# “Q” Factor and Transient Response of an RLC - Eleg. 2111

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```					                      Eleg. 2111 Laboratory 6
“Q” Factor and Transient Response of an RLC Series Circuit
Pre-Lab

I. The “ Q ” factor (or quality factor)

The “ Q ” factor is a measure of how much energy is lost in a circuit or device
when it is driven by a sinusoidal signal. Resonant devices such as microwave
cavities and sharply tuned circuits are intended to behave as energy storage
devices, gradually accumulating a significant store of energy when excited by
small signals at the resonant frequency. Capacitors or inductors are intended to
store electromagnetic energy and are made as loss-free as possible. The “ Q ”
factor is defined to be equal to 2 times the ratio of the peak energy stored to the
energy dissipated in one cycle. A high “ Q ” factor means high energy storage at
low loss. Clearly, its value will depend upon the frequency. For a series RLC
circuit resonant at a radian frequency of 0 = 1 LC , application of the definition
leads to the following result for the quality factor at the resonant frequency,
denoted Q0 :

L         1
Q0   =     0
=            ,                     (1
RT           0 RT C

where RT represents the total series resistance, and L is the inductance and C is
the capacitance of the circuit.

This expression assumes that all losses in the circuit can be accounted for by a
single resistance RT . Often, this resistance is not just the DC resistance contained
in the circuit, but is an effective resistance accounting for other losses such as
hysteresis, eddy currents and dielectric losses.

The quality factor is much more than an arbitrary measure of quality for
a given circuit. A great deal of information about the behavior of a tuned RLC
circuit is summarized in this single number. In this laboratory, we will explore
the significance of the “ Q ” factor by measuring it, and the damped natural
frequency, for a particular RLC series circuit. This will be accomplished by
sampling the waveform of the response of the RLC circuit to a step function of
voltage, and carrying out a “least squares fit” of the data to a theoretical
expression for that response.

II. Transient Response of an RLC Series Circuit to a Voltage Step.

Figure 1. shows a series RLC circuit in the form that will be used in the
laboratory. VS will supply a positive square pulse with an amplitude of 5 volts to
simulate a step function of voltage. Rg is the internal signal generator resistance
and has a value of 50 ohms. “L” is a 0.1 H “inductance box” inductor with a
nominal internal resistance, “R” of 130 ohms. The capacitance “C” has a value
of 0.2 µF , and “Ro” is a load resistance of 75 ohms. The resonant frequency of
the circuit is   = 1 LC and the quality factor is given by equation 1 as
0

Q0 =      0 L ( Rg + Ro + R ) . The probe labeled Vo shows the oscilloscope
probe connection, with the scope ground connected to the node marked with the
“ground” symbol.

Rg         R         L         C

V    Vo
Vs
Ro

0

Figure 1. An RLC series circuit with a step voltage input.

We wish to compute the voltage V0 seen across the output resistor Ro when a
voltage step function is applied at Vs. The Laplace transform, which you have
studied in your circuits course, was made for this type of problem. First, the "s"
domain impedances of the inductance and capacitance are expressed as
s L and 1 s C , respectively. Next, the voltage step function Vs is expressed in the
" s" domain as V A s , where V A is the amplitude of the step (which is the open-
circuit voltage of the signal generator). Then the output voltage, Vo , measured
by the probe can be found by voltage division as:

VA            Ro
Vo   =                                      .                      (2
s sL + (Ro + R g + R ) + 1 sC

Multiplying the numerator and denominator by s L gives the result:

V A Ro L
Vo   =                                           .                 (3
s + s (Ro + R g + R ) L + 1 LC
2

Multiplying and dividing equation (3 by (Ro + R g + R ) and rearranging yields:

V A Ro             (Ro + Rg + R ) L
Vo   =                                                  .          (4
Ro + R g + R s 2 + s (Ro + R g + R ) L + 1 LC

By noting that

Ro + R g + R             Ro + R g + R
=     0                  =        0
,             (5
L                        0   L             Q0
equation (4 can be written in terms of Q0 and                            0        as:

V A Ro                                 Q0
Vo    =                                           0
.                                   (6
Ro + R g + R s 2 + s (                0   Q0 ) +                       2
0

Equation (6 can be rearranged to take the form of the Laplace transform of a
damped sine wave. Completing the square in the denominator

