measurements.
• Properly grounding the oscilloscope protects you
  from a hazardous shock and protects your circuits
  from damage.
• Grounding the oscilloscope is necessary for
  safety. If a high voltage contacts the case of an
  ungrounded oscilloscope, any part of the case,
  including knobs that appear insulated, it can
  give you a shock. However, with a properly
  grounded oscilloscope, the current travels
  through the grounding path to earth ground rather
  than through you to earth ground.
• To ground the oscilloscope means to connect it to
  an electrically neutral reference point (such as
  earth ground). Ground your oscilloscope by
  plugging its three-pronged power cord into an
  outlet grounded to earth ground.
• Grounding is also necessary for taking accurate
  measurements with your oscilloscope. The
  oscilloscope needs to share the same ground as
  any circuits you are testing.
• Some oscilloscopes do not require the separate
  connection to earth ground. These oscilloscopes
  have insulated cases and controls, which keeps
  any possible shock hazard away from the user.

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Scope Probes Most passive probes have some
degree of attenuation factor, such as 10X, 100X,
and so on. By convention, attenuation factors,
such as for the 10X attenuator probe, have the X
after the factor. In contrast, magnification
factors like X10 have the X first
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Vertical Controls

• Position and Volts per Division
• The vertical position control lets you move the
  waveform up or down to exactly where you want it
  on the screen.
• The volts per division (usually written
  volts/div) setting varies the size of the
  waveform on the screen. A good general purpose
  oscilloscope can accurately display signal levels
  from about 4 millivolts to 40 volts.
• Often the volts/div scale has either a variable
  gain or a fine gain control for scaling a
  displayed signal to a certain number of
  divisions.

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Input Coupling

• Coupling means the method used to connect an
  electrical signal from one circuit to another.

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Horizontal Controls

• Position and Seconds per Division
• The horizontal position control moves the
  waveform from left and right to exactly where you
  want it on the screen.
• The seconds per division (usually written as
  sec/div) setting lets you select the rate at
  which the waveform is drawn across the screen
  (also known as the time base setting or sweep
  speed). This setting is a scale factor. For
  example, if the setting is 1 ms, each horizontal
  division represents 1 ms and the total screen
  width represents 10 ms (ten divisions). Changing
  the sec/div setting lets you look at longer or
  shorter time intervals of the input signal.

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Trigger Position

• The trigger position control may be located in
  the horizontal control section of your
  oscilloscope. It actually represents "the
  horizontal position of the trigger in the
  waveform record." Horizontal trigger position
  control is only available on digital
  oscilloscopes.
• Varying the horizontal trigger position allows
  you to capture what a signal did before a trigger
  event (called pretrigger viewing).
• Digital oscilloscopes can provide pretrigger
  viewing because they constantly process the input
  signal whether a trigger has been received or
  not. A steady stream of data flows through the
  oscilloscope the trigger merely tells the
  oscilloscope to save the present data in memory.
  I
• n contrast, analog oscilloscopes only display the
  signal after receiving the trigger.

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Trigger Controls (cont)
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Pulse and Rise Time Measurements
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Multimeter tutorial

• A meter is a measuring instrument. An ammeter
  measures current, a voltmeter measures the
  potential difference (voltage) between two
  points, and an ohmmeter measures resistance.
• A multimeter combines these functions, and
  possibly some additional ones as well, into a
  single instrument.

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To measure current, the circuit must be broken to
allow theammeter to be connected in
series Ammeters must have a LOW resistance
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To measure potential difference (voltage), the
circuit is not changed the voltmeter is
connected in parallelVoltmeters must have a
HIGH resistance
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To measure resistance, the component must be
removed from the circuit altogether Ohmmeters
work by passing a current through the component
being tested
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Digital MultimetersDigital meters give an
output in numbers, usually on a liquid crystal
display.Most modern multimeters are digital and
traditional analogue types are destined to become
obsolete.Digital multimeters come in a wide
range of sizes and capability. Everything from
simple 3 ½ digit auto ranging pocket meters to
larger 8 ½ digit bench model with operator or
computer (IEEE488 compatible) settable range
selection
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Function Generator

• An electronic instrument that generates various
  waveforms such as
• Sine wave
• Square wave
• Pulse trains
• Sawtooth
• The amplitude, DC offset, frequency are
  adjustable.

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Function Generators (cont)

• Like multimeters there is a wide variety of
  device offering various
• Amplitude characteristics
• Bandwidth
• Adjustments of rise and fall times
• Modulation capability (AM, FM, Pulse, etc.)

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Power Supply

• This is the device that transfers electric power
  from a source to a load using electronic
  circuits.
• Typical application of power supplies is to
  convert utility's AC input power to a regulated
  voltage(s) required for electronic equipment.
• Depending on the mode of operation of power
  semiconductors PS can be linear or switching.
• In a switched-mode power supply, or SMPS power
  handling electronic components are continuously
  switching on and off with high frequency in order
  to provide the transfer of electric energy. By
  varying duty cycle, frequency or a phase of these
  transitions an output parameter (such as output
  voltage) is controlled. Typical frequency range
  of SMPS is from 20 kHz to several MHz.

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Power Supply (cont)

• Power supplies like many of the other electronic
  instruments, come in many varieties with a wide
  range of capabilities
• Parameters that are Power Supply specific
  include
• Voltage levels
• Current
• Regulation
• Protection
• Output impedance
• Noise (ripple)
• Its the designer (or researcher) responsibility
  to identify the characteristics required.

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Oscilloscope
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Oscilloscope(continue)

• ?DEMO.Lab3a

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Capacitance (continue from before)

• A capacitor simply consists of two conductors
  which are electrically isolated from one
  another.  This means that no current can readily
  flow from one conductor to the other.
• the units of the capacitance must equal one
  coulomb per volt, which is defined to be one
  farad, F
• One Farad one(coulomb/volts)
• 1F1C/V

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RC Circuitsqualitative description
assume the switch is thrown to position B at the
time t   0.  When the switch is at the position
B the circuit consists of the single loop which
contains, starting at point B and moving around
the circuit clockwise, the resistor R, the
capacitor C, and  finally the voltage source
DVs.  In this configuration the voltage source
attempts to push charge  around the circuit in a
clockwise direction (remember that the power
source tries to push current out of its positive
terminal).
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RC Circuitsqualitative description(continue)
After some time at position B, we will throw the
switch to position A.  (The time since the switch
was thrown to position A is called a new time t
.  The prime on a symbol is used to denote
the fact that this is the value of the quantity
under consideration since the switch was thrown
to position A.  It therefore follows that the
switch is thrown to the position A at the time t
0.) After the switch has been thrown to
position A the circuit consists solely of the
resistor and the capacitor, with no voltage
source. In this case there is no external energy
being used to move charges around the circuit
loop.
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RC Circuitsqualitative description(continue)
The behavior of the voltage across the capacitor
as a function of time and the current around the
circuit (and in particular, through the resistor)
as a function of time for the case in which the
switch has been thrown to position B.  (This is
the case of the charging capacitor.) Then after
that is the case in which the switch has been
thrown to position A (the case of the discharging
capacitor).