Series Resonance MCQ Quiz in मल्याळम - Objective Question with Answer for Series Resonance - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

The quality factor Q is defined as the ratio of the resonant frequency to the bandwidth.

\(Q=\frac{{{f}_{r}}}{BW}=\frac{1}{R}\sqrt{\frac{L}{C}}\)

The sharpness of the resonance in RLC series resonant circuit is measured by the quality factor and is explained in the figure shown below:

Observations:

• Less the Bandwidth, more the Quality factor.
• More the Bandwidth, less is the Quality factor.
• For the above figure, B₂ > B₁, so Q₂ < Q₁

Concept:

For a series RLC circuit, the net impedance is given by:

Z = R + j (XL - XC)

XL = Inductive Reactance given by:

XL = ωL

XC = Capacitive Reactance given by:

X_C = 1/ωC

The magnitude of the impedance is given by:

\(|Z|=\sqrt{R^2+(X_L-X_C)^2}\)

The current flowing across the series RLC circuit will be:

\(I=\frac{V}{|Z|}\)

The quality factor is defined as the ratio of the maximum energy stored to maximum energy dissipated in a cycle

\(Q = 2π \frac{{Maximum\;energy\;stored}}{{Total\;energy\;lost\;per\;period}}\)

The quality factor in a series RLC circuit is given by:

\(Q = \frac{1}{R}\sqrt {\frac{L}{C}}\)

\(Q = \frac{{ω L}}{{R}}\)

Calculation:

R = 20 Ω, L = 10 mH

I_max = 2 A, f = 1000 Hz

\(Q = \frac{{2\pi \times 1000 \times 10 \times {{10}^{ - 3}}}}{{20}}\)

= π

= 3.14

Half power frequencies:

Half power frequencies are the frequencies at which current is \(\frac{1}{{\sqrt 2 }}\) times of maximum current.

The half power frequencies are shown in the figure below.

f0 is the resonant frequency

f1 is the lower half power frequency

f2 is the higher half power frequency

The relation between them is, \({f_0} = \sqrt {{f_1}{f_2}}\)

Bandwidth (BW) \(= \frac{{{f_0}}}{Q} = {f_2} - {f_1} = \frac{R}{L}\)

Where Q is the quality factor

\({f_1} = {f_0} - \frac{{BW}}{2}\)

\({f_2} = {f_0} + \frac{{BW}}{2}\)

Calculation:

Resonant frequency (f₀) = 100 kHz

Quality factor (Q) = 50

Bandwidth (BW) = 100/50 = 2 kHz

Lower half-power frequency = 100 – 1 = 99 kHz

Concept:

For a series RLC circuit, the net impedance is given by:

Z = R + j (XL - XC)

At resonance, XL = XC, resulting in the net impedance to be minimum. This eventually results in the flow of the maximum current.

∴ The condition for the resonance is:

X _L = X_C

Calculation:

Given inductive reactance is 100 Ω and capacitive reactance is 200 Ω

To operate the circuit at the resonance then we have to make both the reactance equal.

As the capacitive reactance is more we have to reduce it which means we have to increase the capacitance. (X_c = 1 / 2πfC)

In order to increase the capacitance, another capacitor should be added in parallel to another capacitor. 

Let the initial capacitor be C₁ and added capacitor be C₂ and the net capacitance of these 2 will be the C. 

\(\rm X_{C_1} = 200\; \Omega = \frac{1}{2π fC_1}\)

\(\rm C_1 = \frac{1}{2 π f 200} = \frac{1}{2 π \times 50 \times 200} = 15.9 μ F\)

At resonance

\(\rm X_L = X_C = 100 = \frac{1}{2π fc}\)

\(\rm C = \frac{1}{2 π f 100} = 31.83\ μ F\)

C = C₁ + C₂ = C₂ + 15.9 μF = 31.83 μF

C₂ = 15.9 μF

The value of added capacitor is 15.9 μF.

The maximum voltage drop across the Capacitor and as well Inductor when the circuit operates at the resonance frequency.

The resonant frequency of the RLC Series circuit is given by,

\(f_r=\dfrac{1}{2\pi\sqrt{LC}}\)

While, In a series RLC, maximum voltage across capacitor alone occurs at a frequency slightly below the resonant frequency.

This frequency is given by,

f_c = \(\dfrac{1}{2\pi}\sqrt{{\dfrac{1}{LC}}-\dfrac{R^2}{2L}}\)

Additional Information

At the series resonance frequency, the inductive reactance (XL­) is equal to the capacitive reactance (XC).

The impedance in an RLC series circuit is,

\(Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \)

At the resonance conditions, impedance is equal to resistance.

