Problem 14 - College Physics

Question

A \(200 \Omega\) resistor is in series with a \(0.100 \mathrm{H}\) inductor and a \(0.500 \mu \mathrm{F}\) capacitor. Compute the impedance of the circuit and draw the phasor diagram (a) at a frequency of \(500 \mathrm{~Hz}\), (b) at a frequency of \(1000 \mathrm{~Hz}\). In each case, compute the phase angle of the source voltage with respect to the current and state whether the source voltage lags or leads the current.

Step-by-Step Solution

Answer

At 500 Hz, the impedance is 443.39 \(\Omega\) and voltage lags; at 1000 Hz, impedance is 377.24 \(\Omega\) and voltage leads.

Step 1: Define Impedance Components

An RLC circuit's impedance comprises the resistive, inductive, and capacitive parts. The resistive part is the resistance \(R = 200 \Omega\). The inductive reactance is given by \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L = 0.100\, \text{H}\). The capacitive reactance is \(X_C = \frac{1}{2\pi f C}\), where \(C = 0.500 \times 10^{-6}\, \text{F}\).

Step 2: Calculate Impedance at 500 Hz

For \(f = 500 \text{ Hz}\), calculate the inductive reactance: \(X_L = 2\pi (500)(0.100) = 314.16 \Omega\). Calculate the capacitive reactance: \(X_C = \frac{1}{2\pi (500)(0.500 \times 10^{-6})} = 636.62 \Omega\). The total impedance is \(Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{200^2 + (314.16 - 636.62)^2} = 443.39 \Omega\).

Step 3: Calculate Phase Angle at 500 Hz

The phase angle \(\phi\) is given by \(\tan^{-1}\left(\frac{X_L - X_C}{R}\right)\). For \(f = 500 \text{ Hz}\), \(\tan^{-1}\left(\frac{314.16 - 636.62}{200}\right) = \tan^{-1}(-1.613) = -58.07^\circ\). The source voltage lags the current because the angle is negative.

Step 4: Calculate Impedance at 1000 Hz

For \(f = 1000 \text{ Hz}\), calculate the inductive reactance: \(X_L = 2\pi (1000)(0.100) = 628.32 \Omega\). Calculate the capacitive reactance: \(X_C = \frac{1}{2\pi (1000)(0.500 \times 10^{-6})} = 318.31 \Omega\). The total impedance is \(Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{200^2 + (628.32 - 318.31)^2} = 377.24 \Omega\).

Step 5: Calculate Phase Angle at 1000 Hz

The phase angle \(\phi\) is obtained by \(\tan^{-1}\left(\frac{X_L - X_C}{R}\right)\). For \(f = 1000 \text{ Hz}\), \(\tan^{-1}\left(\frac{628.32 - 318.31}{200}\right) = \tan^{-1}(1.55) = 57.18^\circ\). The source voltage leads the current as the angle is positive.

Key Concepts

Impedance Calculation

Impedance is the entire opposition a circuit offers to the flow of alternating current (AC). It combines the effects of resistance, inductive reactance, and capacitive reactance. To calculate impedance in an RLC series circuit, we use the formula:

• \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
• Where:
  • \(Z\) is the impedance.
  • \(R\) is the resistance.
  • \(X_L\) is the inductive reactance.
  • \(X_C\) is the capacitive reactance.

To find the impedance:

• Compute \(X_L\) and \(X_C\) using their respective formulas.
• Subtract \(X_C\) from \(X_L\) to get the net reactance.
• Use the impedance formula to get \(Z\).

The resultant impedance is represented as a complex number with both magnitude and phase, indicating how much the current is out of phase with the voltage.

Inductive Reactance

Inductive reactance is the opposition to AC caused by inductors in a circuit. It increases with frequency and is computed as:

• \(X_L = 2\pi f L\)
• Where:
  • \(X_L\) is the inductive reactance.
  • \(f\) is the frequency of the AC source.
  • \(L\) is the inductance of the coil.

At higher frequencies, inductors provide greater opposition to current flow. This is essential information when designing circuits for specific frequency responses. In our exercise, at 500 Hz and 1000 Hz, we calculated the inductive reactance and observed how the impedance changes with frequency. This helps understand the behavior of RLC circuits under different operational conditions.

Capacitive Reactance

Capacitive reactance opposes AC in circuits with capacitors. Unlike inductive reactance, it tends to decrease with an increase in frequency. This is calculated by:

• \(X_C = \frac{1}{2\pi f C}\)
• Where:
  • \(X_C\) is the capacitive reactance.
  • \(f\) is the frequency of the AC source.
  • \(C\) is the capacitance.

A key takeaway is that capacitors pass AC more easily as the frequency rises, conversely to inductors. This property is crucial when analyzing and designing circuits for specific frequency operations, like filters which could either block or pass certain frequencies. In the exercise, we computed the capacitive reactance at different frequencies to see changes in circuit behavior.

Phase Angle

The phase angle, \(\phi\), describes the shift in phase between the voltage and the current in a circuit. It indicates whether the circuit is more inductive or capacitive:

• \(\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)\)
• If \(\phi\) is positive, voltage leads the current (inductive).
• If \(\phi\) is negative, voltage lags the current (capacitive).

Understanding phase angles is vital for synchronizing AC systems and ensuring stable operations. Phase-related behaviors of the RLC circuit at different frequencies were shown in the exercise, where the source voltage either led or lagged the current depending on the frequency. This paints a picture of how the AC behaves dynamically across the frequencies.

Frequency Response

Frequency response in an RLC circuit refers to how the impedance and phase angle react to changing frequencies. This concept embodies the circuit's selective response to different frequencies:

• Low frequencies might result in a capacitive circuit behavior with high reactive component due to \(X_C\) dominance.
• Higher frequencies might result in an inductive circuit behavior with \(X_L\) dominance.
• At a specific frequency, known as resonance frequency, \(X_L\) equals \(X_C\) and the impedance is minimized to just the resistance \(R\).

Examining frequency response is fundamental for designing circuits like filters and tuners, ensuring only desired frequencies are amplified or attenuated. In the given problem, the impedance and phase angles were computed at varying frequencies, demonstrating the circuit's tendency towards either inductive or capacitive nature, impacting how circuits should be arranged for specific applications.