Electrical resistance and conductance
The electrical resistance is a measure of the voltage which is required to have a certain strength of electric current flowing through an electrical conductor (resistor) in electrical engineering. As a symbol for electrical resistance is usually R - resistere derived from the Latin word for " resist " - used. The resistance has the SI unit ohm, their unit symbol is the big omega ( Ω ).
Related to the resistance is the electrical resistivity (symbol ρ ). At this size, it is a material constant ( to effect sizes see below). It enables independent of the geometric shape of the executed leader of the resistance property description.
In historical contexts in the article " Ohm's Law " received.
- 2.1 Presentation
- 2.2 conversions
- 2.3 Special Cases
- 2.4 Causes of the complex resistances
- 2.5 Interconnection, equivalent resistance
- 2.6 locus
- 3.1 series
- 3.2 Parallel connection
- 4.1 Negative differential resistance
- 4.2 Positive differential resistance
Ohmic resistance
Basic relationships
An ohmic resistor, an electrical resistor, the resistance value is in the ideal case, regardless of the voltage, current and frequency. At such an ohmic resistance Ohm's Law applies. When plotting the voltage U over the current I in the form of a UI diagram, you get a line through the origin, the relationship is therefore directly proportional to:
Approximation, and with restrictions, an ohmic resistance through a component, in the simplest case a metal wire, can be realized, which usually also simple resistance - see resistance ( device ) - is called.
If current flows through a resistor and voltage drops from the electric power is in accordance with
Converted into heat.
The reciprocal value of the ohmic resistance, ie the proportionality between current and voltage, ie electrical conductance G of a conductor. Thus:
Calculate the resistance of a conductor
The ohmic resistance of a body can be calculated from its geometrical dimensions and material-specific constant, ρ the resistivity calculated.
For carrying a longitudinally straight conductor of constant cross -sectional area A and length l applies:
The resistivity itself is generally dependent on the temperature and possibly other quantities dependent.
Influence Effects
The above- established equations for the direct current resistance of a straight conductor is then replaced for example by
Where the subscript indicates the temperature for which the criteria apply. In table books, the usual reference temperature is 20 ° C. The values depend on the purity as well as thermal and mechanical treatment; the table values are for guidance only.
The influence of temperature on the resistance can be represented in simple cases using the linear temperature coefficient and the temperature difference. Then, describing the relationship by a linear equation
For most applications with metallic materials is not too large temperature ranges, this linear approximation is sufficient; otherwise terms of higher orders are included in the equation. ( An example with addends up to the fourth power see platinum resistance thermometer in the article. )
Depending on whether the resistance with increasing temperature is larger or smaller, a distinction is made between PTC thermistors or PTC ( resistance increases, in principle, for all metals, . Engl positive temperature coefficient) and thermistors or NTC (resistance value decreases; engl negative temperature coefficient. ).
In the measurement and control technology, the temperature dependence of the electrical resistance is exploited, for example, resistance thermometers, thermal anemometers, thermostats or inrush current limiters.
There are also various special alloys, which are characterized by an over wide temperature ranges approximately constant electrical resistivity, as is required for a measurement resistor.
AC resistance
Representation
On a linear, pure resistance R, which is traversed by alternating current, voltage and current have the same phase angle. However, if a frequency-dependent phase shift and change in resistance occurs, is as a percentage of resistance adds a component X which reacts retards voltage or current changes. For a sinusoidal waveform of voltage and current of the quotient of the amplitudes or rms values is referred to as impedance Z. In the complex AC circuit analysis of the impedance is combined with the phase shift angle as impedance or complex resistance.
In another representation, the two components in the complex plane are perpendicular to each other to together,
Herein, R are called effective resistance and reactance X as. The resistance which does not operate phase-shifting is also referred to as a resistive component of the impedance.
Conversions
The voltage and the current as a sinusoidal quantities of frequency or the angular frequency in the complex plane by pointers and illustrated, we obtain
With
By comparing the two representations of obtained
This results in the apparent resistance:
And the phase shift angle between and:
It also referred to
Or
As will be shown below,
Special cases
- For R = 0 gives:
- For X > 0 and;
- For X <0 and.
- For X = 0 gives:
Causes of complex resistances
For a coil with inductance L is considered
Due to a voltage, the current increases with the time. In AC this follows delayed. With the approach in complex notation, and as above is obtained after differentiation
This means that an inductor for sinusoidal alternating quantities acts as a phase shifting reactance. With results.
Likewise, for a capacitor of capacitance C
Due to a current, the voltage increases with time. For AC voltage, this follows delayed. Obtained in complex notation, and after the integration
This means that a capacity for sinusoidal alternating quantities acts as a phase shifting reactance. Here is.
Interconnection, equivalent resistance
As the equivalent resistance of the complex electrical resistance is known which has the same resistance as an electric circuit or part of an electric circuit it replaces. An equivalent resistance can illustrate the behavior of complex electrical assemblies and enable calculation; see also equivalent circuit.
Actually occurring impedances can be described frequently series circuit or parallel circuit of a resistor having an inductance or a capacitance. Which of the images is used is a matter of better approximation to reality as possible with frequency-independent variables and the appropriateness of the mathematical treatment.
