Alternating Current Test

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Alternating Current Test - 48

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Question 1
1 / -0

A waveform has a baseline of $$3 V$$, a duty cycle of $$20$$%, and an amplitude of $$8 V$$. The average voltage value is

A
$$4 V$$

B
$$4.6 V$$

C
$$1.6 V$$

D
$$11 V$$

Solution

Baseline voltage is $$3V$$ and amplitude is $$8V$$.
As we know duty cycle $$ DC=\dfrac{T_H}{T_H+T_L}$$.
In this problem it is given as $$0.20$$, thus we get
$$0.20 = \dfrac{T_H}{T_H+T_L}$$
$$\implies T_L=4T_H$$
Average value of voltage $$V_a=\dfrac{T_H\times8+T_L\times3}{T_H+T_L}=\dfrac{20}{5}=4V$$
$$\therefore$$ Answer is A.
Question 2
1 / -0

A $$6 kHz$$ sinusoidal voltage is applied to a series RC circuit. The frequency of the voltage across the resistor is

A
$$0 Hz$$

B
$$12 kHz$$

C
$$6 kHz$$

D
$$18 kHz$$

Solution

Frequency of applied voltage and voltage or current accross any component are same. As we know,
$$I=\dfrac{V}{Z}$$ and $$Z$$ has nothing to do with frequency
$$\implies$$ $$I$$ and $$V$$ have same frequency.
And also according to KVL $$V=V_1+V_2+V_3$$
$$\implies$$ voltage and current across any component have same frequency i.e $$6kHz$$.
Question 3
1 / -0

A resistor and a capacitor are in series across a $$20 V$$ ac source. Circuit impedance is $$4.33 k \Omega$$. Current flow in the circuit is

A
$$9.2 mA$$

B
$$92 mA$$

C
$$4.6 mA$$

D
$$460 mA$$

Solution

Voltage of the source $$V = 20$$ volts
Circuit impedance $$Z = 4.33 k\Omega = 4.33\times 10^{3}\Omega$$
Thus current in the circuit $$I = \dfrac{V}{Z}$$
$$\therefore$$ $$I = \dfrac{20}{4.33\times 10^3} = 4.6 \times 10^{-3} A$$
$$\implies$$ $$I = 4.6 mA$$
Question 4
1 / -0

In a parallel RC circuit, there is $$100 mA$$ through the resistive branch and $$100 mA$$ through the capacitive branch. The total rms current is

A
$$200 mA$$

B
$$100 mA$$

C
$$282 mA$$

D
$$141 mA$$

Solution

Current in resistive branch $$I_r = 100 mA$$
Current in capacitive branch $$I_c = 100 mA$$
Thus total rms current $$I_{rms} = \sqrt{I_r^2 + I_c^2}$$
$$\therefore$$ $$I_{rms} = \sqrt{100^2 + 100^2}$$
$$\implies$$ $$I_{rms} = \sqrt{2} \times 100 = 141 mA$$
Question 5
1 / -0

A resistor $$R$$, an inductor $$L$$ and capacitor $$C$$ are connected in series to an oscillator of frequency $$n$$. If the resonant frequency is $${ n }_{ r }$$, then the current lags behind voltage, when

A
$$n=0$$

B
$$n < { n }_{ r }$$

C
$$n={ n }_{ r }$$

D
$$n > { n }_{ r }$$

Solution

When reactance of inductance is more than the reactance of condenser, the current lag behind the voltage.
Thus $$\omega L < \dfrac { 1 }{ \omega c }$$ or $$ \omega > \dfrac { 1 }{ \sqrt { LC } }$$
or $$ n > \dfrac { 1 }{ 2\pi \sqrt { LC } }$$ or $$ n > { n }_{ r }$$
$${ n }_{ r }=$$ resonant frequency
Question 6
1 / -0

The rms value of a.c. voltage with peack of $$311 V$$ is

A
$$220 V$$

B
$$311 V$$

C
$$180 V$$

D
$$320 V$$

Solution

Peak value of a.c. voltage, $$V_o = 311$$ volts
The rms value of voltage, $$V_{rms} = \dfrac{V_o}{\sqrt{2}}$$
$$\therefore$$ $$V_{rms} = \dfrac{311}{\sqrt{2}} = \dfrac{311}{1.414} = 220$$ volts
Question 7
1 / -0

