Series and parallel circuits

From Wikipedia, the free encyclopedia

A series circuit with a voltage source (such as a battery, or in this case a cell) and 3 resistors

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single path, so the same current flows through all of the components. Components connected in parallel are connected along multiple paths, so the same voltage 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 across the circuit 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.

Consider a very simple circuit consisting of four light bulbs and one 6 V 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, 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 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 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 called current-coupled or 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's principle characteristic is that it has 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. 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

${\displaystyle I=I_{1}=I_{2}=I_{3}=\cdots =I_{n}}$

In a series circuit, the current is the same for all of the elements.

Resistors

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

${\displaystyle R_{\text{total}}=R_{\text{s}}=R_{1}+R_{2}+\cdots +R_{n}}$

Rs=>Resistance in series

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:

${\displaystyle {\frac {1}{G_{\mathrm {total} }}}={\frac {1}{G_{1}}}+{\frac {1}{G_{2}}}+\cdots +{\frac {1}{G_{n}}}}$.

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

${\displaystyle G_{\text{total}}={\frac {G_{1}G_{2}}{G_{1}+G_{2}}}.}$

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

${\displaystyle L_{\mathrm {total} }=L_{1}+L_{2}+\cdots +L_{n}}$

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 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 ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$, ${\displaystyle M_{23}}$ and ${\displaystyle M_{21}}$, ${\displaystyle M_{31}}$ and ${\displaystyle M_{32}}$. There are also the three self-inductances of the three coils: ${\displaystyle M_{11}}$, ${\displaystyle M_{22}}$ and ${\displaystyle M_{33}}$.
Therefore

${\displaystyle L_{\mathrm {total} }=(M_{11}+M_{22}+M_{33})+(M_{12}+M_{13}+M_{23})+(M_{21}+M_{31}+M_{32})}$

By reciprocity ${\displaystyle M_{ij}}$ = ${\displaystyle M_{ji}}$ 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

Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

${\displaystyle {\frac {1}{C_{\mathrm {total} }}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\cdots +{\frac {1}{C_{n}}}}$.

Switches

Two or more switches in series form a logical AND; the circuit only carries current if all switches are closed. See AND gate.

Cells and batteries

A battery is a collection of electrochemical cells. If the cells are connected in series, the voltage of the battery will be the sum of the cell voltages. For example, a 12 volt car battery 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.

Voltage

In a series circuit the voltage is addition of all the voltage elements.

${\displaystyle V=V_{1}+V_{2}+V_{3}+\dots +V_{n}}$

Parallel circuits

If two or more components are connected in parallel they have the same potential difference (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 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.

Voltage

In a parallel circuit the voltage is the same for all elements.

${\displaystyle V=V_{1}=V_{2}=\ldots =V_{n}}$

Current

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

${\displaystyle I_{\mathrm {total} }=V\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}\right)}$.

Resistors

To find the total resistance of all components, add the reciprocals of the resistances ${\displaystyle R_{i}}$ of each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:

${\displaystyle {\frac {1}{R_{\mathrm {total} }}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}}$.

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

${\displaystyle R_{\mathrm {total} }={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}.}$

This sometimes goes by the mnemonic product over sum.

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

${\displaystyle {\frac {1}{R_{\mathrm {total} }}}=N{\frac {1}{R}}}$.

and therefore to:

${\displaystyle R_{\mathrm {total} }={\frac {R}{N}}}$.

To find the current in a component with resistance ${\displaystyle R_{i}}$, use Ohm's law again:

${\displaystyle I_{i}={\frac {V}{R_{i}}}\,}$.

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

${\displaystyle {\frac {I_{1}}{I_{2}}}={\frac {R_{2}}{R_{1}}}}$.

An old term for devices connected in parallel is multiple, such as a multiple connection for arc lamps.
Since electrical conductance ${\displaystyle G}$ is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads:

${\displaystyle G_{\mathrm {total} }=G_{1}+G_{2}+\cdots +G_{n}}$.

