Editing Series and parallel circuits

{{Short description|Types of electrical circuits}}
{{Use dmy dates|date=July 2019|cs1-dates=y}}
{{Use list-defined references|date=December 2021}}
[[File:Series circuit.svg|thumb|A series circuit with a [[voltage source]] (such as a battery, or in this case a cell) and three resistance units]]

[[Terminal (electronics)|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 (electrical circuits)|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.<ref name="Resnick_1966"/><ref name="Smith_1966"/>

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 [[Duality (electrical circuits)|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 [[Topology (electrical circuits)|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 drop]]s across each component.<ref name="Resnick_1966"/> 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.<ref name="Resnick_1966"/>

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<span class="anchor" id="Zseries"></span><span class="anchor" id="Xseries"></span><span class="anchor" id="Yseries"></span><span class="anchor" id="Bseries"></span>==
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{{Electromagnetism|Network}}
'''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 [[Single point of failure|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<span class="anchor" id="Iseries"></span>===
<math display="block">I = I_1 = I_2 = \cdots = I_n</math>

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).
<math display="block">V = \sum_{i=1}^n V_i = I\sum_{i=1}^n R_i</math>

===Resistance units<span class="anchor" id="Rseries"></span>===
The total resistance of two or more resistors connected in series is equal to the sum of their individual resistances:
[[File:Resistors_in_series.svg|alt=This is a diagram of several resistors, connected end to end, with the same amount of current through each.|border|center|x100px]]
<math display="block">R = \sum_{i=1}^n R_i = R_1 + R_2 + R_3 \cdots + R_n.</math>
Here, the subscript ''s'' in {{math|''R''<sub>s</sub>}} denotes "series", and {{math|''R''<sub>s</sub>}} 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:
<math display="block">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}.</math>

For a special case of two conductances in series, the total conductance is equal to:
<math display="block">G = \frac{G_1 G_2}{G_1 + G_2}.</math>

===Inductors <span class="anchor" id="Lseries"></span>===
[[Inductor]]s follow the same law, in that the total [[inductance]] of non-coupled inductors in series is equal to the sum of their individual inductances:

[[File:Inductors in series.svg|A diagram of several inductors, connected end to end, with the same amount of current going through each.|border|center|x100px]]
<math display="block">L = \sum_{i=1}^n L_i = L_1 + L_2 + L_3 \cdots + L_n.</math>

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 <math>M_{12}</math>, <math>M_{13}</math>, <math>M_{23}</math> and <math>M_{21}</math>, <math>M_{31}</math> and <math>M_{32}</math>. There are also the three self-inductances of the three coils: <math>M_{11}</math>, <math>M_{22}</math> and <math>M_{33}</math>.

Therefore
<math display="block">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)</math>

By reciprocity, <math>M_{ij}</math> = <math>M_{ji}</math> 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<span class="anchor" id="Cseries"></span>===
{{See also|Capacitor#Networks}}

[[Capacitor]]s follow the same law using the reciprocals. The total [[capacitance]] of capacitors in series is equal to the reciprocal of the sum of the [[Multiplicative inverse|reciprocals]] of their individual capacitances:
[[File:Capacitors_in_series.svg|alt=A diagram of several capacitors, connected end to end, with the same amount of current going through each.|border|center|x100px]]
<math display="block">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}.</math>

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

===Switches===
Two or more [[switch]]es in series form a [[Logical conjunction|logical AND]]; the circuit only carries current if all switches are closed. See [[AND gate]].

===Cells and batteries===
A [[Battery (electricity)|battery]] is a collection of [[electrochemical cell]]s. 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<span class="anchor" id="Zparallel"></span><span class="anchor" id="Xparallel"></span><span class="anchor" id="Yparallel"></span><span class="anchor" id="Bparallel"></span><span class="anchor" id="Parallel_circuit_anchor"></span>==
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{{resistors inductors capacitors in series and parallel.svg}}

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 circuit laws#Kirchhoff's current law|Kirchhoff's current law]].

===Voltage<span class="anchor" id="Rparallel"></span>===
In a '''parallel circuit''', the voltage is the same for all elements.
<math display="block">V = V_1 = V_2 = \dots = V_n</math>

===Current===
The current in each individual resistor is found by [[Ohm's law]]. Factoring out the voltage gives
<math display="block">I = \sum_{i=1}^n I_i = V\sum_{i=1}^n{1\over R_i}.</math>

===Resistance units===
To find the total [[Electrical resistance|resistance]] of all components, add the [[Multiplicative inverse|reciprocals]] of the resistances <math>R_i</math> of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:
[[File:Resistors_in_parallel.svg|alt=A diagram of several resistors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">R = \left(\sum_{i=1}^n{1\over R_i}\right)^{-1} = \left({1\over R_1} + {1\over R_2} + {1\over R_3} + \dots + {1\over R_n}\right)^{-1}</math>

For only two resistances, the unreciprocated expression is reasonably simple:
<math display="block">R = \frac{R_1 R_2}{R_1 + R_2} .</math>

This sometimes goes by the mnemonic ''product over sum''.

