When resistors, can be connected in such a way that the same current flows in them, then they are said to be connected in series. The resistors are said to be connected in parallel if the potential difference is the same across each resistor.

For any combination of resistors in a circuit, we can always find a single resistor that can replace the combination and result in the same total current and potential difference.

For example, a string of light bulbs can be replaced by a single, chosen light bulb that can draw the same current and have the same potential difference between its terminals as the original string of bulbs. The resistance of this single resistor is called the equivalent resistance of the combination.

 

 

If the resistances are connected end-to-end, the same current flows through each resistance, as there is no alternative path. Then

where I is the current, V is the potential difference of the battery and R_S is the equivalent resistance of the combination.

Now,

V = V₁+V₂+V₃

IR_S = I₁R₁ + IR₂ + IR₃ Therefore,

R_S = R₁+ R₂ + R₃

 

 

If the resistance are in parallel, the potential difference across each is the same, but the current is not. Then,

I = I₁ + I₂ + I₃

Since charge is not accumulated at a point,

where R_p is the equivalent resistance of the combination. Therefore,

For the special case of only two resistors in parallel, the expression for the equivalent resistance takes on a particularly simple form, i.e.,

Next, consider the following two networks where the resistances are connected in series - parallel combinations.

 

 

The first network can be simplified by replacing the parallel combination of R₂ and R₃ with its equivalent resistance. This is then in series with R₁.

In the second network, the combination of R₂ and R₃ in series forms a simple parallel combination with R₁.

But, not all networks can be reduced to simple series - parallel combinations, and special methods are required.