Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Electrical Engineering SSC JE (Technical)

Electrical Engineering (EE) : Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

The document Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electrical Engineering SSC JE (Technical).
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Resonance

It is the condition when the voltage across a circuit becomes in phase with the current supplied to the circuit.
At resonance, the circuit behaves like a resistive circuit.
Power factor of the circuit at resonance becomes = "1"
The resonance may be classified into two groups

1. Series resonnat circuit

2. Parallel resonant circuit.

Series Resonance (RLC series circuit)

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

The total impedance of series n/w is given by

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
where

XL = ωL

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
At resonance

lm{z} = 0
X– XC = 0

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

where f0 is the frequency of resonance in Hertz
At resonance, the current is I0 = V/R

Variation in Z with respect to 'ω'

Note : For ω < ω0 series RLC behaves like RC capacitive circuit.

ω > ω0 , behaves like RL inductive circuit

ω = ω0 , behaves like resistive circuit. 

  • Impedance of series RLC circuit is minimum at res onance frequ ency ω = ω0  at resonant frequency Z = R 
  • Current in series RLC circuit is maximum at resonance frequency

Selectivity and Bandwidth : At frequency of resonance, the impedance of a series RLC circuit is minimum. Hence the current is maximum. As the frequency of the applied voltage deviates on either side of the series resonant frequency, the impedance increases and the current falls. Figure shows the variation of current I with frequency for small values of R. Thus, a series RLC circuit possesses frequency selectivity. The frequencies f1 and f2 at which the current I falls to (1/√2) times its maximum values I0( = V/R) are called halfpower frequencies of 3 - dB frequencies. The bandwidth (f2-f1) is called the halfpower bandwidth or 3-dB bandwidth or simply bandwidth (BW) of the circuit.

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Selectivity of a resonant circuit is defined as the ratio of resonant frequency to the BW. Thus,

Selectivity = Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

 Q-factor :   Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Q of an inductor L with internal resistor R = ωL/R
  • Q of a capacitor C with effective internal resistor R = 1/ωCR
  • Q of a leaky capacitor which is represented by a capacitor C with a high resistance RP in shunt = ωCRP.

Selectivity increases with decreasing bandwidth For series Resonant circuit at ω = ω0

  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

For series RLC circuit

 BW = ( w2 – w1) =  R/L

Parallel RLC resonnace circuit:

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Admittance of circuit 

  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

At resonance lm (Y) = 0

  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

at resonance
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Z = R
Properties of Second-Order parallel RLC 

Resonant Circuit : A circuit consisting of a parallel connection of a resistor R, an inductor L, and a capacitor C is called a second - order parallel resonant circuit.

The important proporties of such a circuit are as follows:

  • The resonant frequency is    Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
  • Below resonance, the circuit acts like an RL circuit. 
  • Above resonance, the circuit acts like an RC circuit. 
  • If the conductance is zero (the resistnace is infinite) ate resonance the circuit acts like an open circuit. 
  • The bandwidth is 1/RC. 
  • The quality factor QOP is  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Some Conclusions 

  • Y = G + j(B– BC)

at resonant frequency Ymin = G

or Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Current will be minimum

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

  • Power factor will be unity i.e. cos ø = 1. 
  • In parallel resonance net reactive current is equal to zero 
  • Under resonance condition current flowing through inductor or capacitor is greater than total current.
  • This phenomenon is called as current magnification. 
  • Parallel resonant circuit is also called as antiresonant circuit.

Case-2: 

Consider the circuit shown in figure

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Equivalent phasor diagram Fig. (b)

at resonance I1 sinθ1 = I2 sinθ2
In figure (b),

OA = I1 cosθ1

OC = AB = I1 sinθ1

OK = I2 cosθ2

OM = FK = I2 sinθ2
Corresponding to figure (a)

Y = (G1 + G2) + j(BC – BL)
At resonance, BC = BL
Y = G1 + G2
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Current I = VY

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Resonant frequency is given by

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Case-3: 

This is a very important case. Observe the tank circuit shown in figure (6.7).

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Equivalent phasor diagram Fig. (6.7)

In figure 6.7(b),

OA = I1 cosθ

OC = AB = I1 sinθ1
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
and    I1 sinθ1 = I2

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Now at resonance, I = I1 cosθ1

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev 
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev   [From equation (6.28)]
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev    ....(6.29)
Here L/RC is defined as dynamic impedance of tank circuit i.e.

  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

And resonance frequency for tank circuit is given by

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Q.factor for Parallel Resonance Circuit 

  • Q-factor for parallel resonant circuit is defined as the ratio of current flowing through inductor (or capacitor) to total current

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
⇒ Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

⇒  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev    .   ..(6.32)

put  Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev         ...(6.33)

  • In another way Q-factor can be obtained as

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev
⇒ Q = RωC           ...(6.34)

Q-factor of Tank Circuit

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Tank circuit Fig.(6.9) 

  • Q-factor is given by

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev

Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev   ...(6.35)

Anti-resonance Curve 

  • Anti-resonance curve is shown in figure (6.10)

 Chapter 5 Resonance - Notes, Circuit Theory, Electrical Engineering Electrical Engineering (EE) Notes | EduRev     

Fig. (6.10)

 

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