response of rl and rlc circuits - engineering

ELG-2130 Circuit Theory 3-1

LABORATORY 3 RESPONSE OF RL AND RLC CIRCUITS

Overdamped response Underdamped response RL circuit RLC circuit

3.1 Objectives

This laboratory aims at reaching the following objectives:• To experiment and become familiar with circuits containing energy storage

elements.• To measure the step response of first-order circuits.• To measure the step response of second-order circuits and observe the

typical behavior of underdamped, critically-damped and overdampedsystems.

3.2 Response of a first-order circuit

Circuits containing one inductor or one capacitor are characterized by atransient response followed by a steady-state response. That is, if oneapplied a step function on the source of the circuit (equivalent to switching iton), the voltage and the current across or through the other elements of thecircuit will not exhibit the some step behavior. Voltages and currents will takesome time before they reach their respective final and stable valuescorresponding to steady-state. Figure 3.1 shows the characteristic responseof a first-order circuit to a step input function applied at t=0.

Response of a second-order circuit


FIGURE 3.1 Typical response of a first-order circuit.

The response of a first-order circuit can be recognized by its immediatereaction to the step input as the voltage, or the current, immediately starts tovary towards its steady-state value. This can be observed by the sharpchange in the response at t=0 where the voltage, or the current, abruptlychanges from zero to a curve with a positive slope.Three main parameters are usually considered to evaluate thecharacteristics of a first-order circuit response:• The steady-state value which is the magnitude of the voltage, or current,

after the circuit has reached stability.• The response time which corresponds to the period of time required for the

voltage, or the current, to reach and remain within an error margin of +/-5% of its final steady-state value.

• The overshoot which is the magnitude that exceeds the steady-statevalue, usually expressed as a percentage with respect to the steady-statevalue. However, depending on the circuit’s parameters, the overshootmight not be present, making the step response smoother.

Apart from the plot of the curve, the measurement of a first-order circuitresponse then consists in estimating this set of three main parameters thatwill allow to reproduce and quantify the response.

3.3 Response of a second-order circuit

Circuits containing two inductors or two capacitors or one of each alsoexhibit a transient response before they reach steady-state. However, asthese circuits are more complex, their response might take various formsthat mainly depend on the respective values of R, L and C. Figure 3.2 showsthe characteristic responses for a second-order circuit to a step inputfunction applied at t=0.

t

step response

overshoot

steady-state +5%-5%

Tresponse

Response of a second-order circuit


FIGURE 3.2 Typical response of a second-order circuit.

The first characteristic to observe in a second-order circuit response is asmoother transition between a stable signal and one with a slope. Carefullyexamine and compare the transition areas just after t=0 in figure 3.1 andfigure 3.2 to observe this difference.Depending on the settings of R, L and C in the circuit, the response might beunderdamped, critically-damped or overdamped. Basically, the values of R,L and C determine the magnitude of what is called the damping factor, z, ofthe circuit. In general, if z is smaller than 0.7, the circuit is said to beunderdamped and tends to exhibit decreasing oscillations with an initialovershoot that directly depends on the value of z. For a damping factor closeto 0.7, the circuit is considered as being critically-damped and provides afast response with minimal overshoot and no oscillation. But this system ison the limit of oscillations. Finally, if z is larger than 0.7, the circuit isconsidered overdamped and exhibits a relatively slow transition without anyoscillation.Theoretically, the damping factor can be estimated from the equations of thecircuit. However, in an experimental context, the best way to estimate thevalue of the damping factor is to compare the circuit response with a chart ofcharacteristic second-order systems response as provided in figure 3.2. By

Toscillations

steady-state

overdamped

unde

rdam

ped

critically-damped

tsource: J.-Ch. Gilles, P. Decaulne, M. Pélegrin, “Dynamique

de la Commande Linéaire”, Dunod, 1989.

Preparation


comparing the relative magnitude of the overshoot with respect to thesteady-state value, a direct estimation of z is obtained.The other parameters to be estimated to fully characterize a second-ordercircuit response are similar to those of a first-order circuit:• The steady-state value which is the magnitude of the voltage, or current,

after the circuit has reached stability.• The response time which corresponds to the period of time required for the

voltage, or the current, to reach and remain within an error margin of +/-5% of its final steady-state value. This parameter is meaningfull forcritically-damped and overdamped circuits but remains less useful whenoscillations are implied.

