Electrical resistance and conductance - Wikipedia

Content

• 1 Introduction
• 2 Conductors and resistors
• 3 Ohm's Law
• 4 Relationship with resistivity and conductivity
• 5 Resistance measurement
• 6 Typical resistances
• 7 Static and differential resistance
• 8 AC circuits 8.1 Impedance and tolerance
• 8.2 Frequency dependence of resistance

Resistivity of various materials

We can talk about resistivity only in the presence of elements that conduct current, since dielectrics have infinite or close to infinite electrical resistance. In contrast, metals are very good conductors of current. You can measure the electrical resistance of a metal conductor using a milliohmmeter , or an even more accurate microohmmeter. The value is measured between their probes applied to the conductor section. They allow you to check circuits, wiring, windings of motors and generators.

Conductors and resistors

75 ohm resistor as defined by the electronic color code (purple – green – black – gold – red).
An ohmmeter can be used to check this value. Substances in which electricity can flow are called conductors. A piece of conductive material with a specific resistance intended for use in a circuit is called a resistor. Conductors are made from highly conductive materials such as metals, particularly copper and aluminum. On the other hand, resistors are made from a wide variety of materials depending on factors such as the desired resistance, the amount of energy that needs to be dissipated, accuracy, and cost.

Electricity

Electric current is a physical process of directed movement of charged particles under the influence of an electromagnetic field from one pole of a closed electrical circuit to another. Charge-carrying particles can include electrons, protons, ions and holes. Without a closed circuit, no current is possible. Particles capable of carrying electrical charges do not exist in all substances; those in which they exist are called conductors and semiconductors . And substances in which there are no such particles are dielectrics .

It is generally accepted that the direction of current is from plus to minus, while electrons move from minus to plus!

The unit of current measurement is Ampere ( A

).
In formulas and calculations, current strength is denoted by the
letter I. A current of 1 Ampere is generated when a charge of 1 Coulomb (6.241·1018 electrons) passes through a point in an electrical circuit in 1 second.

Let's look again at our water-electricity analogy. Only now let’s take two tanks and fill them with an equal amount of water. The difference between the tanks is the diameter of the outlet pipe.

Let's open the taps and make sure that the flow of water from the left tank is greater (the diameter of the pipe is larger) than from the right. This experience is clear evidence of the dependence of flow speed on pipe diameter. Now let's try to equalize the two flows. To do this, add water (charge) to the right tank. This will give more pressure (voltage) and increase flow rate (current). resistance acts as the pipe diameter .

The experiments carried out clearly demonstrate the relationship between voltage , current and resistance . We'll talk more about resistance a little later, but now a few more words about the properties of electric current.

If the voltage does not change its polarity, plus to minus, and the current flows in one direction, then it is a direct current and, accordingly, a constant voltage . If the voltage source changes its polarity and the current flows in one direction, then in the other, this is already alternating current and alternating voltage . Maximum and minimum values ​​(indicated on the graph as Io

) are
amplitude or peak current values. In home sockets, the voltage changes its polarity 50 times per second, i.e. the current oscillates here and there, it turns out that the frequency of these oscillations is 50 Hertz, or 50 Hz for short. In some countries, for example in the USA, the frequency is 60 Hz.

Ohm's law

Main article: Ohm's law

The current-voltage characteristics of four devices: two are resistors, a diode, and a battery.
The horizontal axis is voltage drop, the vertical axis is Current. Ohm's law holds when the graph is a straight line through the origin. Consequently, two resistors are ohmic
, but there is no diode and battery.

For many materials the current I

through a material, proportional to the voltage
V
applied across it:
I ∝ V { Displaystyle I propto V }
over a wide range of voltages and currents. Therefore, the resistance and conductivity of objects or electronic components made from these materials are constant. This relationship is called Ohm's Law, and materials that obey it are called ohmic

materials. Examples of ohmic components are wires and resistors. The current-voltage graph of an ohmic device consists of a straight line through the origin with a positive slope.

Other components and materials used in electronics do not obey Ohm's law; current is not proportional to voltage, so resistance depends on voltage and current through them. They are called nonlinear

or
non-ohmic
. Examples include diodes and fluorescent lamps. The current-voltage curve of an ohmless device is a curved line.

Basic laws of current flow in linear electrical circuits

Ohm's Law: Defines the relationship between current, voltage and resistance of a passive element (branch, section, circuit). The real form of the law is applicable only for resistance:

Ohm's law in complex form is valid for any passive element:

Kirchhoff's first law: Determines the balance of currents in any node of an electrical circuit. It can be represented as a real

or in complex form

where is the number of branches connected at a given node.

