Resistors in Series

5 Key excerpts on "Resistors in Series"

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  eBook - ePub

  Learn Audio Electronics with Arduino

  Practical Audio Circuits with Arduino Control

  • Charlie Cullen(Author)
  • 2020(Publication Date)
  • Focal Press

    (Publisher)

  limited by the total resistance within it – all electrons will move the same overall distance, regardless of whether they are currently inside (or outside) any particular resistor:

  The image above shows how combining a low and high resistance effectively reduces the overall current flowing in a circuit, where current is measured as the amount of charge (in coulombs) flowing past a given point in one second. As there is less area in the higher resistor for electrons to move, the number of electrons that can pass any given point will be reduced. Thus, in the second example the distance an electron can travel within the lower resistance is also now reduced – there is nowhere else for it to go. A simple analogy would be to think of electron flow through multiple resistors like a traffic jam, where cars on a multiple-lane road move less overall distance because the nearest exit ramp only allows single-lane traffic to leave.

  As free electrons can only enter through the voltage source, in series circuits voltage varies and current is constant.

  Now we know that all voltage drops in a loop will cancel each other out to zero, and also that the current in that loop will be constant. By combining these two elements with Ohm’s Law, we can develop a simple equation for determining the total resistance in a series circuit (Figure 3.5 ).

  Figure 3.5 Calculating total resistance in a series circuit. In the diagrams, examples of series circuits combining two and three resistors are shown. We know that the supply voltage (V S ) is equivalent to all component voltage drops, and that current in the loop is constant. By rearranging the terms, we can derive the total resistance (R

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  MSP430-based Robot Applications

  A Guide to Developing Embedded Systems

  • Dan Harres(Author)
  • 2013(Publication Date)
  • Newnes

    (Publisher)

  Eq. 3.6 gives:

  (3.7)

  or

  (3.8)

  Equation 3.8 demonstrates yet another important principle – that Resistors in Series add. We could simply replace the two resistors in Figure 3.7 with a single resistor equal to the sum of the two, and the current would be the same.

  The behavior of two resistors in parallel can also be determined quite easily, using Kirchoff’s current law and Ohm’s law. Start with the simple circuit of Figure 3.8 .

  Figure 3.8 Two resistors in parallel.

  Since I 1 and I 2 are referenced away from the top node, but I 3 is referenced into that node, the first two currents will be written with negative signs in the node equation:

  (3.9)

  or

  (3.10)

  In this case, both V

  R 1

  and V

  R 2

  must be equal to V B . Making use of Ohm’s law, currents I 1 and I 2 are:

  (3.11)

  If we wished to simplify the circuit and substitute a single equivalent resistance, R eq , for the two parallel resistors, we would draw it as in Figure 3.9 .

  Figure 3.9 Simplified circuit substituting single resistor for two parallel resistors.

  Note that we’ve discussed only resistors in illustrating Kirchoff’s laws. However, the laws apply to any of the components that we will introduce in this chapter.

  There are a number of additional theorems and laws, such as superposition, Thevenin’s equivalent, Norton’s equivalent, and many others, that can be extremely useful in circuit analysis and design. However, armed with just Ohm’s law and Kirchoff’s laws, plus the additional component behavior described in the next sections, the designer of the robotics projects described in this book should be prepared to tackle the design and analysis of passive component circuits.

  Resistors, capacitors, and inductors

  Resistors

  We’ve already seen how resistors behave in circuits in the previous section. They follow the linear relationship of Ohm’s law (Eq. 3.1 ). Before going on to the next topic, there’s a resistor circuit called the resistor divider that is used quite often in electronics design. It is simply two Resistors in Series (Figure 3.10

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  Basic Electricity

  • U.S. Bureau of Naval Personnel(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  fig. 5-5 (B) ), are determined as follows:

  The division of current in a parallel network follows a definite pattern. This pattern is described by Kirchhoff’s current law which is stated as follows: The algebraic sum of the currents entering and leaving any junction of conductors is equal to zero. This law can be stated mathematically as

  where Ia , Ib , etc., are the currents entering and leaving the junction. Currents entering the junction are assumed to be positive, and currents leaving the junction are considered negative. When solving a problem using equation (5-3) , the currents must be placed into the equation with the proper polarity signs attached.

  Example. Solve for the value of I3 in figure 5-6 .

  Figure 5-6.—Circuit for example problem.

