QuestionApril 21, 2025

2-Series RLC circint, R=4 e Omega, L=100 mathrm(mh) and C=150 mu mathrm(F) . If the supply is 200-mathrm(V) 50-mathrm(Hz)_(2) , find:- (a) Total current. (b) Power consumed. (c) Power factor (P.f) (d) Inspantaneous vortage and current at t=0.008 mathrm(Sec) .

2-Series RLC circint, R=4 e Omega, L=100 mathrm(mh) and C=150 mu mathrm(F) . If the supply is 200-mathrm(V) 50-mathrm(Hz)_(2) , find:- (a) Total current. (b) Power consumed. (c) Power factor (P.f) (d) Inspantaneous vortage and current at t=0.008 mathrm(Sec) .
2-Series RLC circint, R=4 e Omega, L=100 mathrm(mh) and C=150 mu mathrm(F) . If the supply is 200-mathrm(V) 50-mathrm(Hz)_(2) , find:- (a) Total current. (b) Power consumed. (c) Power factor (P.f) (d) Inspantaneous vortage and current at t=0.008 mathrm(Sec) .

Solution
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Answer

(a) Total current: 18.25 \, \text{A} ### (b) Power consumed: 1333.06 \, \text{W} ### (c) Power factor: 0.365 ### (d) Instantaneous voltage: 166.19 \, \text{V}, Instantaneous current: 25.81 \, \text{A} at t=0.008 \, \text{Sec} Explanation 1. Calculate the Impedance The impedance Z of a series RLC circuit is given by Z = \sqrt{R^2 + (X_L - X_C)^2}, where X_L = 2\pi f L and X_C = \frac{1}{2\pi f C}. Here, f = 50 \text{ Hz}. - X_L = 2\pi \times 50 \times 0.1 = 31.42 \, \Omega - X_C = \frac{1}{2\pi \times 50 \times 150 \times 10^{-6}} = 21.22 \, \Omega - Z = \sqrt{4^2 + (31.42 - 21.22)^2} = \sqrt{4^2 + 10.2^2} = \sqrt{16 + 104.04} = \sqrt{120.04} \approx 10.96 \, \Omega 2. Calculate Total Current Use Ohm's Law: I = \frac{V}{Z}. - I = \frac{200}{10.96} \approx 18.25 \, \text{A} 3. Calculate Power Consumed Power consumed P = I^2 R. - P = (18.25)^2 \times 4 \approx 1333.06 \, \text{W} 4. Calculate Power Factor Power factor \text{P.f} = \frac{R}{Z}. - \text{P.f} = \frac{4}{10.96} \approx 0.365 5. Calculate Instantaneous Voltage and Current at t=0.008 \, \text{Sec} Instantaneous voltage v(t) = V_m \sin(2\pi ft), where V_m = \sqrt{2} \times 200. - V_m = 282.84 \, \text{V} - v(0.008) = 282.84 \sin(2\pi \times 50 \times 0.008) \approx 282.84 \sin(2.513) \approx 282.84 \times 0.5878 \approx 166.19 \, \text{V} Instantaneous current i(t) = I_m \sin(2\pi ft - \phi), where I_m = \sqrt{2} \times 18.25 and \phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right). - I_m = 25.81 \, \text{A} - \phi = \tan^{-1}\left(\frac{10.2}{4}\right) \approx 1.19 \, \text{rad} - i(0.008) = 25.81 \sin(2\pi \times 50 \times 0.008 - 1.19) \approx 25.81 \sin(2.513 - 1.19) \approx 25.81 \times 0.9999 \approx 25.81 \, \text{A}

Explanation

1. Calculate the Impedance<br /> The impedance $Z$ of a series RLC circuit is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$, where $X_L = 2\pi f L$ and $X_C = \frac{1}{2\pi f C}$. Here, $f = 50 \text{ Hz}$.<br /><br />- $X_L = 2\pi \times 50 \times 0.1 = 31.42 \, \Omega$<br />- $X_C = \frac{1}{2\pi \times 50 \times 150 \times 10^{-6}} = 21.22 \, \Omega$<br />- $Z = \sqrt{4^2 + (31.42 - 21.22)^2} = \sqrt{4^2 + 10.2^2} = \sqrt{16 + 104.04} = \sqrt{120.04} \approx 10.96 \, \Omega$<br /><br />2. Calculate Total Current<br /> Use Ohm's Law: $I = \frac{V}{Z}$.<br />- $I = \frac{200}{10.96} \approx 18.25 \, \text{A}$<br /><br />3. Calculate Power Consumed<br /> Power consumed $P = I^2 R$.<br />- $P = (18.25)^2 \times 4 \approx 1333.06 \, \text{W}$<br /><br />4. Calculate Power Factor<br /> Power factor $\text{P.f} = \frac{R}{Z}$.<br />- $\text{P.f} = \frac{4}{10.96} \approx 0.365$<br /><br />5. Calculate Instantaneous Voltage and Current at $t=0.008 \, \text{Sec}$<br /> Instantaneous voltage $v(t) = V_m \sin(2\pi ft)$, where $V_m = \sqrt{2} \times 200$.<br />- $V_m = 282.84 \, \text{V}$<br />- $v(0.008) = 282.84 \sin(2\pi \times 50 \times 0.008) \approx 282.84 \sin(2.513) \approx 282.84 \times 0.5878 \approx 166.19 \, \text{V}$<br /><br /> Instantaneous current $i(t) = I_m \sin(2\pi ft - \phi)$, where $I_m = \sqrt{2} \times 18.25$ and $\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$.<br />- $I_m = 25.81 \, \text{A}$<br />- $\phi = \tan^{-1}\left(\frac{10.2}{4}\right) \approx 1.19 \, \text{rad}$<br />- $i(0.008) = 25.81 \sin(2\pi \times 50 \times 0.008 - 1.19) \approx 25.81 \sin(2.513 - 1.19) \approx 25.81 \times 0.9999 \approx 25.81 \, \text{A}$
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