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.
Answer:
A
[tex]R = 513.9 \ \Omega[/tex]
B
[tex]\theta = 34.79^o[/tex]
C
[tex]V_c =212.96 \ V[/tex] , [tex]V_L = 168.139 \ V[/tex] , [tex]V_R = 142.35 \ V[/tex]
Explanation:
From the question we are told that
The resistance is [tex]R = 405 \ \Omega[/tex]
The inductance is L = 1.40 H
The capacitance is [tex]C = 3 \mu F = 3 *10^{-6} \ F[/tex]
The original frequency is [tex]f = 60 \ Hz[/tex]
The new frequency is [tex]f_1 = 69 \ Hz[/tex]
Generally the reactance of the inductor is mathematically represented as
[tex]X_L = 2 \pi * f * L[/tex]
=> [tex]X_L = 2 * 3.142 * 60 * 1.40[/tex]
=> [tex]X_L = 527.5 \ \Omega[/tex]
Generally the reactance of the capacitor is mathematically represented as
[tex]X_c = \frac{1}{2 \pi * f * C}[/tex]
[tex]X_c = \frac{1}{2 * 3.142 * 60 * 3*10^{-6}}[/tex]
=> [tex]X_c = 884.6 \ \Omega[/tex]
Generally the impedance of the circuit is mathematically represented as
[tex]Z = \sqrt{[X_L - X_c ]^2 + R^2}[/tex]
=> [tex]Z = \sqrt{[527.5 -884.6 ]^2 + 405^2}[/tex]
=> [tex]Z = 540 \ \Omega[/tex]
Generally when the frequency is changed , the reactance of the inductor becomes
[tex]X_L' = 2 * 3.142 * 69 * 1.4[/tex]
=> [tex]X_L' = 607 \ \Omega[/tex]
Generally when the frequency is changed , the reactance of the capacitor becomes
[tex]X_c' = \frac{1}{2 * 3.142 * 69 * 3*10^{-6}}[/tex]
=> [tex]X_c ' = 768.8 \ \Omega[/tex]
Generally the new resistance is mathematically represented as
[tex]R ^2 = Z^2 - ( X_c' - X_L')^2[/tex]
=> [tex]R ^2 = 540^2 - ( 603 - 768.8 )^2[/tex]
=> [tex]R = \sqrt{264110.4}[/tex]
=> [tex]R = 513.9 \ \Omega[/tex]
Generally the phase angle is mathematically represented as
[tex]\theta = tan^{-1}[\frac{[X_L - X_C]}{R} ][/tex]
=> [tex]\theta = tan^{-1}[\frac{[ 884.6 - 527.5]}{513.92} ][/tex]
=> [tex]\theta = 34.79^o[/tex]
Generally the maximum current flowing through the circuit is mathematically represented as
[tex]I_{max} = \frac{\Delta V}{ Z}[/tex]
=> [tex]I_{max} = \frac{150}{540 }[/tex]
=> [tex]I_{max} = 0.2778 \ A[/tex]
Generally the maximum voltage across the capacitor is
[tex]V_c = I_{max} * X_c'[/tex]
[tex]V_c = 0.2778* 768.8[/tex]
=> [tex]V_c =212.96 \ V[/tex]
Generally the maximum voltage across the inductor is
[tex]V_L = I_{max} * X_L[/tex]
[tex]V_L = 0.2778* 607[/tex]
=> [tex]V_L = 168.139 \ V[/tex]
Generally the maximum voltage across the resistor is
[tex]V_R = I_{max} * R[/tex]
[tex]V_R = 0.2778* 513.9[/tex]
=> [tex]V_R = 142.35 \ V[/tex]
