Problem 31 - College Physics

Question

A series RL circuit has a resistance of \(100 \Omega\) and an inductance of \(100 \mathrm{mH}\) and is driven by a \(120-\mathrm{V}, 60-\mathrm{Hz}\) source. (a) Find the inductive reactance and the impedance of the circuit. (b) How much current is drawn from the source?

Step-by-Step Solution

Answer

Inductive reactance is approximately 37.70 Ω; impedance is 106.91 Ω; current is 1.12 A.

Step 1: Calculate the Inductive Reactance

The inductive reactance, denoted as \(X_L\), is calculated using the formula \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L\) is the inductance. Here, \(f = 60 \, \text{Hz}\) and \(L = 100 \, \text{mH} = 0.1 \, \text{H}\). Therefore, \(X_L = 2 \pi (60)(0.1) = 12 \pi \, \Omega \approx 37.70 \, \Omega\).

Step 2: Calculate the Impedance of the Circuit

The total impedance \(Z\) in a series RL circuit is found using the formula \(Z = \sqrt{R^2 + X_L^2}\), where \(R\) is the resistance and \(X_L\) is the inductive reactance. Here, \(R = 100 \, \Omega\) and \(X_L \approx 37.70 \, \Omega\). Therefore, \(Z = \sqrt{100^2 + 37.70^2} \approx \sqrt{10000 + 1421.29} \approx \sqrt{11421.29} \approx 106.91 \, \Omega\).

Step 3: Calculate the Current Drawn from the Source

The current \(I\) drawn from the source is calculated using Ohm's Law for AC circuits: \(I = \frac{V}{Z}\), where \(V\) is the voltage and \(Z\) is the total impedance. Here, \(V = 120 \, \text{V}\) and \(Z \approx 106.91 \, \Omega\). Therefore, \(I = \frac{120}{106.91} \approx 1.12 \, \text{A}\).

Key Concepts

Inductive Reactance

In alternating current (AC) circuits, inductive reactance is a crucial concept that captures how an inductor resists the flow of current. Inductive reactance, denoted as \(X_L\), depends on the frequency of the AC source and the value of the inductance. It is computed using the formula:

• \(X_L = 2\pi f L\)

where \(f\) is the frequency in hertz and \(L\) is the inductance in henrys.
This relationship shows that as the frequency or the inductance increases, the inductive reactance also increases, thereby opposing more current.
In our exercise, for the series RL circuit with a frequency of 60 Hz and an inductance of 0.1 H, we calculated \(X_L\) to be approximately 37.70 ohms.
A higher reactance implies that the inductor presents more opposition to changes in current at higher frequencies.
Understanding inductive reactance is vital because it affects how an AC circuit behaves, especially its phase relationship between voltage and current.

Impedance

Impedance combines both resistance and reactance in AC circuits. It’s the complete opposition that a circuit presents to the passage of current when voltage is applied. Impedance is represented by the symbol \(Z\) and is measured in ohms (\(\Omega\)).
In a series RL circuit, impedance is calculated as:

• \(Z = \sqrt{R^2 + X_L^2}\)

where \(R\) is resistance, and \(X_L\) is inductive reactance.
This equation highlights that impedance takes into account both the resistive components and the frequency-dependent reactance found in inductors.
In the example, the resistance is 100 ohms, and the inductive reactance is approximately 37.70 ohms. Plugging these into the formula gives us a total impedance of approximately 106.91 ohms.
Unlike simple resistance, impedance expresses the phase delay introduced by the reactive components, showing both magnitude and direction in what is called a phasor form.
Understanding impedance helps with matching loads in an AC circuit, ensuring efficient power transfer.

Ohm's Law for AC Circuits

Ohm's Law is a fundamental principle in electrical engineering and applies to both DC and AC circuits. For AC circuits specifically, it relates the properties of voltage (V), current (I), and impedance (Z) through the formula:

• \(I = \frac{V}{Z}\)

Here, \(I\) is the current in amperes, \(V\) is the voltage in volts, and \(Z\) is the impedance in ohms.
In this form of Ohm's Law, we see that the current is inversely proportional to the impedance for a given voltage.
In our RL circuit example, with a source voltage of 120 volts and an impedance of 106.91 ohms, the current drawn is computed to be approximately 1.12 amperes.
This relationship is essential as it helps predict how much current will flow in an AC circuit under given conditions, an important factor in designing circuits and determining their operating conditions.
Recognizing these variations in Ohm's Law for AC circuits is vital for accuracy in AC power applications, ensuring that circuits are not overloaded and operate safely.