The electrical resistance of an electrical conductor is the opposition to the passage
of an electric current through that conductor. The inverse quantity is electrical conductance,
the ease with which an electric current passes. Electrical resistance shares some conceptual
parallels with the mechanical notion of friction. The SI unit of electrical resistance is the
ohm, while electrical conductance is measured in siemens.
An object of uniform cross section has a resistance proportional to its resistivity and length
and inversely proportional to its cross-sectional area. All materials show some resistance,
except for superconductors, which have a resistance of zero.
The resistance of an object is defined as the ratio of voltage across it to current
through it, while the conductance is the inverse:
For a wide variety of materials and conditions, V and I are directly proportional to each
other, and therefore R and G are constant. This proportionality is called Ohm's law,
and materials that satisfy it are called "Ohmic" materials.
In other cases, such as a diode or battery, V and I are not directly proportional, or
in other words the I–V curve is not a straight line through the origin, and Ohm's law does
not hold. In this case, resistance and conductance are less useful concepts, and more difficult
to define. The ratio V/I is sometimes still useful, and is referred to as a "chordal resistance"
or "static resistance", as it corresponds to the inverse slope of a chord between the
origin and an I–V curve. In other situations, the derivative may be most useful; this is
called the "differential resistance".
Introduction
In the hydraulic analogy, current flowing through a wire 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.
The voltage drop, 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.
Two properties—geometry and material—mostly determine the resistance and conductance of
a wire, resistor, or other element. 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 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. In a similar way, electrons can flow freely
and easily through a copper wire, but cannot as easily flow 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. Conductors and resistors
Substances electricity can flow through 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
Ohm's law is an empirical law relating the voltage V across an element to the current
I through it:
(V is directly proportional to I). This law is not always true: For example, it is false
for diodes, batteries, etc. However, it is true to a very good approximation for wires
and resistors. Materials or objects where Ohm's law is true are called ohmic, whereas
objects that do not obey Ohm's law are non-ohmic. Relation to resistivity and conductivity
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
where is the length of the conductor, measured in metres [m], A is the cross-section area
of the conductor measured in square metres [m²], σ is the electrical conductivity measured
in siemens per meter, and ρ is the electrical resistivity of the material, measured in ohm-metres.
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: . Resistivity is a measure of the material's ability to oppose electric
current. This formula is not exact: 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, because the skin effect inhibits current flow near
the center of the conductor. Then, 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, 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. Resistivity varies with temperature. In semiconductors,
resistivity also changes when light is shining on it. These are discussed 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
Static and differential resistance
Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These
are called non-ohmic or nonlinear, and are characterized by an I–V curve, which is
not a straight line through the origin. Resistance and conductance can still be defined
for non-ohmic elements. However, unlike ohmic resistance, nonlinear resistance is not constant
but varies with the voltage or current through the device; its operating point. There are
two types: Static resistance - This corresponds to the
usual definition of resistance; the voltage divided by the current
. 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 - Differential resistance is the derivative of the voltage with respect
to the current; the slope of the IV curve at a point
. If the IV curve is nonmonotonic, 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
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. But for
a capacitor or inductor, the maximum current flow occurs as the voltage passes through
zero and vice-versa. Complex numbers are used to keep track of both the phase and magnitude
of current and voltage:
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,
is the angular frequency of the AC current, is the imaginary unit.
The impedance and admittance may be expressed as complex numbers that can be broken into
real and imaginary parts:
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. for AC circuits, just as 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.
Another reason is that the resistivity itself may depend on frequency
Energy dissipation and Joule heating
Resistors 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, 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. 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.
The formula for Joule heating is:
where P is the power 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 behavior and explanation, see Electrical resistivity and conductivity.
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. 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: 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. For more details see Thermistor#Self-heating effects.
If the temperature T does not vary too much, a linear approximation is typically used:
where is called the temperature coefficient of resistance, is a fixed reference temperature,
and is the resistance at temperature . The parameter is an empirical parameter fitted
from measurement data. Because the linear approximation is only an approximation, is
different for different reference temperatures. For this reason it is usual to specify the
temperature that was measured at with a suffix, such as , and the relationship only holds
in a range of temperatures around the reference. The temperature coefficient 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, 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, 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. 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 NbSn
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.
See also
References
External links The Notion of Electrical Resistance. Review
of the equations that determine the value of electrical resistance.
Clemson Vehicular Electronics Laboratory: Resistance Calculator