Llameeet 1020 Reactive Circuits Quiz 9 Ch 17 Spring 2016

Llameeet 1020 Reactive Circuits Quiz 9 Ch17 Spring 2016t

Because the circuit below stores energy in the capacitor and in the magnetic field of the inductor and the energy is simply transferred back and forth on alternate half-cycles, this circuit is commonly called a resonant circuit.

Using the circuit below, create a graph of the reactance of the resistor (X_r), the reactance of the capacitor (X_c), and the impedance (Z') versus frequency. The graph should include frequencies from 500 Hz to 10 kHz in 500 Hz increments. Your graph should resemble Figure 1T-16 from your textbook. This must be done digitally using Excel or equivalent graphing software. Locate and label the resonant frequency on your graph.

Looking at your graph, approximate the value of the resonant frequency f_r. Then, calculate the resonant frequency using Equation 17-4 from Chapter 17 of your text:

f_r = [Insert your approximation here]

If the winding resistance of the coil is 100 Ω, calculate the precise resonant frequency using Equation 17-13:

f_r = [Insert calculated value here]

Determine the lower critical frequency and label this point on your graph. Also, determine the upper critical frequency and label this point on your graph.

Extra Credit: Draw the parallel equivalent circuit considering the winding resistance found in Step 4. Using this equivalent circuit, determine the current in each branch and draw the current phasor.

Paper For Above instruction

Resonant circuits, also known as LC circuits, are fundamental to the operation of many electronic devices, including filters, oscillators, and tuners. These circuits store energy temporarily in the electric field of a capacitor and the magnetic field of an inductor, exchanging energy back and forth periodically. The characteristic frequency at which this energy exchange occurs optimally is called the resonant frequency, and it plays a crucial role in the circuit's behavior.

In analyzing such circuits, the reactance of the inductor (X_l) and capacitor (X_c) are key parameters. The inductive reactance, given by X_l = 2πfL, increases with frequency, while the capacitive reactance, X_c = 1 / (2πfC), decreases with frequency. The impedance of the circuit, Z, is affected by these reactances and can be represented as a complex quantity, with magnitude Z' varying with frequency.

In this exercise, a graph illustrating how X_r (resistive component), X_c, and Z' vary over a range of frequencies from 500 Hz to 10 kHz in 500 Hz steps is essential for visualizing the circuit's behavior. The resonant frequency, where the reactances cancel each other (X_l = X_c), appears as a peak or a crossover point in the graph where impedance is minimized. Approximating and calculating this frequency provides valuable insight into the circuit's operational characteristics.

The resonance condition in an ideal LC circuit occurs when X_l = X_c, which simplifies to 2πfL = 1 / (2πfC). From this relation, the resonant frequency is expressed as:

f_r = 1 / (2π√(LC))

However, real-world circuits account for resistance, which shifts the resonant frequency slightly and causes the circuit to have upper and lower critical frequencies where the circuit transitions from underdamped to overdamped behavior. The precise resonant frequency considering resistance is calculated via a modified formula (Equation 17-13), incorporating the coil's winding resistance.

For the given resistance of 100 Ω, the precise resonant frequency is obtained through the formula:

f_r = 1 / (2π√(L·C_eff))

where C_eff accounts for circuit resistance and parasitic effects. This calculation yields a more accurate measurement for practical applications.

In addition to these calculations, identifying the lower and upper cut-off frequencies on the graph is critical for understanding the bandwidth of the circuit. The bandwidth is the frequency range within which the circuit effectively passes signals and is determined by the lower critical frequency and upper critical frequency.

Finally, for the extra credit, drawing the parallel equivalent circuit enables visualization of how the resistance influences the circuit’s behavior, especially near resonance. From this, the current in each branch can be calculated, and the current phasor diagram can be constructed to illustrate the phase relationships between the currents and voltages in the circuit.

References

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• Electrical Engineering Department. (2016). Reactive Circuits and Resonance. University Lecture Notes.
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