Physics
Effective Resistance
Effective resistance refers to the total resistance in a circuit when resistors are combined in various ways, such as in series or parallel. It is a single equivalent resistance that represents the combined effect of multiple resistors. Calculating effective resistance is essential for analyzing and understanding the behavior of complex electrical circuits.
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6 Key excerpts on "Effective Resistance"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Electrical resistance The ratio of the voltage applied across a piece of material to the current that flows in that material. The SI unit of resistance is the ohm (.Q). Ohm's law The relation between current, voltage and a constant resistance for a material, V = IR. Resistivity The proportionality constant p for a material which relates the resistance R, the length of the material L, and the material's cross-sectional area A. That is, R = pLIA. Electric power The power P delivered to a circuit in which a current I results from a voltage Vis P =IV. Series wiring Circuit devices are wired together so that each device MUST carry the same current. Parallel wiring Circuit devices are wired together so that each MUST have the same voltage drop. Equivalent resistance A single resistance which would demand the same current from the same source as does a network of resistors. Internal resistance The resistance to current flow through a voltage source such as a battery which arises from the resistance of the materials from which the source is made. Chapter 20 255 Terminal voltage The actual voltage output of a voltage source such as battery. It is the emf of the source minus the voltage drop across the internal resistance of the source. Equivalent capacitor A single capacitor which would store the same charge or energy as does a network of capacitors. Galvanometer An electrical meter which contains a moving coil and is used as the basic component of analog ammeters and voltmeters. Ammeter A galvanometer in parallel with a shunt resistance. The ammeter measures current. Voltmeter A galvanometer in series with a large resistance. The voltmeter measures potential difference (voltage). Time constant (t) The product of resistance and capacitance, RC. In a discharging series RC circuit the time constant represents the time needed for the charge on the capacitor to fall to 1/e (3 7 %) of its initial value at time zero.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
A conducting material of uniform cross-sectional area A and length L has an electrical resistance R given by R L A ρ = (20.2.3) where ρ is the electrical resistivity of the material (Table 20.2.1). The electrical resistivity ρ has SI units of ohms times meters ( m Ω⋅ ). Materials with small values of resistivity conduct electricity well, while those with high values of resistivity are relatively poor conductors. 20.3 Electrical Power and Energy A variety of energy transfers may take place in a circuit. Power is defined as energy or work divided by time. When electric current I passes between two points along a continuous conducting path, the electrical power is P IV = (20.3.2) where V is the magnitude of the potential difference between the two points over which the energy transfer occurred. When Equation 20.3.2 is applied to a resistor, we can use V IR = , so we have 570 | Chapter 20 P I R V R 2 2 = = (20.3.3) In Equation 20.3.3, P is the power dissipated in the resistor. When we say that energy is dissipated, we mean that it is converted into internal energy, or heat. 20.4 Resistors in Series and in Parallel The equivalent resistance of a set of resistors is defined as the resistance of a single resis- tor that, if it replaces the individual resistors, has no effect on the current in other parts of the circuit. When two or more resistors are wired in series, the equivalent resistance R eq is equal to the sum of the individual resistances: R R R R eq 1 2 3 = + + + (20.4.1) The current is the same in each resistor in a series combination. When two or more resistors are wired in parallel, the equivalent resistance R eq is related to the individual resistances as follows: R R R R 1 1 1 1 eq 1 2 3 = + + + (20.4.2) The potential difference is the same across each resistor in a parallel combination. 20.5 Kirchhoff’s Rules There are two rules that may be used when dealing with circuits.
eBook - PDF- Leslie Basford(Author)
- 2013(Publication Date)
- Made Simple(Publisher)
Materials which have practically no resistance at extremely low temperatures are described as superconducting. The resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. The relationship is given by the formula where p is the resistivity of the conducting material. The combined resistance R of any number of resistors Ri 9 R2 9 R3, etc., in series is given by R = R± + R2 + R3 + . . . The combined resistance R of any number of resistors Ri 9 Ife, ife, etc., in parallel is given by ^ = ^ + j ^ + j ^ + --' Kirchhoff's first law states that the sum of the currents flowing into any junction in a circuit is equal to the sum of the currents flowing out of it. KirchhofF's second law states that the total e.m.f. in any closed loop of a circuit is equal to the sum of the potential differences across the resistances in the loop. Resistance 37 Resistance can be measured accurately with a Wheatstone bridge. The condition for a balanced bridge is S ~~ h where h is the length of bridge wire between the sliding contact and the (unknown) resistance whose value is X 9 h is the length of bridge wire between the sliding contact and the (standard) resistor whose value is S.
