Admittance

In general, when talking about electrical circuits we only refer to certain properties like voltage developed and resistance offered by circuit. Resistance can be defined as the opposition offered by the electrical circuit for current to flow. In this article, we will introduce a new property known as the admittance of the circuit. It is often taken into account when we want to know how easily the circuit allows current to flow through it. It is basically a term contrary to resistance. In this article, we will discuss what is admittance along with its derivation from the impedance. We will also learn about the admittance triangle and how admittance varies in series and parallel combination circuits. We will also list the components of admittance and provide a comparison between admittance and impedance. This information when put to use can be applied at various places which have been discussed through the applications. We will conclude the article with some points and list some frequently asked questions for reference.

What is Admittance?

Admittance is the reciprocal of impedance and can be defined as the measure of how easily a circuit allows current to flow through it. In circuits involving AC, we have to talk about the phase of current in addition it its magnitude. The term impedance was used for measuring the resistance offered by the circuit along with the phase. Thereby we can say that admittance is the reciprocal of impedance rather than resistance because admittance takes into account the phase of current in the circuit. The formal definition of admittance is

The measure of the flow of current permitted by a circuit along with the phase of the current. It is measured in units of Siemens or Mho(Ʊ).

Derivation of Admittance from Impedance

Admittance is denoted by symbol Y and impedance is denoted by symbol Z .Let us relate the two symbols. Note that impedance has a real and an imaginary part so impedance can be denoted by

Z=R+jX

where

Z is impedance (ohms)

R is resistance (ohms)

X is reactance(ohms)

Now we know admittance is reciprocal of impedance so

Y=1/Z =Z^-1

On putting values

Y=1/(R+jX )

Multiplying and dividing by (R-jX )

Hence,

Y=(R-jX )/(R²+X²)

Components of Admittance

Admittance is also a complex quantity and has a real part known as Conductance (G) and imaginary part known as Susceptance (B).

Y=G+jB

On comparing

Y= Admittance (Siemens)

G= Conductance (Siemens) = R/(R²+X²)

B= Susceptance (Siemens) = -X/(R²+X²)

Magnitude of Admittance

|Y|= √(G²+B²)= 1/√(R²+X²)

Phase of Admittance

∠Y= arctan(B/G)= arctan( -X/R)

Admittance Triangle

Admittance triangle is a mathematical concept which is used fir representing the admittance of the circuit. The triangle has the sides which are used to denote admittance (Y), susceptance (B) and conductance (G) of a circuit. Look at the picture to see how these parameters are represented by a triangle.

Admittance Triangle

We can use this triangle to find individual components of circuit

The base of triangle is represented by conductance and is denoted by

Conductance= Y cos(Φ)

Hence

G=(1/Z)* (R/Z)

∴ G= R/(R²+X²)

The perpendicular of triangle is represented by susceptance and is denoted by

Susceptance= Y sin(Φ)

Hence

G=(1/Z)* (X/Z)

∴ G= X/(R²+X²)

We can also represent the phase as

Tan(Φ)= B/G

Also,

Power factor =cos(Φ)=G/Y

Admittance of a Series Circuit

Let us see the admittance of a circuit in series circuit.

Series Combination of Inductance and Resistance

In this circuit, Resistance and Inductance reactance are connected in series as shown in the circuit

Resistance and inductance in series

X_L is inductive reactance

Then admittance

Y=1/(R+jX_L)

Multiplying dividing by (R-jX_L)

Y=(R-jX_L)/(R²+X_L²)

Hence

Y=R/(R²+X_L²) -jX_L/(R²+X_L²)

Also,

Y=G-jB_L= √(G²+B_L²)

This is the admittance for the series combination

Series Combination of Capacitive Reactance and Resistance

In this circuit, Resistance and capacitive reactance are connected in series as shown in the circuit

Capacitive Reactance and Resistance in series

Let us calculate the admittance for series combination

X_c is capacitive reactance

Then admittance

Y=1/(R-jX_c)

Multiplying dividing by (R+jX_c)

Y=(R+jX_c)/(R²+X_c²)

Hence

Y=R/(R²+X_c²) + jX_c/(R²+X_c²)

Also

Y=G+jB_c= √(G²+B_c²)

Admittance of a Parallel Circuit

Let us study the parallel combination and admittance in such case. For this ,we need to consider two branches connected in parallel where one branch is series combination of Capacitive Reactance and Resistance and other branch is Series combination of inductance and resistance

