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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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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. Components connected in series are connected along a single "electrical path", and each component has the same
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
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. Components connected in parallel are connected along multiple paths, and each component has the same
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
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 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 other configurations. In a series circuit, the current that flows through each of the components is the same, and the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
across the circuit is the sum of the individual
voltage drop Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirab ...
s across each component. 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. 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 Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirab ...
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

Series circuits are sometimes referred to as ''current''-coupled or 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. A series circuit has only one path through which its current can flow. Opening or breaking a series circuit at any 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 bulb is replaced.


Current

I = I_1 = I_2 = \cdots = I_n In a series circuit, the current is the same for all of the elements.


Voltage

In a series circuit, the voltage is the sum of the voltage drops of the individual components (resistance units). V = V_1 + V_2 + \dots + V_n = I \left( R_1 + R_2 + \dots + R_n \right)


Resistance units

The total resistance of two or more resistors connected in series is equal to the sum of their individual resistances: R_\text = R_\text = R_1 + R_2 + \cdots + R_n. Here, the subscript ''s'' in denotes "series", and denotes resistance in a series.
Electrical conductance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual paralle ...
presents a reciprocal quantity to resistance. Total conductance of a series circuits of pure resistances, therefore, can be calculated from the following expression: \frac = \frac + \frac + \cdots + \frac. For a special case of two conductances in series, the total conductance is equal to: G_\text = \frac.


Inductors

Inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
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: L_\mathrm = L_1 + L_2 + \cdots + L_n 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. When there are more than two inductors, the mutual inductance between each of them and the way the coils influence each other complicates the calculation. For a larger number of coils the total combined inductance is given by the sum of all mutual inductances between the various coils including the mutual inductance of each given coil with itself, which we term self-inductance or simply inductance. For three coils, there are six mutual inductances M_, M_, M_ and M_, M_ and M_. There are also the three self-inductances of the three coils: M_, M_ and M_. Therefore L_\text = \left(M_ + M_ + M_\right) + \left(M_ + M_ + M_\right) + \left(M_ + M_ + M_\right) By reciprocity, M_ = M_ 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

Capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s follow the same law using the reciprocals. The total
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances: \frac = \frac + \frac + \cdots + \frac. Equivalently using
elastance Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of c ...
(the reciprocal of capacitance), the total series elastance equals the sum of each capacitor's elastance.


Switches

Two or more
switch In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type of ...
es in series form a
logical AND In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...
; the circuit only carries current if all switches are closed. See
AND gate The AND gate is a basic digital logic gate that implements logical conjunction (∧) from mathematical logic AND gate behaves according to the truth table. A HIGH output (1) results only if all the inputs to the AND gate are HIGH (1). If not al ...
.


Cells and batteries

A battery is a collection of
electrochemical cell An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or using electrical energy to cause chemical reactions. The electrochemical cells which generate an electric current are called voltaic o ...
s. If the cells are connected in series, the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
of the battery will be the sum of the cell voltages. For example, a 12 volt
car battery An automotive battery or car battery is a rechargeable battery that is used to start a motor vehicle. Its main purpose is to provide an electric current to the electric-powered starting motor, which in turn starts the chemically-powered internal ...
contains six 2-volt cells connected in series. Some vehicles, such as trucks, have two 12 volt batteries in series to feed the 24-volt system.


Parallel circuits

If two or more components are connected in parallel, they have the same difference of potential (
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
) 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 current law Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
.


Voltage

In a parallel circuit, the voltage is the same for all elements. V = V_1 = V_2 = \dots = V_n


Current

The current in each individual resistor is found by Ohm's law. Factoring out the voltage gives I_\text = I_1 + I_2 + \cdots + I_n = V\left(\frac + \frac + \cdots + \frac\right).


Resistance units

To find the total resistance of all components, add the reciprocals of the resistances R_i of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance: \frac = \frac + \frac + \cdots + \frac. For only two resistances, the unreciprocated expression is reasonably simple: R_\text = \frac . This sometimes goes by the mnemonic ''product over sum''. For ''N'' equal resistances in parallel, the reciprocal sum expression simplifies to: \frac = N \frac. and therefore to: R_\text = \frac. To find the
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
in a component with resistance R_i, use Ohm's law again: I_i = \frac\,. The components divide the current according to their reciprocal resistances, so, in the case of two resistors, \frac = \frac. An old term for devices connected in parallel is ''multiple'', such as multiple connections for arc lamps. Since electrical conductance G is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads: G_\text = G_1 + G_2 + \cdots + G_n. 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

Inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
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: \frac = \frac + \frac + \cdots + \frac. 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 , the equivalent inductor is: \frac = \frac If L_1 = L_2 L_\text = \frac The sign of M 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 is negative, then the parallel inductance is nearly zero or the combination is almost non-inductive. It is assumed in the "tightly coupled" case is very nearly equal to . 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 , which can cause problems. More than three inductors become more complex and the mutual inductance of each inductor on each other inductor and their influence on each other must be considered. For three coils, there are three mutual inductances M_, M_ and M_. This is best handled by matrix methods and summing the terms of the inverse of the L matrix (3×3 in this case). The pertinent equations are of the form: v_ = \sum_ L_ \frac


Capacitors

The total
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
of capacitors in parallel is equal to the sum of their individual capacitances: C_\text = C_1 + C_2 + \cdots + C_n. The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.


