# What is the equivalent resistor of different resistors in series or in parallel if they have a different wattage and different resistance?

What is equivalent resistor of two different resistors put in series or in parallel if they have a different wattage and different resistance?

In practice somtimes we have a resistor with $$\\frac{1}{4}\$$watt , $$\\frac{1}{2}\$$watt, $$\1\$$watt or etc.

Say that wattage of a resistor is $$\\frac{3}{4}\$$watt or $$\\frac{3}{5}\$$watt, or etc...

What is the equivalent resistor of a "$$\w_1\$$ watt $$\r_1\$$ resistance" and "$$\w_2\$$ watt $$\r_2\$$ resistance" resistor put in series or in parallel?

We know that if they have the same wattage $$\w\$$ watt and $$\r_1\$$, $$\r_2\$$ resistance then the equivalent resistor might be:

1. in series $$\w\$$ watt $$\r_1\$$ + $$\r_2\$$ resistance
2. in parallel $$\2w\$$ watt $$\\frac{1}{r_1}\$$ + $$\\frac{1}{r_2}\$$ resistance

Forget the wattage of the resistors to begin with, that's causing confusion. Do the analysis, combine resistors together first.

You can combine resistors in series or parallel. When in series the resistors add together. When in parallel, the calculation is a little more complex. Here are the two formulae you need.

When you combine resistance values together, the power doesn't come into it.

In practise, it's your job as a designer to ensure that you don't execeed the power rating for the resistor so you have to calculate the power dissipation of the resistor and ensure it doesn't exceed that stated for the resistor, so you need to work out the current through the resistor and what voltage is dropped across the resistor. You need to know Ohms Law and Power formula (R=V/I , W =VI), re-arrange, combine if necessary and apply.

If you have a problem with actual power exceeding the maximum power rating for the resistor then there are things you can do to get around that: you can use multiple resistors in series, the current through each is the same but the voltage is shared across the resistors, thereby reducing the actual power dissipated by each resistor.

Alternatively putting resistors in parallel will reduce the current but keep the voltage the same. But you have to ensure the the combined resistance of the multiple resistors (whether in series or parallel) achieves the value you want to achieve.

First of all resistance.

When resistors are connected in series, their resistances sum up.

$$R_S = R_1 + R_2$$

When resistors are connected in parallel, their conductances (Reciprocal of resistane) sum up.

$$R_P = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$$

What about power rating? well the total power dissipated is the sum of the power dissipated in the two resistors.

$$P_{Total} = P_{R1} +P_{R2}$$

Ideally, both resistors would operate at their maximum power, however in practice this usually can't happen. Usually, as the total power applied increases one resistor will reach it's power rating before the other.

The power dissipated in a resistor is defined by.

$$P=IV=I^2R=\frac{V^2}{R}$$

For series circuits it is easier to work in terms of current since all resistors in a series circuit see the same current. This allows us to make the following calculation for a series circuit.

$$\frac{P_{R1}}{P_{R2}} = \frac{I^2R_1}{I^2R_2} = \frac{R_1}{R_2}$$

We can now calculate the total power in a series circuit in terms of the power in each resistor individually.

$$P_{Total} = P_{R1} +P_{R2} = P_{R1} +\frac{P_{R2}}{P_{R1}}P_{R1} = P_{R1} \left(1+ \frac{R_2}{R_1}\right)$$

And similarly.

$$P_{Total} = P_{R2} +P_{R1} = P_{R2} +\frac{P_{R1}}{P_{R2}}P_{R2} = P_{R2} \left(1+ \frac{R_1}{R_2}\right)$$

The maximum total power in the series circuit is determined by the point at which at least one of the resistors reaches it's maximum power.

$$P_{MaxTotal} = \min\left(P_{MaxR1} \left(1+ \frac{R_2}{R_1}\right),P_{MaxR2} \left(1+ \frac{R_1}{R_2}\right)\right)$$

Similar calculations are possible for a parallel circuit, with the main difference being that all resistors see the same voltage rather than the same current. If I'm not mistaken this will result in.

$$P_{MaxTotal} = \min\left(P_{MaxR1} \left(1+ \frac{R_1}{R_2}\right),P_{MaxR2} \left(1+ \frac{R_2}{R_1}\right)\right)$$

Two 1 watt resistors in series can be regarded as a 2 watt resistor if the individual resistance values are the same. If the resistance values are different then this is not true.

If one resistance is much bigger than the other then the power rating of both combined in series tends towards the power rating of the higher resistance resistor because it will always have a bigger volt drop across it.

For resistors in parallel, the power rating of the combined parallel pair tends towards the one with the least resistance because this resistor always takes most of the current.

Only resistors of equal resistance can have their individual wattages added to yield the combined wattage.

In series, the current is the same through each resistor so you can use P = $I^2R$ for each resistor Ri, and of course the total resistance is the sum of the Ri. Power for each resistor must not exceed the rating for each resistor, but it's perfectly legitimate to mix wattages. For example, a 1/4-W 10 ohm resistor can handle 158mA but at that current, a 100 ohm resistor would dissipate 2.5W.

In parallel, the voltage is the same across each resistor so you can use P = $E^2/R$, and ensure that each resistor power rating is not exceeded. A 100 ohm 1/4W resistor can handle 5V, but a 10 ohm resistor would dissipate 2.5W. Of course the parallel resistance is found from the reciprocal of the sum of the reciprocals of each value.

Rx = $\frac{1}{1/R1+..+1/Ri}$

First and foremost, resistors have resistance.

Given a Resistor $R_1$ and another one $R_2$, the total resistance of both is

$$R_{series} = R_1 + R_2$$ $$R_{parallel} = R_1 || R_2 = \frac{R_1 R_2}{R_1 + R_2}$$

You can show that $R_{series} > R_{parallel}$: $$R_1 + R_2 > \frac{R_1 R_2}{R_1 + R_2}$$ $$(R_1 + R_2)² > R_1 R_2$$ $$R_1² + 2 R_1 R_2 + R_2² > R_1 R_2$$ $$R_1² + R_1 R_2 + R_2² > 0$$

The amount of power that they consume depends on the voltage that you apply or the current that runs through them.

$$R = \frac{U}{I}$$

$$P = U * I = R * I² = \frac{U²}{R}$$

Resistance voltage in parallel arms is equal. To find the current on the resistors, the voltage on the resistors can be divided by the resistances to find the currents on the individual resistors. The power value can be found as the product of the voltage and current on the resistor. If these separately calculated power values ​​are lower than the wattage of the resistor, there will be no problem. To summarize, there is no point in calculating 2 resistors. Wattage is calculated separately. If the calculated wattage does not exceed the wattage that the resistor can withstand, there will be no problem.