Parallel Circuit

We will individually analyse the two branches first

For branch A, from previous derived results

Conductance G₁=R₁/(R₁²+X_L²)= R₁/Z₁²

Here Z₁ is impedance in ohms

Inductive susceptance B_L=X_L/(R₁²+X_L²)=X_L/Z₁²

Then Y₁=G₁-jB_L= R₁/Z₁² – jX_L/Z₁²

Similarly for branch B

Conductance G₂=R₂/(R₂²+X_c²)= R₂/Z₂²

Here Z2 is impedance in ohms

Inductive susceptance

B_c=X_c/(R₂²+X_c²)=X_c/Z₂²

Then Y₂=G₂+jB_c= R₂/Z₂² + jX_c/Z₂²

On adding the two admittance in series (parallel admittance is added like series)

Y=Y₁+Y₂

Then Y=G₁-jB_L+ G₂-jB_C

∴ Y=(G₁+G₂)-J(B_L+B_C)

Finally ,Y=(R₁/Z₁²+R₂/Z₂²) -j(X_L/Z₁²-X_C/Z₂²)

This is how we can obtain admittance of the whole circuit.

Difference Between Admittance and Impedance

Let us compare admittance and impedance

	ADMITTANCE	IMPEDANCE
DEFINITION	It is measure of how easily a circuit allows current to flow through it.	It is a measure of the resistance offered by a circuit to current flowing though it.
UNIT	Its SI unit is Siemens of mho.	Its SI unit is ohm.
FORMULA	It is denoted by Y=√(G2+B2)	It is denoted by Z=√(R2+X2)
RELATION WITH OTHER	It is reciprocal of impedance i.e. Y=1/Z.	It is reciprocal of admittance i.e. Z=1/Y.
COMPONENTS	It has two components namely conductance and susceptance.	It has two components namely resistance and reactance.

Applications of Admittance

We have seen how admittance is important for determining the characteristics of a circuit. Let us see its applications

• We know that we can easily calculate the power factor using admittance of the circuit. Hence admittance is often used in power analysis of a circuit. It helps to understand how the reactive power is flowing in the circuit along with its phase. Thereby it helps to determine the stability of system.
• While designing filter, we often use S parameters and require the calculation of impedance through easier methods. Calculation of impedance through admittance is a common method employed for designing and analysing filters in electronics industry.
• Admittance can be used in communication systems especially for transmission lines and antennas. Admittance is used in antenna design as it helps to understand and analyse how efficiently will signal transmission occur.
• Impedance matching is an important phenomena where we use admittance. Impedance matching technique is used to analyse how efficiently is signal being transferred with minimal reflections. Impedance matching is an integral part and is done for almost all components of circuit.
• Admittance has another application in microwave engineering where it is used to determine the nature and behaviour of components like waveguides, filters, and resonators. By measuring the admittance, we can design some high frequency circuits.

Solved Examples of Admittance

Q. For the admittance value of 3+j4 , what will be the impedance of the circuit?

Since impedance(Z)= 1/admittance(Y)

Z=1/(3+j4)

i.e.

Z=(3-4j)/(3²+4²)

∴ Z=(3-4j)/25

Q. Find the phase angle of current if admittance of the series circuit is 5+13j.

For a series circuit with admittance Y=G+jB

tan(Φ)=B/G

On comparing

G=5mho B=13mho

So,tan(Φ)=13/5

∴ Φ=arctan(13/5)

Conclusion

As we have seen that admittance i.e. the reciprocal of impedance is an important quantity used for analysing and designing circuits. The magnitude and phase of impedance are used for determining some important quantities like power reflected in circuits. We have also seen that there are various methods used for calculating the admittance of the circuit depending on whether it is a series or a parallel circuit. We have also seen the difference between admittance and reactance through a tabular format and also analysed the applications of admittance in real-life.

FAQs on Admittance

What is power factor and its formula?

Power factor is a measure of how efficiently power is being transmitted in an AC circuit and it varies between 0 and 1. Power factor is denoted by cos(Φ) where Φ is the phase angle between the voltage and current waveforms.

How are impedance and admittance inter-related?

Impedance is the reciprocal of admittance. While impedance is the resistance to current flow in a circuit, admittance is a measure of how easily current is allowed to flow in a circuit. Mathematically Y=1/Z.

How is admittance different from conductance?

Although admittance and conductance are both used to measure how easily current is allowed to flow in a circuit but conductance doesn’t account for the phase of current. While admittance has a real and an imaginary part, conductance only has a real part denoting the magnitude and no phase.