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

Inductors 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:

${\displaystyle {\frac {1}{L_{\mathrm {total} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}$.

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:

${\displaystyle {\frac {1}{L_{\mathrm {total} }}}={\frac {L_{1}+L_{2}-2M}{L_{1}L_{2}-M^{2}}}}$

If ${\displaystyle L_{1}=L_{2}}$

${\displaystyle L_{\text{total}}={\frac {L+M}{2}}}$

The sign of ${\displaystyle M}$ 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 ${\displaystyle M_{12}}$, ${\displaystyle M_{13}}$ and ${\displaystyle M_{23}}$. This is best handled by matrix methods and summing the terms of the inverse of the ${\displaystyle L}$ matrix (3 by 3 in this case).

The pertinent equations are of the form: ${\displaystyle v_{i}=\sum _{j}L_{i,j}{\frac {di_{j}}{dt}}}$

Capacitors

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

${\displaystyle C_{\mathrm {total} }=C_{1}+C_{2}+\cdots +C_{n}}$.

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 switches in parallel form a logical OR; the circuit carries current if at least one switch is closed. See OR gate.

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. Parallel-connected batteries were widely used to power the valve filaments in portable radios, but they are now rare. 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 we can deduce the rules for combining conductances. For two conductances ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ in parallel, the voltage across them is the same and from Kirchhoff's current law (KCL) the total current is

${\displaystyle I_{\text{eq}}=I_{1}+I_{2}.\ \,}$

Substituting Ohm's law for conductances gives

${\displaystyle G_{\text{eq}}V=G_{1}V+G_{2}V\ \,}$

and the equivalent conductance will be,

${\displaystyle G_{\text{eq}}=G_{1}+G_{2}.\ \,}$

For two conductances ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ 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,

${\displaystyle V_{\text{eq}}=V_{1}+V_{2}.\ \,}$

Substituting Ohm's law for conductance then gives,

${\displaystyle {\frac {I}{G_{\text{eq}}}}={\frac {I}{G_{1}}}+{\frac {I}{G_{2}}}}$

which in turn gives the formula for the equivalent conductance,

${\displaystyle {\frac {1}{G_{\text{eq}}}}={\frac {1}{G_{1}}}+{\frac {1}{G_{2}}}.}$

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

${\displaystyle G_{\text{eq}}={\frac {G_{1}G_{2}}{G_{1}+G_{2}}}.}$

Notation

The value of two components in parallel is often represented in equations by two vertical lines, ∥, borrowing the parallel lines notation from geometry.

${\displaystyle R_{\mathrm {eq} }\equiv R_{1}\|R_{2}\equiv \left(R_{1}^{-1}+R_{2}^{-1}\right)^{-1}\equiv {\frac {R_{1}R_{2}}{R_{1}+R_{2}}}}$

This simplifies expressions that would otherwise become complicated by expansion of the terms. For instance:

${\displaystyle R_{1}\|R_{2}\|R_{3}\equiv {\frac {R_{1}R_{2}R_{3}}{R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}}}$.

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 unit 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: R_total = R_artery + R_arterioles + R_capillaries. The largest proportion of resistance in this series is contributed by the arterioles.

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: 1/R_total = 1/R_a + 1/R_b + ... 1/R_n. R_a, R_b, and 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.

If You Want to Save the Planet, Drop the Campaign Against Capitalism

written by Andrew Glover

Original link:  https://quillette.com/2018/10/29/if-you-want-to-save-the-planet-drop-the-campaign-against-capitalism/?fbclid=IwAR2hcTpn4LgsgBddDPSmJ14HHEb9EAbDQBqEzIZUhbm7lbNwgtA4NBGeyaQ 

 

But if you go carrying pictures of chairman Mao,
You ain’t going to make it with anyone anyhow

—The Beatles, 1968

This month, the Intergovernmental Panel on Climate Change (IPCC) issued a report concluding that it is all but inevitable that overall global warming will exceed the 1.5 degree Celsius limit dictated in the 2015 Paris Agreement. The report also discusses the potentially catastrophic consequences of this warming, which include extreme weather events, an accelerated rise in sea levels, and shrinking Arctic sea ice.