The average half-cycle value of a sine wave with a $$40 V$$ peak is

A
$$25.48 V$$

B
$$6.37 V$$

C
$$14.14 V$$

D
$$50.96 V$$

Solution

The source voltage $$V = V_o \sin wt$$
Average half-cycle value of sine wave $$V_{avg} = \dfrac{\int_o^{T/2} V_o \sin (wt) dt} {T/2}$$
Or $$V_{avg} = \dfrac{V_o} {T/2} \times \int_o^{T/2} \sin wt dt$$
Or $$V_{avg} = \dfrac{V_o} {T/2} \times \dfrac{\cos wt}{w} \bigg|^{T/2}_o$$
Or $$V_{avg} = \dfrac{V_o} {T/2} \times \dfrac{\cos\frac{wT}{2} - 1}{w} $$
Or $$V_{avg} = \dfrac{V_o} {T/2} \times \dfrac{-1 - 1}{w} $$
Or $$V_{avg} = \dfrac{V_o} {\pi } \times (-2) $$
$$\implies$$ $$|V_{avg} | = \dfrac{2V_o}{ \pi}$$
We get $$|V_{avg}| = \dfrac{2\times 40}{\pi} = 25.48$$ volts
Question 8
1 / -0

A sine wave with an rms value of $$12 V$$ is riding on a dc level of $$18 V$$. The maximum value of the resulting waveform is

A
$$6 V$$

B
$$30 $$

C
$$35 V$$

D
$$0 V$$

Solution

Rms value of voltage $$V_{rms} = 12 $$ volts
Thus peak voltage $$V_o = \sqrt{2} V_{rms}$$
$$\therefore$$ $$V_o = 1.414 \times 12 = 17$$ volts
Maximum value of resulting waveform $$V_{max} = V_o + 18$$
$$\implies$$ $$V_{max} = 17+18 = 35$$ volts
Question 9
1 / -0

The instantaneous emf and current equations of an RLC series circuit are
$$e=200\sin { \left( \omega t-\dfrac { \pi }{ 6 } \right) } $$
$$i=20\sin { \left( \omega t+\dfrac { \pi }{ 6 } \right) } $$
The average power consumed per cycle is

A
Zero

B
$$2000 W$$

C
$$1000 W$$

D
$$500 W$$

Solution

Instantaneous power of the circuit, $$P=ei$$
Hence the average power of the circuit $$=\dfrac{\int_0^{2\pi/\omega} Pdt}{2\pi/\omega}$$ $$=\dfrac{\int_0^{2\pi/\omega}4000sin(\omega t-\dfrac{\pi}{6})sin(\omega t+\dfrac{\pi}{6})}{2\pi/\omega}W$$

Using: $$cosA-cosB=-2sin(\dfrac{A+B}{2})sin(\dfrac{A-B}{2})$$

Average power $$=\dfrac{\int_0^{2\pi/\omega}4000\dfrac{1}{2}(cos\dfrac{\pi}{3}-cos2\omega t)dt}{2\pi/\omega}W$$ $$=1000\ W$$
Question 10
1 / -0

An LC resonant circuit contains a capacitor $$400 pF$$ and an inductor $$100 \mu H$$. It is set into oscillations coupled to an antenna. Calculate the wavelength of the radiated electromagnetic wave.

A
$$377 \ mm$$

B
$$377 \ cm$$

C
$$377 \ m$$

D
$$3.77 \ cm$$

Solution

Wavelength $$ =\lambda = \dfrac{Velocity}{Frequency} $$

Frequency of the Resonating LC circuit = $$ \dfrac{1}{2\pi \times \sqrt {LC}} $$

$$ \implies Wavelength = C\times 2 \pi \times \sqrt { LC } $$

$$=3\times { 10 }^{ 8 }\times 2\times 3.14\times \sqrt { 100\times { 10 }^{ -6 }\times 400\times { 10 }^{ -12 } } $$
$$=376.8 m$$

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Re-Attempt Weekly Quiz Competition

Alternating Current Test - 48

Result

Alternating Current Test - 48

Score

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Rank

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SHARING IS CARING

Question 1

$$4 V$$

$$4.6 V$$

$$1.6 V$$

$$11 V$$

Solution

Question 2

$$0 Hz$$

$$12 kHz$$

$$6 kHz$$

$$18 kHz$$

Solution

Question 3

$$9.2 mA$$

$$92 mA$$

$$4.6 mA$$

$$460 mA$$

Solution

Question 4

$$200 mA$$

$$100 mA$$

$$282 mA$$

$$141 mA$$

Solution

Question 5

$$n=0$$

$$n < { n }_{ r }$$

$$n={ n }_{ r }$$

$$n > { n }_{ r }$$

Solution

Question 6

$$220 V$$

$$311 V$$

$$180 V$$

$$320 V$$

Solution

Question 7

$$25.48 V$$

$$6.37 V$$

$$14.14 V$$

$$50.96 V$$

Solution

Question 8

$$6 V$$

$$30 $$

$$35 V$$

$$0 V$$

Solution

Question 9

Zero

$$2000 W$$

$$1000 W$$

$$500 W$$

Solution

Question 10

$$377 \ mm$$

$$377 \ cm$$

$$377 \ m$$

$$3.77 \ cm$$

Solution

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