Series and parallel circuits

# Series and parallel circuits

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Components of an electrical circuit
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

or electronic circuit
Electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow...

can be connected in many different ways
Topology (electronics)
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....

. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

flows through all of the components. Components connected in parallel are connected so the same voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

is applied to each component.

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.

In a series circuit, the current through each of the components is the same, and the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

across the components is the sum of the voltages 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 through each component.

As an example, consider a very simple circuit consisting of four light bulbs and one 6 V battery
Battery (electricity)
An electrical battery is one or more electrochemical cells that convert stored chemical energy into electrical energy. Since the invention of the first battery in 1800 by Alessandro Volta and especially since the technically improved Daniell cell in 1836, batteries have become a common power...

. 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, there is same current through all of them, and the voltage drop is 1.5 V 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 6.0 V across each bulb and they all glow.

In a series circuit, every device must function for the circuit to be complete. One bulb burning out in a series circuit breaks the circuit. In parallel circuits, each light 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 called current-coupled or daisy chain
Daisy chain (electrical engineering)
In electrical and electronic engineering a daisy chain is a wiring scheme in which multiple devices are wired together in sequence or in a ring...

-coupled. The current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

in a series circuit goes through every component in the circuit. Therefore, all of the components in a series connection carry the same current. There is only one path in a series circuit in which the current can flow.

A series circuit's main disadvantage or advantage, depending on its intended role in a product's overall design, is that because there is only one path in which its current can flow, opening or breaking a series circuit at any point causes the entire circuit to "open" or stop operating
Single point of failure
A single point of failure is a part of a system that, if it fails, will stop the entire system from working. They are undesirable in any system with a goal of high availability or reliability, be it a business practice, software application, or other industrial system.-Overview:Systems can be made...

. 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.

### Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

Electrical conductance presents a reciprocal quantity to resistance. Total conductance of a series circuits of pure resistors, therefore, can be calculated from the following expression:
.

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

R=R1+R2+R3.....Rn

### Inductors

Inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s follow the same law, in that the total inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

of non-coupled inductors in series is equal to the sum of their individual inductances:

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 neighbours. 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 , , and , and . There are also the three self-inductances of the three coils: , and .

Therefore

By reciprocity = 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
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s follow the same law using the reciprocals. The total capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

of capacitors in series is equal to the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of the sum of the reciprocals of their individual capacitances:

.

### Switches

Two or more switch
Switch
In electronics, a switch is an electrical component that can break an electrical circuit, interrupting the current or diverting it from one conductor to another....

es in series form a logical AND
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

; the circuit only carries current if all switches are 'on'. See AND gate
AND gate
The AND gate is a basic digital logic gate that implements logical conjunction - it behaves according to the truth table to the right. A HIGH output results only if both the inputs to the AND gate are HIGH . If neither or only one input to the AND gate is HIGH, a LOW output results...

.

### Cells and batteries

A battery
Battery (electricity)
An electrical battery is one or more electrochemical cells that convert stored chemical energy into electrical energy. Since the invention of the first battery in 1800 by Alessandro Volta and especially since the technically improved Daniell cell in 1836, batteries have become a common power...

is a collection of electrochemical cell
Electrochemical cell
An electrochemical cell is a device capable of either deriving electrical energy from chemical reactions, or facilitating chemical reactions through the introduction of electrical energy. A common example of an electrochemical cell is a standard 1.5-volt "battery"...

s. If the cells are connected in series, the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

of the battery will be the sum of the cell voltages. For example, a 12 volt car battery
Car battery
An automotive battery is a type of rechargeable battery that supplies electric energy to an automobile. Usually this refers to an SLI battery to power the starter motor, the lights, and the ignition system of a vehicle’s engine...

contains six 2-volt cells connected in series.

## Parallel circuits

If two or more components are connected in parallel they have the same potential difference (voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable 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.

### Resistors

The current in each individual resistor is found by Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

. Factoring out the voltage gives.

To find the total resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

of all components, add the reciprocals
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of the resistances of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:

.

For only two resistors, the unreciprocated expression is reasonably simple:
This sometimes goes by the mnemonic "product over sum".

