Lsim simulates the (time) response of continuous or discrete linear systems to arbitrary inputs. When invoked without left-hand arguments, lsim plots the response on the screen. Lsim(sys,u,t) produces a plot of the time response of the dynamic system model sys to the input history, t,u. Base Excitation models the behavior of a vibration isolation system. The base of the spring is given a prescribed motion, causing the mass to vibrate. This system can be used to model a vehicle suspension system, or the earthquake response of a structure. Rotor Excitation models the effect of a rotating machine mounted on a flexible floor.
RLC Step Response UNIVERSITY of PENNSYLVANIAStep response of RLC CircuitsGoalsTo build RLC circuits and to observe the transientresponse to a step input. You will study and measure the overdamped, criticallydamped and underdamped circuit response.BackgroundRLC circuits are widely used in a variety of applicationssuch as filters in communications systems, ignition systems in automobiles,defibrillator circuits in biomedical applications, etc. The analysis ofRLC circuits is more complex than of the RC circuits we have seen in theprevious lab. RLC circuits have a much richer and interesting responsethan the previously studied RC or RL circuits. A summary of the responseis given below.Lets assume a series RLC circuit as is shown in Figure1.
The discussion is also applicable to other RLC circuits such as theparallel circuit. Figure 1: Series RLC circuitBy writing KVL one gets a second order differentialequation. The solution consists of two parts:x(t) = x n(t) + x p(t), in which x n(t) is the complementary solution(=solution of the homogeneous differential equation also called the naturalresponse) and a x p(t) is the particular solution (also calledforced response). Lets focus on the complementary solution. The form ofthis solution depends on the roots of the characteristic equation,(1) in which is the damping ratio and is the undamped resonant frequency.The roots of the quadratic equation are equal to,(1b)For the example of the series RLC circuit one has thefollowing characteristic equation for the current i L(t) or v C(t),s 2 + R/L.s + 1/LC =0. (2) Depending on the value of the damping ratio one hasthree possible cases:Case 1: Critically damped response:two equal roots s= s 1= s 2(3)The total response consists of the sum of the complementaryand the particular solution.
The case of a critically damped response toa unit input step function is shown in Figure 2.Case 2: Overdamped response: two realand unequal roots s 1 and s 2(4) Figure 2 shows an overdamped response to a unit inputstep function.Figure 2: Critically and overdamped responseto a unit input step function.Case 3: Underdamped response: two complexroots(5) Figure 3 shows an underdamped response to a unit inputstep function. Figure 3: Underdamped response to a unit inputstep function.Pre-lab assignments1. Review the sections on RLC circuit in textbook(6.3 in Basic Engineering Circuit Analysis, by D.
Prove that the expression for the damping ratioand the undamped resonant frequency for the circuit of Figure 1 is equalto,(6) 3. Assume that C=100nF. Find the values of R and L suchthat = 10 krad/s for the threecases of damping ratio equal to 1, 2 and 0.2.4. For the three cases of damping ratio equal to1, 2 and 0.2 find the expression of the voltage v C(t) over thecapacitor using the values of the capacitor, inductor and resistors calculatedabove. Assume a unit step function v S as the input signal, andinitial conditions v C(0)=0 and i L(0)=0. Plot theresponse for the three cases (preferably using a plotting program suchas MATLAB, Maple or a spreadsheet).In-lab assignmentsA. Equipment:.
1. Agilent Signal Generator. 2. Agilent Scope.
3. Protoboard. 4. Resistor: 5Kohm potentiometer. 5. Capacitors: 100nF. 6.
Inductor 100mH. 7. Box with cables and connectors. 8.
Scope Probe. 9. RLC Meter. 10. Multisim softwareB. Simulate the three RLC circuits using Multisim software for the cases of damping ratio equal to 1, 2 and 0.2 (usethe values of R, L and C found from the pre-laboratory).
