Series and parallel circuits: Difference between revisions

Revision as of 17:47, 30 January 2023

Revision as of 16:48, 22 February 2023

Line 2:

{{Use dmy dates|date=July 2019|cs1-dates=y}}

{{Use list-defined references|date=December 2021}}

[[ImageFile:Series circuit.svg|right|thumb|A series circuit with a [[voltage source]] (such as a battery, or in this case a cell) and 3three 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 participate in the series/parallel networks.

Line 9 ⟶ 10:

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 [[dualityDuality (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 [[topologyTopology (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>==

=={{anchor|Zseries|Xseries|Yseries|Bseries}}Series circuits==

<!-- This"Series sectioncircuit" isredirects linkedhere. from [["Battery (electricity)]]" links here. -->

{{Electromagnetism|Network}}

'''Series circuits''' are sometimes referred to as current-coupled or [[Daisy chain (electrical engineering)|daisy chain]]-coupled. The [[electric 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.

Line 26 ⟶ 27:

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}}Current"></span>===

<math display="block">I = I_1 = I_2 = \cdots = I_n</math>

Line 35 ⟶ 36:

<math display="block">V = V_1 + V_2 + \dots + V_n = I \left( R_1 + R_2 + \dots + R_n \right)</math>

==={{anchor|Rseries}}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:

[[ImageFile:Resistors in series.svg|This is a diagram of several resistors, connected end to end, with the same amount of current through each.]]

<math display="block">R_\text{total} = R_\text{s} = R_1 + R_2 + \cdots + R_n.</math>

{{anchor|Lseries}}

Line 52 ⟶ 53:

[[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:

[[ImageFile:Inductors in series.svg|A diagram of several inductors, connected end to end, with the same amount of current going through each.]]

<math display="block">L_\mathrm{total} = L_1 + L_2 + \cdots + L_n</math>

Line 64 ⟶ 65:

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}}Capacitors"></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:

[[ImageFile:Capacitors in series.svg|A diagram of several capacitors, connected end to end, with the same amount of current going through each.]]

<math display="block">\frac{1}{C_\text{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n}.</math>

Line 84 ⟶ 86:

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 (KCL)|Kirchhoff's current law]].

===Voltage<span class="anchor" id="Rparallel"></span>Voltage===

In a parallel circuit, the voltage is the same for all elements.

<math display="block">V = V_1 = V_2 = \dots = V_n</math>

Line 95 ⟶ 97:

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:

[[ImageFile:Resistors in parallel.svg|A diagram of several resistors, side by side, both leads of each connected to the same wires.]]

<math display="block">\frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}.</math>

Line 121 ⟶ 123:

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>Inductors===

[[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:

[[ImageFile:Inductors in parallel.svg|A diagram of several inductors, side by side, both leads of each connected to the same wires.]]

<math display="block">\frac{1}{L_\text{total}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}.</math>

Line 140 ⟶ 142:

<math display="block">v_{i} = \sum_{j} L_{i,j} \frac{di_j}{dt} </math>

===Capacitors<span class="anchor" id="Cparallel"></span>Capacitors===

The total [[capacitance]] of [[capacitors]] in parallel is equal to the sum of their individual capacitances:

[[ImageFile:Capacitors in parallel.svg|A diagram of several capacitors, side by side, both leads of each connected to the same wires.]]

<math display="block">C_\text{total} = C_1 + C_2 + \cdots + C_n.</math>

Line 156 ⟶ 158:

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>==

=={{anchor|Gparallel|Gseries}}Combining conductances==

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_\text{eq} = I_1 + I_2.</math>

Line 216 ⟶ 218:

==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>

<ref name="Ellerman_1995">{{cite book |title=Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics |chapter=Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics |series=G - Reference, Information and Interdisciplinary Subjects Series |work=The worldly philosophy: studies in intersection of philosophy and economics |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |edition=illustrated |publisher=[[Rowman & Littlefield Publishers, Inc.]] |date=1995-03-21 |isbn=0-8476-7932-2 |pages=237–268 |url=http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |chapter-url=https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237 |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20160305012729/http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |archive-date=2016-03-05 |quote=[…] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the ''[[series sum]]'') of the resistances a&nbsp;+ b. If the resistances are placed in parallel, their compound resistance is the ''[[parallel sum]]'' of the resistances, which is denoted by the [[full colon]] […]}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages)</ref>

<ref name="Ellerman_2004">{{cite web |title=Introduction to Series-Parallel Duality |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |publisher=[[University of California at Riverside]] |date=May 2004 |orig-year=1995-03-21 |citeseerx=10.1.1.90.3666 |url=http://www.ellerman.org/wp-content/uploads/2012/12/Series-Parallel-Duality.CV_.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20190810011716/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf<!-- https://archive.today/20190810080659/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf --> |archive-date=2019-08-10 |quote=The [[parallel sum]] of two positive real numbers x:y&nbsp;= [(1/x)&nbsp;+ (1/y)]⁻¹ arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a [[duality (mathematics)|duality]] between the usual [[series sum|(series) sum]] and the parallel sum. […]}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf] (24 pages)</ref>

Line 226 ⟶ 228:

* {{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]]}}

* {{citationcite 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 -->

==External links==

[[Category:Electrical circuits]]