So, the total applied voltage appears across the resistance.

Concept:

The quality factor is defined as the ratio of the maximum energy stored to the maximum energy dissipated in a cycle.

Q = 2π (Maximum energy stored/total energy lost per period)

Q = Resonant frequency/Bandwidth

In a series RLC, 

Bandwidth = \(\frac{R}{L}\)

Resonant frequency = \(\frac{1}{\sqrt {LC}}\)

The quality factor of the series RLC circuit is given as:

\(Q=\frac{L}{R~\sqrt{LC}}\)

\(Q = \frac{1}{R}\sqrt{\frac{L}{C}}\)

Calculation:

The capacitance of the circuit is increased three times and the frequency is slashed by four times.

C₂ = 3 C₁ and ω₂ = 0.25 ω₁

Resonant frequency \({\omega _0} = \frac{1}{{\sqrt {LC} }}\)

\(\frac{{{\omega _1}}}{{{\omega _2}}} = \sqrt {\frac{{{L_2}{C_2}}}{{{L_1}{C_1}}}} \)

⇒ L₂ = (16/3) L₁

New quality factor

\( = \frac{1}{R}\sqrt {\frac{{\frac{{16}}{3}{L_1}}}{{3{C_1}}}} = \frac{4}{3} \times \frac{1}{R}\sqrt {\frac{{{L_1}}}{{{C_1}}}} \)

⇒ Q₂ = (4/3) Q₁ = 1.33 Q₁

Additional InformationWe can also use the formula of Quality Factor as:

\(Q=\frac{1}{\omega RC}\)

Here also by putting the new values of frequency and capacitor, the change in quality factor can be calculated.

Concept:

Damping Ratio for series R-L-C network

\(ζ = \frac{R}{2}\sqrt {\frac{C}{L}} \)

R = resistance in Ω 

C = capacitance in F

L = Inductance in H

ζ = damping ratio

Calculation:

Given that 

R = 1 Ω  , L = 2 mH, C = 5 μF

C → 1000C

\(ζ = \frac{{{1}}}{2}\sqrt {\frac{{5 \times 1000 \times {{10}^{ - 6}}}}{{2 \times {{10}^{ - 3}}}}} \)

ζ = 0.79 = 0.8 < 1

so the damping is under damping

Additional Information

Concept:

The quality factor is defined as the ratio of the maximum energy stored to maximum energy dissipated in a cycle

\(Q = 2\pi \frac{{Maximum\;energy\;stored}}{{Total\;energy\;lost\;per\;period}}\)

In a series RLC, the quality factor is:

\(Q = \frac{{\omega L}}{R} = \frac{1}{{\omega RC}} = \frac{{{X_L}}}{R} = \frac{{{X_C}}}{R}\)

A COIL is a combination of Inductor in series with a resistor

Application:

The quality factor of an inductive coil is inversely proportional to the coil's resistance.

Concept:

• For a series RLC circuit at resonance, the inductive reactance becomes equal to the capacitive reactance.
• Thus, at the time of resonance, the circuit behaves as a purely resistive circuit.
• The value of resonant frequency is given by:

           \(ω_c=\frac{1}{\sqrt{LC}}~rad/sec\)

Calculation:

For the given question; R = 50 Ω, L = 0.4 H, C = 10 μF

\(ω_c=\frac{1}{\sqrt{LC}}~rad/sec\)

\(ω_c= \;\frac{1}{{\sqrt {0.4 \times 10 \times {{10}^{ - 6}}}}}\)

ω_c = 500 rad/sec

Since the circuit at resonance is purely resistive, the current will be given by:

\(\frac{V}{R}=\frac{100}{50}=2~A\)

Q-factor or Quality factor:

• It is defined as the ratio of 2π times of the peak energy stored in the resonator in a cycle of oscillation to the energy lost per radian of the cycle.
• It is a dimensionless parameter
• It describes how underdamped an oscillator or resonator is.

Analysis:  

Generally, A coil is a combination of resistance (R) and inductance(L)

Q-factor of a coil is

Q = ωL / R 

Power factor of a coil is

cos ∅ = R / Z

Z = √(R² + (ωL)² )

tan ∅ = ωL / R 

As Quality factor increases then tan ∅ increases

That means, ∅ increases

∴ The Power factor of the coil decreases.

Points to remember:

• Q- factor = ω₀ / (ω₂ - ω₁)

          ω₀ = Resonant frequency in rad/sec

          ω2 - ω1 = Band width in rad/ sec

• Q = 1/2ζ 

           ζ = Damping frequency