Problem 14 - University Physics with Modern Physics

Question

You have a \(200-\Omega\) resistor, a \(0.400-\mathrm{H}\) inductor, and a \(6.00-\mu \mathrm{F}\) capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 \(\mathrm{V}\) and an angular frequency of 250 \(\mathrm{rad} / \mathrm{s}\) . (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle \(\phi\) of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

Step-by-Step Solution

Answer

(a) Impedance = 605.7 Ω. (b) Current amplitude ≈ 0.0495 A. (c) Voltage across resistor = 9.9 V, across inductor = 4.95 V. (d) Phase angle ≈ -70.9°, voltage lags current. (e) Phasor diagram shows these relationships.

Step 1: Calculate the impedance of the circuit

To find the impedance of the circuit, we first need the inductive reactance, \( X_L \), and capacitive reactance, \( X_C \). The formulas are:- Inductive reactance: \( X_L = \omega L \)- Capacitive reactance: \( X_C = \frac{1}{\omega C} \)Substitute the given values:\( X_L = 250 \times 0.400 = 100 \, \Omega \)\( X_C = \frac{1}{250 \times 6.00 \times 10^{-6}} = 666.67 \, \Omega \)The impedance \( Z \) of the series RLC circuit is given by:\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]Substitute the known values:\( Z = \sqrt{200^2 + (100 - 666.67)^2} = \sqrt{200^2 + (-566.67)^2} = \sqrt{40000 + 320000.9} \approx 605.7 \, \Omega \).

Step 2: Calculate the current amplitude

Using Ohm's law for AC circuits, the current amplitude \( I \) can be calculated as:\[ I = \frac{V_0}{Z} \]where \( V_0 = 30.0 \, \text{V} \) is the voltage amplitude. Substituting the values:\( I = \frac{30.0}{605.7} \approx 0.0495 \, \text{A} \).

Step 3: Calculate the voltage amplitudes across the resistor and inductor

The voltage amplitude across the resistor, \( V_R \), is given by:\[ V_R = I \times R \]Substitute the values:\( V_R = 0.0495 \times 200 = 9.9 \, \text{V} \).The voltage amplitude across the inductor, \( V_L \), is given by:\[ V_L = I \times X_L \]Substitute the values:\( V_L = 0.0495 \times 100 = 4.95 \, \text{V} \).

Step 4: Determine the phase angle and direction

The phase angle \( \phi \) is calculated by:\[ \tan \phi = \frac{X_L - X_C}{R} \]Substitute the values:\( \tan \phi = \frac{100 - 666.67}{200} = \frac{-566.67}{200} = -2.83335 \).The phase angle is \( \phi = \arctan(-2.83335) \approx -70.9^\circ \).Since the phase angle is negative, the source voltage lags the current.

Step 5: Construct the phasor diagram

In the phasor diagram:- The current phasor is on the positive real axis.- The voltage across the resistor is in phase with the current.- The voltage across the inductor leads the current by \(90^\circ\).- The source voltage phasor is \(-70.9^\circ\), showing it lags the current phasor.The phasors for \( V_R \) and the total voltage should be accurately represented respecting their calculated phase relationships.

Key Concepts

Impedance Calculation

Impedance in an RLC circuit is a crucial parameter because it determines how the circuit will respond to alternating currents. In essence, impedance is the "resistance" to the flow of alternating current (AC). In a series RLC circuit, where a resistor (R), an inductor (L), and a capacitor (C) are arranged in series, the total impedance is not just the sum of their resistances and reactances due to the phase differences between the components.

The impedance (Z) can be thought of as a vector sum. It's calculated using the formula:

• \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
• Inductive reactance ( \( X_L \)) is calculated as \( \omega L \), where \( \omega \) is the angular frequency, and L is the inductance.
• Capacitive reactance ( \( X_C \)) is calculated as \( \frac{1}{\omega C} \), where C is the capacitance.

To compute the impedance, you substitute the reactances into the formula. Remember, for a series RLC circuit, capacitive and inductive reactances are vectors that oppose each other due to their out-of-phase behavior. That’s why we subtract them:

• When \( X_L > X_C \), the circuit behaves more inductively.
• When \( X_C > X_L \), it behaves more capacitively.

Ultimately, impedance gives you a clearer understanding of how the circuit will behave with the alternating current and helps predict how much current will flow through the circuit.

Current Amplitude

The current amplitude in an RLC circuit determines the maximum current level that will pass through the circuit. Amplitude in this context refers to the peak or maximum value of the sinusoidal current waveform. Understanding the current amplitude helps gauge how effectively power is being transmitted through your circuit.

Using Ohm's Law for AC circuits, the current amplitude (\( I \)) is determined by dividing the voltage amplitude (\( V_0 \)) by the impedance (\( Z \)) of the circuit. The formula reads:

• \[ I = \frac{V_0}{Z} \]

What this relationship tells us is that:

• As impedance increases, the current amplitude decreases if the voltage remains constant.
• As impedance decreases, the current amplitude increases.

This inverse relationship originates from the fact that higher impedance offers more "resistance" to the AC, reducing the current amplitude. Therefore, understanding both the voltage and impedance is critical to anticipate how much current will flow, which is essential for the safe and efficient design of electrical systems.

Phasor Diagram Construction

A phasor diagram provides a visual representation of the relationship between the voltages across the components in an RLC circuit and the current flowing through it. Phasors are vectors that represent the magnitudes and phases of sinusoidal functions, making it easier to visualize how different components are interacting in the circuit.

To construct a phasor diagram:

• Begin by drawing the current phasor on the real axis, as it serves as the reference.
• The voltage across the resistor (\( V_R \)) is aligned with the current phasor because it is in phase with the current.
• The voltage across the inductor (\( V_L \)) leads the current by 90 degrees. This phasor is drawn perpendicular to the right of the current phasor.
• The vector sum of these phasors represents the total voltage applied in the circuit, often showcasing a lag or lead depending on the sum phase angle calculated earlier.

Including these phasors in a diagram helps illuminate the timing differences between voltages. Understanding the phasor positions is crucial for predicting the behavior of the circuit, especially in AC power systems, where each component reacts out of phase with the others.