Kirchhoff's second law. Determines the voltage balance in any closed circuit circuit:

where is the number of passive elements in a closed circuit: is the number of sources included in this circuit.

Relationship between sinusoidal current and voltage in passive elements

In resistance, the real form of recording the relationship between current and voltage is determined by Ohm’s law:

In complex form (images) are defined by the expressions:

In resistance, current and voltage are always in phase, i.e.

In inductance, voltage and current are related by the relations:

Let us write these expressions in complex form:

where is the complex inductance resistance; — inductive reactance;

- complex conductivity of inductance; - inductive conductivity.

From formulas (1.20) and (1.21) it is clear that the current in the inductance lags in phase from the applied voltage by an angle .

In a capacitance, voltage and current are related by the following relationships:

In complex form, expressions (1.22) take the form:

where is the complex resistance of the capacitance; — capacitance; — complex conductivity of the capacitance; — capacitive conductivity.

From formulas (1.22) and (1.23) it is clear that the current in the capacitor is ahead of the applied voltage in phase by an angle .

Current and voltage diagrams

For greater clarity, time and vector diagrams are used when analyzing processes and calculating penalties.

A time diagram is a graph of instantaneous values ​​of currents and voltages on a coordinate plane along the real time axis (Fig. 1.4). The initial phase reports from zero of the sine wave (or from the maximum of the cosine wave) to the origin. The initial phase is positive if the counting direction coincides with the direction of the time axis.

The phase shift of voltage and current is measured in the direction from zero voltage to zero current. If this direction coincides with the direction of the time axis, then the phase shift is positive.

A vector diagram is a set of vectors constructed on a complex plane with respect to mutual orientation in phase. The length of the vector is proportional to the amplitude of the vibration.

All vectors rotate counterclockwise in time at a speed of . Fig. 1.5 shows a vector diagram corresponding to the timing diagram shown in Fig. 1.4.

The initial phase is measured from the real axis to the vector. The phase shift is measured from the current vector to the voltage vector. The initial phase and phase shift are positive if their counting direction coincides with the direction of vector rotation (counterclockwise).

Relationship with resistivity and conductivity

Main article: Electrical 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, resistance is inversely proportional to cross-sectional area; for example, a thick copper wire has less resistance than an otherwise identical thin copper wire. In addition, for a given material, resistance is proportional to length; for example, a long copper wire has a higher resistance than an otherwise identical short copper wire. The resistance p and conductivity gram of a conductor with a uniform cross-section can therefore be calculated as

р = ρ ℓ А, gram = σ А ℓ. { displaystyle { begin {align} R & = rho { frac { ell} {A}}, [5pt] G & = sigma { frac {A} { ell}}. end {aligned }}}

where ℓ { displaystyle ell } is the length of the conductor, measured in meters (m), A

is the cross-sectional area of ​​the conductor measured in square meters (m2), σ (sigma) is the electrical conductivity measured in Siemens per meter (S m−1) and ρ () is the electrical resistivity (also called
electrical resistivity
) of the material, measured in ohm meters (Ohm m). Resistivity and conductivity are constants of proportionality and therefore depend only on the material from which the wire is made, and not on its geometry. Resistivity and conductance are reciprocal: ρ = 1 / σ { Displaystyle rho = 1 / sigma } . Resistivity is a measure of a material's ability to resist electrical current.

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

Another situation for which this formula is not accurate is with alternating current (AC) because the skin effect prevents current from flowing near the center of the conductor. For this reason, geometric

cross-section is different from
the effective
cross-section in which current actually flows, so the resistance is higher than expected. Likewise, if two conductors next to each other carry alternating current, their resistance increases due to the proximity effect. In industrial frequency networks, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation,[3] or large power cables carrying more than several hundred amperes.

The resistivity of different materials varies greatly: for example, the conductivity of Teflon is about 1030, several times lower than the conductivity of copper. Roughly speaking, this is because metals have a large number of "delocalized" electrons that are not stuck in any one place, so they can move freely over long distances. In an insulator such as Teflon, each electron is tightly bound to one molecule, so it takes a lot of force to remove it. Semiconductors lie between these two extremes. Read more in the article: Electrical resistivity and conductivity. For electrolyte solutions, see the article: Electrical conductivity (electrolytic).

Resistivity depends on temperature. In semiconductors, resistivity also changes when exposed to light. See below.

Resistance and conductivity

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The directional movement of electric charges in any conductor is prevented by its molecules and atoms. The quantity characterizing the resistance of an electrical circuit to the passage of electric current is called electrical resistance ( resistance ).

