Resistors in Series

As shown above, when the n- resistors are connected in series connection, then the total resistance will be the summation of individual resistance.

Proof : 

Let’s say, three resistors are connected in series with one voltage source V. And current flowing through the circuit is I.

If V1, V2 and V3 are the voltage drop across each element then applying the KVL in the loop,

V – V1 – V2 – V3 = 0

⇒  V = V1 + V2 + V3

⇒ V = (I x R1 + I x R2 + I x R3)

⇒ V = I (R1 + R2 + R3) —— (1)

Let’s say, Req is the equivalent resistance of the three resistors.

If three resistors are replaced by the equivalent resistance, then the equivalent circuit can be represented as follows:

Applying the  Kirchhoff’s Voltage Law (KVL) for the equivalent circuit,

V – (I x Req ) = 0

⇒ V = I x Req —— (2)

Equating the equation (1) and (2),

Req = R1 + R2 +R3

Or in general, when n- number of resistors are connected in series then the total or equivalent resistance can be given as

Req = R1 + R2 + R3 + —– + Rn

Resistors in Parallel :

The resistors are said to be connected in parallel when their terminals are connected to the same node.

For example, as shown in the figure, the three resistors are connected in the parallel, as one of their terminals is connected to the node A, while the second terminals are connected to the node B.

If n- resistors are connected in parallel then the equivalent resistance can be given by the following expression.

Proof:

As shown in the figure, three resistors are connected in parallel with one voltage source V.

Here, the current through each resistor will be different. But as they are connected in parallel, the voltage across each resistor will be the same.

If V1, V2 and V3 is the voltage across the resistor R1, R2 and R3 respecitvely then

V = V1 = V2 = V3 ——(1)

If the I is the total current supplied by the voltage source, then applying the KCL at the top node,

I = I1 +I2 + I3 ——- (2)

Now, according to the Ohm’s law,

I1 = V1/ R1

I2 = V2 / R2

I3 = V3 / R3

Using, equation (1) and (2)

I = V x  { (1 / R1) + (1/ R2) + (1/ R3) } ——— (3)

If three resistors are replaced by the equivalent resistance then the equivalent circuit can be represented as follows:

∴ V = I x Req

⇒ I = V / Req  ———(4)

Comparing equation (3) and (4)

1 / Req = { (1 / R1) + (1/ R2) + (1/ R3) }

Or, in general, when n- resistors are connected in parallel, then equivalent resistance Req can be given by the following expression.

1 / Req = { (1 / R1) + (1/ R2) + (1/ R3) + —–  + (1/ Rn)}

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