Problem 5 - College Physics

Question

You are designing an amplifier circuit that will operate in the frequency range from \(20 \mathrm{~Hz}\) to \(20,000 \mathrm{~Hz}\). For the design to work, the reactance of a particular inductor in the circuit cannot exceed \(100 \Omega\). What is the largest inductance that can be used?

Step-by-Step Solution

Answer

The largest inductance that can be used is approximately 0.796 H.

Step 1: Understanding Reactance Formula

The reactance of an inductor is given by the formula \( X_L = 2\pi f L \), where \( X_L \) is the inductive reactance, \( f \) is the frequency, and \( L \) is the inductance. We know from the problem that the maximum allowable reactance is \( 100 \Omega \).

Step 2: Identifying Key Values

We are given the frequency range from \( 20 \, \text{Hz} \) to \( 20,000 \, \text{Hz} \), and the maximum reactance \( X_L \) as \( 100 \Omega \). To find the largest inductance, we will use the lowest frequency \( 20 \, \text{Hz} \) because reactance increases with frequency.

Step 3: Substituting Values into Formula

Substitute the values into the reactance formula: \( 100 = 2\pi (20) L \). Simplifying this gives the equation for \( L \): \( L = \frac{100}{2\pi(20)} \).

Step 4: Calculating the Inductance

Solve for \( L \): \[ L = \frac{100}{2 \cdot 3.1416 \cdot 20} = \frac{100}{125.664} \approx 0.796 \text{ H} \].

Step 5: Verifying the Solution

Check if such L value meets the reactance condition at \( 20 \text{Hz} \): \( X_L = 2 \pi (20) (0.796) = 100 \Omega \). Indeed, this satisfies the condition.

Key Concepts

Inductive Reactance

Inductive reactance is the property of an inductor that resists changes in current. Understanding this concept is crucial for designing circuit elements in electronics. It is quantified using the formula \( X_L = 2\pi f L \), where \( X_L \) is the inductive reactance, \( f \) is the frequency, and \( L \) is the inductance. This formula tells us how reactance varies with frequency and inductance. The higher the frequency, the larger the reactance. Therefore, when designing circuits, engineers must consider the operating frequency range and limit the reactance to ensure proper functionality. In practical terms, this means selecting an inductor with the right value of \( L \) to keep \( X_L \) within desired bounds across the specified frequency range. This balance is key in applications such as amplifier circuits, where specific frequency characteristics are essential for performance.

Frequency Range

The frequency range is a fundamental consideration in amplifier circuit design. In our particular problem, we are dealing with a range between \(20\, \text{Hz}\) to \(20,000\, \text{Hz}\). This range covers the typical human hearing spectrum. The lower limit corresponds to deep bass sounds, while the upper limit includes high pitches. When designing the circuit, engineers need to ensure that all components, like inductors, respond appropriately across this entire range.
When considering inductive reactance, the lowest frequency within the range often dictates the maximum inductance allowed to avoid exceeding the reactance limit. In our case, using the lowest frequency ensures we evaluate the worst-case scenario for most current flow resistance by the inducer, ensuring the circuit functions correctly across the frequencies it needs to.

Inductor Selection

Selecting the right inductor is a critical step in designing an amplifier circuit. It involves choosing a component with a specific inductance value that ensures the circuit meets its performance criteria throughout its operational frequency range. An inductor essentially stores energy in a magnetic field and offers resistance to changes in current through inductive reactance.
For our amplifier circuit task, the reactance was not to exceed \(100\, \Omega\). Consequently, using the lowest operation frequency (\(20\, \text{Hz}\)), we derived the largest inductance value permissible, roughly \(0.796\, \text{H}\). This selection process has to perfectly balance between containing reactance and complying with design specifications such as size, efficiency, and cost, which are practical concerns in engineering.

Electrical Engineering Concepts

Electrical engineering often deals with abstract concepts that must be precisely understood for effective design and innovation. In circuit design scenarios, concepts like resistance, reactance, impedance, and frequency all play interconnected roles.

• Resistance (\(R\)) is straightforward, opposing the flow of current.
• Reactance (\(X\)) is more nuanced since it varies with frequency and results from inductors and capacitors.
• Impedance (\(Z\)) is a combination of resistance and reactance, and plays a core role in how circuits handle alternating current.

By thoroughly understanding these concepts, engineers can craft solutions like our amplifier circuit which operates effectively within set frequency limits while observing reactance constraints. This ensures both the reliability and the effectiveness of electrical systems in real-world applications.