You plan to change the frequency of the generator in this...

You plan to change the frequency of the generator in this circuit to produce a phase angle of smaller magnitude. The resistor is still 175$\Omega$ , the inductor is 90.0 $\mathrm{mH}$ , the capacitor is 15.0$\mu \mathrm{F}$ , and the rms volt- age is 120.0 $\mathrm{V}$ (a) Should you increase or decrease the frequency? Explain. (b) Find the frequency that gives a phase angle of $-22.5^{\circ} .$ (c) What is the rms current in the circuit at the frequency found in part (b)?

You plan to change the frequency of the generator in this circuit to produce a
phase angle of smaller magnitude. The resistor is still 175$\Omega$ , the
inductor is 90.0 $\mathrm{mH}$ , the capacitor is 15.0$\mu \mathrm{F}$ , and the rms volt-
age is 120.0 $\mathrm{V}$ (a) Should you increase or decrease the frequency?
Explain. (b) Find the frequency that gives a phase angle of $-22.5^{\circ} .$ (c) What is the rms current in the circuit at the frequency found in part (b)?

Key Concepts

AC Circuit Analysis

AC circuit analysis involves the examination of circuits powered by alternating current, where voltage, current, and impedance are represented as phasors. This approach uses complex numbers to account for both magnitude and phase differences, enabling a comprehensive understanding of how circuit elements interact under sinusoidal steady-state conditions.

RLC Circuit Theory

RLC circuits consist of resistors, inductors, and capacitors whose combined behavior determines the overall impedance of the circuit. The individual contributions of resistance, inductive reactance, and capacitive reactance result in a frequency-dependent response, making these circuits ideal for studying resonance and phase shift phenomena.

Frequency Dependence of Reactance

Inductive reactance increases with frequency while capacitive reactance decreases with frequency. This inverse relationship is crucial in determining the net reactance of an RLC circuit, and it directly affects the phase angle between voltage and current. Adjusting the frequency changes the dominance of either inductive or capacitive behavior in the circuit.

Phase Angle

The phase angle in an AC circuit quantifies the difference in phase between the applied voltage and the resulting current. In RLC circuits, the phase angle is influenced by the relative values of the inductive and capacitive reactances. A smaller magnitude phase angle indicates a closer match between the voltage and current phases, and adjustments in frequency can be used to achieve a desired phase shift.

Impedance

Impedance is the total opposition to AC current flow, combining resistance and reactance into a single complex quantity. It determines both how much current flows for a given voltage and the phase difference between the voltage and current. Calculating impedance in an RLC circuit is essential for understanding the circuit’s behavior at different frequencies.

Transcript

00:01 In the first item, we want to increase the phase angle from minus 39 .3 degrees to minus 22 .5 degrees.

00:09 Remember that the tangent of the phase angle is equal to xl minus xc divided by the resistance.

00:16 We want that angle to increase.

00:19 And by increasing in this particular situation, the tangent of the phase angle will be closer to zero.

00:27 So we want to decrease this quantity as a wall by only, massing with the frequency.

00:32 Note that in this circuit we have xc bigger than xl.

00:38 So this is a capacitive circuit.

00:40 The capacitance dominates the circuit.

00:42 If we want to decrease the difference between xl and xc, we must decrease xc while increasing xl.

00:51 Remembering that xl is equal to 2 pi times the frequency times the inductance and xc is equal to 1 divided by 2 pi times the frequency times the capacitance, we can see that by increasing the frequency, we do exactly what we want to do.

01:09 By increasing f, we increase xl while decreasing xc.

01:16 And so, what happens is that xl minus xc will decrease as wanted.

01:23 Then we should increase the frequency.

01:26 In the second item, we have to find the value of the new frequency.

01:30 Note that we can do that by using this relation between the tangent of the phase and the properties of the circuit.

01:39 So we get the tangent of the phase being equals to 2 pi fl, f prime here, minus and also phi prime here, 1 divided by 2 pi f prime times c...

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