On closer inspection, however, any capacitor has a small inductive component, such as a coil also has a capacitive component. Even a piece of wire must exactly be described by R, C and L; see also line covering. This is evident in particular when the components come with their geometric dimensions in the range of the wavelength of the applied AC voltage; then they have a non-negligible both inductive and a capacitive component. You may be the resonant circuit, an example is the antenna mentioned here. The ends may be seen as capacitor plates, the "wire" between them as a coil.
Become an ohmic resistor and a reactance connected together, so in complex notation below, of rules for series and parallel circuit can be applied.
Become a capacitive and an inductive impedance connected together, there arises at sufficiently small resistive load, a resonant circuit; the series and parallel circuit and the further consequences to be dealt with under this heading.
Locus
A demonstration tool for the analysis and description of circuits with AC resistors is the locus.
Complex variables can be represented by phasors in the complex plane. If the complex quantity is a function of a (real) parameter and if this parameter is varied, the tip of the pointer moves. A line through all possible pointer peaks is called locus.
The images show loci of impedance as a function of frequency for the specified circuits. In an RL or RC series circuit comprising a resistor is independent of the frequency and of the active component of the impedance of the frequency -independent. At the corresponding parallel combination of active and reactive component of the impedance are shown, both of the frequency -dependent.
Series and parallel circuit
Series
If resistors are connected in series, then add the resistors:
Can illustrate you look at this in two resistors which differ only in length.
The series gives a resistance body length. Then:
Parallel connection
With the parallel connection of n resistors, the conductance or the reciprocal resistances add:
Alternative spelling:
Formula for two resistors in parallel:
This relationship is illustrated on the parallel connection of two resistors, which only differ in their cross-sectional area.
This gives a resistance from the total cross section, ie the following applies:
And therefore
Is a parallel connection of equal resistors with equal values exist () so the total resistance can be calculated by dividing the standard resistor by the number of resistors in the circuit:
Differential resistance
In non-linear current - voltage characteristics - such as for example diodes - the ratio for each pair of current-voltage is different. In this case, Ohm's Law does not apply and we can not speak of an ohmic resistance. However, small changes in voltage are approximately proportional to the small current changes. The ratio of small voltage change and the associated change of current at a certain voltage is referred to as differential resistance. In a graph is plotted in which it corresponds to the slope of the tangent at the point under consideration of the characteristic.
Negative differential resistance
The differential resistance can be negative in a part of the characteristic curve, so that the current decreases with increasing voltage or the current increases with decreasing voltage.
In the picture which is in the range UP
Positive differential resistance
For positive differential resistance, the current increases with increasing voltage. All real existing circuit elements have in some of their characteristics, but always for very high values , a positive differential resistance. Most of the elements in the circuit technology have an exclusively positive differential resistance.
Examples: real resistance, diode, Zener diode, all semiconducting ceramics.
The electrical resistance in the particle model
The physical description uses the idea that the valence electrons in a metal behave as a gas ( electron gas ). In the simplest model, the metal forms a positively charged homogeneous volume in which the electrons can move freely. In this volume the atomic cores are embedded, consisting of the atomic nucleus and the electrons more strongly bound to the lower, fully occupied shells.
The electrons move without external electrical voltage in disordered metal (see Brownian motion). By putting a voltage on, the free electrons are accelerated by the electric field in the direction of the field lines. Flowing an electrical current.
On their way through the metal it comes to elastic collisions of electrons with other electrons, the atomic cores and phonons. In doing so the electron energy at their collision partners from, are scattered and re-accelerated by the electric field. The electrons are decelerated continuously by this interaction and it turns to a medium flow rate.
The in these shocks transmitted to the atomic cores or phonon energy leads to greater self-oscillation around its equilibrium position, its temperature increases. Of stronger vibrations, the cross -sectional area increased for possible collisions, the number increases with increasing temperature and increase the resistance can ( PTC ). The conduction process in thermistors can not be fully explained by this model, as it is here with increasing temperature to a significant charge carrier generation, which overlie the procedure just described.
At very high temperatures, in which the atoms of the material to be ionized (plasma), any substance is electrically conductive, since the previously bound electrons are now available for the charge transport. Conversely, metals and oxides are known, the electrical resistance disappears at very low temperatures below a specific critical temperature: superconductor having no ohmic resistance in direct current, the current flows at this low temperature with no loss.
By the thermal motion of the electrons results in a temperature-dependent noise current, which is referred to as a resistance noise.
Hall effect
The Hall resistance is the ratio of voltage to current of a Hall element with a given magnetic flux density, which voltage appears across the current density. It characterizes the Hall element or the magnetic flux density, however, has nothing to do with the electrical resistance of this Hall element.
The quantum Hall effect manifested by the fact that at low temperatures and strong magnetic fields the voltage occurring perpendicular to the current density does not increase linearly as the classical Hall effect with the flux density, but in stages. This phenomenon leads to a universal constant of nature, the " von Klitzing constant" of the resistance dimension. Since the Von Klitzing constant can be measured relatively easily, it was proposed to use them as normal for measurements of electrical resistance.