Determine the phase constant and the impedance of the RLC...

Determine the phase constant and the impedance of the RLC circuit shown in the figure when the frequency of the time-varying emf is $1.00 \mathrm{kHz}$ $C=100 . \mu \mathrm{F}, L=10.0 \mathrm{mH},$ and $R=100 . \Omega$.

Key Concepts

RLC Circuit

An RLC circuit is a network that includes a resistor, an inductor, and a capacitor working together to influence the behavior of AC signals. The interplay of these elements leads to frequency-dependent phenomena such as resonance, where the circuit can exhibit peak current or voltage responses, and determines the phase relationship between voltage and current.

Impedance

Impedance represents the total opposition to the flow of alternating current in a circuit, combining both resistive and reactive effects. It is a complex quantity that accounts for how both the resistor and the energy-storage components (inductor and capacitor) affect the amplitude and phase of current relative to the applied voltage.

Reactance

Reactance is the portion of impedance that arises due to energy storage in inductors and capacitors, rather than energy dissipation as in resistors. Inductive reactance increases with frequency, while capacitive reactance decreases with frequency, influencing the overall phase and magnitude of the circuit's impedance.

Phase Constant (Phase Angle)

The phase constant, or phase angle, describes the phase difference between the voltage across and the current through an AC circuit. It is determined by the relative sizes of resistive and reactive components, indicating whether the circuit current leads or lags the voltage.

Frequency

Frequency is the rate at which an AC voltage oscillates, and it plays a critical role in determining the values of inductive and capacitive reactances in a circuit. Changes in frequency can significantly alter the balance between energy storage and energy dissipation, impacting both impedance and phase relationships.

Transcript

0:00 Hi there.

00:01 So for this problem, we have, we need to determine the phase constant and the impedance for this rls circuit that is shown in this figure when the frequency is equal to one kilohertz and the capacitance is equal to 100.

00:40 Micropharets and the inductance is equal to 10 millie henries and finally the resistance is equal to 100 um so to calculate the phase um the phase constant we use the following equation that is that the the tangent of the phase constant.

01:13 Well, we apply the tangent of minus one to both sides, and we obtain that that is the tangent of minus one of the reactance for the inductor minus the reactance for the capacitance divided by the resistance of the resistor.

01:31 So first thing that we need to obtain is the reactance for both the inductor and the capacitor.

01:37 So for the aerobtems of the inductor, we know that that is 2 times pi times the frequency times the inductance.

01:49 So we just need to simply substitute those values.

01:53 In here, we have 1 kilohertz for the frequency, and the inductance is equal to 10 millie henries.

02:05 So putting this into the calculator, we obtain a reactance of 62 .8 oms.

02:24 So that's the reactance for the inductor.

02:29 And now we pass to calculate the reactance of the capacitor.

02:34 And we know that that is 1 over 2 pi times the frequency times the capacitance.

02:41 So we just simply substitute those values in here.

02:44 So we have 2 times pi times 1 kilohertz times the capacitance, which in this case is 100 micro ferrets.

03:06 So from this, we obtain a value of 1 .59 oms...

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