Series and parallel circuits: Difference between revisions

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{{Electromagnetism|Network}}

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

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===Voltage===

In a series circuit, the voltage is the sum of the voltage drops of the individual components (resistance units).

<math display="block">V = V_1 + V_2 + \dotssum_{i=1}^n + V_nV_i = I \left(sum_{i=1}^n R_1 + R_2 + \dots + R_n \right)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_\text{total}R = R_\textsum_{si=1}^n R_i = R_1 + R_2 + R_3 \cdots + R_n.</math>

[[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_\text{total} = R_\text{s} = R_1 + R_2 + \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.

==== 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 = \fracleft(\sum_{i=1}^n{G_1\text{totalover G_i}\right)^{-1} = \fracleft({1}{\over G_1} + \frac{1}{\over G_2} + {1\cdotsover G_3} + \fracdots + {1\over G_n}\right)^{G_n-1}.</math>

For a special case of two conductances in series, the total conductance is equal to:

<math display="block">G_\text{total}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_L = \mathrmsum_{totali=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.

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Therefore

<math display="block">L_\text{total}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.

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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]]

<math display="block">C = \fracleft(\sum_{i=1}^n{C_1\text{totalover C_i}\right)^{-1} = \fracleft({1}{\over C_1} + \frac{1}{\over C_2} + {1\cdotsover C_3} + \fracdots + {1\over C_n}\right)^{C_n-1}.</math>

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

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

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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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===Current===

The current in each individual resistor is found by [[Ohm's law]]. Factoring out the voltage gives

<math display="block">I_\text{total}I = I_1 + I_2 + \cdotssum_{i=1}^n + I_nI_i = V\left(\fracsum_{i=1}{R_1} + \frac^n{1}{R_2} + \cdotsover + \frac{1R_i}{R_n}\right).</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 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]]

<math display="block">R = \fracleft(\sum_{i=1}^n{R_1\text{totalover R_i}\right)^{-1} = \fracleft({1}{\over R_1} + \frac{1}{\over R_2} + {1\cdotsover R_3} + \fracdots + {1\over R_n}\right)^{R_n-1}.</math>

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

For only two resistances, the unreciprocated expression is reasonably simple:

<math display="block">R_\text{total}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_\text{total}R} = N \frac{1}{R}.</math>

and therefore to:

<math display="block">R_\text{total}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:

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==== 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_G = \textsum_{totali=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.

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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]]

<math display="block">L = \fracleft(\sum_{i=1}^n{L_1\text{totalover L_i}\right)^{-1} = \fracleft({1}{\over L_1} + \frac{1}{\over L_2} + {1\cdotsover L_3} + \fracdots + {1\over L_n}\right)^{L_n-1}.</math>

[[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">\frac{1}{L_\text{total}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}.</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">\frac{1}{L_\text{total}}L = \frac{L_1L_2 - M^2}{L_1 + L_2 - 2M}{L_1L_2 - M^2}</math>

If <math>L_1 = L_2</math>

<math display="block"> L_\text{total}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.

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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]]

<math display="block">C_C = \textsum_{totali=1}^n C_i = C_1 + C_2 + C_3 \cdots + C_n.</math>

[[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_\text{total} = C_1 + C_2 + \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.

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

Substituting Ohm's law for conductances gives

<math display="block">G_\text{eq}G V = G_1 V + G_2 V</math>

and the equivalent conductance will be,

<math display="block">G_\text{eq}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_\text{eq}V = V_1 + V_2.</math>

Substituting Ohm's law for conductance then gives,

<math display="block">\frac{I}{G_\text{eq}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_\text{eq}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_\text{eq}G = \frac{G_1 G_2}{G_1 + G_2}.</math>

For three conductances in series,

<math display="block">G_\text{eq}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_\mathrm{eq}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>

If {{math|''n''}} components are in parallel, then

<math display="block">R_\text{eq} = \left(\sum_i^n {R_i}^{-1}\right)^{-1}</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-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==