In keeping with the well-established trend, political conservatives generally have exhibited skepticism of these newly published IPCC conclusions. That includes U.S. President Donald Trump, who told 60 Minutes, “We have scientists that disagree with [anthropogenic global warming]. You’d have to show me the [mainstream] scientists because they have a very big political agenda.” On Fox News, a commentator argued that “the planet has largely stopped warming over the past 15 years, data shows—and [the IPCC report] could not explain why the Mercury had stopped rising.” Conservative YouTuber Ian Miles Cheong declared flatly that:

"Climate change is a hoax invented by neo-Marxists within the scientific community to destabilize the world economy and dismantle what they call “systems of oppression” and what the rest of us call capitalism."

This pattern of conservative skepticism on climate change is so well-established that many of us now take it for granted. But given conservatism’s natural impulse toward protecting our heritage, one might think that conservatives would be just as concerned with preserving order in the natural environment as they are with preserving order in our social and political environments. Ensuring that subsequent generations can live well is ordinarily a core concern for conservatives.

To this, conservatives might (and do) counter that they are merely pushing back against environmental extremists who seek to leverage the cause of global warming as a means to expand government, eliminate hierarchies of wealth, and reorganize society along social lines. And while most environmentally conscious citizens harbor no such ambitions, there is a substantial basis for this claim. Indeed, some environmentalists are forthright in seeking to implement the principles of “ecosocialism.” Meteorologist and self-described ecosocialist Eric Holthaus, for instance, responded to the IPCC report by declaring that:

"The world's top scientists just gave rigorous backing to systematically dismantle capitalism as a key requirement to maintaining civilization and a habitable planet."

One of the most prominent voices in this space has been Canadian writer Naomi Klein, whose 2015 book, This Changes Everything: Capitalism vs the Climate, argued that capitalism must be dismantled for the world to avert catastrophe. While I am sympathetic with some of the critiques that Klein directs at corporations and “free market fundamentalism,” her argument doesn’t hold water—because mitigating climate risks is a project whose enormous scope, cost and complexity can only be managed by regulated capitalist welfare states. Moreover, it’s difficult to see how she isn’t simply using the crisis of climate change as a veneer to agitate for her preferred utopian socio-economic system. As has been pointed out by Jonathan Chait of New York magazine, Klein appears to be adapting a mirror image of the same strategy she critiqued in her previous book, The Shock Doctrine, wherein she claimed that cynical politicians, pundits and corporations seize on crises to lock in economic restructuring along radical free market principles.

Simply put, describing the call for climate action in economically or politically revolutionary terms is always going to be counterproductive, because the vast majority of ordinary people in most countries don’t want a revolution. Environmentalists such as Klein are correct, however, in their more limited claim that market mechanisms alone can’t prevent global warming, since such mechanisms don’t impute the environmental costs associated with the way we produce goods and live our lives. Without some means of capturing the social price of environmentally destructive practices—resource extraction, in particular—we will invariably embrace wasteful and damaging practices.

Consider, for instance, the vast quantities of natural gas that are flared at oil wells simply because it’s seen as too costly to build gas pipelines to these facilities. This is a context in which we’d urge government to exercise its regulatory power; or to impose some kind of pricing mechanism that, either by carrot or stick, incentivizes the capture of the flared gas. Public policy has a necessary role in guiding capitalist decision makers toward the long-term sustainability of the environment. Unfortunately, this outcome is hard to achieve in a political environment characterized by tribalism, polarization and blame-shifting.

It is true that when it comes to climate change, the political left is more closely grounded in science than the right (even if both sides often tend to deny inconvenient truths more generally). But the left also has proven to be blinkered when it comes to appropriate responses, a tendency that has seeped into the latest IPCC report. While it’s not surprising that the report advocates support for renewable energy, its authors fail to acknowledge the warming effect that scaled up renewable-energy generation would have on land use due to their low energy density (think of the enormous footprint of solar farms). Likewise, the pro-environmental left’s distaste for nuclear power persists, despite its status as a geographically dense, safe, virtually carbon-free energy source.

