Parallel Circuits

The parallel circuit is one in which several resistance are connected across one another in such away that one terminal of each is connected to form a junction point while the remaining ends are also joined to form another junction

1. Resistors in Parallel

Consider a parallel circuit shown in the Fig. 1

In the parallel connection shown, the three resistances R₁, R₂and R₃ are connected in parallel and combination is connected across a source of voltage 'V'.

In parallel circuit current passing through each resistance is different. Let total current drawn is say ' I ' as shown. There are three paths for this current, one through R₁, second through R₂ and third through R₃. Depending up on the values of R₁, R₂ and R₃the appropriate fraction of total current passes through them. These individual currents are shown as I_1, I₂and I₃.While the voltage across the tow ends of each resistances R₁, R₂ and R₃ is the same and equals the supply voltage V.

Fig. 1

       Now let us study current distribution. Apply Ohm's law to each resistance.
                                        V= I₁R₁ ,    V= I₂R₂    , V = I₃R₃
                                         I₁= V/R₁        , I₂= V/R₂          I₃= V/R₃
                                          I = I₁+ I₂+I₃ = V/R₁ + V/R₂ + V/R₃
= V{ 1/R₁ + 1/R₂ + 1/R₃ )              .................... (1)
       For overall circuit if Ohm's law is applied,
                                           V = I R_eq

I = V/R_eq ......................(2)

Where R_eq= Total or equivalent resistance of the circuit.

Comparing the tow equations,

Where R is the equivalent resistance of the parallel combination.

In general if 'n' resistance are connected in parallel,

1.1 Conductance (G) :

It is known that, 1/R = G (Conductance) hence,

.^.. G = G₁+G₂ +G₃ +.............+ G_n ......For parallel circuits

Important result :

Now if n = 2, tow resistance are in parallel then,

The formula is directly used hereafter, for tow resistance in parallel.

1.2 Characteristic of Parallel Circuits

1) The same potential difference gets across all the resistance in parallel.
2) The total current gets divided into the number of paths equal to the number of resistances in parallel. The total current is always sum of all the individual currents.
I = I₁+ I₂+I₃+ ..........+ I_n
3) The reciprocal of the equivalent resistance of a parallel circuit is equal to the sum of the reciprocal of the individual resistances.
4) The equivalent resistances is the smallest of all the resistances.
R< R₁ , R < R₂ ,.............., R < R_n
5) The equivalent conductance is the arithmetic addition of the individual conductances.
Key point : The equivalent resistance is smaller than the smallest of all the resistances connected in parallel.

2. Inductors in Parallel

Consider the Fig. 1(a). Tow inductors L₁ and L₂ are connected in parallel. The current flowing through L₁ and L₂ are i₁and i₂respectively. The voltage developed across L₁ and L₂ are V_L1 and V_L2 respectively. the equivalent circuits shown in Fig. 2(b).

Fig. 2

For inductor we have,

       For parallel combination,
                            V = V_L1 = V_L2 = and
                             i = i₁+ i₂

That means, reciprocal of equivalent inductance of the parallel combination is the sum of reciprocals of the individual inductances

For n inductances in parallel,

3. Capacitors in Parallel

Consider the Fig. 3(a). Tow capacitors C₁ and C₂ are connected in parallel. The current flowing through C₁ and C₂ are i₁and i₂ respectively and voltages developed across are V_C1 and V_C2 respectively.

The equivalent circuit is shown in the Fig. 3(b).