Electrical Resonance MCQ Quiz - Objective Question with Answer for Electrical Resonance - Download Free PDF

Concept

The resonant frequency of a parallel RLC circuit is given by:

\(f_o={1\over 2\pi \sqrt{LC}}\)

where, f_o = Resonant frequency

L = Inductance

C = Capacitance

Calculation

Given, L = 1 H

C = 1 μF = 1 × 10^-6 F

\(f_o={1\over 2\pi \sqrt{1\times 1\times 10^{-6}}}\)

\(f_o=\rm \frac{1}{2\pi \times 10^{-3}}Hz\)

Quality factor

In a resonant series circuit, the quality factor (Q) is a measure of how underdamped the system is and how sharp the resonance is.

It is given by:

\(QF={1\over R}\sqrt{L\over C}={\omega_oL\over R}\)

From the above expression, it is observed that the quality factor is inversely proportional to capacitance.

If the capacitance in a series RLC circuit is increased, the Q-factor will decrease.

Concept

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

\(QF=R{\sqrt{C\over L}}\)

where, R = Resistance

C = Capacitance

L = Inductance

Explanation

From the above expression, the quality factor is directly proportional to the resistance.

If the resistance (R) in a parallel RLC circuit increases, the quality factor (Q) will increase.

Parallel RLC circuit

The inductive current is given by:

\(I_L={V\over X_L}\)

The capacitive current is given by:

\(I_C={V\over X_C}\)

It is given that I_L > I_C

\({V\over X_L}>{V\over X_C}\)

X_C > X_L

The correct answer is option 1.

Concept

In a resonant series circuit, the quality factor (Q) is a measure of how underdamped the system is and how sharp the resonance is.

It is given by:

\(QF={ω_o\over BW}\)

where, ω_o = Resonance frequency

BW = Bandwidth

Calculation

Given, QF = 100

ω_o = 1 MHz

\(100={10^6\over BW}\)

BW = 1kHz

We have RMS value of source

\({{\rm{V}}_{{\rm{RMS}}}} = \frac{{{{\rm{V}}_{\rm{m}}}}}{{\sqrt 2 }} = 150\)

Then, \({{\rm{V}}_{\rm{L}}} = \sqrt {{\rm{V}}_{{\rm{RMS}}}^2 - {\rm{V}}_{\rm{R}}^2} = \sqrt {{{150}^2} - {{120}^2}}\)

\(\Rightarrow {{\rm{V}}_{\rm{L}}} = 90{\rm{V}}\)

Now,  \({\rm{I}} = \frac{{{{\rm{V}}_{\rm{R}}}}}{{\rm{R}}} = \frac{{120}}{{1{\rm{k}}}} = 120 {\rm{mA}}\)

Again \({{\rm{V}}_{\rm{L}}} = {\rm{I}} \times{\rm{\omega L}}\)

\(\begin{array}{l} \Rightarrow 90 = 0.12 \times 500 \times {\rm{L}}\\ \Rightarrow {\rm{L}} = \frac{{90}}{{0.12 \times 500}} = 1.5{\rm{H}} \end{array}\)

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

In a parallel RLC, \(Q = \frac{R}{{{\rm{\omega L}}}} = \omega RC = \frac{R}{{{X_L}}} = \frac{R}{{{X_C}}}\)

It is defined as, resistance to the reactance of reactive element.

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

\(Q=\frac{{{f}_{r}}}{BW}\)

For parallel RLC Circuit \({{\rm{Q}}_1} = {\rm{R}}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}\)

For Series RLC Circuit \({{\rm{Q}}_2} = \frac{1}{{\rm{R}}}\sqrt {\frac{{\rm{L}}}{{\rm{C}}}}\)

Concept:

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

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

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

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

At resonance, the circuit acts as pure resistive and the impedance equals the resistance.

Calculation:

Resonance frequency (f) = 5000 / 2π Hz

Resonant frequency (ω) = 2πf = 5000 rad/sec

Impedance at resonance (Z) = R = 56 Ω

Quality factor = 25

\(Q = \frac{{\omega L}}{R}\)

\( \Rightarrow 25 = \frac{{5000 \times L}}{{56}}\)

⇒ L = 0.28 H

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

\( \Rightarrow 25 = \frac{1}{{56}}\sqrt {\frac{{0.28}}{C}} \)

⇒ C = 0.143 μF

Parallel Resonance:

The input admittance is given by:

\(Y_{in} = {1 \over R}+j(\omega C-{1\over \omega L})\)

\(Y_{in} = G+j(B_C-B_L)\)

where, Y_in = Admittance

G = Conductance

B_C = Capacitive susceptance

B_L = Inductive susceptance

At resonance, BC = BL 

Hence, Yin = G and is minimum.

