UNIT- 5

RESONANT CIRCUITS

Q.1)Define quality factor and bandwidth. Also establish the relationship between them in a series

resonance circuit. JUNE 2014, JUNE 2013, JUNE 2015

Ans.: Quality factor : The Q factor of series resonant circuit is nothing but the Q factor of inductor or

capacitor in series resonant circuit at resonant frequency. At resonance the reactance of inductor is same

as that of capacitor magnitude wise. Hence the energy stored by both the elements would be same. Q

factor of series resonant circuit is denoted by Q0. At resonance, Q0 is given by

Thus above expression gives quality factor of series resonant circuit in terms of resonant circuit

elements.

Band width: Band width of series RLC circuit is defined as the width of resonant curve upto frequency

at which the power in the circuit is half of its maximum value. At frequency of resonance, the maximum

current I0 is given by,

This current is maximum as impedance is minimum at resonance. Hence at resonant frequency, power

in the circuit is maximum and given by, P0 = Pmax = (I0)2 .R Now, half of maximum power is given

by,

So at the frequencies, where power in the circuit is half of its maximum value, current becomes 1/√ 2

times or 0.707 times of its maximum value. The frequencies at which power in the circuit is half of its

maximum value are called half power frequencies. The band width of series resonant circuit is as shown

in Fig.

At resonant frequency, power in circuit is given by, P0 = Pmax = (I0)2 .R At frequency, power

in circuit is half and it is given by, P’ = ½ (I0)2 . R and Similarly at frequency f2 power in circuit is half

and is given by, P’ = ½ (I0)2 . R Thus, f1 is called lower half-power frequency and f2 is called upper

half power frequency. According to definition of bandwidth, the bandwidth of series RLC circuit is

given by, Bandwidth = (f2-f1)Hz

The half-power frequencies are also referred as 3 dB frequencies or 3 dB points because the

power at these frequencies is 3dB less than that at the resonance. Let us now determine these 3dB

frequencies and evaluate bandwidth and selectively for series resonant circuit.

Relationship between Q factor and Bandwidth The current in a series RLC circuit is given by

equation,

Thus, from above equation it is clear that at half-power frequencies f1 and f2, the reactive part of

impedance of series RLC circuit is equal to resistive part of impedance. We can write,

Selectivity of a resonant circuit is defined as the ability of a circuit to discriminate or distinguish

between desired and undesired frequencies. Selectivity is also defined as the ratio of resonant frequency

to the bandwidth of resonant circuit.

Thus, selectivity of series resonant circuit is directly proportional to the quality factor of circuit at

resonant frequency. So selectivity of the resonant circuit depends on Q0. If Q0 is very high, amplitude

response curve becomes sharper effectively decreasing bandwidth. If Q0 is very small, bandwidth

increases making amplitude response curve flater.

Thus, bandwidth of series resonant circuit is inversely proportional to quality factor Q0 of the circuit at

resonant frequency.

Q.2) A series resonance circuit with R = 17Ω. L = 0.1H and C= 50µF has an applied voltage V=50

0ے ° with a variable frequency. Find the resonant frequency, the value of frequency at which

maximum voltage occurs across inductor and the value of frequency at which maximum voltage

occurs across capacitor. JUNE 2015, JAN.2014

Q.3) Show that resonant frequency of series resonance circuit is equal to the geometric mean of two

half power frequencies. JAN.2015

Thus, f1 is called lower half-power frequency and f2 is called upper half power frequency. According to

definition of bandwidth, the bandwidth of series RCL circuit is given by, Bandwidth = (f2-f1) Hz The

half-power frequencies are also referred as 3 dB frequencies or 3 dB points because the power at these

frequencies is 3 dB less than that at the resonance. Let us now determine these 3 dB frequencies and

evaluate bandwidth and selectively for series resonant circuit.

Thus, from above equation it is clear that at half-power frequencies f1 and f2, the reactive part of

impedance of series RCL circuit is equal is equal to resistive part of impedance. Above equation is

quadratic in ω, which gives two values of ω as ω1 and ω2 whose half-power frequencies (i.e. f1 and f2)

are as shown in Fig. 20. We can write,

Q.4) In the circuit given below in Fig. an inductance of 0.1 H having a Q of 5 is in parallel with a

capacitor. Determine the value of capacitance and coil resistance at resonant frequency of 500 rad/sec.

JAN.2015

Ans.: The given circuit is practical parallel resonant circuit. The antiresonant frequency in terms of the

Q-factor is given by

Q.5) Determine the R-L-C parallel circuit parameters whose response curve is as shown fig. What are

the new values of ωr and band width if c is increased 4 times? JAN.2013

Q.6) In the case of a series resonant circuit with frequency variation, obtain expressions for

i) ωC at which maximum voltage occurs across C

ii) ωL at which maximum voltage occurs across L and show that ω1>ωC JAN.2014

Q.7) Derive the expression for the resonant frequency of the circuit shown in Fig. Also show that the

circuit will resonate at all frequencies if RL = RC = √ L/C JAN.2014, JUNE 2015

The two branches connected in parallel will produce resonance when the resultant current through

combination, i.e. I, is in phase with voltage V as shown in Fig. 10(b). The condition of parallel

resonance is that the impedance of the parallel combination is purely resistive. The condition for

resonance can be derived as follows : The admittance of branch containing L and RL is given by

where far is frequency of resonance. The values of RL and RC are in general vary small. RL is ohmic

resistance of coil or inductor and RC is leakage and dielectric loss resistance of capacitor. Both these are

unavoidable but actually both are very small. If the two resistances are selected such that then the

imaginary term in the equation (1) becomes zero. Thus the reactance is zero. For all frequencies, the

circuit is under unity power factor condition. With the above mentioned condition, antiresonance is

possible at all the frequencies in the circuit.