Inductors in AC Circuits: Videos & Practice Problems

Inductors in AC circuits exhibit unique behaviors where the voltage leads the current by 90 degrees, as described by the equation for voltage across an inductor: $I^{\max }\omega L\cos (\omega t+\frac{\pi }{2})$. The inductive reactance, akin to resistance, is given by $X=\omega L$. Understanding these relationships is crucial for analyzing AC circuits effectively.

Inductors in AC Circuits

Inductors play a crucial role in alternating current (AC) circuits, influencing how voltage and current interact. In an AC circuit, the current oscillates over time, described by the equation I(t) = I_{max} \sin(\omega t), where I_{max} is the maximum current and \omega is the angular frequency. The voltage across an inductor is determined by Faraday's law, expressed as V_L = L \frac{dI}{dt}, where L is the inductance and \frac{dI}{dt} is the rate of change of current.

Using calculus, the voltage across an inductor in an AC circuit can be represented as V_L(t) = I_{max} \omega L \cos(\omega t + \frac{\pi}{2}). This indicates that the voltage across the inductor leads the current by 90 degrees, meaning that the voltage reaches its peak before the current does. In contrast, the voltage across a resistor is in phase with the current, represented by V_R(t) = I_{max} R \cos(\omega t). The phase difference between the voltage and current in an inductor is a key characteristic of inductive reactance.

The maximum voltage across an inductor can be calculated as V_{max} = I_{max} \omega L, where \omega L is known as the inductive reactance, analogous to resistance in resistors. The units for inductive reactance are ohms, similar to resistance.

To illustrate these concepts, consider an example where an AC power source provides a maximum voltage of 120 volts at a frequency of 60 hertz. If the maximum current in the circuit is 5 amps, we can find the inductance of the inductor. According to Kirchhoff's loop rule, the maximum current can be expressed as I_{max} = \frac{V_{max}}{X_L}, where X_L = \omega L. Rearranging gives us L = \frac{V_{max}}{\omega I_{max}}.

First, we calculate the angular frequency: \omega = 2\pi f = 2\pi \times 60 \text{ Hz} \approx 377 \text{ s}^{-1}. Substituting the known values into the inductance formula yields:

L = \frac{120 \text{ V}}{377 \text{ s}^{-1} \times 5 \text{ A}} \approx 0.064 \text{ H}.

This example highlights the relationship between voltage, current, and inductance in AC circuits, emphasizing the importance of understanding inductors in electrical engineering.

Inductors and Graphs

In this example, we analyze the relationship between voltage, current, and inductive reactance in an AC circuit involving an inductor. The peak voltage of the AC source is determined by observing the maximum voltage across the inductor, which, according to Kirchhoff's loop rule, matches the source voltage. In this case, the peak voltage, denoted as \( V_{\text{max}} \), is 10 volts.

Next, we calculate the frequency of the AC source. The graph provides a time interval that represents one complete cycle, allowing us to determine the period \( T \). Here, the period is found to be 0.1 seconds. The frequency \( f \) can then be calculated using the formula:

\[ f = \frac{1}{T} \]

Substituting the period, we find:

\[ f = \frac{1}{0.1 \, \text{s}} = 10 \, \text{Hz} \]

Finally, we address the inductive reactance \( X_L \) of the circuit. The maximum current through the inductor is given as 2.5 amps. The relationship between the maximum voltage, maximum current, and inductive reactance is expressed by the formula:

\[ X_L = \frac{V_{\text{max}}}{I_{\text{max}}} \]

Substituting the known values, we have:

\[ X_L = \frac{10 \, \text{V}}{2.5 \, \text{A}} = 4 \, \Omega \]

This example illustrates the straightforward application of fundamental AC circuit principles, including the relationships between voltage, current, frequency, and inductive reactance.

Will a frequency f = 60 Hz or ω = 75 s⁻¹ produce a larger max current in an inductor connected to an AC source?

• A. f = 60 Hz
• B. ω = 75 s^-1
• C. Both produce the same amount of maximum current
• D. More information is required.

Answer: (B)

Here’s what students ask on this topic:

What is the role of inductors in AC circuits?

In AC circuits, inductors play a crucial role by opposing changes in current. The voltage across an inductor leads the current by 90 degrees, which means the voltage reaches its peak before the current does. This phase difference is described by the equation for voltage across an inductor: $V_L=I_{\max }\bullet \omega \bullet L\bullet \cos (\omega t+\frac{\pi }{2})$. This behavior is essential for understanding how inductors affect the overall impedance and phase relationships in AC circuits.

How do you calculate the inductive reactance in an AC circuit?

The inductive reactance (X_L) in an AC circuit is calculated using the formula: $X_L=\omega \bullet L$, where $\omega$ is the angular frequency and $L$ is the inductance. The angular frequency is given by $\omega =\; 2\pi \bullet f$, where $f$ is the frequency in hertz. Inductive reactance is measured in ohms (Ω) and represents the opposition that an inductor offers to the AC current.

Why does the voltage lead the current in an inductor in an AC circuit?

The voltage leads the current in an inductor in an AC circuit because of the inductor's property of opposing changes in current. According to Faraday's Law, the voltage across an inductor is proportional to the rate of change of current through it. Mathematically, this is expressed as $V_L=L\bullet \frac{\mathrm{dI}}{\mathrm{dt}}$. As a result, the voltage reaches its maximum value before the current does, creating a phase difference of 90 degrees. This phase shift is crucial for understanding the behavior of AC circuits containing inductors.

How do you determine the inductance of an inductor in an AC circuit?

To determine the inductance (L) of an inductor in an AC circuit, you can use the relationship between the maximum voltage (V_max), maximum current (I_max), and angular frequency (ω). The formula is: $L=\frac{V_{\max }}{\omega }$. First, calculate the angular frequency using $\omega =\; 2\pi \bullet f$, where $f$ is the frequency in hertz. Then, plug in the values for V_max, I_max, and ω to find the inductance.

What is the difference between inductive reactance and resistance?

Inductive reactance (X_L) and resistance (R) both oppose the flow of current, but they do so in different ways. Resistance is a property of resistors and is constant, regardless of the frequency of the current. It is measured in ohms (Ω) and follows Ohm's Law: $V=I\bullet R$. Inductive reactance, on the other hand, depends on the frequency of the AC current and the inductance of the inductor. It is given by $X_L=\omega \bullet L$, where $\omega$ is the angular frequency. Unlike resistance, inductive reactance causes a phase shift between voltage and current.