Additional Information

Concept:

In series LCR circuits, Resonance is a condition in which the inductive reactance and capacitive reactance are equal and lie opposite in phase, so cancel out each other. Only, resistance is left as impedance.

The frequency at which the series LCR circuit goes into resonance is:

\(f = \frac{1}{{2π }}\sqrt {\frac{1}{{LC}}}\)

At resonance, the impedance of the series LCR circuit is minimum and hence the current is maximum.

A series LCR circuit is as shown:

The phasor diagram is as shown below:

At resonance, X_L = X_C, i.e. the voltage across the inductor and capacitor are equal and opposite. The resultant phasor diagram at resonant frequency will, therefore, be:

The impedance - frequency curve is shown below:

In parallel resonant circuit, inductive reactance (XL) is equal to capacitive reactance (XC). The circuit becomes pure resistive.

At resonance XL = XC.

At parallel resonance, impedance becomes maximum.

At parallel resonance line current Ir = IL cos ϕ

\(\frac{V}{{{Z_r}}} = \frac{V}{{{Z_1}}}X\frac{R}{{{Z_1}}}or\frac{1}{{{Z_r}}} = \frac{R}{{Z_L^2}}\;\)

\(\frac{1}{{{Z_r}}} = \frac{{\frac{R}{L}}}{C} = \frac{{CR}}{L}\left( {as\;Z_L^2 = \frac{L}{C}} \right)\)

Therefore, the circuit impedance will be given as

Parallel RLC circuit:

The characteristic equation is given as,

\({s^2} + \frac{1}{{RC}}s + \frac{1}{{LC}} = 0\)

The bandwidth of a parallel RLC network is given as:

\({\rm{BW}} = \frac{{{{\rm{\omega}}_{\rm{r}}}}}{{\rm{Q}}}=\frac{1}{RC}\)

Observations:

• The bandwidth is inversely proportional to the resistance, i.e. as the resistance increases, the bandwidth decreases.
• The bandwidth is independent of inductance. Hence the bandwidth of the circuit remains the same if L is increased.
• At resonance, the imaginary part of the net impedance becomes zero, which makes the input impedance a real quantity.
• Also, at resonance, the input impedance for a parallel resonant circuit attains a maximum value. This is opposite to a series resonance where at resonance, the impedance attains a minimum value.

Concept:

RLC series circuit:

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:

XL = 1/ωC

ω = 2 π f

ω = angular frequency

f = linear frequency

The magnitude of the impedance is given by:

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

\(V = \sqrt {V_1^2 + {{\left( {{V_2} - {V_3}} \right)}^2}} \)

Where, V₁ = voltage across the resistor

V₂ = voltage across the inductor

V3 = voltage across the capacitor

V = resultant voltage

The current flowing across the series RLC circuit will be:

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

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

∴\(I=\frac{V}{R}\)

Calculation:

V1 = 100 V, V2 = 50 V, V3 = 50 V 

V2 = V3 (As data provided in the question)

So, the circuit is in series resonance.

V = V1 = 100 V

The source voltage is 100 V

Concept:

The graph between impedance Z and the frequency of the parallel RLC circuit:

Here,

f1 is the lower cutoff frequency

f2 is the upper cutoff frequency

fr is the resonant frequency

BW is the bandwidth

Formula:

BW = f2 – f1

\({f_1} = {f_r} - \left( {\frac{{BW}}{2}} \right)\)

\({f_2} = {f_r} + \left( {\frac{{BW}}{2}} \right)\)

Calculation:

Given

Resonant frequency fr = 20 kHz

Bandwidth = 2 kHz

The upper cutoff frequency is given as:

\({f_2} = {f_r} + \left( {\frac{{BW}}{2}} \right)\)

\({f_2} = {20kHz} + \left( {\frac{{2kHz}}{2}} \right)\)

f₂ = 21 kHz

The selectivity/sharpness of the resonance amplifier is measured by the quality factor and is explained in the figure shown below:

Selectivity of a resonant circuit is defined as the ratio of resonant frequency f_r to the half power bandwidth.

Selectivity = resonance frequency/3-dB Bandwidth
\( = \frac{{{f_r}}}{{{f_2} - {f_1}}}\)

Important Points

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

\(Q=\frac{{ω}_{r}}{BW}\)

ωr = Resonant frequency

B.W. = Bandwith of the amplifier