Concept of Impedance & Admittance

Impedance

AC circuits contain both resistance and reactance that are combined together to give a total impedance (Z) that limits current flow around the circuit.
Impedance is not equal to the algebraic sum of the resistive and reactive ohmic values as a pure resistance and pure reactance are 90 degree out of phase with each other.
But we can use this 90 degree phase difference as the sides of a right angled triangle, called an impedance triangle, with the impedance being the hypotenuse as determined by Pythagoras theorem.
This geometric relationship between resistance, reactance and impedance can be represented visually by the use of an impedance triangle as shown.

Impedance, which is the vector sum of the resistance and reactance, has not only a magnitude (Z) but it also has a phase angle, which represents the phase difference between the resistance and the reactance.

Resistance, denoted R, is a measure of the extent to which a substance opposes the movement of electrons among its atoms.
The more easily the atoms give up and/or accept electrons, the lower the resistance, which is expressed in positive real numbers ohms.
Resistance is observed with alternating current and also with direct current (DC).
Examples of materials with low resistance, known as electrical conductors, include copper, silver, and gold.
High-resistance substances are called insulators and include materials such as polyethylene, mica, and glass.
A material with an intermediate levels of resistance is classified as Semiconductor . Examples are Silicon, germanium, and GaAs(Gallium Arsenide).

Reactance, denoted X, is an expression of the extent to which an electronic component, circuit, or system stores and releases energy as the current and voltage fluctuate with each AC cycle.
Reactance is expressed in imaginary number ohms.
It is observed for AC, but not for DC.
When AC passes through a component that contains reactance, energy might be stored and released in the form of a magnetic field, in which case the reactance is inductive (denoted +jX_L); or energy might be stored and released in the form of an electric field, in which case the reactance is capacitive (denoted –jX_C).
Reactance is conventionally multiplied by the positive square root of -1, which is the unit imaginary number called the j operator, to express Z as a complex number of the form R + jX_L (when the net reactance is inductive) or R – jX_C (when the net reactance is capacitive).
Where X_L = ωL, ω is the frequency in rad/sec, and ω = 2πf , f is the frequency in Hz & X_C =1/ωC

The illustration shows a coordinate plane modified to denote complex-number impedances.
Resistance appears on the horizontal axis, moving toward the right.(The left-hand half of this coordinate plane is not normally used because negative resistances are not encountered in common practice.)
Inductive reactance appears on the positive imaginary axis, moving upward.
Capacitive reactance is depicted on the negative imaginary axis, moving downward.
As an example, a complex impedance consisting of 4 ohms of resistance and +j5 ohms of inductive reactance is denoted as a vector from the origin to the point on the plane corresponding to 4 + j5.

In series circuits, resistances and reactance are add together independently.
Suppose a resistance of 100.00 ohms is connected in a series circuit with an inductance of 10.000μHenry. At 4.0000 MHz, the complex impedance is:

Z_RL = R + jX_L = 100.00 + j251.33

If a capacitor of 0.001nF is put in place of the inductor , the resulting complex impedance at 4.0000 MHz is:

Z_RC = R – jX_C = 100.00 – j39.789

If all three components are connected in series, then the reactances add, yielding a complex impedance of:

Z_RLC = 100 + j251.33 – j39.789 = 100 + j211.5

Parallel RLC circuits are more complicated to analyze than are series circuits.
To calculate the effects of capacitive and inductive reactance in parallel, the quantities are converted to inductive susceptance and capacitive susceptance.
Susceptance is the reciprocal of reactance.
Susceptance combines with conductance , which is the reciprocal of resistance, to form complex admittance, which is the reciprocal of complex impedance.

Admittance is defined as a measure of how easily a circuit or device will allow current to flow through it.
Admittance is the reciprocal (inverse) of impedance.
The term impedance is introduced which have the same function of resistance but have both magnitude and phase.
Its real part is resistance, and the imaginary part is reactance, which came from the impeding mechanism.
When looking at admittance vs impedance, admittance is the inverse (i.e. the reciprocal) of impedance.
Therefore it has the opposite function of impedance.
Impedance consists of real part (resistance) and imaginary part(reactance) .
The symbol for impedance is the Z symbol, and the symbol for admittance is the Y symbol.

Admittance is also a complex number as impedance which is having a real part, conductance(G) and imaginary part, Susceptance (B).

Admittance Triangle is formed by admittance (Y), susceptance (B) and conductance (G) as shown below

When a circuit consist of Resistance and Inductance reactance in series is considered as shown below.

When the circuit consist of Resistance and Capacitive reactance in series is considered as shown below.

A circuit which consist of two branches say A and B are considered as shown in figure below. ‘A’ comprises of an inductive reactance, X_L and a resistance, R₁ and ‘B’ comprises of a capacitive reactance, X_C and a resistance , R₂.
The voltage , V is applied to the circuit.

For Branch A

For Branch B