Question

The discussion of \(\mathrm{RL}, \mathrm{RC},\) and \(\mathrm{RLC}\) circuits in this chapter has assumed a purely resistive resistor, one whose inductance and capacitance are exactly zero. While the capacitance of a resistor can generally be neglected, inductance is an intrinsic part of the resistor. Indeed, one of the most widely used resistors, the wire-wound resistor, is nothing but a solenoid made of highly resistive wire. Suppose a wire-wound resistor of unknown resistance is connected to a DC power supply. At a voltage of \(V=10.0 \mathrm{~V}\) across the resistor, the current through the resistor is 1.00 A. Next, the same resistor is connected to an AC power source providing \(V_{\mathrm{rms}}=10.0 \mathrm{~V}\) at a variable frequency. When the frequency is \(20.0 \mathrm{kHz},\) a current, \(I_{\mathrm{rms}}=0.800 \mathrm{~A},\) is measured through the resistor. a) Calculate the resistance of the resistor. b) Calculate the inductive reactance of the resistor. c) Calculate the inductance of the resistor. d) Calculate the frequency of the AC power source at which the inductive reactance of the resistor exceeds its resistance.

Short answer

In summary, the unknown resistor has a resistance of 10.0 Ω when connected to the DC power source; when connected to the AC power source, its impedance is 12.5 Ω, and its inductive reactance is 7.5 Ω. The inductance of the resistor is approximately 5.96 × 10^(-6) H. The frequency at which the inductive reactance exceeds the resistance is approximately 267.2 kHz.

Step-by-step solution

Calculate the resistance of the resistor when connected to the DC power source

To calculate the resistance, we can use Ohm's law: \(V = IR\). We are given that the voltage across the resistor is \(V = 10.0\) V, and the current is \(I = 1.00\) A. Plugging in the values, we get: R = \(\frac{V}{I} = \frac{10.0}{1.00}\) = 10.0 \(\Omega\) So, the resistance of the resistor is 10.0 \(\Omega\).

Calculate the impedance of the resistor connected to the AC power source

The impedance, Z, of a resistor and inductor connected in series can be calculated using the formula: \(Z = \sqrt{R^2 + X_L^2}\) We have the resistance, R = 10.0 \(\Omega\), and the AC current, \(I_{\mathrm{rms}} = 0.800\) A. We also know that the AC voltage is \(V_{\mathrm{rms}} = 10.0\) V. Using Ohm's law for impedance, we get: \(Z = \frac{V_{\mathrm{rms}}}{I_{\mathrm{rms}}} = \frac{10.0}{0.800} = 12.5\) \(\Omega\)

Calculate the inductive reactance of the resistor

Now that we have the impedance, Z, and resistance, R, we can calculate the inductive reactance, \(X_L\), using the equation: \(Z^2 = R^2 + X_L^2 \Rightarrow X_L = \sqrt{Z^2 - R^2} = \sqrt{12.5^2 - 10.0^2} = \sqrt{56.25} = 7.5\) \(\Omega\) So, the inductive reactance is 7.5 \(\Omega\).

Calculate the inductance of the resistor

We can find the inductance, L, using the formula for inductive reactance: \(X_L = 2\pi fL \Rightarrow L = \frac{X_L}{2\pi f}\) We are given the frequency, \(f = 20.0\) kHz = \(20.0 \times 10^3\) Hz. Plugging in the values, we get: \(L = \frac{7.5}{2\pi (20.0 \times 10^3)}\) = \(5.96 \times 10^{-6}\) H So, the inductance of the resistor is approximately \(5.96 \times 10^{-6}\) H.

Calculate the frequency at which the inductive reactance exceeds the resistance

We are asked to find the frequency at which the inductive reactance, \(X_L\), is equal to the resistance, R. Using the formula for inductive reactance, we can write: \(X_L = R \Rightarrow 2\pi f L = R \Rightarrow f = \frac{R}{2\pi L}\) Using the obtained values for R and L, we have: \(f = \frac{10.0}{2\pi (5.96 \times 10^{-6})} \approx 267.2\times 10^3\) Hz So, the frequency at which the inductive reactance of the resistor exceeds its resistance is approximately 267.2 kHz.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in electronics that relates the voltage, current, and resistance of an electric circuit. It is expressed with the equation \( V = IR \), where \( V \) is the voltage across the resistor, \( I \) is the current flowing through it, and \( R \) is the resistance.

This simple relationship allows us to calculate any one of these three quantities if we know the other two. Ohm's Law is especially helpful when analyzing circuits and solving electrical problems. It's important to note that Ohm's Law applies to linear, ohmic devices, where the current changes directly with voltage.

Impedance

In AC circuits, the concept of impedance extends Ohm's Law by taking into account not just resistance, but also reactance, which results from inductors and capacitors. Impedance, denoted as \( Z \), is the total opposition that a circuit presents to alternating current and is measured in ohms.

The formula for impedance when resistance \( R \) and inductive reactance \( X_L \) are present in series is \( Z = \sqrt{R^2 + X_L^2} \).

Impedance accounts for the phase shift between voltage and current caused by the reactive components. Like resistance, higher impedance results in less current flow for a given AC voltage.

Inductive Reactance

Inductive reactance \( X_L \) is the opposition that an inductor presents to the change in current. It occurs only in AC circuits and is a key factor in impedance calculations.

The inductive reactance increases with frequency and the value of the inductance. The formula \( X_L = 2\pi fL \) relates inductive reactance, frequency \( f \), and inductance \( L \).

Unlike resistance, which dissipates energy, inductive reactance temporarily stores and then releases energy in the magnetic field of the inductor, causing a phase shift between voltage and current.

Inductance

Inductance, symbolized by \( L \), is the property of an electrical component to oppose changes in current. It is measured in henrys (H) and relates to the physical characteristics of the inductor, such as the number of coils and the material's permeability.

Formally, inductance is the ratio of the change in magnetic flux to the change in current. It quantifies how effectively a wire winding can generate magnetic fields and influence the electrical circuit.

Inductors are crucial in applications requiring current regulation, filtering, and energy storage.

Wire-Wound Resistor

A wire-wound resistor is a specific type of resistor constructed by winding a resistive wire around a core. This design naturally incorporates some inductance, due to its similarity to a coil or solenoid.

Wire-wound resistors are appreciated for their precision and ability to handle large amounts of power. However, their inherent inductance can affect high-frequency AC circuits, making them act like an inductor alongside their resistive properties.

These resistors are often used where high accuracy and stability are required, such as in laboratory equipment and high-power electronics.