Saturday, August 25, 2012

Resistance in series & parallel

Resistance in Series and parallel

So far, we have discussed up to ohm’s law. By using ohm’s law we will discuss about the connected resistors.
There are two different methods to connect resistors

  1. Series connection
  2. Parallel connection

First we will discuss about the series connection of resistors,

Resistance in series:

In series connection, the current goes through each and every resistor is equal since there are no other parallel paths. As there is no parallel path, electrons have only a single path to flow. Therefore, the number of electrons flow through a one resistor during a specific time period must be equal to the number of electrons flow through in other series resistors.

But as the resistors are connected in series and voltage is given to the two ends of the combination (series connected resistor combination), voltage difference between a single resistor is not equal to the supply voltage. According to figure 4.1 we can see that the supply voltage is equal to the sum of the voltage between each resistor. The voltage between a resistor is depend on the value of the resistance. To calculate this voltage, we can use the ohm’s law. As the current is equal through every resistor, the voltage difference is directly proportional to the resistance of the resistor in a series connection.

                                                                                  figure 4.1                                                              
Applying ohm’s law to every resistor
V1 = IR1  ---------- (1)
V2 = IR2  ----------(2)
V3 = IR3  ----------(3)

And also we can write that,

V = V1 + V2 + V3

From (1),(2),(3)

V = IR1   + IR2  + IR3
V = I (R1 + R2 + R3) ---------(4)

If we consider about a one single resistor which can replace for this set of series connected resistor combination, we call it as the equivalent or total resistance. Here we consider about a single resistor with ‘R’ resistance as the equivalent resistance of this combination.

Then applying ohm’s law to the equivalent resistance,
V = IR ----------(5)

(4) = (5)
IR = I (R1 + R2 + R3)
R = R1 + R2 + R3

This can be apply to ‘n’ number of resistors in series
R = R1 + R2 + R3 + ………. + Rn

Resistance in parallel: 

In parallel connections we can see that the resistors are connected in such a way that voltage difference between each and every resistor is equal. In this case the current is divided. Current through every branch is differs according to the resistance in each branch. We can use the ohm’s law to find this values.

figure 4.2

From ohm’s law,
V = IR
I = (V/R) -------(A)

And also we can write that

I = I1  + I 2 + I3

Similar to the series connection, we define a equivalent resistance ‘R’

Now using (A)

V/R = (V/R1) + (V/R2) + (V/R3)
1/R = 1/R1 + 1/R2+ 1/R3

For n numbers of resistors in parallel

1/R = 1/R1 + 1/R2+ 1/R3 +………..+ 1/Rn

Special case:

When the resistance of every resistor is equal (say ‘r’ ) and n number of resistors are connected,

  • For series connections,    R =  nr
  • For parallel connections, R = r/n

Where R is the equivalent resistance.

  • For parallel resistance, it’s easy to use the following derived equation for two resistors

                                               R  = ( R1R2 / (R1+ R2) )

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