Problem 55 - University Physics with Modern Physics

Question

In an \(L-R-C\) series circuit the magnitude of the phase angle is \(54.0^{\circ},\) with the source voltage lagging the current. The reactance of the capacitor is \(350 \Omega,\) and the resistor resistance is 180\(\Omega .\) The average power delivered by the source is 140 \(\mathrm{W} .\) Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

Step-by-Step Solution

Answer

(a) 621.08 Ω, (b) 1.62 A, (c) 776.83 V

Step 1: Understand the Problem

We have an L-R-C series circuit with a given phase angle, capacitive reactance, resistance, and average power. Our tasks are to find the inductive reactance, the rms current, and the rms voltage of the source.

Step 2: Find the Inductive Reactance

The phase angle \( \phi \) is related to the inductive and capacitive reactance by the equation \( \tan \phi = \frac{X_L - X_C}{R} \). Given \( \phi = 54.0^{\circ} \), \( X_C = 350 \Omega \), and \( R = 180 \Omega \), we have: \[ \tan 54.0^{\circ} = \frac{X_L - 350}{180} \] Solving for \( X_L \), we find: \[ X_L = 180 \times \tan 54.0^{\circ} + 350 \approx 621.08 \Omega \]

Step 3: Calculate the RMS Current

The average power \( P_{avg} \) in the circuit, for an AC circuit, is given by \( P_{avg} = I_{rms}^2 R \cos \phi \). We know \( P_{avg} = 140 \, W \), \( R = 180 \, \Omega \), and \( \cos \phi = \cos 54.0^{\circ} \). So, \[ I_{rms}^2 = \frac{140}{180 \cos 54.0^{\circ}} \] This gives: \[ I_{rms} = \sqrt{\frac{140}{180 \times \cos 54.0^{\circ}}} \approx 1.62 \, A \]

Step 4: Calculate the RMS Voltage

The rms voltage \( V_{rms} \) of the source is given by Ohm’s law for AC circuits: \( V_{rms} = I_{rms} \times Z \), where \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). Using \( R = 180 \Omega \), \( X_L = 621.08 \Omega \), and \( X_C = 350 \Omega \), we have: \[ Z = \sqrt{180^2 + (621.08 - 350)^2} \approx 479.4 \Omega \] Then, \[ V_{rms} = 1.62 \times 479.4 \approx 776.83 \, V \]

Key Concepts

Inductive Reactance

In an L-R-C series circuit, the inductive reactance is a key component. It opposes the change in current through an inductor and is denoted as \(X_L\). Inductive reactance occurs because of the inductor's ability to store energy as a magnetic field. The formula for calculating inductive reactance is \(X_L = \omega \cdot L\), where \(\omega\) is the angular frequency and \(L\) is the inductance. However, in exercises where you need to find \(X_L\), you'll often encounter it within the context of the phase relationship between voltage and current. In a given circuit problem, the inductive reactance can be determined using phase angle data and the relationship \(\tan \phi = \frac{X_L - X_C}{R}\), as shown in the problem's solution. Understanding this will help in finding not just the reactance, but also in analyzing how inductive elements affect the behavior of AC circuits.

RMS Current

RMS stands for root mean square, and it's a statistical measure used in AC circuits to represent an equivalent DC current value. The RMS current is crucial as it indicates how much heating effect an AC will produce, the same as a DC current. The formula for RMS current \(I_{rms}\) in an L-R-C circuit takes into account power and resistance: \[ I_{rms}^2 = \frac{P_{avg}}{R \cos \phi} \]Where \(P_{avg}\) is the average power, \(R\) is the resistance, and \(\cos \phi\) represents the power factor (the cosine of the phase angle).After solving for \(I_{rms}\), you receive a clear measurement of current flow that considers both resistive and reactive elements of the circuit, crucial for ensuring devices are rated appropriately for use with AC power.

RMS Voltage

The RMS voltage is another critical parameter in AC circuits that helps to compare AC with DC voltage. It shows the effective value of the AC voltage that would produce the same power in a resistor as a DC circuit.For calculating RMS voltage \(V_{rms}\) in an AC circuit, Ohm’s law is used in combination with the circuit's total impedance \(Z\):\[ V_{rms} = I_{rms} \times Z \]Where \(Z\) is given by \( \sqrt{R^2 + (X_L - X_C)^2} \), considering resistance \(R\), inductive reactance \(X_L\), and capacitive reactance \(X_C\).RMS voltage helps in verifying the safe operational levels of electrical components and ensuring they function effectively without risk of overheating.

Phase Angle in AC Circuits

The phase angle \(\phi\) in AC circuits is a measure of the time difference between the peak of the voltage waveform and the peak of the current waveform. It is vital in power calculations and dictates the behavior of reactive components.When a phase angle is present, voltage and current are out of sync by some degree, which affects how the circuit's impedance and power factor are calculated. The phase angle is directly related to the difference between inductive and capacitive reactance through the formula:\[ \tan \phi = \frac{X_L - X_C}{R} \]A positive phase angle means the voltage lags behind the current, typical in circuits with predominant inductive reactance.Understanding phase angles is essential for optimizing circuit performance, especially for minimizing power losses and ensuring that reactive power is appropriately managed.