The whole issue has become a sort of microcosm of the blind spots and dogmas embraced by both sides. As Jonathan Haidt argues, conservatives tend to be skeptical of top-down governance, preferring to focus on smaller nested structures that are less ambitious in scope, and hence easier to manage. This general principle takes form in conservative philosopher Roger Scruton’s approach to environmentalism, which argues that activism on issues such as climate change should be undertaken by communities at the local level, rather than by national (or international) bureaucrats and politicians—because the local level is where “people protect things which they know and love, things which are necessary for their life, and which will elicit in them the kind of disposition to make sacrifices, which, after all, is what it’s all about.”

While Scruton’s environmentalism gives us a reason to protect our local environments, the reality is that the effects of many environmentally damaging practices are not just experienced locally. A community may be motivated to protect a nearby forest from logging because it forms part of their love of home, but greenhouse gas emissions are displaced and dispersed into the shared atmosphere, contributing to global atmospheric degradation. Because of this, any approach that dismisses broader policy initiatives is unlikely to succeed in bringing down global carbon emissions. But at the very least, Scruton’s analysis awakens us to the reality that such policies will gain popular support only if they are justified and implemented in a manner that takes into consideration the views and sentiments of conservatives and liberals alike. Wind and solar farms will face less opposition if local communities get a greater say in where they are located. And while carbon taxes are effective in reducing emissions in some jurisdictions, conservatives will usually oppose them unless they are structured in a revenue-neutral manner, by legislating them alongside equivalent reductions in income tax, for instance.

Environmentalists also should acknowledge that some conservative objections to large-scale, top-down global instruments such as the Paris Agreement are perfectly legitimate. The provisions in such treaties typically are non-binding and require the good faith of all signatories. With many authoritarian countries seemingly misleading the rest of the world about their levels of economic activity, it’s not unreasonable to assume they would do the same when it comes to reporting carbon emissions. Moreover, those countries without the means to enforce reductions in carbon emissions domestically can’t be regarded as reliable participants in a global agreement to voluntarily decarbonize their economies.

This isn’t to say we shouldn’t be discussing climate change at a global level, or that international agreements don’t have any value. But environmentalists’ tendency to treat these documents as holy writ comes off as naïve, and thereby tends to undermine their cause.

Overall, our best hope for dealing with the emissions of developing countries is likely to assist them in managing their energy infrastructure so as to bypass high-emissions technologies. China, despite often being lauded for the amount of renewable energy it produces, now emits more carbon dioxide than the U.S. and Europe combined. With technologies such as large-scale solar generation becoming cost competitive with coal, progress is possible, but far from guaranteed without Western support.

These measures aren’t revolutionary. But that’s the point: In the environmental sector, just as in every other arena, there’s an opportunity cost to adopting revolutionary postures—since these revolutionaries tend to make more enemies than allies. If this project is really about saving the planet, rather than destroying capitalism, cooling the earth will mean cooling our rhetoric as well.

Andrew Glover is a sociologist who tweets at @theandrewglover.

Electrical resistance and conductance

From Wikipedia, the free encyclopedia

The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S). 

 

The resistance of an object depends in large part on the material it is made of—objects made of electrical insulators like rubber tend to have very high resistance and low conductivity, while objects made of electrical conductors like metals tend to have very low resistance and high conductivity. This material dependence is quantified by resistivity or conductivity. However, resistance and conductance are extensive rather than bulk properties, meaning that they also depend on the size and shape of an object. For example, a wire's resistance is higher if it is long and thin, and lower if it is short and thick. All objects show some resistance, except for superconductors, which have a resistance of zero.

The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while the conductance (G) is the inverse:

${\displaystyle R={V \over I},\qquad G={I \over V}={\frac {1}{R}}}$

For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constants (although they will depend on the size and shape of the object, the material it is made of, and other factors like temperature or strain). This proportionality is called Ohm's law, and materials that satisfy it are called ohmic materials.

