# Series and parallel circuits

A series circuit with a voltage source (such as a battery, or in this case a cell) and 3 resistors

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single path, so the same current flows through all of the components.[1][2] Components connected in parallel are connected along multiple paths, so the same voltage is applied to each component.[3]

A circuit composed solely of components connected in series is known as a series circuit; likewise, one connected completely in parallel is known as a parallel circuit.

In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component.[1] In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.[1]

Consider a very simple circuit consisting of four light bulbs and one 6 V battery. If a wire joins the battery to one bulb, to the next bulb, to the next 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 four light bulbs are connected in series, there is the same current through all of them, and the voltage drop is 1.5  V across each bulb, which may not be sufficient to make them glow. If the light bulbs are connected in parallel, the currents through the light bulbs combine to form the current in the battery, while the voltage drop is across each bulb and they all glow.

In a series circuit, every device must function for the circuit to be complete. One bulb burning out in a series circuit breaks the circuit. In parallel circuits, each light bulb has its own circuit, so all but one light could be burned out, and the last one will still function.

## Series circuits

Series circuits are sometimes called current-coupled or daisy chain-coupled. The current in a series circuit goes through every component in the circuit. Therefore, all of the components in a series connection carry the same current.

A series circuit's principal characteristic is that it has only one path in which its current can flow. Opening or breaking a series circuit at any point causes the entire circuit to "open" or stop operating. For example, if even one of the light bulbs in an older-style string of Christmas tree lights burns out or is removed, the entire string becomes inoperable until the bulb is replaced.

### Current

${\displaystyle I=I_{1}=I_{2}=I_{3}=\cdots =I_{n}}$

In a series circuit, the current is the same for all of the elements.

### Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

${\displaystyle R_{\text{total}}=R_{\text{s}}=R_{1}+R_{2}+\cdots +R_{n}}$

Rs=>Resistance in series

Electrical conductance presents a reciprocal quantity to resistance. Total conductance of a series circuits of pure resistors, therefore, can be calculated from the following expression:

${\displaystyle {\frac {1}{G_{\mathrm {total} }}}={\frac {1}{G_{1}}}+{\frac {1}{G_{2}}}+\cdots +{\frac {1}{G_{n}}}}$.

For a special case of two resistors in series, the total conductance is equal to:

${\displaystyle G_{\text{total}}={\frac {G_{1}G_{2}}{G_{1}+G_{2}}}.}$

### 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:

${\displaystyle L_{\mathrm {total} }=L_{1}+L_{2}+\cdots +L_{n}}$

However, in some situations, it is difficult to prevent adjacent inductors from influencing each other, as the magnetic field of one device coupled with the windings of its neighbours. This influence is defined by the mutual inductance M. For example if two inductors are in series, there are two possible equivalent inductances 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 ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$, ${\displaystyle M_{23}}$ and ${\displaystyle M_{21}}$, ${\displaystyle M_{31}}$ and ${\displaystyle M_{32}}$. There are also the three self-inductances of the three coils: ${\displaystyle M_{11}}$, ${\displaystyle M_{22}}$ and ${\displaystyle M_{33}}$.

Therefore

${\displaystyle L_{\mathrm {total} }=(M_{11}+M_{22}+M_{33})+(M_{12}+M_{13}+M_{23})+(M_{21}+M_{31}+M_{32})}$

By reciprocity ${\displaystyle M_{ij}}$ = ${\displaystyle M_{ji}}$ 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 mutual 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 the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

${\displaystyle {\frac {1}{C_{\mathrm {total} }}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\cdots +{\frac {1}{C_{n}}}}$.

### Switches

Two or more switches in series form a logical AND; the circuit only carries current if all switches are closed. See AND gate.

### Cells and batteries

A battery is a collection of electrochemical cells. If the cells are connected in series, the voltage of the battery will be the sum of the cell voltages. For example, a 12 volt car battery contains six 2-volt cells connected in series. Some vehicles, such as trucks, have two 12 volt batteries in series to feed the 24-volt system.

### Voltage

In a series circuit the voltage is addition of all the voltage elements.

${\displaystyle V=V_{1}+V_{2}+V_{3}+\dots +V_{n}}$