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.
This article will use "component" to refer to a two-terminal "object" that participates in the series/parallel networks.
The two preceding statements are equivalent, except for exchanging the role of voltage and current.
[1] Consider a very simple circuit consisting of four light bulbs and a 12-volt automotive battery.
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.
Total conductance of a series circuits of pure resistances, therefore, can be calculated from the following expression:
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.
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.
The formula is easily extended to any number of series coils with mutual coupling.
Some vehicles, such as trucks, have two 12 volt batteries in series to feed the 24-volt system.
If two or more components are connected in parallel, they have the same difference of potential (voltage) across their ends.
The same voltage is applied to all circuit components connected in parallel.
For N equal resistances in parallel, the reciprocal sum expression simplifies to:
The components divide the current according to their reciprocal resistances, so, in the case of two resistors,
is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors is simply:
The relations for total conductance and resistance stand in a complementary relationship: the expression for a series connection of resistances is the same as for parallel connection of conductances, and vice versa.
Inductors follow the same law, in that the total inductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:
If the inductors are situated in each other's magnetic fields, this approach is invalid due to mutual inductance.
If the polarity of one coil is reversed so that M is negative, then the parallel inductance is nearly zero or the combination is almost non-inductive.
It is assumed in the "tightly coupled" case M is very nearly equal to L. However, if the inductances are not equal and the coils are tightly coupled there can be near short circuit conditions and high circulating currents for both positive and negative values of M, which can cause problems.
Parallel-connected batteries were widely used to power the valve filaments in portable radios.
Some solar electric systems have batteries in parallel to increase the storage capacity; a close approximation of total amp-hours is the sum of all amp-hours of in-parallel batteries.
From Kirchhoff's circuit laws the rules for combining conductance can be deducted.
Two disposable zinc cells in series might power a flashlight or remote control at 3 volts; the battery pack for a hand-held power tool might contain a dozen lithium-ion cells wired in series to provide 48 volts.
Series circuits were formerly used for lighting in electric multiple units trains.
Series circuits for train lighting were superseded, first by motor-generators, then by solid state devices.
Series resistance can also be applied to the arrangement of blood vessels within a given organ.
The total resistance of this parallel arrangement is expressed by the following equation: 1/Rtotal = 1/Ra + 1/Rb + ... + 1/Rn.
Ra, Rb, and Rn are the resistances of the renal, hepatic, and other arteries respectively.
