National Communications Forum 2000 Session T01A

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Circuit Analysis II - EE4114
Dr. Olga Korostynska weeks 1-6 C2-056, ECE
Dept. email olga.korostynska_at_ul.ieDr.
Sinead O'Keeffe weeks 7-12 C2-052, ECE
Dept. email sinead.o'keeffe_at_ul.ie
Exam 80 Tutorials 20
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Logarithms

• Basic Relationships
• Examining the relationships between the variables
  of the logarithmic function. The Mathematical
  expression
• N (b)X
• States that the number N is equal to the base
  b taken to the power x.
• The value of x could be determined using
  logarithms
• X log10 N
• The natural logarithm ( ln ) has a base e, where
  e 2.7183
• If a conversion from one base to the other is
  required
• logex 2.3 log10x

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Logarithms

• Some of the most common applications of the
  logarithmic function
• Plotting the response of a system for a range of
  values that may otherwise be impossible or
  unwieldy with a linear scale.
• Levels of power, voltage, and the like, can be
  compared without dealing with very large or very
  small numbers that often cloud the true impact of
  the difference in magnitudes.
• A number of systems respond to outside stimuli in
  a nonlinear logarithmic manner. The result is a
  mathematical model that permits a direct
  calculation of the response of the system to a
  particular input signal.
• The response of a cascaded or compound system can
  be rapidly determined using logarithms if the
  gain of each stage is known on a logarithmic
  basis.

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Logarithms

• Graphs
• Semilog graphs have only one log scale, the other
  being linear.
• Spacing of the log scale is determined by taking
  the log (base 10) of the number.
• Scaling starts with 1, since the log10 1 0.
• Keep in mind that almost 50 of the width of one
  log interval (1 to 10) is represented by 3 rather
  than by the 5 of a linear scale.
• The spacing between 1 and 10, 10 and 100, 100 and
  1000, and so on, will be the same.

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Properties of Logarithms

• Characteristics of logarithms
• The common or natural logarithm of the number 1
  is 0
• log10 1 0
• The log of any number less than 1 is a negative
  number
• log10 0.1 ?1
• The log of the product of two numbers is the sum
  of the logs
• log10 ab log10 a log10 b
• The log of the quotient of two numbers is the
  difference of the logs
• log10 a/b log10 a ? log10 b
• The log of a number taken to a power is equal to
  the product of the power and the log of the
  number
• log10 an n log10 a

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Decibels

• Two levels of power can be composed using a unit
  of measure called the bel, defined as
• To provide a unit of measure of less magnitude, a
  decibel is defined
• The result is the following important equation,
  which compares power levels P2 and P1 in decibels

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Decibels

• Voltage Gain
• Decibels are also used to provide a comparison
  between voltage levels. By substituting the
  basic power equation into the equation which
  compares levels of P2 and P1 in decibels.
• Modern VOMs and DMMS have a dB scale designed to
  provide an indication of power ratios referenced
  to a standard level of 1mW at 600?.
• The accuracy of the reading is only if the load
  has the characteristics of the rating on the
  meter.

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Decibels

• The Human Auditory Response
• Most frequent application of the decibel scale is
  in the communication and entertainment
  industries.
• The human ear does not respond in a linear
  fashion to changes in source power levels,
  meaning that doubling the audio power level from
  1/2 W to 1 W does not result in a doubling of the
  loudness for the human ear.
• The human ear responds in a logarithmic fashion
  to changes in audio power levels.
• To double the sound level received by the human
  ear, the power rating of the acoustical source
  (in watts) must be increased by a factor of 10.

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Filters

• Any combination of passive (R, L and C) and/or
  active (transistors or operational amplifiers)
  elements, designed to select or reject a band of
  frequencies is called a filter.
• Electronic filters were first introduced in
  1910-1920 when radio became popular a device was
  required that removed unwanted static from the
  signal to improve reception, these early devices
  used passive components, i.e. resistors,
  capacitors and inductors and as such were
  referred to as passive filters.
• With the introduction of active circuit
  elements, op-amps, active filters were developed.
• Passive filters contain only resistors,
  capacitors, and inductors.
• Active filters employ transistors or op-amps in
  addition to resistors and capacitors.

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• Filter circuits are used in a wide variety of
  applications. In the field of telecommunication,
  band-pass filters are used in the audio frequency
  range (0 kHz to 20 kHz) for modems and speech
  processing.
• High-frequency band-pass filters (several
  hundred MHz) are used for channel selection in
  telephone central offices.
• Data acquisition systems usually require
  anti-aliasing low-pass filters as well as
  low-pass noise filters in their preceding signal
  conditioning stages.
• System power supplies often use band-rejection
  filters to suppress the 60-Hz line frequency and
  high frequency transients.
• In addition, there are filters that do not
  filter any frequencies of a complex input signal,
  but just add a linear phase shift to each
  frequency component, thus contributing to a
  constant time delay. These are called all-pass
  filters.
• At high frequencies (gt 1 MHz), all of these
  filters usually consist of passive components
  such as inductors (L), resistors (R), and
  capacitors (C). In the lower frequency range (1
  Hz to 1 MHz), however, the inductor value becomes
  very large and the inductor itself gets quite
  bulky, making economical production difficult. In
  these cases, active filters become important.

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High Pass Filter, Frequency Response
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Band-Pass Filter, Frequency Response
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Band-Reject (Stop-Band) Filter, Frequency Response
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• In general, all filters belong to the four broad
  categories los-pass, high-pass, pass-band,
  stop-band.
• For each form there are critical frequencies
  that define the regions of pass-bands and
  stop-bands.
• Any frequency in the pass-band will pass through
  to the next stage with at least 70.7 of the
  maximum output voltage.
• For some stop-band filters, the stop-band is
  defined by conditions other than the 0.707 level.
  In fact, for many stop-band filters, the
  condition that V01/1000Vmax (corresponding with
  60 dB) is used to define the stop-band region,
  with the pass-band continuing to be defined by
  the 0.707 level.
• The resulting frequencies between the two
  regions are then called the transition
  frequencies and establish the transition region.