RESPONSE OF RL AND RLC
CIRCUITS
LABORATO RY 3
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 overdamped
systems.
3.2 Response of a first-order circuit
Circuits containing one inductor or one capacitor are characterized by a
transient response followed by a steady-state response. That is, if one
applied a step function on the source of the circuit (equivalent to switching it
on), the voltage and the current across or through the other elements of the
circuit will not exhibit the some step behavior. Voltages and currents will take
some time before they reach their respective final and stable values
corresponding to steady-state. Figure 3.1 shows the characteristic response
of a first-order circuit to a step input function applied at t=0.
ELG-2130 Circuit Theory
3-1
Response of a second-order circuit
FIGURE 3.1
Typical response of a first-order circuit.
step response
overshoot
+5%
-5%
steady-state
t
Tresponse
The response of a first-order circuit can be recognized by its immediate
reaction to the step input as the voltage, or the current, immediately starts to
vary towards its steady-state value. This can be observed by the sharp
change in the response at t=0 where the voltage, or the current, abruptly
changes from zero to a curve with a positive slope.
Three main parameters are usually considered
characteristics of a first-order circuit response:
to
evaluate
the
• 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-state
value, usually expressed as a percentage with respect to the steady-state
value. However, depending on the circuit’s parameters, the overshoot
might not be present, making the step response smoother.
Apart from the plot of the curve, the measurement of a first-order circuit
response then consists in estimating this set of three main parameters that
will 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 also
exhibit a transient response before they reach steady-state. However, as
these circuits are more complex, their response might take various forms
that mainly depend on the respective values of R, L and C. Figure 3.2 shows
the characteristic responses for a second-order circuit to a step input
function applied at t=0.
ELG-2130 Circuit Theory
3-2
Response of a second-order circuit
Typical response of a second-order circuit.
damp
ed
Toscillations
under
FIGURE 3.2
critically-damped
steady-state
rd
ov e
am
ped
t
source: J.-Ch. Gilles, P. Decaulne, M. Pélegrin, “Dynamique
de la Commande Linéaire”, Dunod, 1989.
The first characteristic to observe in a second-order circuit response is a
smoother transition between a stable signal and one with a slope. Carefully
examine and compare the transition areas just after t=0 in figure 3.1 and
figure 3.2 to observe this difference.
Depending on the settings of R, L and C in the circuit, the response might be
underdamped, critically-damped or overdamped. Basically, the values of R,
L and C determine the magnitude of what is called the damping factor, z, of
the circuit. In general, if z is smaller than 0.7, the circuit is said to be
underdamped and tends to exhibit decreasing oscillations with an initial
overshoot that directly depends on the value of z. For a damping factor close
to 0.7, the circuit is considered as being critically-damped and provides a
fast response with minimal overshoot and no oscillation. But this system is
on the limit of oscillations. Finally, if z is larger than 0.7, the circuit is
considered overdamped and exhibits a relatively slow transition without any
oscillation.
Theoretically, the damping factor can be estimated from the equations of the
circuit. However, in an experimental context, the best way to estimate the
value of the damping factor is to compare the circuit response with a chart of
characteristic second-order systems response as provided in figure 3.2. By
ELG-2130 Circuit Theory
3-3
Preparation
comparing the relative magnitude of the overshoot with respect to the
steady-state value, a direct estimation of z is obtained.
The other parameters to be estimated to fully characterize a second-order
circuit 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 for
critically-damped and overdamped circuits but remains less useful when
oscillations are implied.
• The overshoot with respect to the steady-state value. This parameter is
important for underdamped systems as a large overshoot might result in
the saturation of some electronic components. On the other hand, the
overshoot level cannot be measured for circuits having a large damping
factor.
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 fine
tuning of circuit designs.
3.4 Preparation
In order to prepare the experiments of this laboratory, complete the following
steps before you arrive in the laboratory. People who are responsible for the
laboratory might require to see your preparation before you can start the
manipulations.
• 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.
ELG-2130 Circuit Theory
3-4
Preparation questions
3.5 Preparation questions
The following preparation questions refer to different sections of the
laboratory (see following pages).
3.5.1 Measurements on a first-order circuit
Considering the first-order circuit of figure 3.3 which has a total resistance of
500Ω (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 total
resistance.
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 total
resistance.
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 on
both graphs.
5)Considering that the source, Vs(t), is now a sinusoidal waveform of 200
Hz, estimate the phase shift on VR(t) with respect to the source (assuming
that Vs(t) has a null phase). You might use a graphical representation of
Vs(t) and VR(t) to obtain this estimate by applying the technique
introduced in laboratory 2.
3.5.2 Measurements on a second-order circuit
Considering th second-order circuit of figure 3.4 which has a total resistance
of 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 the
source, Vs. Assume that the capacitor has no internal resistance and that
RL has 2/5 of the total resistance.
2)Compute the same responses when the total resistor is 2000Ω instead of
500Ω (considering that the potentiometer has been reajusted such that
Rv+RL=2000Ω). The internal resistance of the capacitor is negligible and
that of the inductor is constant.
3)Plot the responses using a computer program (such as Matlab) for each
value of the total resistance and conclude on the nature of the response in
each case (overdamped, critically damped or underdamped). Estimate the
response time in each case.
