Series and parallel circuits

Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and itself can participate in a series or parallel topology. Whether a two-terminal "object" is an electrical component (e.g. a resistor) or an electrical network (e.g. resistors in series) is a matter of perspective. This article will use "component" to refer to a two-terminal "object" that participates in the series/parallel networks.

Components connected in series are connected along a single "electrical path", and each component has the same electric current through it, equal to the current through the network. The voltage across the network is equal to the sum of the voltages across each component.^[1]^[2]

Components connected in parallel are connected along multiple paths, and each component has the same voltage across it, equal to the voltage across the network. The current through the network is equal to the sum of the currents through each component.

The two preceding statements are equivalent, except for exchanging the role of voltage and current.

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. Many circuits can be analyzed as a combination of series and parallel circuits, along with other configurations.

In a series circuit, the current that flows through each of the components is the same, and the voltage across the circuit is the sum of the individual voltage drops 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 flowing through each component.^[1]

Consider a very simple circuit consisting of four light bulbs and a 12-volt automotive 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, the same current flows through all of them and the voltage drop is 3 volts 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 12 volts across each bulb and they all glow.

In a series circuit, every device must function for the circuit to be complete. If one bulb burns out in a series circuit, the entire circuit is broken. 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 referred to as current-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 has only one path through 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 faulty bulb is replaced.

Current

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

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

Voltage

In a series circuit, the voltage is the sum of the voltage drops of the individual components (resistance units). $V=\sum _{i=1}^{n}V_{i}=I\sum _{i=1}^{n}R_{i}$

Resistance units

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

$R=\sum _{i=1}^{n}R_{i}=R_{1}+R_{2}+R_{3}\cdots +R_{n}.$ Here, the subscript s in $R s$ denotes "series", and $R s$ denotes resistance in a series.

Conductance

Electrical conductance presents a reciprocal quantity to resistance. Total conductance of a series circuits of pure resistances, therefore, can be calculated from the following expression: $G=\left(\sum _{i=1}^{n}{1 \over G_{i}}\right)^{-1}=\left({1 \over G_{1}}+{1 \over G_{2}}+{1 \over G_{3}}+\dots +{1 \over G_{n}}\right)^{-1}.$

For a special case of two conductances in series, the total conductance is equal to: $G={\frac {G_{1}G_{2}}{G_{1}+G_{2}}}.$

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:

$L=\sum _{i=1}^{n}L_{i}=L_{1}+L_{2}+L_{3}\cdots +L_{n}.$

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 neighbors. 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 is termed 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 $L=\left(M_{11}+M_{22}+M_{33}\right)+\left(M_{12}+M_{13}+M_{23}\right)+\left(M_{21}+M_{31}+M_{32}\right)$

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 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:

$C=\left(\sum _{i=1}^{n}{1 \over C_{i}}\right)^{-1}=\left({1 \over C_{1}}+{1 \over C_{2}}+{1 \over C_{3}}+\dots +{1 \over C_{n}}\right)^{-1}.$

Equivalently using elastance (the reciprocal of capacitance), the total series elastance equals the sum of each capacitor's elastance.

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.

Parallel circuits

If two or more components are connected in parallel, they have the same difference of potential (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applied to all circuit components connected in parallel. The total current is the sum of the currents through the individual components, in accordance with Kirchhoff's current law.