- To study the step response of second order circuits.
- To understand the difference between overdamped, critically damped and underdamped responses.
- To determine theoretically and experimentally the damped natural frequency in the under-damped case.
- Function generator
- Digital multimeter (DMM)
Second-order circuits are RLC circuits that contain two energy storage elements. They can be represented by a second-order differential equation. A characteristic equation, which is derived from the governing differential equation, is often used to determine the natural response of the circuit. The characteristic equation usually takes the form of a quadratic equation, and it has two roots s1 and s2.
When these roots are rewritten as,
then the natural response of the circuit is determined by:
- , there are two real and distinct roots Overdamped, as shown in Figure 5 – 1 (a) and (d).
- , there are two real equal roots Critically damped, as shown in Figure 5 – 1 (b) and (e).
- , there are two complex roots Underdamped, as shown in Figure 5 – 1 (c) and (f).
Figure 5 – 1 Second order circuits natural responses
For all circuits, C = 0.01 uF, L = 100 mH.
A. Step voltage input
- For both circuits in Figure 5 – 2, write the characteristic equation.
- Calculate the resistance range for R for the following cases:
- Over-damped response,
- Critically damped response,
- Under-damped response.
- Plot or sketch the response due to a step voltage input, when:
For the circuit in Figure 5 – 2 (a),
- R = 22 kΩ
- R = 6.3 kΩ
- R = 2.2 kΩ
For the circuit in Figure 5 – 2 (b),
- R = 680 Ω
- R = 1.6 kΩ
- R = 4.7 kΩ
Figure 5 – 2 Circuits with step input
B. Square wave input
- Set R = 470 Ω (for the circuit in Figure 5 – 3 (a)) or R = 22 kΩ (for the circuit in Figure 5 – 3 (b)). Calculate α, ωo, and ωd.
- Plot or sketch the output voltage due to a square wave input with a frequency of 400 Hz and amplitude of 4 V.
Figure 5 – 3 Circuits with square wave input
Build and simulate the circuits in Figure 5 – 3 using Multisim. Set the input voltage to with a frequency of 400 Hz. Display on the oscilloscope. Compare this result with that from PREPARATION step B.
On the function generator use the same square wave input settings as in SIMULATION. Build the circuits shown in Figure 5 – 3. Complete the measurements described below.
A. Natural responses
- Use a resistor box, and set R at the values given below. Use the DMM to check the resistance values before connecting them into the circuit.
For Figure 5 – 3 (a)
- R = 22 kΩ,
- R = 6.3 kΩ,
- R = 2.2 kΩ.
For Figure 5 – 3 (b)
- R = 680 Ω,
- R = 1.6 kΩ,
- R = 4.7 kΩ.
- On the oscilloscope, connect Ch1 to the input and Ch2 to the output so that both the input and the output are displayed on the screen.
- For each case, save the screen image with the associated measurements for both the input and the output on to a USB drive. (Follow the steps as explained in Experiment #4 to do this.)
- For each case, indicate if the output response is overdamped, critically damped or underdamped.
B. Damped natural frequency measurement
- Set R = 470 Ω for the circuit in Figure 5 – 3 (a) or R = 22 kΩ for the circuit in Figure 5 – 3 (b), save the screen image for both the input and the output, and compare it with the results from PREPARATION and SIMULATION.
- Zoom in on the output curve so that at least two whole oscillations (ripples) of the output from the beginning of an output cycle are displayed. Use the cursors to measure the time period Td between the first two peaks (or between two zero phases). ωd is calculated using:
Prepare your report as per the guidelines given in APPENDIX III.