3.10 Combination of Resistors - Series and Parallel

The current through a single resistor R across which there is a potential difference V is given by Ohm’s law I = V/R. Resistors are sometimes joined together and there are simple rules for calculation of equivalent resistance of such combination.

3.10.1 Series Combination

FIGURE 3.13 A series combination of two resistors R1 and R2.

Two resistors are said to be in series if only one of their end points is joined (Fig. 3.13). If a third resistor is joined with the series combination of the two (Fig. 3.14), then all three are said to be in series. Clearly, we can extend this definition to series combination of any number of resistors.

FIGURE 3.14 A series combination of three resistors R1, R2, R3.

Consider two resistors R1 and R2 in series. The charge which leaves R1 must be entering R2. Since current measures the rate of flow of charge, this means that the same current I flows through R1 and R2. By Ohm’s law:

Potential difference across R1 = V1 = IR1, and

Potential difference across R2 = V2 = IR2.

The potential difference V across the combination is V1+V2. Hence,

V = V1+ V2 = I (R1 + R2) (3.36)

This is as if the combination had an equivalent resistance R eq , which by Ohm’s law is

R eq = V/I = (R1 + R2) (3.37)

If we had three resistors connected in series, then similarly

V = I R1 + I R2 + I R3 = I (R1+ R2+ R3). (3.38)

This obviously can be extended to a series combination of any number n of resistors R1, R2....., Rn. The equivalent resistance Req is

Req = R1 + R2 + .. .. + Rn (3.39)

3.10.2 Parallel Combination

Two or more resistors are said to be in parallel if one end of all the resistors is joined together and similarly the other ends joined together (Fig. 3.15).

FIGURE 3.15 Two resistors R1 and R2 connected in parallel.

Consider now the parallel combination of two resistors (Fig. 3.15). The charge that flows in at A from the left flows out partly through R1 and partly through R2. The currents I, I1, I2 shown in the figure are the rates of flow of charge at the points indicated. Hence,

I = I1 + I2 .. .. (3.40)

The potential difference between A and B is given by the Ohm’s law applied to R1

V = I1 R1 .. .. (3.41)

Also, Ohm’s law applied to R2 gives

V = I2 R2 .. .. (3.42)

∴ I = I1 + I2 = V/R1 + V/R2 = V(1/R1 + 1/R2) (3.43)

If the combination was replaced by an equivalent resistance Req , we would have, by Ohm’s law

I = V/Req (3.44)

Hence, 1/ R eq = 1/R1 +1/ R2 .. .. (3.45)

We can easily see how this extends to three resistors in parallel (Fig. 3.16).

FIGURE 3.16 Parallel combination of three resistors R1, R2 and R3.

Exactly as before

I = I1 + I2 + I3 .. .. (3.46)

and applying Ohm’s law to R1, R2 and R3 we get,

V = I1 R1, V = I2 R2, V = I3 R3 .. .. (3.47)

So that I = I 1+ I2 + I3 = V(1/R1 + 1/R2 + 1/R3) .. .. (3.48)

An equivalent resistance Req that replaces the combination, would be such that

I = V/Req .. .. (3.49)

and hence 1/R eq = 1/R1 + 1/R2 + 1/R3 .. .. (3.50)

We can reason similarly for any number of resistors in parallel. The equivalent resistance of n resistors R1, R2 .. .. ,Rn is

1/Req= 1/R1 + 1/R2 +... + 1/Rn (3.51

These formulae for equivalent resistances can be used to find out currents and voltages in more complicated circuits. Consider for example, the circuit in Fig. (3.17), where there are three resistors R1,
R2 and R3.

FIGURE 3.17 A combination of three resistors R1, R2 and R3. R2, R3 are in parallel with an equivalent resistance Req^ 23. R1 and Req^ 23 are in series with an equivalent resistance Req^ 123.

R2 and R3 are in parallel and hence we can replace them by an equivalent Req^23 between point B and C with

1/Req^23 = 1/R2 +1/R3

or, Req^23 = R2R3/(R2 + R3) (3.52)

The circuit now has R1 and Req^23 in series and hence their combination can be replaced by an equivalent resistance with

Req^123 = Req^23 + R1 (3.53)

If the voltage between A and C is V, the current I is given by

I = V/Req^123 = V/R1 + [R2R3/(R2 + R3)]

= V × (R2 + R3)/(R1R2 + R1R3 + R2R3) (3.54)