What is admittance – Admittance definition, theory and formulas

Admitance definition (permeability) – the inverse of impedance, the total electrical conductivity in AC circuits.

where: Y – admittance, Z – impedance.

Admittance is a complex number, its real part is conductance G and imaginary part is susceptance B:

Y = G + jB

The modulus of admittance is given by the formula:

The SI unit of conductance, susceptance and admittance modulus is simens.

In general, admittance is a function of frequency. For direct current its imaginary part is equal to zero and real part is equal to conductance.

Analogous to the concept of conductance in a resistive circuit is the concept of admittance

for an RLC circuit. Admitance is defined as the inverse of impedance.

It is usually denoted by the letter Y, whereby:

Y=1/Z

The permittivity of the capacitor is equal to:

Coil:

On the other hand, the admittance of a resistor is equal to its conductance:

Similarly, the inverse of the reactance X is specially called susceptance. The value of susceptance for a capacitor is equal to:

Whereas for the coil:

The nodal method follows directly from Kirchhoff’s current equations written for all independent nodes in the circuit. The current of each branch of the circuit is expressed in terms of the node potentials. It has been shown that any linear RLC circuit can be described by a matrix equation of nodal potentials of the form:

in which Y is a node matrix of dimension N x N, where N is the number of independent nodes in the circuit, V is a vector of independent node potentials of dimension N and Izr is a vector of source currents that constitute the forcing. The node matrix Y is defined in the form:

The elements Yii located on the principal diagonal matrix Y are called the eigen admittances of the i-th node. For RLC circuits without controlled sources, the eigen admittance of the i-th node is equal to the sum of the admittances of all the branches included in the i-th node. The elements Yij located outside the main diagonal are the mutual admittances between the i-th node and the j-th node. The mutual admittance of two nodes is equal to the admittance connecting these nodes taken with a minus sign. The mutual admittance of the i-th and j-th node is the same as that of the j-th and i-th node, i.e., Yij = Yji . The admittance matrix Y for RLC circuits without controlled sources is thus a symmetric matrix.

Admittance in series

Admittance in parallel

Admittance circuits – two-port network

Definition of two-port network

A two-port network is a four-terminal element having two pairs of ordered terminals, of which one pair is the input and the other pair is the output The designation of a two-port network with the returns of the terminal currents and voltages marked is shown in the following figure.

The condition of equality of currents must be satisfied for the input and output of the two-port network :

as shown in the figure. The current and voltage signals on the input side will be marked with the indicator 1, and on the output side – with the indicator 3. We will conventionally assume that both currents: on the input and on the output are facing the rectangle denoting the two-port network.

Depending on the elements that make up the circuit, the two-port network can be linear (when all the circuit elements are linear) or nonlinear. In the following discussion, we will limit ourselves to linear two-port network only. The two-port network will be called passive, if it does not produce energy but only takes it from the power source and transforms it in a specific way. A two-port network consisting only of passive elements R, L, C and M is always a passive two-port network. A passive two-port network is capable of storing and dissipating the energy drawn from the source, and can also give it away, but at any time instant t this energy cannot exceed the energy drawn. A two-port network that does not meet the above conditions is an active two-port network (energy generator).

Admittance equation

If the voltages of both gates U1 and U2 are taken as independent variables, the two-port network will assume an admittance description that can be expressed in the form.

The matrix Y is called the admittance matrix and the parameters of this matrix have the interpretation of operator admittances.

References:

https://encyklopedia.naukowy.pl/Admitancja

https://elektrotechnikapila.files.wordpress.com/2012/06/teoria_obwodow.pdf

http://www2.wt.pw.edu.pl/~clucyk/ep/ep14r6.pdf

https://esezam.okno.pw.edu.pl/mod/book/view.php?id=27&chapterid=425