Resistance is designated by the letter R , the unit of measurement is 1 ohm.

To evaluate the electrical properties of a conductor material, resistivity ρ is used - this is the resistance of a conductor with a length of 1 m and a cross-sectional area of ​​1 mm2.

The resistance of a conductor depends on the material (resistivity), length L and cross-section S , as well as temperature. In metal conductors, the resistance increases with increasing temperature.

Electrical conductivity of a substance (conductivity) is the property of a substance to conduct electric current. This is the reciprocal of resistance, denoted g, measured in siemens, See.

Depending on electrical conductivity, all substances are divided into:

- conductors (conduct electric current well) - metals, solutions of salts, acids, alkalis, coal, graphite. In electrical engineering, aluminum and copper are used to make wires;

- dielectrics (insulators) - practically do not conduct electric current. these include: air, gases, mica, plastics, porcelain, varnishes, enamels, rubber, etc.

- semiconductors - in conductivity they occupy an intermediate position between conductors and dielectrics (silicon, germanium)

Ohm's law

Ohm's law expresses the relationship between emf (voltage), resistance and current.

Ohm's law for a circuit section

The current strength in a section of an electrical circuit is equal to the voltage at the terminals of this section divided by its resistance.

Ohm's law for an electric circuit (closed)

The current strength in a closed circuit is directly proportional to the emf and inversely proportional to the total resistance of the circuit.

The total resistance is equal to the sum of the external R and internal r resistances. Internal resistance is the resistance of the electric current source.

To measure the current in a circuit, an ammeter ; it is connected to the circuit in series (the current circuit is broken and at the point of the break the ends of the wires are connected to the ammeter terminals).

A voltmeter is used to measure voltage ; it is connected in parallel.

Connection diagram for ammeter and voltmeter

Work and power of electric current

Passing through a conductor, electric current does work, which is usually called electrical energy.

Electrical energy ( W ) or work ( A ) performed by electric current is equal to the product of voltage, the current in the circuit and the time it travels.

Work is measured in joules, J , electrical energy is usually measured in kilowatt-hours, kWh , because The joule is a very small unit of measurement.

1 kW*h=3600000 J

Power is the work produced (or consumed) in one second.

Power is designated by the letter P and is measured in watts, W.

are used to measure power , electric meters are used to measure electrical energy .

Thermal effect of current. Joule-Lenz law.

When electric current passes through a conductor, the conductor heats up as a result of collisions with atoms. The amount of heat generated in a conductor during the passage of electric current is determined by the Joule–Lenz law.

The amount of heat generated, Q, is directly proportional to the square of the current, the resistance of the conductor and the time the current passes through the conductor.

The amount of heat is measured in joules, J.

Kirchhoff's law

The sum of currents entering a node is equal to the sum of currents leaving the node.

A node is a point. in which three or more wires are connected.

Terms used in the electric power industry

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Recommended pages:

Typical resistances

See also: Electrical resistivity of elements (data page) and Electrical resistivity and conductivity

AC circuits

Impedance and tolerance

Main articles: Electrical impedance and Reception

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

you ( t ) = r e ( U 0 ⋅ e j ω t ), i ( t ) = r e ( i 0 ⋅ e j ( ω t + φ ) ), Z _ = U _ i _ , Y _ = i _ U _ { Displaystyle u (t) = { mathfrak {Re} } left (U_ {0} cdot e ^ {j omega t} right), quad i (t) = { mathfrak {Re} } left (I_ {0 } cdot e ^ {j ( omega t + varphi)} right), quad { underline {Z}} = { frac { underline {U}} { underline {I }}}, quad { underline {Y}} = { frac { underline {I}} { underline {U}}}} Voltage (red) and current (blue) versus time (horizontal axis) for a capacitor (top) and an inductor (bottom). Since the amplitude of the current and voltage of the sine wave are the same, the absolute value of the resistance is 1 for both the capacitor and the inductor (in any units on the graph). On the other hand, the phase difference between current and voltage is -90° for a capacitor; Therefore the complex phase of the capacitor resistance is -90°. Similarly, the phase difference between current and voltage is +90° for an inductor; therefore, the complex phase of the inductor's impedance is +90°.

Where:

• t
  time,
• u (t)
  and
  Eto)
  are voltage and current respectively as a function of time,
• U0
  and
  I0
  indicate the voltage amplitude of the corresponding current,
• ω { displaystyle omega } is the angular frequency of the alternating current,
• φ { displaystyle varphi } displacement angle,
• U
  ,
  I
  ,
  Z
  , and
  Y
  are complex numbers,
• Z
  is called resistance,
• Y
  is called tolerance,
• Re indicates the real part,
• j = − 1 { displaystyle j = { sqrt {-1))} is the imaginary unit.

