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

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Series circuits are sometimes referred to as 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 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

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

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).

$${\displaystyle V=V_{1}+V_{2}+\dots +V_{n}=I\left(R_{1}+R_{2}+\dots +R_{n}\right)}$$

Resistance units

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

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

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:

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

For a special case of two conductances 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 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 ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$, ${\displaystyle M_{23}}$ 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