Physics, 19.11.2020 18:40 maddie7155

What if the frequency is now increased to f = 69 Hz, and we want to keep the impedance unchanged? (A) What new resistance should we use to achieve this goal? R = Ω (B) What is the phase angle (in degrees) between the current and the voltage now? ϕ = ° (C) Find the maximum voltages across each element.

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Answer from: jrcspiderman2003

Complete Question

A series RLC circuit has R = 405 Ω, L = 1.40 H, C = 3 µF. It is connected to an AC source with f = 60.0 Hz and ΔVmax = 150 V. What if the frequency is now increased to f = 69 Hz, and we want to keep the impedance unchanged? (A) What new resistance should we use to achieve this goal? R = Ω (B) What is the phase angle (in degrees) between the current and the voltage now? ϕ = ° (C) Find the maximum voltages across each element.

$R = 513.9 \ \Omega$

$\theta = 34.79^o$

$V_c =212.96 \ V$ , $V_L = 168.139 \ V$ , $V_R = 142.35 \ V$

Explanation:

From the question we are told that

The resistance is $R = 405 \ \Omega$

The inductance is L = 1.40 H

The capacitance is $C = 3 \mu F = 3 *10^{-6} \ F$

The original frequency is $f = 60 \ Hz$

The new frequency is $f_1 = 69 \ Hz$

Generally the reactance of the inductor is mathematically represented as

$X_L = 2 \pi * f * L$

=> X_L = 2 * 3.142 * 60 * 1.40

=> $X_L = 527.5 \ \Omega$

Generally the reactance of the capacitor is mathematically represented as

$X_c = \frac{1}{2 \pi * f * C}$

$X_c = \frac{1}{2 * 3.142 * 60 * 3*10^{-6}}$

=> $X_c = 884.6 \ \Omega$

Generally the impedance of the circuit is mathematically represented as

$Z = \sqrt{[X_L - X_c ]^2 + R^2}$

=> $Z = \sqrt{[527.5 -884.6 ]^2 + 405^2}$

=> $Z = 540 \ \Omega$

Generally when the frequency is changed , the reactance of the inductor becomes

X_L' = 2 * 3.142 * 69 * 1.4

=> $X_L' = 607 \ \Omega$

Generally when the frequency is changed , the reactance of the capacitor becomes

$X_c' = \frac{1}{2 * 3.142 * 69 * 3*10^{-6}}$

=> $X_c ' = 768.8 \ \Omega$

Generally the new resistance is mathematically represented as

R ^2 = Z^2 - ( X_c' - X_L')^2

=> R ^2 = 540^2 - ( 603 - 768.8 )^2

=> $R = \sqrt{264110.4}$

=> $R = 513.9 \ \Omega$

Generally the phase angle is mathematically represented as

$\theta = tan^{-1}[\frac{[X_L - X_C]}{R} ]$

=> $\theta = tan^{-1}[\frac{[ 884.6 - 527.5]}{513.92} ]$

=> $\theta = 34.79^o$

Generally the maximum current flowing through the circuit is mathematically represented as

$I_{max} = \frac{\Delta V}{ Z}$

=> $I_{max} = \frac{150}{540 }$

=> $I_{max} = 0.2778 \ A$

Generally the maximum voltage across the capacitor is

$V_c = I_{max} * X_c'$

V_c = 0.2778* 768.8

=> $V_c =212.96 \ V$

Generally the maximum voltage across the inductor is

$V_L = I_{max} * X_L$

V_L = 0.2778* 607

=> $V_L = 168.139 \ V$

Generally the maximum voltage across the resistor is

$V_R = I_{max} * R$

V_R = 0.2778* 513.9

=> $V_R = 142.35 \ V$

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