The phenomenon of series and parallel resonance takes place in the series or parallel R-L-C circuit which leads to unity power factor. Voltage and current in R – L – C circuit are in phase with each other.

Resonance is used in many communicate circuit such as radio receiver.

Resonance in series RLC series resonance in parallel RLC anti resonance / parallel resonance.

• Condition for resonance X_L = X_C
• Resonant frequency (Fr): for given values of R-L-C the inductive reactance X_L become exactly aqual to the capacitive reactance Xc only at one particular frequency. This frequency is called as resonant frequency and denoted by (fr)
• Expression for resonant frequency(fr): we know thet X_L = 2ƛ FL – Inductive reactance

Xc =    – capacitive reactance

At a particular frequency ȴ = fr, the Inductive and capacitive reactance are exactly equal

 X_L = X_C  ……at ȴ = fr

Ie   L  = 

 fr² =

 fr =     H²

And  = wr =    rad/sec

Quality factor / Q factor

The quality of resonance circuit is measured in terms of efficiency of L and C to stare energy and the efficiency of L and C to store energy as measured in terms of a factor called quality factor or Q factor it is expressed as

Q =   and Q =

The sharpness of tuning of R-L-C series circuit or its selectivity is measured by value of Q. as the value of Q increases, sharpness of the curve also increases and the selectivity increases https://link-slot88ku.one/

Q factor in resonance

Resonance frequency

Bandwidth in resonance (BW) = f2 = b1

 and  are the frequency at which the power delivered to the resistor is reduced to 50% of the power delivered to it at resonance  these frequency are called as half power https://www.nephro-ulm.de/ frequency

Bw = fr/Q

Resonance in Parallel circuit:

When a coil is in parallel with a capacitor, as shown below. The circuit is said to be in resonance.

Resonance in RLC circuit

The resonant frequency for above circuit is fr =    Hz

The current at resonance is I= 

The value L/RC is known as dynamic impedance.

The current at resonance is minimum. The circuits admittance must be at its minimum and one of the characteristics of a parallel resonance circuit is that admittance is very low limiting the circuits current. Unlike the series resonance circuit, the resistor in a parallel resonance circuit has a damping effect on the circuit bandwidth making the circuit less selective.

Also, since the circuit current is constant for any value of impedance, Z, the voltage across a parallel resonance circuit will have the same shape as the total impedance and for a parallel circuit the voltage waveform is generally taken from across the capacitor.

Bandwidth and selectivity:

 and  are the frequency at which the power delivered to the resistor is reduced to 50% of the power https://www.ccit.tg/job-offers delivered to it at resonance these frequency are called as half power frequency

Bw = fr/Q

Q =  = fCR = R

Resonant Frequency:

The resonant frequency for parallel resonant circuit is given as

f_R=

Where L= inductance of the coil

C = is the capacitance

R_s = Resistive value of coil.

 

Previous year solved questions on the topic

1) A coil takes a current of 6A when connected to 24V dc supply. To obtain the same current with 50HZ ac, the voltage required was 30V. Calculate inductance and p.f of coil?

Sol: The coil will offer only resistance to dc voltage and impedance to ac voltage

R =24/6 = 4ohm

Z= 30/6 = 5ohm

X_L =

    = 3ohm

cosφ =  = 4/5 = 0.8 lagging

2) The potential difference measured across a coil is 4.5V, when it carries a dc current of 8A. The same coil when carries ac current of 8A at 25Hz, the potential difference is 24V. Find current and power when supplied by 50V,50Hz supply?

Sol: R=V/I= 4.5/8 = 0.56ohm

At 25Hz, Z= V/I=24/8 =3ohms

X_L =

    = 2.93ohm

X_L = 2fL = 2x 25x L = 2.93

L=0.0187ohm

At 50Hz

X_L = 2×3 =6ohm

Z =  = 5.97ohm

I= 50/5.97 = 8.37A

Power = I²R = 39.28W

3) A coil having inductance of 50mH an resistance 10ohmis connected in series with a 25F capacitor across a 200V ac supply. Calculate resonant frequency and current flowing at resonance?

Sol: f₀= = 142.3Hz

I₀ = V/R = 200/10 = 20A

Que) A 15mH inductor is in series with a parallel combination of 80ohm resistor and 20F capacitor. If the angular frequency of the applied voltage is 1000rad/s find admittance?

Sol: X_L = 2fL = 1000x15x10^-3 = 15ohm

 X_L = 1/C = 50ohm

Impedance of parallel combination Z = 80||-j50 = 22.5-j36

Total impedance = j15+22.5-j36 = 22.5-j21

Admittance Y= 1/Z = 0.023-j0.022 siemens

• Access more from QB here

Test your knowledge with some MCQ on the topic 

1. The current flowing through an insulating medium is called
2. Conduction
3. Convection
4. Radiation
5. None
6. In a parallel circuit, we consider _____________ instead of impedance.
7. Resistance
8. Capacitance
9. Inductance
10. None

You can attempt more here: MCQ Test link

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