For ''N'' equal resistances in parallel, the reciprocal sum expression simplifies to:
<math display="block">\frac{1}{R} = N \frac{1}{R}.</math>
and therefore to:
<math display="block">R = \frac{R}{N}.</math>

To find the [[current (electricity)|current]] in a component with resistance <math>R_i</math>, use Ohm's law again:
<math display="block">I_i = \frac{V}{R_i}\,.</math>

The components divide the current according to their reciprocal resistances, so, in the case of two resistors,
<math display="block">\frac{I_1}{I_2} = \frac{R_2}{R_1}.</math>

An old term for devices connected in parallel is ''multiple'', such as multiple connections for [[arc lamp]]s.

==== Conductance ====
Since electrical conductance <math>G</math> is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors is simply:
<math display="block">G = \sum_{i=1}^n G_i = G_1 + G_2 + G_3 \cdots + G_n.</math>

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<span class="anchor" id="Lparallel"></span>===
[[Inductor]]s 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:
[[File:Inductors_in_parallel.svg|alt=A diagram of several inductors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">L = \left(\sum_{i=1}^n{1\over L_i}\right)^{-1} = \left({1\over L_1} + {1\over L_2} + {1\over L_3} + \dots + {1\over L_n}\right)^{-1}.</math>

If the inductors are situated in each other's magnetic fields, this approach is invalid due to mutual inductance. If the mutual inductance between two coils in parallel is {{mvar|M}}, the equivalent inductor is:
<math display="block">L = \frac{L_1L_2 - M^2}{L_1 + L_2 - 2M}</math>

If <math>L_1 = L_2</math>
<math display="block"> L = \frac{L + M}{2}</math>

The sign of <math>M</math> depends on how the magnetic fields influence each other. For two equal tightly coupled coils the total inductance is close to that of every single coil. If the polarity of one coil is reversed so that {{mvar|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 {{mvar|M}} is very nearly equal to {{mvar|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 {{mvar|M}}, which can cause problems.

More than three inductors become more complex and the mutual inductance of each inductor on each other inductor and their influence on each other must be considered. For three coils, there are three mutual inductances <math>M_{12}</math>, <math>M_{13}</math> and <math>M_{23}</math>. This is best handled by matrix methods and summing the terms of the inverse of the <math>L</math> matrix (3×3 in this case).

The pertinent equations are of the form:
<math display="block">v_{i} = \sum_{j} L_{i,j} \frac{di_j}{dt} </math>

===Capacitors<span class="anchor" id="Cparallel"></span>===
The total [[capacitance]] of [[capacitors]] in parallel is equal to the sum of their individual capacitances:
[[File:Capacitors_in_parallel.svg|alt=A diagram of several capacitors, side by side, both leads of each connected to the same wires.|border|center|x120px]]
<math display="block">C = \sum_{i=1}^n C_i = C_1 + C_2 + C_3 \cdots + C_n.</math>

The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.

===Switches===
Two or more [[switch]]es in parallel form a [[logical disjunction|logical OR]]; the circuit carries current if at least one switch is closed. See [[OR gate]].

===Cells and batteries===
If the cells of a battery are connected in parallel, the battery voltage will be the same as the cell voltage, but the current supplied by each cell will be a fraction of the total current. For example, if a battery comprises four identical cells connected in parallel and delivers a current of 1 [[ampere]], the current supplied by each cell will be 0.25 ampere. If the cells are not identical in voltage, cells with higher voltages will attempt to charge those with lower ones, potentially damaging them.

Parallel-connected batteries were widely used to power the [[Vacuum tube|valve]] filaments in [[portable radio]]s. Lithium-ion rechargeable batteries (particularly laptop batteries) are often connected in parallel to increase the ampere-hour rating. 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.

==Combining conductances<span class="anchor" id="Gparallel"></span><span class="anchor" id="Gseries"></span>==
From [[Kirchhoff's circuit laws]] the rules for combining conductance can be deducted. For two conductances <math>G_1</math> and <math>G_2</math> in ''parallel'', the voltage across them is the same and from Kirchhoff's current law (KCL) the total current is
<math display="block">I = I_1 + I_2.</math>

Substituting Ohm's law for conductances gives
<math display="block">G V = G_1 V + G_2 V</math>
and the equivalent conductance will be,
<math display="block">G = G_1 + G_2.</math>

For two conductances <math>G_1</math> and <math>G_2</math> in series the current through them will be the same and Kirchhoff's Voltage Law says that the voltage across them is the sum of the voltages across each conductance, that is,
<math display="block">V = V_1 + V_2.</math>

Substituting Ohm's law for conductance then gives,
<math display="block">\frac{I}{G} = \frac{I}{G_1} + \frac{I}{G_2}</math>
which in turn gives the formula for the equivalent conductance,
<math display="block">\frac{1}{G} = \frac{1}{G_1} + \frac{1}{G_2}.</math>

This equation can be rearranged slightly, though this is a special case that will only rearrange like this for two components.

