UPDATED. I observe that the problem is easy to solve when the resistors are all of the same resistance and same maximum power handling, but more complex when they are different. In the analysis below, I have gone through the various cases to provide what I think is a complete answer to the question for series, parallel, and combinations.
Unfortunately, the answer depends entirely on the specifications for each resistance, and the way they are connected together (series, parallel, or combinations). There is no one plug-and-play formula for any but some simple cases (mentioned below); analysis is required to ensure that you don’t burn up resistors.
Resistance in series
Resistors connected in series have a total resistance that is the sum of each of the resistors.
If there are n resistors with the same resistance R, the total resistance is therefore R*n.
Resistance in parallel
Resistors connected in parallel have a resistance calculated by:
- Rparallel = 1/(1/R1 + 1/R2 + …)
If there are n resistors with the same resistance R, the total resistance is therefore R/n.
The relationship of voltage (E), current (I), and resistance (R) can be expressed in three algebraically equivalent ways:
- E = I * R
- R = E / I
- I = E / R
Calculating Power Dissipation
For a resistance (single resistor R, or effective resistance of a network), the power is simply
(The latter substitutes an Ohm's Law equation for current, so that power can be determined using only the voltage and resistance.)
Power Handling for Series Resistors
Power handling must be computed for each resistor in order to determine whether its rated maximum is exceeded, or not. First, calculate the total resistance:
Now determine the current for the series group:
For a series of resistors, the current is the same through all resistors, but the voltage depends on the resistance of each resistance:
- Er1 = I / R1
- Er2 = I / R2
Note that the sum of the Er values adds up to E.
The power dissipated for each resistor is determined by:
The total power dissipated by the series network is the sum of Pr1 + Pr2 etc.
You have to be sure that none of the Pr values exceeds the rated maximum for the resistor.
If you have n resistors, each with the same resistance R and maximum power Pr, the total power is
(Note: (I/R) is the voltage drop of each resistor.)
But Ptotal is not all that useful, as it is the power dissipated for a given voltage. Increasing the voltage means having to check all resistors again to ensure that their individual maximum power dissipation is not exceeded.
Power handling for resistors in parallel
Assuming all resistors are connected in parallel, across the power source of voltage E, with resistors of different resistance and power capacity:
(Note: E/R1 is the current dissipated by resistor R1)
The total power dissipated by the parallel network is simply the sum of the power dissipated for each resistor. However, you must ensure that for each resistor, the power dissipated (as calculated above) does not exceed the maximum rating for that resistor.
For a parallel circuit of n resistors, each with the same resistance, and a maximum power dissipation Pr, the total maximum dissipation is
Resistors in series/parallel combinations
Determine the resistance and power for each series or parallel group, then treat the result as a virtual replacement resistor. Continue to analyze the network until you are down to one replacement resistor. Check to make sure that no resistor has a power dissipation greater than the rated maximum.
ANY combination of resistors can be analyzed as combinations of series and parallel networks. If the diagram you start with does not show obvious networks of those types, try redrawing it so that it does.