31. Alternating Current

Inductors in AC Circuits

31. Alternating Current

Inductors in AC Circuits: Videos & Practice Problems

22 of 0

Problem 22Multiple Choice

A water filtration system treated as LC circuit is designed to allow clean water (DC component) to pass through with minimal loss, while efficiently filtering out contaminants (AC components) that fluctuate over time.
The input water consists of a mixture of clean water and contaminants, where: Clean Water (V₁): Represents the steady, desired component that needs to be passed through the filter. Contaminants (V₂): Represent the fluctuating impurities (e.g., dirt, chemicals) that need to be removed, where V₂= sin(ωt) with amplitude and angular frequency ω.
The filtration system consists of two key components:

Coarse Filter (Analogous to Inductor L): The coarse filter resists the flow of large contaminants, effectively filtering out high-frequency contaminants.
Fine Filter (Analogous to Capacitor C): The fine filter allows fine water to pass through while blocking smaller contaminants.
The system is designed with the following assumptions: Negligible resistance (R→0).
Inductive reactance is greater than capacitive reactance (X_L>X_C).
The goal is to determine the behavior of the filtration system, particularly focusing on the following aspects:
(a) Water Flow Through the Filters: Determine the amplitude and phase of the water flowing through the system. This will involve calculating the impedance of the system and the resulting current.
(b) Output Water Quality: Show that the AC component of the output water quality is given by V_2,out=Q/C-V₁, where Q is the charge accumulated on the fine filter and V₁is the DC component (clean water). Determine the amplitude and phase of the AC component of the output contaminants.
(c) Attenuation of Contaminants: Demonstrate that the attenuation of the contaminants (AC components) is greatest when X_L≫X_C. Calculate the ratio of the output to input AC voltage under this condition.
(d) Output Clean Water: Compare the DC component (clean water) in the output with the DC component in the input. Discuss how the system behaves with respect to steady water flow (DC component).

$$\left$(a$\right$)$\frac{V_0}{X_{L}$+$\omega$ C}$\sin\[\left$($\omega$ t+$\frac{\pi}{2}\]\right$)$

$$\left$(b$\right$)$\frac{V_0}{\omega^2LC+1}\[\sin\]\left$($\omega$ t$\right$)$

$\left\)(c$\right$)\(\frac{1}{\omega^2LC+1}$

$(d)V_{1,out}=0$

$$\left$(a$\right$)$\frac{V_0}{X_{L}$+$\frac{1}{\omega C}$}$\sin\[\left$($\omega$ t+$\frac{\pi}{2}\]\right$)$

$$\left$(b$\right$)$\frac{V_0}{\omega^2LC+1}\[\sin\]\left$($\omega$ t$\right$)$

$\left\)(c$\right$)\(\frac{1}{\omega^2LC-1}$

$$\left$(d$\right$)V_{1,out}=0$

$$\left$(a$\right$)$\frac{V_0}{X_{L}$-$\frac{1}{\omega C}$}$\sin\[\left$($\omega$ t-$\frac{\pi}{2}\]\right$)$

$$\left$(b$\right$)$\frac{V_0}{\omega^2LC+1}\[\sin\]\left$($\omega$ t$\right$)$

$\left\)(c$\right$)\(\frac{1}{\omega^2LC+1}$

$$\left$(d$\right$)V_{1,out}=V_{in}$

$$\left$(a$\right$)$\frac{V_0}{X_{L}$-$\frac{1}{\omega C}$}$\sin\[\left$($\omega$ t-$\frac{\pi}{2}\]\right$)$

$$\left$(b$\right$)$\frac{V_0}{\omega^2LC-1}\[\sin\]\left$($\omega$ t$\right$)$

$\left\)(c$\right$)\(\frac{1}{\omega^2LC-1}$

$$\left$(d$\right$)V_{1,out}=V_{in}$