How Resistances Combine Help
When electrical components or devices containing dc resistance are connected together, their resistances combine according to specific rules. Sometimes the combined resistance is more than that of any of the components or devices alone. In other cases the combined resistance is less than that of any of the components or devices all by itself.
Resistances In Series
When you place resistances in series, their ohmic values add up to get the total resistance. This is intuitively simple, and it’s easy to remember.
Resistance in Series Practice Problem
Suppose that the following resistances are hooked up in series with each other: 112 ohms, 470 ohms, and 680 ohms ( Fig. 12-9 ). What is the total resistance of the series combination?
Just add the values, getting a total of 112 + 470 + 680 = 1,262 ohms. You can round this off to 1,260 ohms. It depends on the tolerances of the components—how much their actual values are allowed to vary, as a result of manufacturing processes, from the values specified by the vendor. Tolerance is more of an engineering concern than a physics concern, so we won’t worry about that here.
Resistances In Parallel
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 decreases. Mathematically, the rule is straightforward, but it can get a little messy.
One way to evaluate resistances in parallel is to consider them as conductances instead. Conductance is measured in units called Siemens , sometimes symbolized S. (The word Siemens serves both in the singular and the plural sense). In older documents, the word mho (ohm spelled backwards) is used instead. In parallel, conductances add up in the same way 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 . Conductance in Siemens is the reciprocal of resistance in ohms. This can be expressed neatly in the following two formulas. It is assumed that neither R nor G is ever equal to zero:
G = I/R
R = I/G
Resistance in Parallel Practice Problem
Consider five resistors in parallel. Call them R 1 through R 5 , and call the total resistance R , as shown in the diagram of Fig. 12-10 . Let R 1 = 100 ohms, R 2 = 200 ohms, R 3 = 300 ohms, R 4 = 400 ohms, and R 5 = 500 ohms, respectively. What is the total resistance R of this parallel combination?
Converting the resistances to conductance values, you get G 1 = 1/100 = 0.0100 Siemens, G 2 = 1/200 = 0.00500 siemens, G 3 = 1/300 = 0.00333 siemens, G 4 = 1/400 = 0.00250 siemens, and G 5 = 1/500 = 0.00200 Siemens. Adding these gives G = 0.0100 + 0.00500 + 0.00333 + 0.00250 + 0.00200 = 0.02283 Siemens. The total resistance is therefore R = 1/ G = 1/0.02283 = 43.80 ohms. Because we’re given the input numbers to only three significant figures, we should round this off to 43.8 ohms.
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. In a more general sense, the resistances in Fig. 12-10 combine like this:
R = 1/(1/ R 1 + 1/ R 2 + 1/ R 3 + 1/ R 4 + 1/ R 5 )
If you prefer to use exponents, the formula looks like this:
R = ( R −1 1 + R −1 2 + R −1 3 + R −1 4 + R −1 5 ) −1
These resistance formulas are cumbersome for some people to work with, but mathematically they represent the same thing we just did in the above problem.
Practice problems of these concepts can be found at: Direct Current Practice Problems
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