In other cases, such as a transformer, diode or battery, V and I are not directly proportional. The ratio V/I is sometimes still useful, and is referred to as a "chordal resistance" or "static resistance", since it corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations, the derivative ${\displaystyle {\frac {dV}{dI}}\,\!}$ may be most useful; this is called the "differential resistance".

Introduction

The hydraulic analogy compares electric current flowing through circuits to water flowing through pipes. When a pipe (left) is filled with hair (right), it takes a larger pressure to achieve the same flow of water. Pushing electric current through a large resistance is like pushing water through a pipe clogged with hair: It requires a larger push (electromotive force) to drive the same flow (electric current).

In the hydraulic analogy, current flowing through a wire (or resistor) is like water flowing through a pipe, and the voltage drop across the wire is like the pressure drop that pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow. (Conductance and resistance are reciprocals.)

The voltage drop (i.e., difference between voltages on one side of the resistor and the other), not the voltage itself, provides the driving force pushing current through a resistor. In hydraulics, it is similar: The pressure difference between two sides of a pipe, not the pressure itself, determines the flow through it. For example, there may be a large water pressure above the pipe, which tries to push water down through the pipe. But there may be an equally large water pressure below the pipe, which tries to push water back up through the pipe. If these pressures are equal, no water flows. (In the image at right, the water pressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is mostly determined by two properties:

• geometry (shape), and
• material

Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe. In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.

Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the same shape and size. Similarly, electrons can flow freely and easily through a copper wire, but cannot flow as easily through a steel wire of the same shape and size, and they essentially cannot flow at all through an insulator like rubber, regardless of its shape. The difference between copper, steel, and rubber is related to their microscopic structure and electron configuration, and is quantified by a property called resistivity.

In addition to geometry and material, there are various other factors that influence resistance and conductance, such as temperature; see below.

Conductors and resistors

A 6.5 MΩ resistor, as identified by its electronic color code (blue–green–black-yellow-red). An ohmmeter could be used to verify this value.

Substances in which electricity can flow are called conductors. A piece of conducting material of a particular resistance meant for use in a circuit is called a resistor. Conductors are made of high-conductivity materials such as metals, in particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs.

Ohm's law

The current-voltage characteristics of four devices: Two

resistors, a diode, and a battery. The horizontal axis is

voltage drop, the vertical axis is current. Ohm's law is satisfied

when the graph is a straight line through the origin. Therefore,

the two resistors are ohmic, but the diode and battery are not.

For many materials, the current I through the material is proportional to the voltage V applied across it:

${\displaystyle I\propto V}$

over a wide range of voltages and currents. Therefore, the resistance and conductance of objects or electronic components made of these materials is constant. This relationship is called Ohm's law, and materials which obey it are called ohmic materials. Examples of ohmic components are wires and resistors. The current-voltage (IV) graph of an ohmic device consists of a straight line through the origin with positive slope.

Other components and materials used in electronics do not obey Ohm's law; the current is not proportional to the voltage, so the resistance varies with the voltage and current through them. These are called nonlinear or nonohmic. Examples include diodes and fluorescent lamps. The IV curve of a nonohmic device is a curved line.

Relation to resistivity and conductivity

A piece of resistive material with electrical contacts on both ends.

The resistance of a given object depends primarily on two factors: What material it is made of, and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area; for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire. Also, for a given material, the resistance is proportional to the length; for example, a long copper wire has higher resistance than an otherwise-identical short copper wire. The resistance R and conductance G of a conductor of uniform cross section, therefore, can be computed as

${\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}},\\G&=\sigma {\frac {A}{\ell }}.\end{aligned}}}$

where ${\displaystyle \ell }$ is the length of the conductor, measured in metres [m], A is the cross-sectional area of the conductor measured in square metres [m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m⁻¹), and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: ${\displaystyle \rho =1/\sigma }$. Resistivity is a measure of the material's ability to oppose electric current.

This formula is not exact, as it assumes the current density is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires.