Impedance and conductance can be expressed as complex numbers, which can be broken down into real and imaginary parts:

Z _ = R + j X , Y _ = G + j B { displaystyle { underline {Z)) = R + jX, quad { underline {Y)) = G + jB}

where p

and
gram
resistance and conductance respectively,
X
is reactance, and
B
is susceptibility.
For ideal resistors Z
and
Y
are reduced to
p
and
g
respectively, but for alternating current networks containing capacitors and inductors,
X
and
B
are non-zero.

Z _ = 1 / Y _ { displaystyle { underline {Z)) = 1 / { underline {Y))} for AC circuits, as well as R = 1 / G { Displaystyle R = 1 / G} for DC circuits.

Frequency dependence of resistance

A key feature of AC circuits is that resistance and conductance can vary with frequency, a phenomenon known as universal dielectric response.[8] One of the reasons mentioned above is the skin effect (and related proximity effect). Another reason is that the resistance itself may depend on frequency (see Drude model, deep level traps, resonant frequency, Kramers–Kronig relations, etc.)

Physical meaning of reactance

In coils and capacitors, when voltage is applied, energy accumulates in the form of magnetic and electric fields, which takes some time.

Magnetic fields in alternating current networks change following the changing direction of movement of charges, while providing additional resistance.

In addition, a stable phase shift in voltage and current occurs, and this leads to additional losses of electricity.

Energy dissipation and Joule heating

Main article: Joule heating

Passing a current through a resistive material causes heating in a phenomenon called Joule heating. This picture shows a cartridge heater, heated by Joule heating, glowing red hot.

Resistors (and other elements with resistance) prevent the passage of electrical 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
.

Dissipation of electrical energy is often undesirable, especially in the case of transmission losses in power lines. High voltage transmission helps reduce losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, such as in electric stoves and other electric heaters (also called resistance heaters

). Another example: incandescent light bulbs rely on Joule heating: the filament is heated to such a high temperature that it glows with white-hot thermal radiation (also called incandescence).

Formula for Joule heating:

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

where n

is the power (energy per unit time) converted from electrical energy to thermal energy,
p is
the resistance, and
i is
the current through the resistor.

Dependence of resistance on other conditions

Temperature dependence

Main article: Electrical resistivity and conductivity § Temperature dependence

Near room temperature, the resistivity of metals generally increases with increasing temperature, while the resistivity of semiconductors generally decreases with increasing temperature. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For a detailed description of the behavior and explanation, see Electrical Resistivity and Conductivity.

As a result, the resistance of wires, resistors and other components often changes with temperature. This effect can be undesirable and cause electronic circuit failure at extreme temperatures. However, in some cases the effect is put to good use. When the 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 commonly used in two ways. Firstly, they can be used as thermometers: By measuring the resistance, you can determine the ambient temperature. Secondly, they can be used in conjunction with Joule heating (also called self-heating): if a large current is passed through a resistor, the temperature of the resistor 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. Typically, self-heating can turn a resistor into a nonlinear and hysteretic circuit element. See Thermistor#Self-heating effects for more details.

If temperature T

not much different, a linear approximation is usually used:
R ( T ) = R 0 [ 1 + α ( T − T 0 ) ] { Displaystyle R ( T ) = R_ {0} [ 1+ alpha ( T-T_{0}) ]}
where α { displaystyle alpha } is the temperature coefficient of resistance

, T 0 { displaystyle T_{0)} is a fixed reference temperature (usually room), and R 0 { displaystyle R_{0)} resistance at temperature T 0 { displaystyle T_{0)} . The parameter α {displaystyle alpha} is an empirical parameter selected based on measurement data. Since the linear approximation is only an approximation, α{displaystyle alpha} differs for different reference temperatures. For this reason, it is common to specify the temperature at which α { displaystyle alpha } was measured with a suffix such as α 15 { displaystyle alpha _ {15)} , and this relationship only holds over a range of temperatures around the reference.[9]

The temperature coefficient α { displaystyle alpha } is typically + 3 × 10−3 K−1 to + 6 × 10−3 K−1 for metals near room temperature. For semiconductors and dielectrics it is usually negative and has very different magnitudes.[e]

Strain Dependence

Main article: Load cell

Just as the resistance of a conductor depends on temperature, the resistance of a conductor depends on voltage. By placing a conductor under tension (a form of stress that results in deformation in the form of stretching of the conductor), the length of the stretched section of the conductor increases and its cross-sectional area decreases. Both of these effects contribute to an increase in the resistance of the stressed section of the conductor. Under compression (deformation in the opposite direction), the resistance of the deformed section of the conductor decreases. See the discussion on strain gauges for details of devices created to exploit this effect.

