A sinusoidal voltage of peak value 283 volt and frequency 50 Hz is applied to series in LCR circuit in which R=3 ohm , L=25.48 mH , C=0.000796 F. Find=

a) rms voltage

b) X_L and X_C

C) impedence(Z)

d) peak current and phase angle

e) rms value of current and voltage across circuit elements

f) power dissipated in circuit

g) power factor

h) power input

i) frequency of supply at which resonance occur.

j) impedence at resonant condition

k) the current at resonant condition.

l) power dissipated at resonant condition.

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To solve the problem involving a sinusoidal voltage applied to an LCR circuit, we will break it down step by step. This will help us find the various parameters you need, including RMS voltage, reactance, impedance, current, power, and resonance characteristics. Let's get started!

1. RMS Voltage

The root mean square (RMS) voltage can be calculated from the peak voltage using the formula:

V_rms = V_peak / √2

Given that the peak voltage (V_peak) is 283 volts:

V_rms = 283 / √2 ≈ 200.5 volts

2. Reactance of Inductor (X_L) and Capacitor (X_C)

The reactance of the inductor (X_L) and capacitor (X_C) can be calculated using the following formulas:

• X_L = 2πfL
• X_C = 1 / (2πfC)

Substituting the values:

X_L = 2π(50)(25.48 × 10^-3) ≈ 7.99 ohms

X_C = 1 / (2π(50)(0.000796)) ≈ 3.99 ohms

3. Impedance (Z)

The total impedance (Z) in an LCR circuit can be calculated using the formula:

Z = √(R² + (X_L - X_C)²)

Substituting the known values:

Z = √(3² + (7.99 - 3.99)²) = √(9 + 16) = √25 = 5 ohms

4. Peak Current and Phase Angle

The peak current (I_peak) can be calculated using Ohm's law:

I_peak = V_peak / Z

Substituting the values:

I_peak = 283 / 5 ≈ 56.6 A

The phase angle (φ) can be found using:

φ = tan^-1((X_L - X_C) / R)

φ = tan^-1((7.99 - 3.99) / 3) ≈ 0.93 radians

5. RMS Value of Current and Voltage Across Circuit Elements

The RMS current (I_rms) can be calculated as:

I_rms = I_peak / √2 ≈ 56.6 / √2 ≈ 40 A

The voltage across each element can be calculated as:

• V_R = I_rms * R = 40 * 3 = 120 V
• V_L = I_rms * X_L = 40 * 7.99 ≈ 319.6 V
• V_C = I_rms * X_C = 40 * 3.99 ≈ 159.6 V

6. Power Dissipated in the Circuit

The power dissipated (P) in the circuit can be calculated using:

P = I_rms² * R = 40² * 3 = 4800 W

7. Power Factor

The power factor (PF) is given by:

PF = cos(φ) = cos(0.93) ≈ 0.6

8. Power Input

The total power input (S) can be calculated as:

S = V_rms * I_rms = 200.5 * 40 ≈ 8020 W

9. Frequency of Supply at Resonance

The frequency at which resonance occurs is given by:

f_res = 1 / (2π√(LC))

Substituting the values:

f_res = 1 / (2π√(25.48 × 10^-3 * 0.000796)) ≈ 31.8 Hz

10. Impedance at Resonant Condition

At resonance, the impedance is purely resistive:

Z_res = R = 3 ohms

11. Current at Resonant Condition

The current at resonance can be calculated as:

I_res = V_rms / Z_res = 200.5 / 3 ≈ 66.83 A

12. Power Dissipated at Resonant Condition

The power dissipated at resonance is:

P_res = I_res² * R = 66.83² * 3 ≈ 13,400 W

In summary, we have calculated various parameters of the LCR circuit, including RMS voltage, reactance, impedance, current, power, and resonance characteristics. Each step builds on the previous one, ensuring a