<math display="block">G = \frac{G_1 G_2}{G_1 + G_2}.</math>
For three conductances in series,
<math display="block">G = \frac{G_1 G_2 G_3}{G_1 G_2 + G_1 G_3 + G_2 G_3}.</math>

==Notation==
The value of two components in parallel is often represented in equations by the [[parallel (operator)|parallel operator]], two vertical lines (∥), borrowing the [[Parallel (geometry)#Symbol|parallel lines notation from geometry]].
<math display="block">R \equiv R_1 \parallel R_2 \equiv \left(R_1^{-1} + R_2^{-1}\right)^{-1} \equiv \frac{R_1 R_2}{R_1 + R_2}</math>

This simplifies expressions that would otherwise become complicated by expansion of the terms.  For instance:
<math display="block">R_1 \parallel R_2 \parallel R_3 \equiv \frac{R_1 R_2 R_3}{R_1 R_2 + R_1 R_3 + R_2 R_3} . </math>

==Applications==
A common application of series circuit in consumer electronics is in batteries, where several cells connected in series are used to obtain a convenient operating voltage. 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.  For example, if the supply voltage was 600 volts there might be eight 70-volt bulbs in series (total 560 volts) plus a [[resistor]] to drop the remaining 40 volts.  Series circuits for train lighting were superseded, first by [[motor-generator]]s, then by [[Solid state (electronics)|solid state]] devices.

Series resistance can also be applied to the arrangement of blood vessels within a given organ.  Each organ is supplied by a large artery, smaller arteries, arterioles, capillaries, and veins arranged in series. The total resistance is the sum of the individual resistances, as expressed by the following equation: {{math|1=''R''<sub>total</sub> = ''R''<sub>artery</sub> + ''R''<sub>arterioles</sub> + ''R''<sub>capillaries</sub>}}. The largest proportion of resistance in this series is contributed by the arterioles.<ref name="BRS"/>

Parallel resistance is illustrated by the [[circulatory system]]. Each organ is supplied by an artery that branches off the [[aorta]]. The total resistance of this parallel arrangement is expressed by the following equation: {{math|1=1/''R''<sub>total</sub> = 1/''R''<sub>a</sub> + 1/''R''<sub>b</sub> + ... + 1/''R''<sub>n</sub>}}. {{math|''R''<sub>a</sub>}}, {{math|''R''<sub>b</sub>}}, and {{math|''R''<sub>n</sub>}} are the resistances of the renal, hepatic, and other arteries respectively. The total resistance is less than the resistance of any of the individual arteries.<ref name="BRS"/>

==See also==
{{div col}}
* [[Anti-parallel (electronics)]]
* [[Electrical impedance#Combining impedances|Combining impedances]]
* [[Current divider]]
* [[Equivalent impedance transforms]]
* [[Hydraulic analogy]] 
* [[Network analysis (electrical circuits)]]
* [[Resistance distance]]
* [[Series-parallel duality]]
* [[Series-parallel partial order]]
* [[Series and parallel springs]]
* [[Topology (electrical circuits)]]
* [[Voltage divider]]
* [[Wheatstone bridge]]
* [[Y-Δ transform]]
{{div col end}}

==References==
{{Reflist|refs=
<ref name="Resnick_1966">{{cite book |author-last1=Resnick |author-first1=Robert |author-last2=Halliday |author-first2=David |date=1966 |title=Physics |volume=I and II |edition=Combined international |publisher=[[Wiley (publisher)|Wiley]] |lccn=66-11527 |chapter=Chapter 32 |id=Example 1 }}</ref>
<ref name="Smith_1966">{{cite book |author-last=Smith |author-first=R. J. |date=1966 |title=Circuits, Devices and Systems |publisher=[[Wiley (publisher)|Wiley]] |edition=International |location=New York |lccn=66-17612 |page=21 }}</ref>
<ref name="BRS">{{cite book |series=Board Review Series |title=Physiology |author-first=Linda S. |author-last=Costanzo |page=74 }}</ref>
}}

==Further reading==
* {{cite book |author-last=Williams |author-first=Tim |title=The Circuit Designer's Companion |publisher=[[Butterworth-Heinemann]] |date=2005 |isbn=0-7506-6370-7 }}
* {{cite magazine |url=http://www.edn.com/design/components-and-packaging/4421194/Resistor-combinations--How-many-values-using-1kohm-resistors-- |title=Resistor combinations: How many values using 1K ohm resistors? |magazine=[[EDN magazine]] }}
* {{cite book |chapter-url=https://www.grund-wissen.de/physik/mechanik/festkoerper-fluessigkeiten-gase/fluessigkeiten.html |title=Mechanik der Flüssigkeiten |chapter=Strömungswiderstand |language=de |date=2018-01-04 |author-first=Bernhard |author-last=Grotz }}<!-- analog parallel and serial circuits. Find English source and add to article -->

[[Category:Electrical circuits]]