Template:Short description Template:About In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance and resistance are defined. The SI unit of admittance is the siemens (symbol S); the older, synonymous unit is mho, and its symbol is ℧ (an upside-down uppercase omega Ω). Oliver Heaviside coined the term admittance in December 1887.<ref>Template:Cite journal</ref> Heaviside used Template:Mvar to represent the magnitude of admittance, but it quickly became the conventional symbol for admittance itself through the publications of Charles Proteus Steinmetz. Heaviside probably chose Template:Mvar simply because it is next to Template:Mvar in the alphabet, the conventional symbol for impedance.<ref>Ronald R. Kline, Steinmetz: Engineer and Socialist, p. 88, Johns Hopkins University Press, 1992 Template:ISBN.</ref>
Admittance Template:Mvar, measured in siemens, is defined as the inverse of impedance Template:Mvar, measured in ohms:
<math display="block">Y \equiv \frac{1}{Z}</math>
Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:
<math display="block">Y = G + j B \,,</math>
where
- Template:Mvar is the admittance (siemens);
- Template:Mvar is the conductance (siemens);
- Template:Mvar is the susceptance (siemens); and
- Template:Math, the imaginary unit.
The dynamic effects of the material's susceptance relate to the universal dielectric response, the power law scaling of a system's admittance with frequency under alternating current conditions.
Conversion from impedance to admittanceEdit
Template:Complex Z The impedance, Template:Mvar, is composed of real and imaginary parts, <math display="block">Z = R + jX \,,</math> where
- Template:Mvar is the resistance (ohms); and
- Template:Mvar is the reactance (ohms).
<math display="block">Y = Z^{-1}= \frac{1}{R + jX} = \left( \frac{1}{R^2 + X^2} \right) \left(R - jX\right) </math>
Admittance, just like impedance, is a complex number, made up of a real part (the conductance, Template:Mvar), and an imaginary part (the susceptance, Template:Mvar), thus:
<math display="block">Y = G + jB \,,</math>
where Template:Mvar (conductance) and Template:Mvar (susceptance) are given by:
<math display="block">\begin{align}
G &= \mathrm{Re}(Y) = \frac{R}{R^2 + X^2}\,, \\
B &= \mathrm{Im}(Y) = -\frac{X}{R^2 + X^2}\,.
\end{align}</math>
The magnitude and phase of the admittance are given by:
<math display="block">\begin{align}
\left | Y \right | &= \sqrt{G^2 + B^2} = \frac{1}{\sqrt{R^2 + X^2}} \\
\angle Y &= \arctan \left( \frac{B}{G} \right) = \arctan \left( -\frac{X}{R} \right)\,,
\end{align}</math>
where
- Template:Mvar is the conductance, measured in siemens; and
- Template:Mvar is the susceptance, also measured in siemens.
Note that (as shown above) the signs of reactances become reversed in the admittance domain; i.e. capacitive susceptance is positive and inductive susceptance is negative.
Shunt admittance in electrical power systems modelingEdit
In the context of electrical modeling of transformers and transmission lines, shunt components that provide paths of least resistance in certain models are generally specified in terms of their admittance. Each side of most transformer models contains shunt components which model magnetizing current and core losses. These shunt components can be referenced to the primary or secondary side. For simplified transformer analysis, admittance from shunt elements can be neglected. When shunt components have non-negligible effects on system operation, the shunt admittance must be considered. In the diagram below, all shunt admittances are referred to the primary side. The real and imaginary components of the shunt admittance, conductance and susceptance, are represented by Template:Math and Template:Mvar, respectively.<ref>Template:Cite book</ref>
Transmission lines can span hundreds of kilometers, over which the line's capacitance can affect voltage levels. For short length transmission line analysis, which applies to lines shorter than Template:Convert, this capacitance can be ignored and shunt components are not necessary in the model. Lines from Template:Convert, generally considered to be in the medium-line category, contain a shunt admittance governed by<ref>J. Glover, M. Sarma, and T. Overbye, Power System Analysis and Design, Fifth Edition, Cengage Learning, Connecticut, 2012, Template:ISBN, Chapter 5 Transmission Lines: Steady-State Operation</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">Y=yl=j\omega Cl\,,</math> where
- Template:Mvar is the total shunt admittance;
- Template:Mvar is the shunt admittance per unit length;
- Template:Mvar is the length of the transmission line; and
- Template:Mvar is the capacitance of the line.
See alsoEdit
ReferencesEdit
<references />Template:Authority control