• The overshoot with respect to the steady-state value. This parameter isimportant for underdamped systems as a large overshoot might result inthe saturation of some electronic components. On the other hand, theovershoot level cannot be measured for circuits having a large dampingfactor.

Two supplementary parameters might be estimated depending on the nature(damping) of the response:• The rise time which is defined as the period of time required for the

response to go from 10% to 90 % of its steady-state value.• The period of oscillations which can be measured on underdamped

responses.The two latter parameters are not very widely used in practice except for finetuning of circuit designs.

3.4 Preparation

In order to prepare the experiments of this laboratory, complete the followingsteps before you arrive in the laboratory. People who are responsible for thelaboratory might require to see your preparation before you can start themanipulations.• Carefully read the introduction notes that describe safety rules to follow in

the laboratory.• Read and understand the sections in your course notes on first and

second order circuits.• Carefully read sections 3.2 and 3.3 describing the characteristic responses

of first and second order circuits.• Read and understand the experimental procedure below.• Examine all circuits that will be used for this laboratory and answer all

preparation questions.

Preparation questions


3.5 Preparation questions

The following preparation questions refer to different sections of thelaboratory (see following pages).

3.5.1 Measurements on a first-order circuitConsidering the first-order circuit of figure 3.3 which has a total resistance of500Ω (Rv+RL=500Ω):1)Determine the response of the inductor in the circuit, VL(t), to a 1V step

function applied on the source, Vs. Assume that RL has 2/5 of the totalresistance.

2)Estimate the response time of this circuit.3)Determine the response of the resistor in the circuit, VR(t), to the same 1V

step function on the source. Assume that Rv has 3/5 of the totalresistance.

4)Plot the responses, VL(t) and VR(t), of this circuit to the 1V step function.Preferably use the same scales and clearly show the initial conditions onboth graphs.

5)Considering that the source, Vs(t), is now a sinusoidal waveform of 200Hz, estimate the phase shift on VR(t) with respect to the source (assumingthat Vs(t) has a null phase). You might use a graphical representation ofVs(t) and VR(t) to obtain this estimate by applying the techniqueintroduced in laboratory 2.

3.5.2 Measurements on a second-order circuitConsidering th second-order circuit of figure 3.4 which has a total resistanceof 500Ω (Rv+RL=500Ω):1)Determine the responses on the resistor, the inductor and the capacitor,

VR(t), VC(t) and VL(t), in the circuit to a 1V step function applied on thesource, Vs. Assume that the capacitor has no internal resistance and thatRL has 2/5 of the total resistance.

2)Compute the same responses when the total resistor is 2000Ω instead of500Ω (considering that the potentiometer has been reajusted such thatRv+RL=2000Ω). The internal resistance of the capacitor is negligible andthat of the inductor is constant.

3)Plot the responses using a computer program (such as Matlab) for eachvalue of the total resistance and conclude on the nature of the response ineach case (overdamped, critically damped or underdamped). Estimate theresponse time in each case.

Parts and equipments required


3.6 Parts and equipments required

• 1 dual-channel oscilloscope• 1 function generator• 1 digital multimeter• 1 potentiometer (0-10 kΩ)• 2 capacitors: 0.1 µF and 0.22 µF• 1 inductor: 100 mH

3.7 Experimental part

After having completed the analysis of all circuits by answering thepreparation questions, perform the manipulations described in the followingsections and validate your results by comparing them with the theoreticalvalues that you obtained. Don’t forget to also complete the analysis sectionrelated to each experiment.

3.7.1 Measurements on a first-order circuit

FIGURE 3.3 First-order circuit (RL).

1)The 100 mH inductor is made of a long winding of copper wire.Consequently it has an internal resistance which cannot be neglected.Measure the internal resistance of the inductor, RL, with a digitalohmmeter.

2)Build the circuit shown in figure 3.3 using a potentiometer for Rv. Adjustthe potentiometer such that the total resistance is equal to 500Ω when thepotentiometer is connected in series with the inductor, that is:Rv+RL=500Ω. Use a 1V peak square wave (with a minimum value =0V) toreproduce a series of step functions on the source, Vs(t). Use a sourcefrequency of 200 Hz.

3)Using the oscilloscope, observe the waveform of VR(t) and focus on oneperiod of the approximate square wave that you obtain. Plot the response

100 mHRL

Rv+-Vs

++

-

-

VL

VR

inductor

Experimental part


that you observe with respect to time. Estimate the steady-state value andthe response time of VR(t).