Dependence on illumination

Main articles: Photoresistor and Photoconductivity

Some resistors, especially those made from semiconductors, exhibit photoconductivity

, which means their resistance changes when light shines on them.
That's why they are called photoresistors
(or
light-dependent resistors
). This is a common type of light detector.

Originals, values ​​and images of sinusoidal currents and voltages

Sinusoidal current and voltage in real form (originals) are written as

where are the vibration amplitudes; — initial phases of oscillations.

Average and effective values ​​of a sinusoidally varying quantity. The average value of current and voltage is taken to be their average value over half a period. Average current

Likewise

RMS current value

Likewise

Instantaneous complex values ​​(images, symbols) of sinusoidal current and voltage:

where is the complex current amplitude; — complex current (complex effective current value): — complex voltage amplitude; — complex voltage (complex effective value of voltage).

Superconductivity

Main article: Superconductivity

Superconductors are materials that have exactly zero resistance and infinite conductivity because they can have V = 0 and I 0. This also means that there is no Joule heating, or in other words, no electrical dissipation. Therefore, if a superconducting wire is turned into a closed circuit, current will flow through it indefinitely. Superconductors require cooling to temperatures of about 4 K with liquid helium for most metal superconductors such as niobium-tin alloys, or cooling to temperatures of about 77 K with liquid nitrogen for expensive, brittle and delicate ceramic high-temperature superconductors. However, there are many technological applications of superconductivity, including superconducting magnets.

see also

• Electronic portal

• Conduction quantum Von Klitzing constant (its inverse)

Footnotes

1. The resistivity of copper is about 1.7 × 10−8 Ωm. See
   Ritter (2004).[4]
2. For the new Energizer E91 AA alkaline battery, the internal resistance changes from 0.9 ohms at –40°C to 0.1 ohms at +40°C.[6]
3. A 60 W light bulb (120 V power in the US) draws an RMS current of 60 W/120 V = 500 mA, so its resistance is 120 V/500 mA = 240 ohms. The resistance of a 60 W light bulb in Europe (230 V network) is 900 Ohms. Filament resistance depends on temperature; These values ​​refer to the moment when the filament is already heated and the light is already on.
4. 100,000 ohms for contact with dry skin, 1000 ohms for contact with wet or damaged skin. High voltage destroys the skin, reducing resistance to 500 ohms. Other factors and conditions are also important. For more information, see Electrical Shock article, and NIOSH 98-131.[7]
5. See Electrical resistivity and conductivity for the table. The temperature coefficient of resistivity is similar to, but not identical to, the temperature coefficient of resistivity. The slight difference is due to thermal expansion and changes in resistor dimensions.

Recommendations

1. ^ a b
   Brown, Forbes T. (2006).
   Dynamics of Engineering Systems
   . CRC Press. paragraph 43. ISBN 978-0-8493-9648-9.
2. ^ a b
   Kaiser, Kenneth L. (2004).
   Handbook of Electromagnetic Compatibility
   . CRC Press. pp. 13–52. ISBN 978-0-8493-2087-3.
3. Fink and Beaty (1923). "Standard Guide for Electrical Engineers". Nature
   (11th ed.).
   111
   (2788): 17–19. Bibcode:1923Natura.111..458R. Doi:10.1038/111458a0. HDL:2027/mdp.39015065357108. S2CID 26358546.
4. Ritter, Bridget. "Facts 2004". hypertextbook.com
   .
5. McDonald, John D. (2016). Design of electrical substations
   (Second ed.). CRC Press. pp. 363ff. ISBN 978-1-4200-0731-2.
6. Battery Internal Resistance (PDF) (Report). Energizer Corp.
7. "Electrocution Deaths of Workers" (PDF). National Institute for Occupational Safety and Health. Publication No. 98-131. Retrieved November 2, 2014.
8. Zhai, Chunpu; Gan, Yixiang; Hanaor, Dorian; Proust, Gwenall (2018). "Voltage-dependent electric transport and its universal scaling in bulk materials." Letters on Extreme Mechanics
   .
   22
   : 83–88. arXiv:1712.05938. doi:10.1016/j.eml.2018.05.005. S2CID 51912472.
9. Ward, M.R. (1971). Electrical engineering
   . McGraw-Hill. pp. 36–40.