Series and parallel circuits: Difference between revisions

Revision as of 19:22, 13 March 2024

Revision as of 05:02, 23 August 2024

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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>===

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[[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 = R_\textsum_{si=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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==== 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===

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[[File: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_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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[[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 = \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>

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

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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===

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[[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 = \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>

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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[[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 = \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>

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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[[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_C = \textsum_{totali=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.

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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]], 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==

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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"/>

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''_total = 1/''R''_a + 1/''R''_b + ... + 1/''R''_n}}. {{math|''R''_a}}, {{math|''R''_b}}, and {{math|''R''_n}} 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==

* [[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]]

==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]]