Alternating Current Test

Directions For Questions

Consider the parallel resonant circuit shown in adjacent figure. One branch contains an inductor of inductance $$L$$ and a small ohmic resistance $$R$$, whereas the other branch contains a capacitor of capacitance $$C$$. The circuit is fed by a source of alternating emf
$$E={ E }_{ 0 }{ e }^{ i\omega t }={ E }_{ 0 }\sin { \omega t } $$
The impedance of inductor branch, $${ Z }_{ 1 }=R+j\omega L$$
The impedance of capacitor branch, $${ Z }_{ 2 }=\left( 1/j\omega C \right) $$
Net impedance $$Z$$ of the two parallel branches is given by
$$\cfrac { 1 }{ Z } =\cfrac { 1 }{ { Z }_{ 1 } } +\cfrac { 1 }{ { Z }_{ 2 } } =\cfrac { 1 }{ R+j\omega L } +j\omega C=\cfrac { R }{ { R }^{ 2 }+{ L }^{ 2 }{ \omega }^{ 2 } } +j\omega \left[ C-\cfrac { L }{ { R }^{ 2 }+{ L }^{ 2 }{ \omega }^{ 2 } } \right] $$
The current flowing in the circuit
$$I=\cfrac { E }{ Z } =E\left[ \cfrac { R }{ { R }^{ 2 }+{ L }^{ 2 }{ \omega }^{ 2 } } +j\omega \left[ C-\cfrac { L }{ { R }^{ 2 }+{ L }^{ 2 }{ \omega }^{ 2 } } \right] \right] $$
For resonance to occur, the current must be in phase with the applied emf. For this, the reactive component of current should be zero, ie
$$\omega \left( C-\cfrac { L }{ { R }^{ 2 }+{ L }^{ 2 }{ \omega }^{ 2 } } \right) =0$$
This gives resonant angular frequency
$$\quad { \omega }_{ r }=\sqrt { \cfrac { 1 }{ LC } -\cfrac { { R }^{ 2 } }{ { L }^{ 2 } } } $$
At parallel circuit resonance, the impedance is maximum and current is minimum. Parallel resonant circuit is sometimes called the anti-resonance in order to distinguish from series resonance.

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Alternating Cur...

Question 1

$$13.86$$ sec.

$$6.48$$ sec.

$$3.24$$ sec.

$$20.52$$ sec.

Question 2

$$100\sqrt{2}\Omega$$

$$200\Omega$$

$$100\Omega$$

$$200\sqrt{2}\Omega$$

Question 3

$$\sqrt { \cfrac { 3 }{ 5 } } $$

$$\sqrt { \cfrac { 5 }{ 3 } } $$

$$3/5$$

$$5/3$$

Question 4

84.5

95

65

66

Question 5

reactance

admittance

inductance

susceptance

Question 6

$$342\ W$$

$$305\ W$$

$$209\ W$$

$$242\ W$$

Question 7

$$V=100\ volt, I=2\ amp$$

$$V=100\ volt, I=5\ amp$$

$$V=1000\ volt, I=2\ amp$$

$$V=300\ volt, I=1\ amp$$

Question 8

$$50\Omega $$

$$200\Omega $$

$$270\Omega $$

none of these

Question 9

$$\sqrt\frac{17}{2}A$$

$$\sqrt\frac{2}{17}A$$

$$\sqrt\frac{3}{\sqrt2}A$$

$$3\sqrt2A$$

Question 10

Neither the average power nor the average current is zero

Average voltage is zero but the average power is non zero

Both the average power and average current is zero

Average power is zero, but the average current is non zero

User

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