ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
ECE 231 Laboratory Exercise 8. Time Domain Response 2nd Order Circuits
Laboratory Group (Names) ______
OBJECTIVES
- Observe and calculate the time domain response of a 2nd order circuit.
- Gain experience in plotting circuit response.
- Gain experience in observing the time domain response of a 2nd order circuit on an oscilloscope.
EQUIPMENTREQUIRED
- ECE 231Circuit Board (In Stock room) – use the 10K potentiometer and 0.1 f capacitor on board
- Three BNC cables (one for the out square wave signal from the signal generator and two for the circuit input/ output voltages to the oscilloscope)
- One lot of clip leads and/or jumper wires
- 100 mH to 400 mH inductor box (stockroom)
- Signal generator. Use square wave output for channel 1 input to the oscilloscope and to the circuit input. Set the signal generator to 10 Vp-p at 100 Hz or slower. Make sure you set the signal generator output to high impedance – 1 Mohm not the 50 ohm. The input impedance to the oscilloscope is 1 Mohm.
- Two channel oscilloscope. Use channel 1 for the output from the signal generator and channel 2 for the output (voltage across the capacitor).
BACKGROUND
The time domain response of a 2nd order circuit is important to understanding the transient behavior of many physical systems. There are three cases that will be examined in this laboratory experiment. They are the under damped, critically damped, and over damped cases. You will be using a series circuit similar to the one used when determining the frequency response of a circuit in Experiment 7 except that the output of the circuit will be taken across the capacitor not the resistor. Equations (6), (9), and (12) below show the basic form of the circuit behavior for the threepossible solutions to the 2nd order system differential equation (2) that describes behavior of the circuit shown in Figure 2. These equations are derived beginning with a loop equation of the circuit shown in Figure 2. By knowing the boundary conditions (initial and final values), the coefficients of the defining equations can be determined. The bandwidth of the input square wave should be at least 5 times the time for the critically damped response to reach steady state.
The experiment will explore the response of the circuit to a step input. This will be done by applying a square wave to the circuit and observing the output response on an oscilloscope When applying a square wave to the circuit the frequency of the square wave must be lower in frequency (longer in time) than the time response you are trying to observe. Usually five time constants of the critically damped response is sufficient.
The response of the voltage across the capacitor in this experiment is similar to driving a car over a speed bump and observing the response of the shock absorber system. An elevator control system is an example of a critically damped or over damped control system. Many structural systems are under damped and so are many mechanical systems. A mechanical scale may be designed for any of the three types of responses. The pneumatic closure on a door is normally over damped. The damping ratio is defined as the cosine of the angle between the natural frequency vector,ωn, and the real axisin the complex plain such that ωn . The units are radians/second not Hz (cycles per second) See Figure 1.
(source: MIT Dept. of Mechanical Engineering)
Figure 1. Complex Plane where ζ is the dampening coefficient.
You will need to construct the circuit shown in Figure 2 for this exercise. R1 will vary from 1k to 10K and L1 will vary from 100 mH to 400 mH.
Figure 2. Circuit that you will construct to demonstrate 2nd order system operation.
The simulation results of an underdamped case are shown on the oscilloscope.
The theoretical derivations of the equations that predict the behavior of a series RLC circuit are shown below.
Let’s start by writing the general differential loop equation in the time domain for the series (not a parallel) circuit shown in Fig. 2. Start by writing a loop (mesh) equation using Kirchhoff’s voltage law around the loop. The result is equation (1). Voltage rises are negative and voltage drops are positive.
(1)
Now take the derivative of (1) and rearrange it to get equation (2)
(2)
Transfer equation (2) to the frequency domain using the Laplace Transform where
(3)
Where (refer to Figure 1)
and (4)
Where is called the damping ratio and where is the natural frequency of the circuit
Roots of the general solution are
(5)
The time domain solutions to the Laplacian Equation (3) has three solutions we are interested in depending upon the forcing function and the initial boundary condition.
Over Damped solution 1 (two non-equal real roots)
(6)
(7)
(8)
The time domain simulation of the over damped solution is shown in Figure 3.
Computer simulation
Overdamped Oscilloscope trace
Figure 3. Circuit simulation of overdamped case using National Instrument,s Multisim softwareor Texas Instrument’s Tina Software. R = 10K
Critically Damped solution 2 (two equal real roots)
(9)
(10)
(11)
The time domain simulation of the critically damped solution is shown in Figure 4.
Computer simulation
Critically Damped Oscilloscope trace
Figure 4. Circuit simulation of critically damped case using National Instrument’s Multisim software or Texas Instrument’s Tina Software. R = 2.2 K and L =100 mH and C= 0.1 µF
Under Damped solution 3 ( two complex conjugate roots)
(12)
(13)
(14)
where
(15)
The time domain simulation of the underdamped solution is shown in Figure 5.
Computer simulation
Underdamped Oscilloscope trace
Figure 5. Circuit simulation of underdamped case using National Instrument’s Multisim software or Texas Instrument’s Tina Software, R =100 Ω to 1KΩ
PROCEDURE
- Construct the circuit shown in Figure 1. You are going to vary the value of R1(the potentiometer) from 10 K (overdamped case- two real roots)to approximately 2.2 K (critically damped case- two equal roots) to 1K (underdamped case- complex conjugate roots). Verify these numbers by solving equation (5). The oscilloscope curve shown in Figure 3 is for R1 = 10 K.
- Connect your signal generator (set to a square wave output) to the input of your circuit. Select a reasonable input such as 5 volts. Adjust the frequency of the square wave until you can observe the response of the circuit. This should be about 100 Hz. Observe the output on the oscilloscope by connecting the oscilloscope across the capacitor. Record and tabulate all of your settings and readings. For good characterization of the circuit behavior, the square wave pulse width must be sufficiently long in order for the circuit to approximately reach a steady state condition. For this circuit a 100 Hz pulse width square wave would be a good starting point. Set the oscilloscope vertical sensitivity to 5V per division and the horizontal sensitivity to approximately 1 ms per division.
- Plot your results on linear graph paper or take a photo of the oscilloscope image. Measure the damped frequency of oscillations using the vertical cursors. Compare it to the value calculated using equation (4) and (15)
- Measure the time for the signal to reach the first peak (Tp) of the underdamped curve and the frequency of the underdamped oscillations using the oscilloscope cursors. The following equation can be obtained by taking the derivative of equation (12) and setting it equal to zero.
- Write a professional comprehensive lab report using a word processor. Show your results and include a comprehensive conclusion. There are lots of sample lab reports on the internet.
- What kind of errors did you get between what you calculated and what you measured?
- What is the comparison between the oscillations observed when R1 was set to 1k and n as calculated using equation (4)?
- Note that the series resistance of the inductor may have to be included in the calculation.
How does this exercise help you to understand the behavior of mechanical or structural systems? A good reference site is Bucknell University’s website which covers 2nd order system theory, and it has interactive videos.
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