Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) PDF Download

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)

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

The total impedance of series n/w is given by

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
where

XL = ωL

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
At resonance

lm{z} = 0
X– XC = 0

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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.

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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

Selectivity = Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

 Q-factor :   Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • 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

  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

For series RLC circuit

 BW = ( w2 – w1) =  R/L

Parallel RLC resonnace circuit:

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Admittance of circuit 

  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

At resonance lm (Y) = 0

  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

at resonance
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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    Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
  • 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  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Some Conclusions 

  • Y = G + j(B– BC)

at resonant frequency Ymin = G

or Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • Current will be minimum

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

  • 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

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Current I = VY

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonant frequency is given by

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Case-3: 

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

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Equivalent phasor diagram Fig. (6.7)

In figure 6.7(b),

OA = I1 cosθ

OC = AB = I1 sinθ1
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
and    I1 sinθ1 = I2

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Now at resonance, I = I1 cosθ1

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) 
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)   [From equation (6.28)]
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)    ....(6.29)
Here L/RC is defined as dynamic impedance of tank circuit i.e.

  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

And resonance frequency for tank circuit is given by

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

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

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
⇒ Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

⇒  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)    .   ..(6.32)

put  Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)         ...(6.33)

  • In another way Q-factor can be obtained as

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)
⇒ Q = RωC           ...(6.34)

Q-factor of Tank Circuit

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Tank circuit Fig.(6.9) 

  • Q-factor is given by

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)   ...(6.35)

Anti-resonance Curve 

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

 Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)     

Fig. (6.10)

 

The document Resonance | Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electrical Engineering SSC JE (Technical).
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FAQs on Resonance - Electrical Engineering SSC JE (Technical) - Electrical Engineering (EE)

1. What is resonance in electrical engineering?
Ans. Resonance in electrical engineering refers to the phenomenon where the reactive components of an AC circuit cancel each other out, resulting in a significant increase in current or voltage. It occurs when the inductive reactance and capacitive reactance in the circuit are equal, causing the circuit to resonate at a specific frequency.
2. How does resonance affect electrical circuits?
Ans. Resonance can have both positive and negative effects on electrical circuits. On the positive side, it can be used to amplify signals, as in the case of radio receivers and antennas. On the negative side, resonance can lead to excessive current flow and voltage spikes, which can damage or even destroy components in the circuit.
3. What are the applications of resonance in electrical engineering?
Ans. Resonance finds various applications in electrical engineering. Some of the common applications include the tuning of radio receivers, designing filters, impedance matching, and power factor correction. Resonance is also utilized in the design of electrical transformers and inductive heating systems.
4. How is resonance calculated in electrical circuits?
Ans. Resonance in electrical circuits is calculated using the formula: Resonant frequency (f) = 1 / (2π√(LC)) Where L is the inductance of the circuit and C is the capacitance. By adjusting the values of L and C, the resonant frequency can be controlled.
5. What are the dangers of resonance in electrical circuits?
Ans. Resonance can pose several dangers in electrical circuits. It can lead to excessive current flow, which can cause overheating and damage to components such as wires, transformers, and capacitors. Resonance can also cause voltage spikes, which may exceed the rated voltage of the circuit, leading to equipment failure or even electrical shocks. It is therefore crucial to design circuits that can withstand potential resonance effects and to implement proper protection measures.
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