SERIES AND PARALLEL CIRCUITS

Electrical circuit components can be connected together in one of two ways: 'series' or 'parallel'. These two names describe the method of attaching the components, that is one after the other or next to each other. If two or more circuit components are connected end to end like a daisy chain, it is said they are connected in ''series''. A series circuit is a single path for current through all of its components. If two or more circuit components are connected like the rungs of a ladder it is said they are connected in ''parallel''. A parallel circuit is a different path for current through each of its components. A parallel circuit provides the same voltage across all its components.
As an example, consider a very simple circuit consisting of two lightbulbs and one 9 V battery. If a wire joins the battery to one bulb, to the next bulb, then back to the battery, in one continuous loop, the bulbs are said to be in series. If each bulb is wired to the battery in a separate loop, the bulbs are said to be in parallel. If the two lightbulbs are connected in series the same current flows in both of them; each lightbulb experiences about 4.5 V. If the two lightbulbs are connected in parallel the currents flowing in the two lightbulbs combine to form the current flowing in the battery; each lightbulb experiences 9 V.

Series circuits

Series circuits are sometimes called ''cascade''-coupled or daisy chain-coupled. The current that flows in a series circuit has to flow through every component in the circuit. Therefore, all components in a series connection carry the same current.

Resistors

To find the total resistance of all the components, add the individual resistances of each component:

A diagram of several resistors, connected end to end, with the same amount of current going through each.

R_mathrm{total} = R_1 + R_2 + cdots + R_n

for components in series with resistances

R_1

R_2

, etc. To find the current

I

, use Ohm's law:
:

I = rac{V}{R_mathrm{total}}

.
To find the voltage across a component with resistance

R_i

, use Ohm's law again:
:

V_i = IR_i ,

where

I

is the current, as calculated above. The components divide the voltage according to their resistances, so, in the case of two resistors,
:

rac{V_1}{V_2} = rac{R_1}{R_2}

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

A diagram of several inductors, connected end to end, with the same amount of current going through each.

L_mathrm{total} = L_1 + L_2 + cdots + L_n

However, in some situations it is difficult to prevent adjacent inductors from influencing each other, as the magnetic field of one device couples with the windings of its neighbours. This influence is defined by the mutual inductance M. For example, if you have two inductors in series, there are two possible equivalent inductances:
:

L_mathrm{total} = (L_1 + M) + (L_2 + M),

or
:

L_mathrm{total} = (L_1 - M) + (L_2 - M),

,
depending on how the magnetic fields of both inductors influence each other.
When there are more than two inductors, the mutual inductance between each of them and the way the coils influence each other complicates the calculation. For a larger number of coils the total combined inductance is given by the sum of all mutual inductances between the various coils including the mutual inductance of each given coil with itself, which we term self-inductance or simply inductance. For three coils, there are six mutual inductances

M_{12}

M_{13}

M_{23}

and

M_{21}

M_{31}

and

M_{32}

. There are also the three self-inductances of the three coils:

M_{11}

M_{22}

and

M_{33}

.
Therefore
:

L_mathrm{total} = (M_{11} + M_{22} + M_{33}) + (M_{12} + M_{13} + M_{23}) + (M_{21} + M_{31} + M_{32})

By reciprocity

M_{ij}

M_{ji}

so that the last two groups can be combined. The first three terms represent the sum of the self-inductances of the various coils. The formula is easily extended to any number of series coils with mututal coupling. The method can be used to find the self-inductance of large coils of wire of any cross-sectional shape by computing the sum of the mutual inductance of each turn of wire in the coil with every other turn since in such a coil all turns are in series.

Capacitors

Capacitors follow a different law. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

A diagram of several capacitors, connected end to end, with the same amount of current going through each.

rac{1}{C_mathrm{total}} = rac{1}{C_1} + rac{1}{C_2} + cdots + rac{1}{C_n}

.
The working voltage of a series combination of identical capacitors is equal to the sum of voltage ratings of individual capacitors provided that equalizing resistors are used to ensure equal voltage division. This is all because of Ohm's law.

Parallel circuits

If two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. Hence, the same voltage is applicable to all circuit components connected in parallel. The total current ''I'' is the sum of the currents through the individual components, in accordance with Kirchhoff's Current Law. The current in each individual resistor is found by Ohm's Law. Factoring out the voltage gives
:

I_mathrm{total} = Vleft(rac{1}{R_1} + rac{1}{R_2} + cdots + rac{1}{R_n}
ight)

Notation

The parallel property can be represented in equations by two vertical lines

parallel

(as in geometry) to simplify the equations.