Question

Figure \(6.12\) shows a series \(R L C\) circuit driven by a sinusoidal \(A C\) voltage source whose value is \(120 \angle 0^{\circ}\) volts. The impedance of the inductor in this circuit is \(Z_{L}=j 2 \pi f L\), where \(j\) is \(\sqrt{-1}, f\) is the frequency of the voltage source in hertz, and \(L\) is the inductance in henrys. The impedance of the capacitor in this circuit is \(Z_{C}=-j \frac{1}{2 \pi f C^{*}}\) where \(C\) is the capacitance in farads. Assume that \(R=100 \Omega, L=0.1 \mathrm{mH}\), and \(C=\) \(0.25 \mathrm{nF}\). The current I flowing in this circuit is given by Kirchhoff's Voltage Law to be $$ \mathbf{I}=\frac{120 \angle 0^{\circ} \mathrm{V}}{R+j 2 \pi f L-j \frac{1}{2 \pi f^{C}}} $$ a. Calculate and plot the magnitude of this current as a function of frequency as the frequency changes from \(100 \mathrm{kHz}\) to \(10 \mathrm{MHz}\). Plot this information on both a linear and a log-linear scale. Be sure to include a title and axis labels. b. Calculate and plot the phase angle in degrees of this current as a function of frequency as the frequency changes from \(100 \mathrm{kHz}\) to \(10 \mathrm{MHz}\). Plot this information on both a linear and a log-linear scale. Be sure to include a title and axis labels. c. Plot both the magnitude and phase angle of the current as a function of frequency on two sub-plots of a single figure. Use log-linear scales.

Short answer

In conclusion, we analyzed an RLC circuit driven by a sinusoidal AC voltage source and calculated the total impedance, complex current, magnitude and phase angle of the current. By plotting the magnitude and phase angle of the current as functions of frequency, we were able to visualize how the current changes with frequency and observe the properties of the RLC circuit in both linear and log-linear scales.

Step-by-step solution

Calculate Total Impedance Z

Using the given values of resistance, inductance and capacitance, we can calculate the total impedance \(Z\) in the circuit: \(Z = R + j 2 \pi fL - j \frac{1}{2 \pi fC}\) Here, \(R = 100 \Omega\), \(L = 0.1mH\), and \(C = 0.25nF\).

Solve for Complex Current I

Next, we will calculate the complex current \(\mathbf{I}\) as a function of frequency using the given formula: \(\mathbf{I} = \frac{120 \angle 0^{\circ} V}{R + j 2 \pi fL - j \frac{1}{2 \pi fC}}\)

Calculate Magnitude and Phase Angle of the Current

We can express the complex current as a complex number in polar form: \(\mathbf{I} = |I| \angle \Theta^\circ\). To calculate the magnitude \(|I|\) and phase angle \(\Theta^\circ\) at any frequency, we can use the following formulas: \( |I| = \frac{120}{\sqrt{R^2 + (2 \pi fL - \frac{1}{2 \pi fC})^2}}\) and \(\Theta^\circ = \arctan{\frac{2 \pi fL - \frac{1}{2 \pi fC}}{R}}\)

Plot Magnitude and Phase Angle as Functions of Frequency

To visualize the results, we need to create the following plots: 1. Magnitude of the current as a function of frequency (linear and log-linear scale) with title and axis labels. 2. Phase angle of the current as a function of frequency (linear and log-linear scale) with title and axis labels. 3. A single figure with two sub-plots for magnitude and phase angle of the current in log-linear scales. For this step, use a software tool like Python, MATLAB, or other data visualization tool to plot the magnitude and phase angle of the current as functions of frequency given the formulas for magnitude and phase angle. Remember to clearly label the plots with titles, axis labels, and legends.

Key concepts

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) is a fundamental principle in circuit analysis that states the sum of all voltages around any closed loop in a circuit equals zero. This law reflects the conservation of energy where the total amount of voltage provided by sources is equal to the voltage drops across the components.

Let's use this principle to analyze an RLC circuit, which consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series with an AC voltage source. According to KVL, the voltage across the resistor plus the voltage across the inductor plus the voltage across the capacitor must equal the voltage of the AC source. In the exercise, this is applied to derive the equation for the complex current I flowing through the circuit.

For students to understand KVL better, it's important to grasp that voltages across inductors and capacitors in AC circuits are 'out of phase' with the current due to their reactive properties, which is why the impedance of these components includes the imaginary unit 'j'. The voltage across each component is not just an amplitude, but also has a phase angle, leading to the use of complex arithmetic in these calculations.

Impedance Calculation

Impedance calculation is essential for analyzing any AC circuit's behavior. Impedance, denoted as Z, combines the resistance (R) and the reactance (X) of a circuit element into a single measure and is expressed in ohms (Ω). The impedance of an RLC circuit is complex, meaning it has both a real part (resistance) and an imaginary part (reactance), which is frequency-dependent.

In our exercise, the total impedance Z of the series RLC circuit is calculated with the following formula: \[ Z = R + j 2 \pi fL - j \frac{1}{2 \pi fC} \.\] Here, j represents the square root of -1, reflecting the right angle phase shift between resistance and reactance. Inductive reactance increases with frequency (\(j2\pi fL\)), while capacitive reactance decreases with frequency (\(-j\frac{1}{2\pi fC}\)).

By understanding these relationships, students can better envision how components behave differently at various frequencies. A resistor's impedance is purely real and constant with frequency, an inductor's impedance increases linearly with frequency, and a capacitor's impedance decreases inversely with frequency.

Complex Current Analysis

Complex current analysis is indispensable when dealing with AC circuits because the current varies sinusoidally with time. Current's complexity arises from its magnitude and phase relative to the voltage source. By representing the current in the complex form, circuit analysis can accommodate phase shifts due to reactive components.

In our RLC circuit example, the complex current \( \mathbf{I} \) is found using the following formula: \[ \mathbf{I} = \frac{120 \angle 0^{\circ} \mathrm{V}}{R + j 2 \pi fL - j \frac{1}{2 \pi fC}} \.\] The numerator represents the voltage source with a phase angle of 0°. The denominator is the total impedance Z of the circuit. The complex current I then has a magnitude and phase that can be determined using the magnitude-phase form \( \mathbf{I} = |I| \angle \Theta^\circ \).

For students, visualizing this current as a vector on the complex plane can assist in understanding how the current's magnitude and direction (phase) change with different frequencies. The vector's length is the current's magnitude, and the angle it makes with the horizontal axis is the phase angle.

Frequency Response Plot

A frequency response plot is a powerful graphical tool used to visualize how an RLC circuit responds to a range of frequencies. It shows the magnitude and phase of the output signal, such as voltage or current, as a function of frequency.

In our case, we're interested in plotting the magnitude and phase angle of the complex current I. Such plots reveal important characteristics such as resonance frequencies, bandwidth, and stability of the circuit. The exercise instructs plotting these characteristics on both linear and log-linear scales. A linear scale plots values with equal spacing, whereas a log-linear scale uses a logarithmic scale for frequency, helping to compress a wide range of frequencies and emphasize behavior at lower frequencies.

Creating both types of plots provides a comprehensive view of the circuit's behavior. The linear scale is intuitive, while the log-linear scale is better for observing the circuit's response over several orders of magnitude in frequency. These plots help students correlate mathematical expressions of current with visually interpretable graphs, greatly enhancing their understanding of AC circuit operation and design.