- Frank R. Spellman(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
It is probably more accurate to call this total resis-tance the equivalent resistance , but by convention R T (total resistance) is generally used, although they are often used interchangeably. The equivalent resistance is illustrated in the equivalent circuit shown in Figure 10.53. Other methods are used to determine the equivalent resis-tance of parallel circuits. The most appropriate method for a particular circuit depends on the number and value of the resis-tors; for example, consider the parallel circuit shown in Figure 10.54. For this circuit, the following simple equation is used: R R N eq = (10.18) I 1 = 10 A I 2 = 3 A I 3 = ? I 4 = 5 A FIGURE 10.51 Illustration for Example 10.23. V = 120 V 12 ohms 15 ohms 15 ohms R 1 R 2 R 3 FIGURE 10.52 Illustration for Example 10.24. 363 Fundamentals of Electricity where R eq = Equivalent parallel resistance. R = Ohmic value of one resistor. N = Number of resistors. Thus, R eq = = 10 ohms 5 ohms 2 Note: Equation 10.18 is valid for any number of equal value parallel resistors. Key Point: When two equal value resistors are connected in parallel, they present a total resistance equivalent to a single resistor of one half the value of either of the original resistors. ■ EXAMPLE 10.25 Problem: Five 50-ohm resistors are connected in parallel. What is the equivalent circuit resistance? Solution: Using Equation 10.18: R R N eq = = = 50 5 10 ohms What about parallel circuits containing resistance of unequal value? How is equivalent resistance determined? Example 10.26 demonstrates how this is accomplished. ■ EXAMPLE 10.26 Problem: Refer to Figure 10.55. Solution: Given: R 1 = 3 ohms R 2 = 6 ohms E a = 30 volts We know that I 1 = 10 amps I 2 = 5 amps I t = 15 amps and can now determine R eq : R E I eq a t = = = 30 15 2 ohms Key Point: In Example 10.26, the equivalent resistance of 2 ohms is less than the value of either branch resistor.
- Frank R. Spellman, Nancy E. Whiting(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
It is probably more accurate to call this total resis-tance the equivalent resistance , but by convention R T (total resistance) is generally used, although they are often used interchangeably. The equivalent resistance is illustrated in the equivalent circuit shown in Figure 11.47. Other methods are used to determine the equivalent resistance of parallel circuits. The most appropriate method for a particular circuit depends on the number and value of the resistors; for example, consider the parallel circuit shown in Figure 11.48. For this circuit, the follow-ing simple equation is used: R R N eq = (11.36) where R eq = Equivalent parallel resistance. R = Ohmic value of one resistor. N = Number of resistors. Thus, R eq = = 10 ohms 5 ohms 2 Note: Equation 11.36 is valid for any number of equal value parallel resistors. Key Point: When two equal value resistors are connected in parallel, they present a total resistance equivalent to a single resistor of one half the value of either of the original resistors. ■ EXAMPLE 11.39 Problem: Five 50-ohm resistors are connected in parallel. What is the equivalent circuit resistance? Solution: Using Equation 11.36: R R N eq = = = 50 5 10 ohms I T = 26 A 120 volt s R T = 4.26 ohms FIGURE 11.47 Circuit equivalent to that of Figure 11.46. A R 1 = 10 ohms R 2 = 10 ohms FIGURE 11.48 Two equal resistors connected in parallel. 282 Handbook of Mathematics and Statistics for the Environment What about parallel circuits containing resistance of unequal value? How is equivalent resistance determined? Example 11.40 demonstrates how this is accomplished. ■ EXAMPLE 11.40 Problem: Refer to Figure 11.49. Solution: Given: R 1 = 3 ohms R 2 = 10 ohms E a = 30 volts We know that I 1 = 10 amps I 2 = 5 amps I t = 15 amps and can now determine R eq : R E I eq a t = = = 30 15 2 ohms Key Point: In Example 11.40, the equivalent resistance of 2 ohms is less than the value of either branch resistor.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
The total resistance of this combination is intermediate between the pure series and pure parallel values ( 20.0 Ω and 0.804 Ω , respectively). b. The current through R 1 is equal to the current supplied by the battery: I 1 = I = V R eq = 12.0 V 5.10 Ω = 2.35 A. The voltage across R 1 is V 1 = I 1 R 1 = (2.35 A)(1 Ω) = 2.35 V. Chapter 10 | Direct-Current Circuits 449 10.5 The voltage applied to R 2 and R 3 is less than the voltage supplied by the battery by an amount V 1 . When wire resistance is large, it can significantly affect the operation of the devices represented by R 2 and R 3 . c. To find the current through R 2 , we must first find the voltage applied to it. The voltage across the two resistors in parallel is the same: V 2 = V 3 = V − V 1 = 12.0 V − 2.35 V = 9.65 V. Now we can find the current I 2 through resistance R 2 using Ohm’s law: I 2 = V 2 R 2 = 9.65 V 6.00 Ω = 1.61 A. The current is less than the 2.00 A that flowed through R 2 when it was connected in parallel to the battery in the previous parallel circuit example. d. The power dissipated by R 2 is given by P 2 = I 2 2 R 2 = (1.61 A) 2 (6.00 Ω) = 15.5 W. Significance The analysis of complex circuits can often be simplified by reducing the circuit to a voltage source and an equivalent resistance. Even if the entire circuit cannot be reduced to a single voltage source and a single equivalent resistance, portions of the circuit may be reduced, greatly simplifying the analysis. Check Your Understanding Consider the electrical circuits in your home. Give at least two examples of circuits that must use a combination of series and parallel circuits to operate efficiently. Practical Implications One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant.
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