Reactance and impedance

Introduction

An LCR series circuit is a circuit that contains resistance, capacitance and inductance in series across an alternating current. 

Reactance

The inertia against the flow of electrons in an electrical circuit is called reactance. It is also called an imaginary resistance of an electrical circuit. It is measured in ohm and denoted as R.

The motion of electrons generates the current. The force against this motion is called reactance. This reactance is generated by inductance or capacitance. The reactance resists the changing current in a magnetic field and changing voltage in an electric field. It is measured in ohm and denoted as X.

Types of Reactance

There are two types of reactance

• Capacitive reactance (XC)
• Inductance reactance (XL)

Capacitive reactance

The capacitor is a device that passes alternating current. The normal circuit resistance present in any conductor opposes the alternating current, but capacitors oppose the current flow. At the given voltage and frequency, the electrons return to the plate and storage is limited. This storage ability is called the capacitance of the capacitor. As the capacitance increases, the electrons on the plate also increase. Thus, the current flow increases.

The opposition effect by capacitor increases when the frequency increases. Hence, it is inversely proportional. This is called capacitive reactance. The capacitive reactance decreases when the frequency increases. At a given frequency, the capacitive reactance decreases with the increased capacitance.

It is measured in ohm and denoted as XC

XC = 1/2fC

where, Xc = capacitive reactance in ohms

f – frequency in Hertz

C – capacitance in farads

2 – constant

Phasor term of reactance

X = XL + XC

Inductive reactance

When the current flowing through an inductor continuously reverses itself in a source, the effect of magnitude is greater than the direct current. When the amount of inductance (L) and reverse current increase, the opposition of the inertia effect will increase. This opposing force that the inductor presents to the flow of alternating current is called inductive reactance. It is measured in ohms and represented as XL.

The induced voltage is directly proportional to the magnetic lines of force in the conductor. When the rate of frequency is increased, the electromotive force will increase. The induced voltage increases with the increased inductance. When the ampere-turns are higher, the emf will be greater. Reactance will increase when the frequency and inductance increase.

Inductive reactance (XL) = 2fL

Where XL = inductive reactance in ohms

 2 = constant

f = frequency in Hertz

L= inductance in Henry

Reactance in LCR circuit

In an LCR series circuit, the capacitive and inductance reactance have opposing effects. The capacitive reactance causes the current to lead the voltage. In contrast, the inductance reactance causes the current to lag voltage. When both are combined in the circuit, the capacitive and inductance reactance difference will be the resulting effect. This is called reactance.

X = XL – XC  (or)   X = XC – XL

Problem

A circuit contains an inductor of 120 H in series with a capacitor of 0.001 F and an operating frequency of 3 MHz. What is the value of net reactance?

Solution

Given , L = 120 H, C = 0.001 F  , f = 3 MHz

We know , (XL) = 2fL

XL = 2 * 3.14  * 4 MHz * 120 H 

   =  3014

XC = 1/ 2fC

  = 1/ (2 * 3.14 * 4 MHz * 0.001 F )

              = 39.80

X = XL – XC

   =3014 – 39.80 

   = 2974.2

The net reactance is 2974.2

Impedance

Impedance is a property that opposes the flow of electrons in an electrical circuit. It is a combination of both reactance and resistance. It is represented as Z and is measured in ohm.

Phasor term of impedance

Z = R + jX

Impedance in LCR circuit

The combined opposition of both resistance and reactance is called impedance. It is denoted by Z. The impedance cannot be calculated by adding the resistance and reactance. As it is an AC circuit, the R, C, L will step in and reach maximum values simultaneously.

The larger the reactance than the resistance, the more the phase difference is near 90°. The smaller the reactance than the resistance, the more the phase difference is near 0°.

The resistance and reactance can be found using the construction of a right-angle triangle.

The magnitude of the current depends on the frequency in the LCR circuit. When Z is maximum, I will be minimum.

Impedance , Z = 1 / R2 + (L – 1/C)2

Resonance

The system with the tendency to oscillate at a greater amplitude at some particular frequencies is termed resonance. The particular frequency at which the system can oscillate normally is called natural frequency.

Resonant frequency

The frequency with increased amplitude is called the resonant frequency.

R = 1 / LC 

Example : Swing

Initially, the swing oscillates in natural frequency. When someone pushes the swing forcefully, then the amplitude of the oscillation increases, resulting in resonant frequency.

Resonance of LCR series circuit

The amplitude will be maximum at the resonant frequency. Resonance is determined when both the L and C are in the circuit.

The current amplitude is given as Im = ( Vm/Z )

 At resonance,  Im would be maximum and Z would be minimum.

Im=  Vm/(R2 + ( XC – XL )2 

So , 1/C =L

R= ( 1/ LC ) , which is the resonant frequency.

Conclusion

We can provide the opposition force against the current flowing through the electrical circuit by reactance and resistance. Impedance is used to maintain the current flow in a circuit. Hence, both play an important role in circuit theory. Moreover, resonance is important in establishing a stable frequency in an electrical circuit to produce AC signals.