Effective Resistance

4 Key excerpts on "Effective Resistance"

• 

  eBook - ePub

  Engineering Science

  • W. Bolton(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  Chapter 9 and this reviews the principles and then extends them to more difficult problems.

  11.2.1 Resistors in series

  For series resistors the equivalent resistance is the sum of the resistances of the separate resistors:

  equivalent resistance Re = R + R2 + . . .

  where R 1 , R 2 , etc. are the resistances of the separate resistors. As an illustration of the reduction technique, consider the series circuit shown in Figure 11.1(a) . The equivalent resistance R e of the two resistors R 1 and R 2 is R 1 + R 2 and thus we can reduce the circuit to that in Figure 11.1(b) . The circuit current I can then be obtained for this reduced circuit, being E/R e and hence is E/(R 1 + R 2 ).

  Figure 11.1 Circuit reduction

  11.2.2 Resistors in parallel

  For resistors in parallel the equivalent resistance Re is given by:

  and hence, for two resistors:

  As an illustration of the circuit reduction technique involving parallel resistors, consider the circuit shown in Figure 11.2(a) . This circuit is shown in a number of alternative but equivalent forms, all involving a battery of e.m.f. E and two parallel resistors. The equivalent resistance R e for the two parallel resistors is given by the above equation as Re = R1 R2 / (R1 + R2 ). Thus the current I drawn from the voltage source E is E/R e = E (R 1 + R 2 )/R1 R2 .

  Figure 11.2 (a) Versions of the same parallel circuit, (b) the equivalent circuit

  11.2.3 Circuits with series and parallel resistors

  For a circuit consisting of both series and parallel components, we can use the above techniques to simplify each part of the circuit in turn and so obtain a simple equivalent circuit.

  As an illustration, consider the circuit in Figure 11.3(a) . As a first step we can reduce the two parallel resistors to their equivalent, thus obtaining circuit (b) with R p = R 2 R 3 / (R 2 + R 3 ). We then have R 1 in series with R p and so can obtain the equivalent resistance R e = R 1 = R 2 R 3 /(R 2 + R 3 ) and circuit (c). Thus the current I = E /[R 1 + R 2 R 3 /(R 2 + R 3

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  eBook - ePub

  Power System Fundamentals

  • Pedro Ponce, Arturo Molina, Omar Mata, Luis Ibarra, Brian MacCleery(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  1

  Linear Electric Circuits

   

  1.1 Passive elements in electric circuits

  An active element is defined as an element that is capable of furnishing an average power greater than zero to a particular device. However, a passive element cannot supply an average power that is greater than zero over an infinite time interval.

  The resistor is the simplest passive element and is based on the statement of the fundamental relationship called Ohm’s law. It states that the voltage across conducting materials is directly proportional to the current flowing through it,

  (1.1)

  v = R i ,

  where the constant of proportionality R is called resistance. Its unit is called ohm, which is 1 [V / A ], and it is abbreviated by the Greek letter

  Ω

  .

  Equation (1.1 ) is a linear equation; therefore it is considered to be the definition of a linear resistor. The linear resistor is an idealized circuit element; the voltage-current ratios of these physical devices are constant only within certain ranges of current, voltage, or power, and also depend on temperature and other external factors.

  Resistors can be connected in series or parallel so the total resistance can be calculated according to the following expressions: series connection (R = R1 + R2 + R3 + ...), parallel connection (1 / R = 1 / R1 + 1 / R2 + 1 / R3 +...). Figure 1.1 shows a series connection.

  Another passive element used in electric circuits is the capacitor. The capacitance C is defined as the following voltage-current relationship

  (1.2)

  i = C

  dv

  dt

  ,

  where v and i are functions of time that satisfy the conventions for a passive element. Therefore the unit of capacitances is an ampere-second per volt, or coulomb per volt. So a farad [F ] is defined as one coulomb per volt and used as unit of capacitance.

  A capacitor consists of two conducting surfaces separated by a thin insulating material with a very large resistance, in which electric charge may be stored.

  Some of the characteristics of the capacitor are obtained from Equation (1.2

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  An Introduction to Electrical Science

  • Adrian Waygood(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  thousands of such shocks! And, rather surprisingly, his results apparently compare remarkably well with what we know today about the electrical resistance of various materials.

  Resistance (symbol: R ) is, to some extent, dependent upon the quantity of free electrons available as charge carriers within a given volume of material, and the opposition to the drift of those free electrons due to the obstacles presented by fixed atomic structure and forces within that material.

  For example, conductors have very large numbers of free electrons available as charge carriers and, therefore, have low values of resistance. On the other hand, insulators have relatively few free electrons in comparison with conductors, and, therefore, have very high values of resistance.

  But resistance is also the result of collisions between free electrons drifting through the conductor under the influence of the external electric field, and the stationary atoms. Such collisions represent a considerable reduction in the velocity of these electrons, with the resulting loss of their kinetic energy contributing to the rise of the conductor’s temperature. So it can be said that the consequence of resistance is heat .

  The consequence of resistance is heat.

  Resistance, therefore, can be considered to be a useful property as it is responsible for the operation of incandescent lamps, heaters, etc. On the other hand, resistance is also responsible for temperature increases in conductors which result in heat transfer away from those conductors into their surroundings – we call these energy losses , which, of course, are undesirable.

  We can modify the natural resistance of any circuit by adding resistors

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  Electric Circuits and Signals

  • Nassir H. Sabah(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Like ideal sources, ideal resistors, capacitors, and inductors are abstractions. Not only are their values constant, but each is assumed to model solely the field attribute it represents. Thus, ideal resistors only dissipate energy; they do not store electric or magnetic energy. Ideal capacitors only store electric energy; they do not store magnetic energy, nor do they dissipate energy. Ideal inductors only store magnetic energy; they do not store electric energy, nor do they dissipate energy. Practical circuit elements depart from this ideal to varying degrees, depending on their physical realization and operating conditions. They all have limitations on the maximum current or voltage they can handle, which must be taken into consideration in circuit design by ensuring that the specified ratings are not exceeded. The values of practical circuit elements may be considered constant over a limited operating range and may change over a prolonged period. For example, the heat generated by an appreciable current in a resistor changes its temperature and, hence, its resistance. A magnetic field, and hence some inductance, is associated with the current through a resistor. A capacitance is associated with the electric field due to the voltage difference between the ends of a current-carrying resistor. Similarly, energy is dissipated by the finite resistance of the coil of an inductor, and a capacitance is associated with the voltage drops between the ends of a coil as well as between adjacent turns of the coil. The dielectric of a capacitor is never perfect and dissipates charge and energy, and an inductance is associated with the leads of a capacitor. These parasitic effects are generally negligible at low frequencies but become significant at high-enough frequencies.
• Ideal circuit elements are basic , in the sense that an ideal circuit element cannot be modeled in terms of other ideal circuit elements. On the other hand, practical circuit elements are modeled in terms of a combination of ideal circuit elements. For example, a coil is modeled at low frequencies by a combination of an ideal inductor and an ideal resistor.
• As is true of macroscopic systems in general, energy and charge must be conserved in all electric circuits.

Summary of Main Concepts and Results

• Current is the rate of flow of electric charge.
• The voltage, or electric potential difference, between two points is the difference in electric potential energy, per unit positive charge, between these points.
• Power is the rate at which energy is delivered or absorbed.
• Conservation of energy implies conservation of power.
• If current i is in the direction of a voltage drop v , then the power p = vi represents power absorbed by the element through which i flows. Conversely, if i is in the direction of a voltage rise v then the power p represents power delivered by the element through which i