Admittance

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See also: Admission

In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). The SI unit of admittance is the siemens. Oliver Heaviside coined the term in December 1887.

$Y = Z^{-1} = 1/Z \,$

where

Y is the admittance, measured in siemens

Z is the impedance, measured in ohms

Note that the synonymous unit mho, and the symbol ℧ (an upside-down Omega Ω), are also in common use.

Then as

$Z = R + iX \,$

$Y = Z^{-1}= \frac{1}{R+iX} = \left( \frac{1}{R+iX} \right) \cdot \left( \frac{R-iX}{R-iX} \right) = \left( \frac{R}{R^2+X^2} \right) + i\left(\frac{-X}{R^2+X^2}\right)$

Admittance, just like Impedance, is therefore a complex number, made up of a real part (the conductance, G), and an imaginary part (the susceptance, B), shown by the equation:

$Y = G + i B \,$

$Y = G + iB = \left( \frac{R}{R^2+X^2} \right) + i \left( \frac{-X}{R^2+X^2} \right)$

Then G (Conductance) and B (Susceptance) are given by:

$G = Re(Y) = \left( \frac{R}{R^2+X^2} \right)$

$B = Im(Y) = \left( \frac{-X}{R^2+X^2}\right)$

The magnitude of admittance is given by:

$\left | Y \right | = \sqrt {G^2 + B^2} = \frac {1} {\sqrt {R^2 + X^2} } \,$

where

G is the conductance, measured in siemens

B is the susceptance, measured in siemens

In mechanical systems (particularly in the field of haptics), an admittance is a dynamic mapping from force to motion. In other words, an equation (or virtual environment) describing an admittance would have inputs of force and would have outputs such as position or velocity. So, an admittance device would sense the input force and "admit" a certain amount of motion.

Similar to the electrical meanings of admittance and impedance, an impedance in the mechanical sense can be thought of as the "inverse" of admittance. That is, it is a dynamic mapping from motion to force. An impedance device would sense the input motion and "impede" the motion with some force.

An example of these concepts is a virtual spring. The equation describing a spring is Hooke's Law,

$F = -kx \,$

If the input to the virtual spring is the spring displacement, x, and the output is the force that the virtual spring applies, F, then the virtual spring would be classified as an impedance. If the input to the virtual spring is the force applied to the spring, F, and the output is the spring displacement, x, then the virtual spring would be classified as an admittance.