Use a square wavewith 1Vpp (i.e. Amplitude of 0.5V with offset of 0.5V - use the functiongenerator in EWB) and frequency of 200 Hz as input voltage. Compare thewaveforms with the one you calculated in the pre-lab.
Make a print out.2. Get the components L and C you will need to buildthe RLC circuit. A real inductor consists of a parasitic resistor (dueto the windings) in series with an ideal inductor as shown in Figure 4.Measure the value of the inductor and the parasitic resistance R Lusing an RLC meter and record these in your notebook.
Measure also thevalue of the capacitor. For the resistors use a 5 kOhm potentiometer.Figure 4: Model of an inductor3. Build the series RLC circuit of Figure 5, usingthe values for L and C found in the pre-lab corresponding to the dampingratio of 1, 2 and 0.2. Figure 5: RLC circuit: (a) R TOT includesall resistors in the circuit; (b) showing the different resistors in thecircuit.The total resistor R TOT of the circuitconsists of three components: R T which is the output resistanceof the function generator (50 Ohm), the parasitic resistor R Land the actual resistor R. First calculate the required resistor R suchthat the total resistor corresponds to the one found in the pre-lab foreach case. Fill out a table similar to the one shown below.Damping ratio120.2R T (Ohm).R L (Ohm).R tot (Ohm).R (Ohm).4. Measure the response of each case.Case 1: critically damped response.a.
Set the potentiometer to the value R calculatedabove corresponding to a damping ratio of 1.b. Set the function generator to 1Vpp with an offsetvoltage of 0.5V and a frequency of 200 Hz. Display this waveform on theoscilloscope. Measure the voltage over the capacitor and display the waveformv C(t)on the scope. Measure its characteristics: risetime, V min,V max, and Vpp. Make also a print out of the display.
Comparethe measured results with the one from the pre-lab and the simulations. Case 2: overdamped response.a. Set the potentiometer to the value R calculatedabove corresponding to a damping ratio of 2.
Measure and display the responseover the capacitor and make a print out. Determine the rise time, min andmax value of the voltage v C.b. Calculate one of the time constants of the expression(4). Usually one of the time constants is considerablylarger than the other one which implies that the exponential with the smallesttime constant dies out quickly.
You can make use of this to find the largesttime constant. Measure two points on the graph (v1,t1) and (v2,t2) as shownin Figure 6.
![Square Wave Excitation And Response Square Wave Excitation And Response](/uploads/1/2/5/5/125558063/266513036.png)
Choose t1 sufficiently away from the origin so that one ofthe exponentials has decayed to zero. You can than make use of the followingrelationship to find the time constant:(7) in which Vf is the final value of the exponential (valueat the time t=infinite). The expression you derived in the last lab: t =trise/2.2is a special case of the above expressions (i.e. V1=0.1Vmax; v2=0.9Vmax).Figure 6: method to measure the time constant.Case 3: underdamped responsea.
Set the potentiometer corresponding to the valueR calculated above corresponding to a damping ratio of 0.2. Measure anddisplay the response over the capacitor and make a print out.
Determineits characteristics: voltage and time of the first peak, voltage and timeof the second peak. Make a print out.b.
![Response Response](http://mriquestions.com/uploads/3/4/5/7/34572113/5444977_orig.gif)
Determine the value of tand w dfrom the measured waveform (See Figure 3). Use the expression (7) to determinethe value of the time constant ( t =1/ s ).5. Vary the potentiometer and observe the behavior ofthe response (display the voltage over the capacitor). Notice when theoutput goes from underdamped to critically damped and overdamped. In general,a critically damped response is preferred because it does not give overshootor 'ringing' and has a fast rise time.
An overdamped response has a slowerrise time than the other responses, while the underdamped response risesthe fastest, but also give a lot of overshoot which is not desired. Recordyour observations in you lab notebook.References:J. Irwin, 'Basic Engineering Circuit Analysis,'5th edition, Prentice Hall, Upper Saddle River, NJ, 1996.