Goals
To build RLC circuits and to observe the transient response to a step input. You will study and measure the overdamped, critically damped and underdamped circuit response.
Background
RLC circuits are widely used in a variety of applications such as filters in communications systems, ignition systems in automobiles, defibrillator circuits in biomedical applications, etc. The analysis of RLC circuits is more complex than of the RC circuits we have seen in the previous lab. RLC circuits have a much richer and interesting response than the previously studied RC or RL circuits. A summary of the response is given below.
Lets assume a series RLC circuit as is shown in Figure 1. The discussion is also applicable to other RLC circuits such as the parallel circuit.
By writing KVL one gets a second order differential equation. The solution consists of two parts:
(1b)
Case 1: Critically damped response: two equal roots s= s_{1}= s_{2}
Case 2: Overdamped response: two real and unequal roots s_{1} and s_{2}
Case 3: Underdamped response: two complex roots
Prelab assignments
1. Review the sections on RLC circuit in textbook (6.3 in Basic Engineering Circuit Analysis, by D. Irwin).
2. Prove that the expression for the damping ratio and the undamped resonant frequency for the circuit of Figure 1 is equal to,
4. For the three cases of damping ratio equal to 1, 2 and 0.2 find the expression of the voltage v_{C}(t) over the capacitor using the values of the capacitor, inductor and resistors calculated above. Assume a unit step function v_{S} as the input signal, and initial conditions v_{C}(0)=0 and i_{L}(0)=0. Plot the response for the three cases (preferably using a plotting program such as MATLAB, Maple or a spreadsheet).
Inlab assignments
A. Equipment:
1. Simulate the three RLC circuits using Multisim software for the cases of damping ratio equal to 1, 2 and 0.2 (use the values of R, L and C found from the prelaboratory). Use a square wave with 1Vpp (i.e. amplitude of 0.5V with offset of 0.5V  use the function generator in EWB) and frequency of 200 Hz as input voltage. Compare the waveforms with the one you calculated in the prelab. Make a print out.
2. Get the components L and C you will need to build
the RLC circuit. A real inductor consists of a parasitic resistor (due
to the windings) in series with an ideal inductor as shown in Figure 4.
Measure the value of the inductor and the parasitic resistance R_{L}
using an RLC meter and record these in your notebook. Measure also the
value of the capacitor. For the resistors use a 5 kOhm potentiometer.
3. Build the series RLC circuit of Figure 5, using the values for L and C found in the prelab corresponding to the damping ratio of 1, 2 and 0.2.
The total resistor R_{TOT} of the circuit consists of three components: R_{T} which is the output resistance of the function generator (50 Ohm), the parasitic resistor R_{L} and the actual resistor R. First calculate the required resistor R such that the total resistor corresponds to the one found in the prelab for each case. Fill out a table similar to the one shown below.







.  .  . 

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Case 1: critically damped response.
b. Set the function generator to 1Vpp with an offset voltage of 0.5V and a frequency of 200 Hz. Display this waveform on the oscilloscope. Measure the voltage over the capacitor and display the waveform v_{C}(t)on the scope. Measure its characteristics: risetime, V_{min}, V_{max}, and Vpp. Make also a print out of the display. Compare the measured results with the one from the prelab and the simulations.
b. Calculate one of the time constants of the expression (4). Usually one of the time constants is considerably larger than the other one which implies that the exponential with the smallest time constant dies out quickly. You can make use of this to find the largest time constant. Measure two points on the graph (v1,t1) and (v2,t2) as shown in Figure 6. Choose t1 sufficiently away from the origin so that one of the exponentials has decayed to zero. You can than make use of the following relationship to find the time constant:
Case 3: underdamped response
b. Determine the value of t
and w_{d}
from the measured waveform (See Figure 3). Use the expression (7) to determine
the value of the time constant (t=1/s).
References:
J. D. Irwin, "Basic Engineering Circuit Analysis,"
5th edition, Prentice Hall, Upper Saddle River, NJ, 1996.
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Created by Jan Van der Spiegel: April 15, 1997.
Updated by Sid Deliwala on Jan 11, 2013.