Problem 48 - Physics

Question

Concept Questions (a) An ac circuit contains only a resistor and a capacitor in series. Is the phase angle \(\phi\) between the current and the voltage of the generator positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}} ?\) (b) An ac circuit contains only a resistor and an inductor in series. Is the phase angle \(\phi\) positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the inductive reactance \(X_{\mathrm{L}}\) ? Account for your answers. Problem A series circuit has an impedance of \(192 \Omega\), and the phase angle is \(\phi=-75.0^{\circ} .\) The circuit contains a resistor and either a capacitor or an inductor. Find the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) or the inductive reactance \(X_{\mathrm{L}},\) whichever is appropriate.

Step-by-Step Solution

Answer

The resistance is 49 Ω, and the capacitive reactance is 184 Ω.

Step 1: Understand the Conceptual Framework

For an AC circuit with a resistor and a capacitor, the current leads the voltage, making the phase angle \( \phi \) negative. The impedance \( Z \) is calculated as \( Z = \sqrt{R^2 + X_C^2} \), where \( X_C \) is the capacitive reactance. On the other hand, for a resistor and inductor in series, the current lags the voltage, yielding a positive \( \phi \), with impedance given by \( Z = \sqrt{R^2 + X_L^2} \), where \( X_L \) is the inductive reactance.

Step 2: Identify Type of Reactance from Phase Angle

Given the phase angle \( \phi = -75.0^{\circ} \), which is negative, indicates the circuit contains a resistor and a capacitor.

Step 3: Use Impedance and Phase Angle to Find Components

We are given \( Z = 192 \Omega \) and \( \phi = -75.0^{\circ} \). We use the formula \( \tan \phi = \frac{X_C}{R} \). First, solve for \( X_C = R\tan \phi \) using \( \tan(-75.0^{\circ}) \approx -3.732 \).

Step 4: Express Impedance Equation

We know \( Z^2 = R^2 + X_C^2 \). Substituting \( X_C = R \times (-3.732) \) gives \[ Z^2 = R^2 + (R \times (-3.732))^2 \].

Step 5: Calculate Resistance and Reactance

Substitute \( Z = 192 \Omega \) into the impedance formula: \[ 192^2 = R^2 + (R \times (-3.732))^2 \]. Simplifying, \[ 36864 = R^2 + (13.91)R^2 \]. This results in: \[ 36864 = 14.91R^2 \]. Solve for \( R \): \[ R = \sqrt{\frac{36864}{14.91}} \approx 49 \Omega \]. Then \( X_C = 49 \times (-3.732) \approx -183.9 \Omega \) but we consider the magnitude as \( 184 \Omega \).

Step 6: Round and Finalize the Answer

Rounding \( R \) and \( X_C \) to a reasonable precision, we find that the resistance \( R = 49 \Omega \) and the capacitive reactance \( X_C = 184 \Omega \).

Key Concepts

Impedance in AC Circuits

In an AC circuit, impedance plays a crucial role just like resistance does in a DC circuit. However, impedance is more comprehensive because it incorporates not just resistance, but also reactance. Impedance, denoted as \( Z \), is a complex number that represents the total opposition that a circuit offers to the flow of alternating current. The formula for impedance in a series circuit containing a resistor \( R \) and a reactance \( X \) (either capacitive or inductive) is:

• \( Z = \sqrt{R^2 + X^2} \)

Impedance is measured in ohms (\( \Omega \)). For circuits with a capacitor or inductor, the reactance changes with frequency, adding a dynamic element to the impedance.
Understanding impedance helps in designing circuits that can effectively handle signals of various frequencies without causing undesirable changes in voltage or current.
In a resistor-capacitor (RC) circuit, impedance includes the capacitive reactance \( X_C \), while in a resistor-inductor (RL) circuit, it encompasses inductive reactance \( X_L \).

Phase Angle in AC Circuits

The phase angle \( \phi \) in an AC circuit is an indicator of the phase difference between the voltage and the current. This phase angle is important because it affects how power is transmitted and utilized in a circuit. The value of \( \phi \) is derived from the relative positions of the voltage and current sinusoidal waveforms:

• A negative phase angle indicates that the current leads the voltage, which is typical in RC circuits where the reactance is capacitive.
• A positive phase angle, on the other hand, signifies that the current lags behind the voltage, which is common in RL circuits with inductive reactance.

The mathematical expression relating phase angle to reactance and resistance is \( \tan \phi = \frac{X}{R} \).
This relationship helps us determine the nature of reactance present in the circuit just by observing the sign and magnitude of the phase angle.
Thus, knowing the phase angle is key in defining power factors and ensuring efficient energy transmission.

Reactance in AC Circuits

Reactance is the part of impedance that enables an AC circuit to store and release energy. It is a crucial concept because it determines how capacitors and inductors influence current flow in AC circuits. Reactance is measured in ohms and can be either capacitive or inductive:

• **Capacitive Reactance (\( X_C \)):** Related to capacitors, it is calculated as \( X_C = \frac{1}{2 \pi f C} \), where \( f \) is frequency and \( C \) is capacitance. Capacitive reactance decreases with higher frequency, indicating that capacitors are more "open" to AC at high frequencies.
• **Inductive Reactance (\( X_L \)):** Associated with inductors, it is given by \( X_L = 2 \pi f L \), where \( L \) is inductance. Inductive reactance increases with frequency, showing that inductors oppose higher frequency AC.

Understanding reactance helps in calculating the overall impedance in circuits and predicting how components will behave at different frequencies.
Reactance is essential for applications like tuning radios or filtering signals, where it defines how signal frequencies are processed.