Another situation for which this formula is not exact is with alternating current (AC), because the skin effect inhibits current flow near the center of the conductor. For this reason, the geometrical cross-section is different from the effective cross-section in which current actually flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC current, their resistances increase due to the proximity effect. At commercial power frequency, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation, or large power cables carrying more than a few hundred amperes.

What determines resistivity?

The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about 10³⁰ times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal has large numbers of "delocalized" electrons that are not stuck in any one place, but free to move across large distances, whereas in an insulator (like teflon), each electron is tightly bound to a single molecule, and a great force is required to pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electrical resistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity (electrolytic).

Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed to light. See below.

Measuring resistance

An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use four-terminal sensing.

Typical resistances

Component	Resistance (Ω)
1 meter of copper wire with 1 mm diameter	0.02
1 km overhead power line (typical)	0.03
AA battery (typical internal resistance)	0.1
Incandescent light bulb filament (typical)	200–1000
Human body	1000 to 100,000

Static and differential resistance

The IV curve of a non-ohmic device (purple). The static resistance at point A is the inverse slope of line B through the origin. The differential resistance at A is the inverse slope of tangent line C.

The IV curve of a component with negative differential resistance, an unusual phenomenon where the IV curve is non-monotonic.

Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These are called non-ohmic or non-linear, and their I–V curves are not straight lines through the origin.

Resistance and conductance can still be defined for non-ohmic elements. However, unlike ohmic resistance, non-linear resistance is not constant but varies with the voltage or current through the device; i.e., its operating point. There are two types of resistance

• Static resistance (also called chordal or DC resistance) – This corresponds to the usual definition of resistance; the voltage divided by the current

${\displaystyle R_{\mathrm {static} }={\frac {V}{I}}\,}$.

It is the slope of the line (chord) from the origin through the point on the curve. Static resistance determines the power dissipation in an electrical component. Points on the IV curve located in the 2nd or 4th quadrants, for which the slope of the chordal line is negative, have negative static resistance. Passive devices, which have no source of energy, cannot have negative static resistance. However active devices such as transistors or op-amps can synthesize negative static resistance with feedback, and it is used in some circuits such as gyrators.

• Differential resistance (also called dynamic, incremental or small signal resistance) – Differential resistance is the derivative of the voltage with respect to the current; the slope of the IV curve at a point

${\displaystyle R_{\mathrm {diff} }={\frac {dV}{dI}}\,}$.

If the IV curve is nonmonotonic (with peaks and troughs), the curve has a negative slope in some regions—so in these regions the device has negative differential resistance. Devices with negative differential resistance can amplify a signal applied to them, and are used to make amplifiers and oscillators. These include tunnel diodes, Gunn diodes, IMPATT diodes, magnetron tubes, and unijunction transistors.

AC circuits

Impedance and admittance

The voltage (red) and current (blue) versus time (horizontal axis) for a capacitor (top) and inductor (bottom). Since the amplitude of the current and voltage sinusoids are the same, the absolute value of impedance is 1 for both the capacitor and the inductor (in whatever units the graph is using). On the other hand, the phase difference between current and voltage is −90° for the capacitor; therefore, the complex phase of the impedance of the capacitor is −90°. Similarly, the phase difference between current and voltage is +90° for the inductor; therefore, the complex phase of the impedance of the inductor is +90°.

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their phases. For example, in an ideal resistor, the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a capacitor or inductor, the maximum current flow occurs as the voltage passes through zero and vice versa (current and voltage are oscillating 90° out of phase, see image at right). Complex numbers are used to keep track of both the phase and magnitude of current and voltage:

${\displaystyle V(t)={\text{Re}}(V_{0}e^{j\omega t}),\quad I(t)={\text{Re}}(I_{0}e^{j\omega t}),\quad Z={\frac {V_{0}}{I_{0}}},\quad Y={\frac {I_{0}}{V_{0}}}}$

where:

• t is time,
• V(t) and I(t) are, respectively, voltage and current as a function of time,
• V₀, I₀, Z, and Y are complex numbers,
• Z is called impedance,
• Y is called admittance,
• Re indicates real part,
• ${\displaystyle \omega }$ is the angular frequency of the AC current,
• ${\displaystyle j={\sqrt {-1}}}$ is the imaginary unit.

