Question

A coil has a resistance of 48.0 \(\Omega\). At a frequency of 80.0 Hz the voltage across the coil leads the current in it by 52.3\(^\circ\). Determine the inductance of the coil.

Short answer

The inductance of the coil is 0.123 H.

Step-by-step solution

Understanding the Problem

We are given a coil with a resistance \( R = 48.0 \, \Omega \) and a phase angle \( \theta = 52.3^\circ \) between voltage and current at a frequency \( f = 80.0 \, \text{Hz} \). We need to find the coil's inductance \( L \).

Converting Phase Angle to Radians

Convert the phase angle from degrees to radians since it's more convenient for calculations. Use the formula: \[\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\]For \( \theta = 52.3^\circ \), \( \theta = \frac{\pi}{180} \times 52.3 = 0.912 \, \text{radians} \).

Understanding Impedance and Inductive Reactance

The total impedance \( Z \) of the coil can be determined from the phase angle \( \theta \) using the relation:\[\tan(\theta) = \frac{X_L}{R}\]where \( X_L \) is the inductive reactance. We solve for \( X_L \) from the tangent of the phase angle.

Calculating Inductive Reactance (\(X_L\))

Using \( \tan(\theta) = \frac{X_L}{R} \), solve for \( X_L \):\[X_L = R \times \tan(\theta)\]Substituting values, \[ X_L = 48.0 \, \Omega \times \tan(0.912) = 48.0 \, \Omega \times 1.279 = 61.84 \, \Omega \].

Determining Inductance (\(L\))

Using the formula for inductive reactance, \( X_L = 2\pi f L \), solve for \( L \):\[L = \frac{X_L}{2\pi f}\]Substituting the known values, \[L = \frac{61.84}{2\pi \times 80.0} = \frac{61.84}{502.65} = 0.123 \, \text{H} \].

Conclusion

The inductance of the coil is \( L = 0.123 \, \text{H} \).

Key concepts

Impedance

Impedance is a key concept when dealing with AC circuits. It is similar to resistance in a DC circuit but is more complex as it combines both resistance (R) and reactance (X).
In an AC circuit, impedance is represented as a complex number because it has both a magnitude and a phase component. The magnitude tells us how much the circuit impedes the flow of AC electricity, while the phase component indicates how the current and voltage waveforms are out of step with each other. This is crucial in understanding how components like coils work.

• Impedance is represented by the letter "Z".
• Measured in ohms (Ω), just like resistance.
• It can also convey how much a given component might "twist" the voltage and current out of alignment.

In this exercise, the coil's impedance links directly to the phase angle, which is the measure of how much the voltage leads over the current. The greater the impedance, the more the current lags behind the voltage.

Phase Angle

When you hear the term phase angle, think about timing differences in AC waveforms. In this context, it's the angle of lead or lag between the voltage and the current.
The phase angle is particularly important because it shows us how out of synchrony the current is in relation to the voltage.

• Phase angle is denoted by the symbol \(\theta\).
• Measured in degrees or radians. For calculations, converting degrees to radians is often necessary using the conversion \(\theta = \frac{\pi}{180} \times \text{degrees}\).
• It shows the "delay" or "lead" the current has, which reflects on how the coil (or any reactance) behaves in the circuit.

Understanding how to convert and use the phase angle is essential for impedance and reactance calculations, especially when determining properties like a coil's inductance. In your exercise, we saw a phase angle of 52.3° transformed to roughly 0.912 radians.

Inductive Reactance

Inductive reactance is closely related to how a coil behaves in an AC circuit. It is a measure of the coil's opposition to the change in current and is dependent on both the frequency of the AC and the inductance itself.
Inductive reactance is defined mathematically as \(X_L = 2\pi f L\), where \(f\) is the frequency of the alternating current, and \(L\) is the inductance of the coil.

• Inductive reactance is represented by the symbol \(X_L\).
• Also measured in ohms (Ω), like impedance and resistance.
• Essentially, the higher the frequency or inductance, the greater the inductive reactance.

In the original exercise, you calculated the inductive reactance \(X_L\) to be 61.84 Ω. This was found by using the relationship \(X_L = R\tan(\theta)\), connecting it to the known coil resistance and the phase angle. This step was crucial for ultimately calculating the coil's inductance. By understanding and calculating \(X_L\), you learn how much the coil "affects" the AC current flow.