Measurement of Inductance MCQ Quiz in मल्याळम - Objective Question with Answer for Measurement of Inductance - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

• Maxwell’s bridge is used for the measurement of inductance
• Hay’s bridge is used for the measurement of inductance of high ‘Q’ coils.
• Schering bridge is used for the measurement of capacitance and loss angle
• Wien bridge is used for the measurement of frequency

Important Points

Here R₁ and L₁ are the unknown resistance and inductance of the coil.

At bridge balance condition,

Z₁Z₄ = Z₂Z₃

\( \Rightarrow \left( {{R_1} + j\omega {L_1}} \right)\left( {\frac{{{R_4} + j\omega {c_4}}}{{{R_4} + \frac{1}{{j\omega {c_4}}}}}} \right) = {R_2}{R_3}\)

\( \Rightarrow \left( {{R_1} + j\omega {L_1}} \right)\left( {\frac{{{R_r}}}{{{R_4}j\omega {C_4} + 1}}} \right) = {R_2}{R_3}\)

\( \Rightarrow {R_2}{R_3}{R_4}\;j\omega {c_4} + {R_2}{R_3} = {R_1}{R_4} + j\omega {L_1}{R_1}\)

\( \Rightarrow {R_1} = \frac{{{R_2}{R_3}}}{{{R_4}}},\;{L_1} = {R_2}{R_3}{C_4}\)

From the above expression, it is clear that unknown inductance (L₁) is in terms of known capacitance C₄. 

Hay’s bridge:

• In Hay’s bridge, a resistance is connected in series with a standard capacitor.
• Hay’s bridge is used for the measurement of the inductance of coils with high-quality factors.

Maxwell’s inductance-capacitance bridge:

In Maxwell’s inductance bridge resistance is connected in parallel with the standard capacitor.

Wein bridge:

The circuit diagram of the Wein bridge oscillator is shown below:

The phase angle criterion for oscillation is that the total phase shift around the circuit must be 0°.

The total phase shift around circuit 0° occurs when the bridge gets balanced, which is at resonance.

∴ The frequency of oscillation is given by:

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

Schering Bridge:

Schering bridge is used to measure the dissipation factor and capacitance.

The detectors commonly used for AC bridges are:

Headphones: These are widely used as detectors at audio frequencies of 250 Hz and over up to 3 or 4 kHz. They are most sensitive detectors for this frequency range.

When working at a single frequency a tuned detector normally gives the greatest sensitivity and discrimination against harmonics in the supply.

Vibration galvanometers: These are extremely useful for power and low audio frequency ranges. These are manufactured to work at various frequencies ranging from 5 Hz to 1000 Hz but are most commonly used below 200 Hz as below this frequency they are most sensitive than the headphones.

Tuneable amplifier detectors: The transistor amplifier can be tuned electrically and thus can be made to respond to a narrow bandwidth at the bridge frequency. This detector can be used, over a frequency range of 1 kHz to 100 kHz.

Analysis:

For the figure shown, under the balanced condition

\({Z_x}\left( {\frac{{400}}{{\left( {j\omega } \right)\left( {0.05} \right) \times {{10}^{ - 6}} \times 400 + 1}}} \right) = 200 \times 750\;\)

\({Z_x} = \frac{{750}}{2}\left( {1 + j\;\left( {0.2} \right)\left( \omega \right) \times {{10}^{ - 4}}} \right)\)

(Rx + jωLx) = 375 + j 75 × 10-4 ω

Now compare real and imaginary term

Rx = 375 Ω

Lx = 7.5 mH

Explanation:

Hay’s bridge is used for measurement of inductance of coils with high quality factor i.e. inductive impedance with large phase angle.

Important Points

Q-Meter:

The instrument which measures the quality factor of the storage element that is inductor or capacitor at certain frequencies is known as the Q-meter.

The quality factor is the indication of energy storage and energy dissipation in the Transient element due to the dissipation element.

Q-factor for Capacitance,

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

Q-factor for Inductance,

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

Application:

Given,

R = 10 Ω

C = 65 pF

f = 1 MHz

R_in = 0.02 Ω

Let Quality factor is (Q_f1) without insertion resistance and is given by,

\(Q_{f1}=\frac{1}{\omega RC}=\frac{1}{2\pi \times 10^6 \times 10 \times 65\times 10^{-12}}=244.9\)

When insertion resistance (R_in) is used then-new Quality factor becomes (Q_f2) and it is given by

\(Q_{f2}=\frac{1}{\omega (R+R_{in})C}=\frac{1}{2\pi \times 10^6 \times (10+0.02) \times 65\times 10^{-12}}=244.4\)

Now percentage error (E) obtain by the above will be

\(E=\frac{Q_{f1}-Q_{f2}}{Q_{f1}} \times 100=\frac{244.9-244.4}{244.9}\times 100 = 0.2\ \%\)

Concept:

At bridge balance condition,

\(\left( {{R_1} + j\omega {L_1}} \right)\left( {\frac{{\left( {{R_3}} \right)\left( {\frac{1}{{j\omega {C_3}}}} \right)}}{{{R_3} + \frac{1}{{j\omega {C_3}}}}}} \right) = {R_2}{R_4}\)

\(\left( {{R_1} + j\omega {L_1}} \right)\left( {\frac{{{R_3}}}{{1 + j\omega {R_3}{C_3}}}} \right) = {R_2}{R_4}\)

\(\left( {{R_1} + j\omega {L_1}} \right){R_3} = {R_2}{R_4}\left( {1 + j\omega {R_3}{C_3}} \right)\)

\( \Rightarrow {R_1}{R_3} + j\omega {L_1}{R_3} = {R_2}{R_4} + j\omega {R_3}{C_3}{R_2}{R_4}\)

By comparing both the sides,

\({R_1} = \frac{{{R_2}{R_4}}}{{{R_3}}} = \frac{{500 \times 400}}{{2000 }} = 100\;{\rm{\Omega }}\)

\({L_1} = {C_3}{R_2}{R_4} = 0.5 \times {10^{ - 6}} \times 500 \times 400 = 0.1\;H\)

The quality factor, \(Q = \frac{{\omega {L_1}}}{{{R_1}}}\) 

\(= \frac{{2\pi \times 50 \times 0.1}}{{100}} = 0.3141\)

The voltage magnification of an electromagnetic system, particularly in the context of resonant circuits or components (like inductors and capacitors), is often associated with the term 'Quality Factor', also known as the Q factor.

In the context of resonant circuits, the Q factor can also be interpreted as the ratio of the reactive power (which is maximum at resonance) to the active power in the circuit. When referring to a coil (inductor) in such a circuit, the Q factor of the coil, which is determined by the ratio of its inductive reactance to resistance, tells how well it can establish a voltage across its terminals that is larger than the voltage applied to it -- thus, one might consider the Q factor as an indicator of 'voltage magnification'

 Additional Information

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

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

\(Q = \frac{Resonant ~frequency}{Bandwidth}\)

In a parallel RLC, 

Bandwidth = \(\frac{1}{RC}\)

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

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

\(Q=\frac{RC}{\sqrt{LC}}\)

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