  Solution: First the currents are given proper signs.

  these currents are placed into equation (5-3) with the proper signs as follows:

  Basic equation: Substitution: Combining like terms:

  thus, I3 has a value of 2 amperes, and the negative sign shows it to be a current leaving the junction.

  Example. Using figure 5-7 , solve for the magnitude and direction of I3 :

  Solution:

  Thus, I3 is 2 amperes, and its positive sign shows it to be a current entering the junction.

  Figure 5-7.—Circuit for example problem.

  PARALLEL RESISTANCE

  The preceding discussion of current introduced certain principles involving the characteristics and effects of resistance in parallel circuits. A detailed explanation of the characteristics of parallel resistances will be considered in this section. The explanation will commence with a simple parallel circuit. Various methods used to determine the total resistance in parallel circuits will be described.

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  Electronic Servicing and Repairs

  • Trevor Linsley(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Fig. 8.10 . Since all resistors are now in series

  R T = R 1 + R 2 + Rp

  R T = 3Ω + 6Ω + 1Ω= 10Ω

  Figure 8.9 A series/parallel circuit.

  Figure 8.10 Equivalent series circuit.

  Thus, the circuit may be represented by a single equivalent resistor of value 10Ω as shown in Fig. 8.11 . The total current flowing in the circuit may be found by using Ohm’s law 1= V/Ros

  The potential differences across the individual resistors are:

  V 1 = 1 × R 1 = 1A × 3Ω =3V

  V 2 = 1 × R 2 = 1A × 6Ω =6V

  V P = 1 × R P = 1A × 1Ω =1V

  Since the same voltage acts across all branches of a parallel circuit the same p.d. of one volt will exist across each resistor in the parallel branch R 3 ,R 4 , and R 5 . Six volts will be dropped across R 2 and three volts across R 1 .

  Figure 8.11 Single equivalent resistor for Fig. 8.9 .

  EXAMPLE 2

  Determine the total resistance and the current flowing through each resistor for the circuit shown in Fig. 8.12 . By inspection, it can be seen that R 1 and R 2 are connected in series while R 3 is connected in parallel across R 1 and R 2 The circuit may be more easily understood if we redraw it as in Fig. 8.13 . For the series branch, the equivalent resistor can be found from

  R S = R 1 + R 2

  R S = 3Ω + 3Ω=6Ω

  Figure 8.12 A series/parallel circuit for Example 2.

  Figure 8.13 Equivalent circuit for Example 2.

  Figure 8.13 may now be represented by a more simple equivalent circuit as shown in Fig. 8.14 . Since the resistors are now in parallel, the equivalent resistance may be found from

  The total current is

  Let the current flowing through resistor R 3 be called l 3

  Figure 8.14 Simplified equivalent circuit for Example 2.

  Let the current flowing through both resistors R 1 and R 2 as shown in Fig. 8.13 , be called l s ;

  CAPACITORS IN COMBINATION

  Capacitors, like resistors, may be joined together in various combinations of series or parallel connections, see Fig. 8.15 . The equivalent capacitance C T

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  A Practical Introduction to Electrical Circuits

  • John E. Ayers(Author)
  • 2024(Publication Date)
  • CRC Press

    (Publisher)

  CDR may be used to find how a total current splits among parallel resistances. It states that the fraction of total current flowing in one parallel resistor is equal to the equivalent resistance for the other resistors divided by the sum of the equivalent resistance for the other resistors and the resistance for the branch in question.

  Some configurations or resistors may not be simplified by the use of parallel and series combinations. These configurations generally contain wye-connected or delta-connected combinations of resistors, and an example is the Wheatstone bridge. Circuits of this type may be simplified by using delta–wye or wye–delta transformations.

  The principle of superposition is a useful tool for solving some circuit problems. It states that when a linear system is driven by more than one independent source, the total response is the sum of the individual responses associated with each of the independent sources. Here, the total response is a voltage or a current, and we can find it by summing the individual responses, each determined by leaving one independent source active while deactivating all others.

  Problems

  • Problem 1.1. Consider the electrical circuit of Figure P1.1 containing a battery and two resistors.
    Long Description for Figure P1.1

    Circuit involving the series connection of a battery and two resistors in a single mesh. The positive terminal of a 9 V battery is connected to one terminal of a resistor having the color code red, yellow, brown, gold. The other terminal of this resistor is connected to a second resistor with the color code brown, blue, brown, gold. The other terminal of this resistor is connected to the negative terminal of the 9 V battery.

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