Prelab

• Calculate the resistance, R₁ , required for circuit 8.1 to behave as though critically damped.
  • At t=0, the switch closes for sufficient time to allow for a steady state condition in the circuit.
  • The circuit then opens (square wave).
• Calculate the roots of the quadratic equation for each of the three cases and write out the general form of the solution for the step and natural response.
• Bring a USB memory key to record your oscilloscope images.

• Optional: Simulate the circuit using PSpice to confirm your calculation.
  • Use a pulse source of sufficient on-time to charge your circuit to steady state (~100Hz).
  • Measure the response of the voltage across the capacitor in each case.
    • You may use a value 1/10 R_crit and 10 R_crit to simulate turning the potentiometer.
  • View the response after the pulse source switches to the off state.
  • Simulate a sufficient time period to view the response.

Purpose

To observe the three classes of damping most commonly encountered in engineering, and how to design for or predict particular behavior in circuits.

Theory

Much like in the capacitor and inductor introductory lab, the circuits in this lab have a transient response. The voltage and current across the components are time variant, and dependent on the nature of the particular circuit to which they are connected. To build on the principles of Lab 7, these circuits will have a non-first-order response. The following circuits illustrate cases in which the complex responses are a result of the combined effects of both the capacitor and inductor influencing each other.

Resistors have a simple "Real" resistance, but capacitors and inductors have a frequency dependent "Complex" resistance, in time-varying circuits, the term impedance is used as an umbrella term for the resistance of all components. Used in this context, the word Real refers to the Real numbers, and Complex, the Complex numbers.

Just as in the last lab, we will simulate a step response (switch turning on and off) with a signal of an appropriate frequency to allow settling and steady-state between cycles. The step response allows for the demonstration of the natural response. The natural response is different than a forced response, one where there is a continuous AC signal driving the circuit.

There are several properties which are relative each other at a given frequency, and are considered characteristic of a given circuit:

• Resonant Frequency - ω_o or f_o : The frequency of oscillation in the circuit if only complex impedances were present.
  • ω_o = 1/sqrt(LC), and f_o = ω_o /2π, measured in radians per second, or Hertz.
• Damping Coefficient or "neper frequency" - α : The degree of decay in the signal per unit time.
  • α = R/(2L) for a series circuit, or 1/(2RL) for a parallel, measured in radians per second, or Hertz. R refers to the Real Resistance only.
• Damping Ratio - ζ : A measure of the damping ratio relative to the resonant frequency.
  • ζ = α/ω_o, this is a unitless parameter where:
    • ζ<1 for an underdamped, ζ>1 for an overdamped, and ζ=1 for a critically damped circuit.
• Overshoot: The magnitude of the signal crossing above/below the desired level.
• Settling Time: The period of time required to reduce the "ringing" below a certain threshold range.

As mentioned above, the resonant frequency occurs when the total impedance is at a minimum. Since capacitor impedance, and inductor impedance act in opposite directions, when they are of equal magnitude, the resulting vector will be zero. This serves as the basis for calculating the resistance for a critically damped circuit in which ζ=1, implying that α = ω_o. The procedure will first calculate the frequency at which the impedance of the capacitor and inductor is identical and opposite, and then the resistance which will cause this frequency in the circuit.

If α = ω_o Then:

1/sqrt(LC) = R/(2L) for a series circuit

or

1/(2RL) for a parallel circuit.

Solving for R will give the Real resistance (due to resistors only) to critically damp the circuit. A similar calculation can be used to find any parameter missing given the remaining two. Additionally, for any desired damping ratio, you can solve for a set of parameters that will behave as desired.

Results

1. Describe what class of damping is shown in each of the images for circuit 8.1 along side each image.
2. Did the calculated R₁ value create a critically damped circuit? How can you tell from the image?
   • What was the actual value and why might the value differ?
3. Bonus [0.5 ea] Show on the underdamped image (you may draw on the image): overshoot and settling time. Provide by inspection a value for each.

Questions

1. What is the effect of increasing the resistance on damping in a series RLC circuit? and decreasing? (as shown in lab)
2. What is the effect of increasing the resistance on damping in a parallel RLC circuit? and decreasing? (not shown in lab)

No formal report is required for this lab.

Include:

• Titlepage.
• Oscilloscope images with a description of what type of damping is shown.
• The calculated and measured value of R_crit with explanation of any difference.
• Answer the questions posed.