ELG-2130 Circuit Theory
3-5
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 the
preparation questions, perform the manipulations described in the following
sections and validate your results by comparing them with the theoretical
values that you obtained. Don’t forget to also complete the analysis section
related to each experiment.
3.7.1 Measurements on a first-order circuit
FIGURE 3.3
First-order circuit (RL).
inductor
100 mH
RL
+
Vs
+
-
VL
-
+
VR
-
Rv
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 digital
ohmmeter.
2)Build the circuit shown in figure 3.3 using a potentiometer for Rv. Adjust
the potentiometer such that the total resistance is equal to 500Ω when the
potentiometer is connected in series with the inductor, that is:
Rv+RL=500Ω. Use a 1V peak square wave (with a minimum value =0V) to
reproduce a series of step functions on the source, Vs(t). Use a source
frequency of 200 Hz.
3)Using the oscilloscope, observe the waveform of VR(t) and focus on one
period of the approximate square wave that you obtain. Plot the response
ELG-2130 Circuit Theory
3-6
Experimental part
that you observe with respect to time. Estimate the steady-state value and
the response time of VR(t).
4)Increase the potentiometer resistance value by 400Ω. Observe the new
response on the resistor and plot this curve. Estimate the new steadystate value and the new response time of VR(t).
5)Reajust the potentiometer to its initial value such that Rv+RL=500Ω. Then
decrease the potentiometer resistance value by 200Ω (e.g. 600Ω less
than in step 4). Observe again the new response on the resistor and plot
this curve. Estimate the new steady-state value and the new response
time of VR(t).
6)Reajust the potentiometer to its initial value such that Rv+RL=500Ω, and
switch the source to a sinusoidal waveform of 200 Hz. Display
simultaneously Vs(t) and VR(t). PLot the curves and estimate the phase
shift between Vs(t) and VR(t) using the technique introduced in laboratory
2.
ANALYSIS:
1)Compare the measured responses to a step input function with the
theoretical ones by comparing their steady-state values and their
response times. Explain any significant discrepancy.
2)Discuss the effect of the potentiometer resistance value on the steadystate magnitude and on the response time. Explain the observed
variations based on your theoretical solution for the circuit.
3)Compare the measured phase shift in the response to a sinusoidal
waveform with the theoretical one obtained in the preparation. Explain any
significant discrepancy.
ELG-2130 Circuit Theory
3-7
Experimental part
3.7.2 Measurements on a second-order circuit
FIGURE 3.4
Second-order circuit (RLC).
inductor
100 mH
RL
Rv
+
Vs
+
-
VL
-
+
Vc
0.22 µF
-
1)Measure the internal resistance of the inductor, RL, with a digital
ohmmeter.
2)Build the circuit shown in figure 3.4 using a potentiometer for Rv. Adjust
the potentiometer such that the total resistance is equal to 500Ω when the
potentiometer is connected in series with the inductor, that is:
Rv+RL=500Ω. Use a 1V peak square wave (with a minimum value =0V) to
reproduce a series of step functions on the source, Vs(t). Use a source
frequency of 200 Hz.
3)Using the oscilloscope, observe the waveforms of Vs(t) and VC(t) focusing
on one period of the signal. Plot the response that you observe with
respect to time.
4)Determine the nature of the response (underdamped, critically damped or
overdamped) that you obtain and estimate the response time.
5)Reduce the potentiometer resistance value by 200Ω. Observe the new
response on VC(t) and plot this curve. Examine the effect of the resistor on
the characteristics of the response, especially on the response time and
the frequency of oscillations.
6)Reajust the potentiometer such that Rv+RL=2000Ω. Observe the new
response on VC(t) and plot this curve.
7)Determine the nature of the new response (underdamped, critically
damped or overdamped) that you obtain and estimate the response time.
Examine the effect of the resistor on the nature and the characteristics of
the response.
8)By changing the resistance value on the potentiometer, determine
experimentally the value of the resistor, Rv, that brings the circuit in a
critically damped configuration. Observe the response on VC(t), plot this
curve and estimate the response time.
9)Keep the setting you obtained for the critically-damped configuration and
only replace the capacitor for one of 0.1 µF. Observe the response on
VC(t), plot this curve and examine the effect of the value of the capacitor
on the nature of the response.
ELG-2130 Circuit Theory
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Experimental part
10)Keep the setting you obtained for the critically-damped configuration and
bring back the 0.22 µF capacitor in the circuit. Switch the source to a
sinusoidal waveform of 200 Hz with a 1V peak magnitude. Observe Vs(t)
and VC(t). Plot these two curves together and examine the time shift
between signals. Estimate the magnitude of the time shift between the
source and the capacitor voltage using the technique discussed in
laboratory 2.
ANALYSIS:
1)Compare the measured step responses with the theoretical ones. Justify
any significant discrepancy.
2)Discuss the effect of the potentiometer resistance value on the nature of
the 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.
ELG-2130 Circuit Theory
3-9
SITE
University of Ottawa
ELG-2130B
Circuit Theory
Laboratory Report # 3
Response of RL and RLC circuits
Presented to
Dr. P. Payeur
By:
Names
Students #
Team #: _______________
Date: ____________________