Question

A series combination of a \(22.0-\mathrm{mH}\) inductor and a \(145.0-\Omega\) resistor are connected across the output terminals of an ac generator with peak voltage \(1.20 \mathrm{kV},\) (a) \(\mathrm{At}\) \(f=1250 \mathrm{Hz},\) what are the voltage amplitudes across the inductor and across the resistor? (b) Do the voltage amplitudes add to give the source voltage (i.e., does \(\left.V_{R}+V_{L}=1.20 \mathrm{kV}\right) ?\) Explain. (c) Draw a phasor diagram to show the addition of the voltages.

Short answer

Question: In a series RL circuit, the peak source voltage is \(1.20 \mathrm{kV}\), the resistance of the resistor is \(42 \Omega\), and the inductor has an inductance of \(150 \mathrm{mH}\). The frequency of the AC source is \(60 \mathrm{Hz}\). Calculate the voltage amplitudes across the inductor and resistor, and verify that they add up to the source voltage. Draw a phasor diagram to explain the voltage addition. Answer: The voltage amplitude across the inductor is \(V_L = 1048.82 \mathrm{V}\), and across the resistor is \(V_R = 521.94 \mathrm{V}\). Yes, their sum equals the source voltage \((V_R + V_L = 1.20 \mathrm{kV})\). The phasor diagram shows the addition of these voltages as vectors in the complex plane, with \(V_R\) being a horizontal vector and \(V_L\) being a vertical vector, adding up to form a right-angled triangle with the source voltage as the hypotenuse.

Step-by-step solution

Calculating the Impedance of the Circuit

To find the impedance of the circuit, we will first calculate the inductive reactance of the inductor \((X_L)\) using the formula: \(X_L = 2\pi fL\), where \(f\) is the frequency and \(L\) is the inductance. Then, we will find the impedance \((Z)\) using the formula: \(Z = \sqrt{R^2 + X_L^2}\), where \(R\) is the resistance of the resistor.

Calculating the Current in the Circuit

To find the current in the circuit, we will use Ohm's law. Since the given voltage is the peak voltage, we will first convert it to the rms voltage using the formula: \(V_{rms} = \frac{V_{peak}}{\sqrt{2}}\). Then, we will find the current \((I)\) using: \(I = \frac{V_{rms}}{Z}\)

Calculating Voltage Amplitudes across Inductor and Resistor

To find the voltage amplitudes across the inductor and resistor \((V_L, V_R)\), we will use: \(V_L = IX_L\) and \(V_R = IR\). Lastly, we'll convert these rms voltages into peak voltages by multiplying with \(\sqrt{2}\).

Verifying if Voltage Amplitudes Add to Give Source Voltage

We will check whether the sum of \(V_L\) and \(V_R\) equals the source voltage, i.e., \(V_{R} + V_{L} = 1.20 \mathrm{kV}\).

Drawing Phasor Diagram

Lastly, we will draw a phasor diagram to show the addition of the voltages. We will represent \(V_R\) and \(V_L\) as vectors in the complex plane and show their addition in the diagram.