4)Increase the potentiometer resistance value by 400Ω. Observe the newresponse on the resistor and plot this curve. Estimate the new steady-state value and the new response time of VR(t).

5)Reajust the potentiometer to its initial value such that Rv+RL=500Ω. Thendecrease the potentiometer resistance value by 200Ω (e.g. 600Ω lessthan in step 4). Observe again the new response on the resistor and plotthis curve. Estimate the new steady-state value and the new responsetime of VR(t).

6)Reajust the potentiometer to its initial value such that Rv+RL=500Ω, andswitch the source to a sinusoidal waveform of 200 Hz. Displaysimultaneously Vs(t) and VR(t). PLot the curves and estimate the phaseshift between Vs(t) and VR(t) using the technique introduced in laboratory2.

ANALYSIS:

1)Compare the measured responses to a step input function with thetheoretical ones by comparing their steady-state values and theirresponse times. Explain any significant discrepancy.

2)Discuss the effect of the potentiometer resistance value on the steady-state magnitude and on the response time. Explain the observedvariations based on your theoretical solution for the circuit.

3)Compare the measured phase shift in the response to a sinusoidalwaveform with the theoretical one obtained in the preparation. Explain anysignificant discrepancy.

Experimental part


3.7.2 Measurements on a second-order circuit

FIGURE 3.4 Second-order circuit (RLC).

1)Measure the internal resistance of the inductor, RL, with a digitalohmmeter.

2)Build the circuit shown in figure 3.4 using a potentiometer for Rv. Adjustthe potentiometer such that the total resistance is equal to 500Ω when thepotentiometer is connected in series with the inductor, that is:Rv+RL=500Ω. Use a 1V peak square wave (with a minimum value =0V) toreproduce a series of step functions on the source, Vs(t). Use a sourcefrequency of 200 Hz.

3)Using the oscilloscope, observe the waveforms of Vs(t) and VC(t) focusingon one period of the signal. Plot the response that you observe withrespect to time.

4)Determine the nature of the response (underdamped, critically damped oroverdamped) that you obtain and estimate the response time.

5)Reduce the potentiometer resistance value by 200Ω. Observe the newresponse on VC(t) and plot this curve. Examine the effect of the resistor onthe characteristics of the response, especially on the response time andthe frequency of oscillations.

6)Reajust the potentiometer such that Rv+RL=2000Ω. Observe the newresponse on VC(t) and plot this curve.

7)Determine the nature of the new response (underdamped, criticallydamped or overdamped) that you obtain and estimate the response time.Examine the effect of the resistor on the nature and the characteristics ofthe response.

8)By changing the resistance value on the potentiometer, determineexperimentally the value of the resistor, Rv, that brings the circuit in acritically damped configuration. Observe the response on VC(t), plot thiscurve and estimate the response time.

9)Keep the setting you obtained for the critically-damped configuration andonly replace the capacitor for one of 0.1 µF. Observe the response onVC(t), plot this curve and examine the effect of the value of the capacitoron the nature of the response.

100 mHRL

+-Vs

+

-Vc 0.22 µF

Rv

inductor

+ -VL

Experimental part


10)Keep the setting you obtained for the critically-damped configuration andbring back the 0.22 µF capacitor in the circuit. Switch the source to asinusoidal waveform of 200 Hz with a 1V peak magnitude. Observe Vs(t)and VC(t). Plot these two curves together and examine the time shiftbetween signals. Estimate the magnitude of the time shift between thesource and the capacitor voltage using the technique discussed inlaboratory 2.

ANALYSIS:

1)Compare the measured step responses with the theoretical ones. Justifyany significant discrepancy.

2)Discuss the effect of the potentiometer resistance value on the nature ofthe response (damping) and on the response time.

3)Discuss the effect of the capacitor value on the nature of the response.4)Compare the behavior of second-order circuits with that of first-order

circuits. Analyze the way they responde to variations of their parameters,R, L and/or C.

SITEUniversity of Ottawa

ELG-2130BCircuit Theory

Laboratory Report # 3

Response of RL and RLC circuits

Presented toDr. P. Payeur

By:

Team #: _______________

Date: ____________________

Names Students #

response of rl and rlc circuits - engineering

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