Series and parallel circuits: Difference between revisions

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===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]]

[[File: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 = \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.

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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==