Lab 2 - RLC AC Circuits

Lab 2 - RLC AC Circuits

Problem

To investigate the sinusoidal steady-state behavior of simple passive circuits and to illustrate the relationship between the frequency domain and time-domain representations of AC voltages and currents.

Introduction

The presence of an inductor in an electric circuit gives the current an electrical “inertia” since inductors try to resist changes in the flow of current. The presence of a capacitor in a circuit means that charge can flow into one side of the capacitor to be stored there. Later on, this charge flows back into the circuit and restores electric current as the capacitor discharges. These two properties of inertia and energy storage are analogous to the inertia and energy storage of a mass-spring combination. In a mechanical system, friction causes damping and in electric circuits resistance causes damping.

If an electrical or mechanical system is driven by a periodic external force whose frequency matches the natural frequency of oscillations, then the system is said to be “in resonance” and the amplitude of the oscillations can grow very large.

In a series RLC circuit, as the value of the resistor is increased, the magnitude of the resonance decreases due to the increased dampening. As the magnitude of the resonance decreases, the Q decreases but the bandwidth increases. The bandwidth indicates the range of frequencies that will pass through a circuit. Conversely, the higher the Q, the smaller the range of frequencies is passed. The image below shows the relationship between Q and Bandwidth. The values are specific to this circuit, but the trends are typical of RLC circuits.

Your introduction should discuss Q, Bandwidth, and how they relate to the resonant frequency.

The relationship between frequency and current in RLC circuits should be discussed.

What a bandpass filter is, how it is used, and how the results of this lab can help with filter design (as in Q3)

What dB (decibels) are, and how they used to compare power values.

Results

Part 1:

1. Compare the expected and experimental resonant frequencies.
2. Does KVL hold based on the voltages across each element?
3. What does the voltage value across the capacitor and inductor together say about the phase?
4. What is the phase shift in V_R using V_S as a reference?
5. Draw the phasor diagram for the circuit at the resonant frequency.

Part 2:

1. Find power dissipated in each resistor at f₁ & f₂ versus V_Rmax.
2. Find the Bandwidth & Q for each resistor where Q = f₀/(f₂-f₁).

Questions

1. Graph the relationship between |Z_eq| and frequency for both resistors. What is the current at f = 0 Hz (DC) and at f = ∞ Hz ?
2. Find the Resistance value that gives a Q of 1.6.
3. Design a filter similar to the lab for 10kHz with an input impedance of R=1kΩ and a bandwidth of 1kHz. Check your results with PSpice and include the graph of V_R.
   • For a bandwidth of 1kHz centered on 10kHz, all frequencies less than 9.5kHz and greater than 10.5kHz should be at least 3 dBV down
     • 70.7% of V_in or 50% of P_o
   • Desireable frequencies are
   • The bandwidth is given by R/L (in radians) or R/(2πL) Hertz.
   • Simulate using AC Sweep from 5kHz to 15kHz.
   • What is the Q value of this filter?
   • Draw a schematic for your filter, showing the value of the inductor and capacitor chosen attached to the proposed load resistor.
   • Include your plot of V vs Frequency for the filter you designed showing the 3 dBV down line and the frequencies below and above w₀ that will be filtered.
   • This question is worth [4] towards the lab mark, it is challenging, and will require a bit calculation and a few PSpice simulations.