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Resistances in series
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When you place resistances in series, their ohmic values simply add together to get the total resistance. This is easy to see intuitively, and it s quite simple to remember.
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Suppose the following resistances are hooked up in series with each other: 112 , and 680 . What is the total resistance of the series combination (Fig. 4-8) , 470
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4-8 Three resistors in series (Problem 4-14).
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Just add the values, getting a total of 112 + 470 + 680 1262 . You might round this off to 1260 . It depends on the tolerances of the components how precise their actual values are to the ones specified by the manufacturer.
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Resistances in parallel
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When resistances are placed in parallel, they behave differently than they do in series. In general, if you have a resistor of a certain value and you place other resistors in parallel with it, the overall resistance will decrease. One way to look at resistances in parallel is to consider them as conductances instead. In parallel, conductances add, just as resistances add in series. If you change all the ohmic values to siemens, you can add these figures up and convert the final answer back to ohms. The symbol for conductance is G. This figure, in siemens, is related to the resistance R, in ohms, by the formulas: G R 1/R, and 1/G
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Consider five resistors in parallel. Call them R1 through R5, and call the total resistance R as shown in the diagram Fig. 4-9. Let R1 100 , R2 200 , R3 300 , R4 400 and R5 500 respectively. What is the total resistance, R, of this parallel combination Converting the resistances to conductance values, you get G1 1/100 0.01 siemens, G2 1/200 0.005 siemens, G3 1/300 0.00333 siemens, G4 1/400 0.0025 siemens, and G5 1/500 0.002 siemens. Adding these gives G 0. 01 + 0. 005 + 0.00333 + 0.0025 + 0.002 0.0228 siemens. The total resistance is therefore R 1/G 1/0.0228 43.8 .
Resistances in series-parallel 75
4-9 Five resistors in parallel, R1 through R5, give a total resistance R. See Problems 4-15 and 4-16.
When you have resistances in parallel and their values are all equal, the total resistance is equal to the resistance of any one component, divided by the number of components.
Problem 4-16
Suppose there are five resistors R1 through R5 in parallel, as shown in Fig. 4-9, all having a value of 4.7K . What is the total resistance, R You can probably guess that the total is a little less than 1K or 1000 . So you can convert the value of the single resistor to 4,700 and divide by 5, getting a total resistance of 940 . This is accurate to two significant figures, the 9 and the 4; engineers won t usually be worried about the semantics, and you can just say 940 .
Division of power
When combinations of resistances are hooked up to a source of voltage, they will draw current. You can easily figure out how much current they will take by calculating the total resistance of the combination and then considering the network as a single resistor. If the resistances in the network all have the same ohmic value, the power from the source will be evenly distributed among the resistances, whether they are hooked up in series or in parallel. If there are eight identical resistors in series with a battery, the network will consume a certain amount of power, each resistor bearing 1/8 of the load. If you rearrange the circuit so that the resistors are in parallel, the circuit will dissipate a certain amount of power (a lot more than when the resistors were in series), but again, each resistor will handle 1/8 of the total power load. If the resistances in the network do not all have identical ohmic values, they divide up the power unevenly. Situations like this are discussed in the next chapter.
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