CHAPTER 5 RESISTANCES IN SERIES AND PARALLEL

5.1       SERIES CIRCUITS

The term ‘in series’ means that two or more circuits are supplied one after the other in any single circuit, as shown diagrammatically in Figure 5.1.

Figure 5.1 SERIES RESISTANCES

Since there is only one single path from the power source through the circuits and back again, the same current flows through all.  The voltage, or ‘pressure’, is reduced by resistance according to Ohm’s Law: each circuit element causes a ‘voltage drop’ across it, very similar to the ‘loss of head’ due to fluid flow in a hydraulic system.  Also the sum of the individual volt-drops is equal to the applied voltage.

By Ohm’s Law the volt-drop V₁ across the load R₁ is

                                                                 V₁ =I x R₁

and similarly                                            V₂ = I x R₂, I being the same for each of the cases.

Therefore:                              V₁ + V₂ + …. = IR₁ + IR₂  + ….

                                                                     = I(R₁ + R₂  + .… )

But the sum of the individual volt-drops V₁ + V₂ + …. = V (the applied voltage).

By Ohm’s Law for the equivalent circuit  V = IR.

Hence:                                                   IR =I(R₁+R₂+ ....)

                                           or                   R =R₁+R₂+ ....

 

That is to say, the total resistance of a series circuit is equal to the sum of all the individual resistances.

It is evident that the failure of any single component in a series circuit interrupts the supply to all; also that each element of load must work at a reduced voltage.  For these reasons the series arrangement of loads is seldom used in power circuits.

5.2       PARALLEL CIRCUITS

The term ‘in parallel’ means that the circuits are so arranged that there is a separate path through each, as shown in Figure 5.2.

Figure 5.2 PARALLEL RESISTANCES

The voltage applied to every circuit element is the same throughout.  The total current divides between the circuits according to the resistance of each element, so that the current flowing through each individual circuit is less than the total, and the sum of the currents flowing through the individual elements is equal to the total available current.

By Ohm’s Law the current I₁ flowing through the load R₁ is:

I1	=	V
		R1

. and similarly

I2	=	V
		R2

                                          

But the sum of the currents flowing in each individual circuit

I1

+

I2

+

….

=

I  (the total current)

Hence:

I	=	V (	1	+	1	+	….)
			R1		R2

or	I	=	1	+	1	+	….
	V		R1		R2

But by Ohm’s Law for the equivalent circuit  V	=	IR	or	I	=	1
				V		R

\	1	=	1	+	1	+	….
	R		R1		R2

That is to say, the inverse of the equivalent resistance of a set of parallel circuits is equal to the sum of the inverses of each individual resistance.

It is evident that for the power engineer the parallel circuit has two important practical advantages.  First, the failure of any element of load has no effect on the rest; they continue to receive a supply at the correct voltage and to draw the current which each individually requires.  Second, all apparatus is supplied at the same voltage.  Consequently, the parallel circuit is used almost exclusively for power supply in industrial plant.

It should be observed in passing that the characteristic of the series circuit, in which resistances in series have the effect of reducing the voltage at different points of the circuit, finds wide practical application in electronic apparatus such as radio, control, and ‘solid-state’ measuring equipment.

5.3       SUMMARY

If R is the single resistance equivalent to a number of individual resistances R₁, R₂, R₃, …, then:

(a)      if R₁, R₂ etc. are in series,

                                                                     R  =   R₁   +   R₂   +   R₃   +   .…

(b)      if R₁, R₂ etc. are in parallel,

1	=	1	+	1	+	1	+	….
R		R1		R2		R3