Switches

Two or more
switch In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type of ...
es in parallel form a
logical OR In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
; the circuit carries current if at least one switch is closed. See
OR gate The OR gate is a digital logic gate that implements logical disjunction. The OR gate returns true if either or both of its inputs are true; otherwise it returns false. The input and output states are normally represented by different voltage lev ...
.


Cells and batteries

If the cells of a battery are connected in parallel, the battery voltage will be the same as the cell voltage, but the current supplied by each cell will be a fraction of the total current. For example, if a battery comprises four identical cells connected in parallel and delivers a current of 1 ampere, the current supplied by each cell will be 0.25 ampere. If the cells are not identical, cells with higher voltages will attempt to charge those with lower ones, potentially damaging them. Parallel-connected batteries were widely used to power the
valve A valve is a device or natural object that regulates, directs or controls the flow of a fluid (gases, liquids, fluidized solids, or slurries) by opening, closing, or partially obstructing various passageways. Valves are technically fitting ...
filaments in portable radios. 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

From
Kirchhoff's circuit laws Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirc ...
we can deduce the rules for combining conductances. For two conductances G_1 and G_2 in ''parallel'', the voltage across them is the same and from Kirchhoff's current law (KCL) the total current is I_\text = I_1 + I_2. Substituting Ohm's law for conductances gives G_\text V = G_1 V + G_2 V and the equivalent conductance will be, G_\text = G_1 + G_2. For two conductances G_1 and G_2 in series the current through them will be the same and Kirchhoff's Voltage Law tells us that the voltage across them is the sum of the voltages across each conductance, that is, V_\text = V_1 + V_2. Substituting Ohm's law for conductance then gives, \frac = \frac + \frac which in turn gives the formula for the equivalent conductance, \frac = \frac + \frac. This equation can be rearranged slightly, though this is a special case that will only rearrange like this for two components. G_\text = \frac. For three conductances in series, G_\text = \frac.


Notation

The value of two components in parallel is often represented in equations by the parallel operator, two vertical lines (∥), borrowing the parallel lines notation from geometry. R_\mathrm \equiv R_1 \parallel R_2 \equiv \left(R_1^ + R_2^\right)^ \equiv \frac This simplifies expressions that would otherwise become complicated by expansion of the terms. For instance: R_1 \parallel R_2 \parallel R_3 \equiv \frac . If components are in parallel, then R_\text = \left(\sum_i^n ^\right)^


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 An electric multiple unit or EMU is a multiple-unit train consisting of self-propelled carriages using electricity as the motive power. An EMU requires no separate locomotive, as electric traction motors are incorporated within one or a number ...
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-generators, then by 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: . The largest proportion of resistance in this series is contributed by the arterioles. Parallel resistance is illustrated by the
circulatory system The blood circulatory system is a system of organs that includes the heart, blood vessels, and blood which is circulated throughout the entire body of a human or other vertebrate. It includes the cardiovascular system, or vascular system, tha ...
. Each organ is supplied by an artery that branches off the
aorta The aorta ( ) is the main and largest artery in the human body, originating from the left ventricle of the heart and extending down to the abdomen, where it splits into two smaller arteries (the common iliac arteries). The aorta distributes o ...
. The total resistance of this parallel arrangement is expressed by the following equation: . , , and 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.


See also

* Network analysis (electrical circuits) *
Topology (electrical circuits) The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with t ...
*
Wheatstone bridge A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. The primary benefit of the circuit is its ability to provid ...
*
Y-Δ transform The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like t ...
*
Voltage divider In electronics, a voltage divider (also known as a potential divider) is a passive linear circuit that produces an output voltage (''V''out) that is a fraction of its input voltage (''V''in). Voltage division is the result of distributing the inp ...
*
Current divider In electronics, a current divider is a simple linear circuit that produces an output current (''I''X) that is a fraction of its input current (''I''T). Current division refers to the splitting of current between the branches of the divider. The cu ...
* Combining impedances * Equivalent impedance transforms * Resistance distance * Series-parallel duality * Series-parallel partial order * Series and parallel springs *
Hydraulic analogy The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes in play in ...
*
Anti-parallel (electronics) In electronics, two anti-parallel or inverse-parallel devices are connected in parallel but with their polarities reversed. One example is the TRIAC, which is comparable to two thyristors connected back-to-back (in other words, reverse paralle ...


References


Further reading

* * * {{citation , 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


External links


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Video ''“What's a Circuit, Series and Parallel (ElectroBOOM101–005)”''
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