The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts:

${\displaystyle Z=R+jX,\quad Y=G+jB}$

where R and G are resistance and conductance respectively, X is reactance, and B is susceptance. For ideal resistors, Z and Y reduce to R and G respectively, but for AC networks containing capacitors and inductors, X and B are nonzero.

${\displaystyle Z=1/Y}$ for AC circuits, just as ${\displaystyle R=1/G}$ for DC circuits.

Frequency dependence of resistance

Another complication of AC circuits is that the resistance and conductance can be frequency-dependent. One reason, mentioned above is the skin effect (and the related proximity effect). Another reason is that the resistivity itself may depend on frequency (see Drude model, deep-level traps, resonant frequency, Kramers–Kronig relations, etc.)

Energy dissipation and Joule heating

Running current through a material with high resistance creates heat, in a phenomenon called Joule heating. In this picture, a cartridge heater, warmed by Joule heating, is glowing red hot.

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, electrical energy is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called Joule heating (after James Prescott Joule), also called ohmic heating or resistive heating.

The dissipation of electrical energy is often undesired, particularly in the case of transmission losses in power lines. High voltage transmission helps reduce the losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, for example in electric stoves and other electric heaters (also called resistive heaters). As another example, incandescent lamps rely on Joule heating: the filament is heated to such a high temperature that it glows "white hot" with thermal radiation (also called incandescence).

The formula for Joule heating is:

${\displaystyle P=I^{2}R}$

where P is the power (energy per unit time) converted from electrical energy to thermal energy, R is the resistance, and I is the current through the resistor.

Dependence of resistance on other conditions

Temperature dependence

Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For the detailed As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usually platinum, while a thermistor is made of ceramic or polymer.)

Resistance thermometers and thermistors are generally used in two ways. First, they can be used as thermometers: By measuring the resistance, the temperature of the environment can be inferred. Second, they can be used in conjunction with Joule heating (also called self-heating): If a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general, self-heating can turn a resistor into a nonlinear and hysteretic circuit element.

If the temperature T does not vary too much, a linear approximation is typically used:

${\displaystyle R(T)=R_{0}[1+\alpha (T-T_{0})]}$

where ${\displaystyle \alpha }$ is called the temperature coefficient of resistance, ${\displaystyle T_{0}}$ is a fixed reference temperature (usually room temperature), and ${\displaystyle R_{0}}$ is the resistance at temperature ${\displaystyle T_{0}}$. The parameter ${\displaystyle \alpha }$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, ${\displaystyle \alpha }$ is different for different reference temperatures. For this reason it is usual to specify the temperature that ${\displaystyle \alpha }$ was measured at with a suffix, such as ${\displaystyle \alpha _{15}}$, and the relationship only holds in a range of temperatures around the reference.

The temperature coefficient ${\displaystyle \alpha }$ is typically +3×10⁻³ K⁻¹ to +6×10⁻³ K⁻¹ for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.

Strain dependence

Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain. By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect.

Light illumination dependence

Some resistors, particularly those made from semiconductors, exhibit photoconductivity, meaning that their resistance changes when light is shining on them. Therefore, they are called photoresistors (or light dependent resistors). These are a common type of light detector.

Superconductivity

Superconductors are materials that have exactly zero resistance and infinite conductance, because they can have V=0 and I≠0. This also means there is no joule heating, or in other words no dissipation of electrical energy. Therefore, if superconductive wire is made into a closed loop, current flows around the loop forever. Superconductors require cooling to temperatures near 4 K with liquid helium for most metallic superconductors like niobium–tin alloys, or cooling to temperatures near 77K with liquid nitrogen for the expensive, brittle and delicate ceramic high temperature superconductors. Nevertheless, there are many technological applications of superconductivity, including superconducting magnets.