For N equal resistors in parallel, the reciprocal sum expression simplifies to:.

and therefore to:.

To find the current in a component with resistance , use Ohm's law again:.

The components divide the current according to their reciprocal resistances, so, in the case of two resistors,.

An old term for devices connected in parallel is multiple, such as a multiple connection for arc lamp
Arc lamp
"Arc lamp" or "arc light" is the general term for a class of lamps that produce light by an electric arc . The lamp consists of two electrodes, first made from carbon but typically made today of tungsten, which are separated by a gas...

s.

Since electrical conductance is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads:
.

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
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s follow the same law, in that the total inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:

.

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 M, the equivalent inductor is:

If

The sign of depends on how the magnetic fields influence each other. For two equal tightly coupled coils the total inductance is close to that of each single coil. If the polarity of one coil is reversed so that 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 M is very nearly equal to 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 M, which can cause problems.

More than three inductors becomes 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 , and . This is best handled by matrix methods and summing the terms of the inverse of the matrix (3 by 3 in this case).

The pertinent equations are of the form:

### Capacitors

Capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s follow the same law using the reciprocals. The total capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

of capacitors in parallel is equal to the sum of their individual capacitances:

.

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
Switch
In electronics, a switch is an electrical component that can break an electrical circuit, interrupting the current or diverting it from one conductor to another....

es in parallel form a logical OR
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

; the circuit carries current if at least one switch is 'on'. See OR gate
OR gate
The OR gate is a digital logic gate that implements logical disjunction - it behaves according to the truth table to the right. A HIGH output results if one or both the inputs to the gate are HIGH . If neither input is HIGH, a LOW output results...

.

### 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 contains four cells connected in parallel and delivers a current of 1 ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

, the current supplied by each cell will be 0.25 ampere. Parallel-connected batteries were widely used to power the valve
Vacuum tube
In electronics, a vacuum tube, electron tube , or thermionic valve , reduced to simply "tube" or "valve" in everyday parlance, is a device that relies on the flow of electric current through a vacuum...

filaments in portable radios but they are now rare.

## Combining conductances

From Kirchhoff's circuit laws
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...

we can deduce the rules for combining conductances. For two conductances and in parallel the voltage across them is the same and from Kirchoff's Current Law the total current is

Substituting Ohm's law for conductances gives

and the equivalent conductance will be,

For two conductances and 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,

Substituting Ohm's law for conductance then gives,

which in turn gives the formula for the equivalent conductance,

This equation can be rearranged slightly, though this is a special case that will only rearrange like this for two components.

## Applications

Series circuits were formerly used for lighting in electric multiple unit
Electric multiple unit
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 of the carriages...

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
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

to drop the remaining 40 volts. Series circuits for train lighting were superseded, first by motor-generator
Motor-generator
A motor-generator is a device for converting electrical power to another form. Motor-generator sets are used to convert frequency, voltage, or phase of power. They may also be used to isolate electrical loads from the electrical power supply line...

s, then by solid state
Solid state (electronics)
Solid-state electronics are those circuits or devices built entirely from solid materials and in which the electrons, or other charge carriers, are confined entirely within the solid material...

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: Rtotal = Rartery + Rarterioles + Rcapillaries. The largest proportion of resistance in this series is contributed by the arterioles.

Parallel resistance is illustrated by the circulatory system
Circulatory system
The circulatory system is an organ system that passes nutrients , gases, hormones, blood cells, etc...

. Each organ is supplied by an artery that branches off the aorta
Aorta
The aorta is the largest artery in the body, originating from the left ventricle of the heart and extending down to the abdomen, where it branches off into two smaller arteries...

. The total resistance of this parallel arrangement is expressed by the following equation: 1/Rtotal = 1/Ra + 1/Rb + ... 1/Rn. Ra, Rb, and Rn 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.

• network analysis (electrical circuits)
Network analysis (electrical circuits)
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this...

• Wheatstone bridge
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. Its operation is similar to the original potentiometer. It was invented by Samuel Hunter Christie in 1833 and...

• Y-Δ transform
• Voltage divider
• Current divider
• Combining impedances
• Equivalent impedance transforms
Equivalent impedance transforms
An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network...