Series and parallel circuits: Difference between revisions

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{{Short description|noneTypes of electrical circuits}}

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

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[[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>==

<!-- "Series circuit" redirects here. "Battery (electricity)" links here. -->

{{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.

{{Short description|none}}

===Current<span class="anchor" id="Iseries"></span>===

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

connectedIn ina series iscircuit, equalthe tocurrent is the sumsame offor theirall individualof the resistances:elements.

===Voltage===

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

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 seriesResistors_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>

{{anchor|Lseries}}

Here, the subscript ''s'' in {{math|''R''_s}} denotes "series", and {{math|''R''_s}} denotes resistance in a series.

==== CondsadfasdasuctanceConductance ====

[[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>

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<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>

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[[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 seriesCapacitors_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]]

[[File: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">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>

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

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

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===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 parallelResistors_in_parallel.svg|alt=A diagram of several resistors, side by side, both leads of each connected to the same wires.|border|center|x120px]]

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

<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>

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===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 parallelInductors_in_parallel.svg|alt=A diagram of several inductors, side by side, both leads of each connected to the same wires.|border|center|x120px]]

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

<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>

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===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 parallelCapacitors_in_parallel.svg|alt=A diagram of several capacitors, side by side, both leads of each connected to the same wires.|border|center|x120px]]

[[File: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 = \sum_{i=1}^n C_i = C_1 + C_2 + C_3 \cdots + C_n.</math>

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==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>

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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-generatormotor–generator]]s, then by [[Solid 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''_total = ''R''_artery + ''R''_arterioles + ''R''_capillaries}}. The largest proportion of resistance in this series is contributed by the arterioles.<ref name="BRS"/>

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==See also==

{{div col}}

* [[Anti-parallel (electronics)]]

* [[Electrical impedance#Combining impedances|Combining impedances]]

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* [[Wheatstone bridge]]

* [[Y-Δ transform]]

{{div col end}}

==References==