More than one resistances in series and parallel can be connected, in addition to that more than two resistances in series and parallel can also be connected in combination. Resistors can be connected together in an unlimited number of series and parallel combinations to form complex resistive circuits.

But what if we want to connect various resistors together in “BOTH” parallel and series combinations within the same circuit to produce more complex resistive networks, how do we calculate the combined or total circuit resistance, currents and voltages for these resistive combinations.

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# Resistances in Series and Parallel Combinations

Resistor circuits that combine series and parallel resistors networks together are generally known as **Resistor Combination** or mixed resistor circuits.

The method of calculating the circuits equivalent resistance is the same as that for any individual series or parallel circuit and hopefully we now know that resistors in series carry exactly the same current and that resistors in parallel have exactly the same voltage across them. Here we will discuss mainly about series and parallel combination.

## Resistances in Series

Suppose you have, three resistors, R_{1}, R_{2} and R_{3} and you connect them end to end as shown in the figure below, then it would be referred as **resistances in series**.

In case of series connection, the equivalent resistance of the combination, is sum of these three electrical resistances. That means, resistance between point A and D in the figure below, is equal to the sum of three individual resistances.

The current enters in to the point A of the combination, will also leave from point D as there is no other parallel path provided in the circuit.

Now say this current is I. So this current I will pass through the resistance R_{1}, R_{2} and R_{3}. Applying Ohm’s law, it can be found that voltage drops across the resistances will be V_{1} = IR_{1}, V_{2} = IR_{2} and V_{3} = IR_{3}. Now, if total voltage applied across the combination of **resistances in series**, is V. Then obviously

Since, sum of voltage drops across the individual resistance is nothing but the equal to applied voltage across the combination.

Now, if we consider the total combination of resistances as a single resistor of electric resistance value R, then according to Ohm’s law, V = IR ………….(2)

Now, comparing equation (1) and (2), we get

So, the above proof shows that equivalent resistance of a combination of resistances in series is equal to the sum of individual resistance. If there were n number of resistances instead of three resistances, the equivalent resistance will be

## Resistances in Parallel

Let’s three resistors of resistance value R_{1}, R_{2} and R_{3} are connected in such a manner, that right side terminal of each resistor are connected together as shown in the figure below, and also left side terminal of each resistor are also connected together.

This combination is called **resistances in parallel**. If electric potential difference is applied across this combination, then it will draw a current I (say).

As this current will get three parallel paths through these three electrical resistances, the current will be divided into three parts. Say currents I_{1}, I_{1} and I_{1} pass through resistor R_{1}, R_{2} and R_{3} respectively.

Where total source current

Now, as from the figure it is clear that, each of the **resistances in parallel**, is connected across the same voltage source, the voltage drops across each resistor is same, and it is same as supply voltage V (say). Hence, according to Ohm’s law,

Now, if we consider the equivalent resistance of the combination is R. Then,

Now putting the values of I, I_{1}, I_{2} and I_{3} in equation (1) we get,

The above expression represents equivalent resistance of resistor in parallel. If there were n number of resistances connected in parallel, instead of three resistances, the expression of equivalent resistance would be

### Resistances in Series and Parallel Combinations Conclusion

After going through the above portion of resistances in series and parallel we can now establish a resistances in series and parallel combinations. I hope you enjoy when reading this article, thank you.