Resistances in Series-Parallel

Sets of resistors, all having identical ohmic values, can be connected together in parallel sets of series networks, or in series sets of parallel networks. By doing this, the total power-handling capacity of the resistance can be greatly increased over that of a single resistor.

Sometimes, the total resistance of a series-parallel network is the same as the value of any one of the resistors. This is always true if the components are identical, and are in a network called an n-by-n matrix. That means, when n is a whole number, there are n parallel sets of n resistors in series (following figure A), or else there are n series sets of n resistors in parallel (following figure B). Either arrangement
gives the same practical result.

Series-parallel resistances.
  At A, sets of series resistors are connected in parallel.
  At B, sets of parallel resistances are connected in series. These examples show symmetrical n-by-n matrices with n = 3.

Engineers and technicians sometimes use series-parallel networks to obtain resistances with large power-handling capacity. A series-parallel array of n by n resistors will have n2 times that of a single resistor. Thus, a 3 × 3 series-parallel matrix of 2 W resistors can handle up to 32 × 2 = 9 × 2 = 18 W, for example. A 10 × 10 array of 1-W resistors can dissipate up to 100 W. The total power handling capacity is multiplied by the total number of resistors in the matrix. But this is true only if all the resistors have the same ohmic values, and the same power-dissipation ratings.

It is unwise to build series-parallel arrays from resistors with different ohmic values or power ratings. If the resistors have values and/or ratings that are even a little nonuniform, one of them might be subjected to more current than it can withstand, and it will burn out. Then the current distribution in the network can change so a second component fails, and then a third. It’s hard to predict the current and power distribution in an array when its resistor values are all different.

If you need a resistance with a certain power-handling capacity, you must be sure the network can handle at least that much power. If a 50-W rating is required, and a certain combination will handle 75 W, that’s fine. But it isn’t good enough to build a circuit that will handle only 48 W. Some extra tolerance, say 10 percent over the minimum rating needed, is good, but it’s silly to make a 500-W network using far more resistors than necessary, unless that’s the only convenient combination given the parts available.

Nonsymmetrical series-parallel networks, made up from identical resistors, can increase the power handling capability over that of a single resistor. But in these cases, the total resistance is not the same as the value of the single resistors. The overall power-handling capacity is always multiplied by the total number of resistors, whether the network is symmetrical or not, provided all the ohmic values are identical. In engineering work, cases sometimes arise where nonsymmetrical networks fit the need.