## Step-Transient-Response with RLC Circuits

Step-Transient-Response with RLC Circuits

1. Introduction
The purpose of this lab is to observe and control the oscillatory time response of R-L-C
circuits. While at this point you have not been exposed to the new energy storage circuit
element, the Inductor, this Lab will give you the chance to see it first in an actual circuit.
Then subsequently we will consider the theory of R-L-C circuits in the class room.
In this lab we examine R-C, R-L and R-L-C circuits driven to a time-variation response
by a sudden step change in the drive input signal produced by a Signal Generator and use
a Scope to observe and measure the response. So in block diagram form the structure of
the Lab setup is shown in Figure 13-1, as was used in Lab 12.
Figure 13-1 Block Diagram of Transient Response Setup
Again we will use a repetitive ‘train’ of many pulse events, each separated by a given
amount of time. Figure 13-2 depicts such a ‘train’ of step events.
(b) ‘Train’ of Steps
Figure 13-2 Step Transients; Repeated ‘Train’ of Events
The real difference in this lab is that we introduce a new energy-storage circuit element,
called the inductor. The inductor has a symbol and circuit constraint depicted in Figure
13-3:
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Inductor Symbol: Constraint: vL = L
diL
dt
Figure 13-3 Inductor Circuit Symbol and Circuit Constraint
An inductor in a resistor-inductor circuit causes a 1st–order differential equation response
and hence like a capacitor-resistor circuit it too has an exponential characteristic with a
time constant, tau, where now tau = L/Rth. So here in this Lab the inductor is first
experienced experimentally, then afterward we will discuss it in class.
And then in this Lab we make the huge change of having BOTH a capacitor AND an
inductor connected within the same circuit. This means there will be two different
energy storage elements in the same circuit. Marvelous new things happen !!!
2.1 Step Transients in a Pulse Train
Here select a signal generator frequency of 100 Hz. This means the repeat time T =
(1/100) sec = 10 millisecond.
On the Scope, the ‘trigger’ control is set to trigger on the channel (CH1) connected to
the signal generator AND the Scope trigger polarity is set for a ‘positive’ transition,
meaning the transition for triggering starts when there is a small value that transitions to a
more positive value. This trigger point corresponds to the ‘t = 0’ transition in Figures 13-
2 and 13-3 above.
The GROUND of the Signal Generator MUST connect to the ground of the Scope,
otherwise you short-circuit the Signal Generator and get no output.
Make a square-wave pulse train of 100 Hz frequency with your Signal Generator and
measure the pulses with your Scope. Use your Scope probe to make these measurements.
TAKE CARE for the SCOPE GAIN SETTING and allow for any Scope probe
attenuation. Also take care for the ‘grounds’. The connections from the Signal Generator
include both a signal and a corresponding ground. This is to re-acquaint you with the
system you used in Lab 12.
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2.2 Capacitor R-C Circuit, Step Response
Now construct a simple R-C circuit, such as depicted in Figure 13-4. Use a resistor of
1kohm and capacitor of 1 microfarad. Note: keep the Signal Generator Pulse Frequency
at 100Hz . [This is a similar circuit to that used in Lab 12.]
Figure 13-4 Simple R-C Circuit Connected to Signal Generator and Scope
Use two scope probes (CH1 and CH2) to observe the input and output voltages, vin(t)
and vout(t), of your circuit. Be sure to have the scope probe ground at the signal
generator ground directly connect your square-wave pulse Signal Generator output to
your R-C circuit input AND to the Scope CH1 input probe. And set the generator output
magnitude to be 5 volts at 100Hz as measured by the Scope.
Task (13-2-1) Scope Measurement of Capacitive Circuit Transient Response
With this connection of two Scope probes and simultaneous display of both channels on
the Scope the input-output relationship can be observed.
(i) Sketch and label the circuit transient drive, vin(t), and response, vout(t), as measured by
the scope.
(ii) Does the output voltage (= capacitor voltage) get to an equilibrium value?
a) Equal to the ‘high’ 5 volt value of the square-wave, or
b) does it exhibit some other equilibrium when the input is ‘high’, if so explain.
(iii) What is the measured time-constant for your circuit; use the 5-tau rule to determine
(iv) How well does the measured time-constant value agree with the theoretical value?
2.3 Inductor R-L Circuit, Step Response
Now change your above circuit (see Section 2.2 above) and replace the capacitor with an
inductor, see Figure 13-5. Chose an inductance value of about 1mH, any value in the
range of about 500µH to 3mH is okay. ALSO, change the resistor value to 47 ohms.
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Figure 13-5 Resistive Load on Simple R-C Circuit with Two Scope Probes
The reason to set the numerical values of the R and L components as above is to adjust
the time-constant, tau = L/R, of the circuit so that the response fits conveniently within
the time of the same 100Hz frequency of the drive pulse. This frequency corresponds to
a time of only 5 milliseconds in the ‘high’ state, and hence the response needs to fit
within this time if the frequency is held constant. Adjust the frequency of the pulse
source and scope as necessary if the response does not fit within the 5 millisec time of the
100Hz pulse.
Take Care: Our real inductors are comprised of copper wire that has a finite resistance.
Hence the actual model for our inductor is an inductance in series with a series resistance,
RS. The equivalent circuit for the actual inductor is in Figure 13-6. This added resistance
value can be estimated with an ohm-meter measurement.
Figure 13-6 Actual Inductor with Internal Series Resistance RS
Task (13-3-1) Scope Measurement of Inductive Circuit Transient Response
With this connection of two Scope probes and simultaneous display of both channels on
the Scope the input-output relationship can be observed.
(i) Sketch and label the circuit transient drive, vin(t), and response, vout(t), as measured by
the scope.
(ii) Does the output voltage (= inductor voltage) get to an equilibrium value?
a) Equal to the ‘high’ 5 volt value of the square-wave, or
b) does it exhibit some other equilibrium when the input is ‘high’, if so explain.
(iii) What is the measured time-constant for your circuit; use the 5-tau rule to determine
(iv) How well does the measured time-constant value agree with the theoretical value?
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2.4 Combined R-L-C Circuit, Step Response
Now change your above circuit (see Section 2.3 above) and add a capacitor in series with
the inductor, Figure 13-7. Here we will again start with a value of R = 47 ohms.
Figure 13-7 Parallel R-L-C Circuit
With this circuit that combines to different forms of energy storage, electric in the
capacitor and magnetic in the inductor, perhaps a new form of transient response may be
observed. However, the form of this new response will depend upon the circuit
component values.
Task (13-4-1) Scope Measurement of Parallel R-L-C Circuit Transient Response
With this connection of two Scope probes and simultaneous display of both channels on
the Scope the input-output relationship can be observed.
(i) Sketch and label the circuit transient drive, vin(t), and response, vout(t), as measured by
the scope.
(ii) Does the output voltage (= L-C voltage) get to an equilibrium value?
a) Equal to the ‘high’ 5 volt value of the square-wave, or
b) does it exhibit some other equilibrium when the input is ‘high’, if so explain.
(iii) What is the measured time-constant for your circuit; use the 5-tau rule to determine