V A Ro                                                      Q0
Vo    =                                                                0
,   (7
Ro + R g + R           (s +   0       2Q0 ) +
2
(        2
0
2
0    4Q02      )
2       2
and multiplying and dividing by                  0       0   4Q02 results in the form

V A Ro           (       Q0 )
2                2
4Q02
Vo     =                            0                                        0                0
.       (8
Ro + R g + R       2
0
2
0    4Q02 (s +      0   2Q0 ) +
2
(       2
0
2
0   4Q02       )
By setting = 0 2Q0 and d =                        0   1 1 4Q02 equation (8 can be made to
take on the simpler appearance,

V A Ro (2   d )
Vo    =                                           d
.                                   (9
Ro + R g + R            (s +       )   2
+        2
d

From a table of Laplace transforms, output voltage in the time domain is found to
be,

V A Ro (2   d )
v o (t ) = u (t )                     e               t
Sin(                 d   t),                                       (10
Ro + R g + R

where u (t ) is the unit step function.

Since the circuit is time invariant1, applying the step function of voltage at time t 0
instead of time zero simply delays the solution by t 0 to give the solution

V A Ro (2   d )
v o (t t 0 ) = u (t t 0 )                         e               (t   t0 )
Sin(              d   (t       t 0 )) .               (11
Ro + R g + R

1
The circuit itself does not change with time.
Substitution of nominal component values from the circuit of Figure 1. yields the
following values:

0   = 1         LC          = 7071 rad sec
Q0    =           L (Ro + R g + R ) = 2.77
(12
0

=   0       2Q0        = 1275

d   =       0    1 1 4Q02                  = 6955 rad sec

Substitution of these values in equation (11 yields

v o (t t 0 ) = u (t t 0 )(0.539 )e          1275(t t 0 )
Sin(6955(t t 0 )) .   (13

These values will be used as a first estimate of the behavior of the RLC series
circuit. This estimate will be needed when fitting the laboratory data to the form
of equation (11 to obtain measured values for      and d , and, by use of the
relations of (12 , values for Q0 and         0   .

Laboratory

Required Parts and Equipment

1.   One proto-board.
2.   One inductance box (variable inductance), set to 0.1 H.
3.   One 0.2 µF capacitor.
4.   One 75 ohm resistor.
5.   One signal generator with DC offset capability.
6.   One cathode ray oscilloscope.
7.   One digital storage scope with floppy disk capability.

Required Information

1.   User manual for signal generator.
2.   User manual for cathode ray oscilloscope.
3.   User manual for digital storage oscilloscope.
4.   Tektronix TDS 3012 – Storing a waveform to diskette (included as the last
page of this laboratory.)

Laboratory Procedure

I. Signal generator set-up

1. Turn on both the signal and the cathode ray oscilloscope. Adjust the CRO for DC
coupling and set the ground level in the center of the screen.
2. Set the signal generator for a symmetric square-wave output. Attach the
oscilloscope probes across the generator and set the generator voltage to a peak-
to-peak voltage of 5 volts.
3. Turn on the DC offset of the generator by pulling out the DC offset control and
increase the DC offset until the minimum square wave voltage is at zero (that is,
until the square wave voltage swings between zero and +5 volts).

II. Circuit set-up

1. Before connecting the circuit of Figure 1, measure the actual value of the
resistance of the 0.1 H inductance-box inductor, and of the resistor Ro, for later
use in data reduction and in writing the laboratory report.
2. Taking care not to disturb the signal generator settings, connect the circuit shown
in Figure 1. Keep your circuit compact, using the shortest wires and most direct
connections possible. Because of the fast rise-time of the square wave and the
circuit transient, frequencies much higher than 100 Hertz are present. If you can
affect the oscilloscope trace by moving the wires in your circuit, you may have a
problem.
3. Connect the oscilloscope probe across the resistor Ro as shown in Figure 1.
4. Adjust the oscilloscope time-base so that one full horizontal sweep (one screen
width) corresponds to about 4 or 5 milliseconds. Set the trigger source to internal
and channel 1, and set the mode to normal. Then adjust the channel 1
volts/division and the trigger level until the appearance of the oscilloscope
waveform resembles the graph shown in Figure 2.

Figure 2. RLC Transient response.

III. Data collection

1. Once you have verified that your circuit set-up is correct, disconnect the cathode
ray oscilloscope and ask your instructor for one of the digital storage
oscilloscopes (you may have to share use of these scopes).
2. Set the digital storage scope for a vertical sensitivity of 200 mv div and a time
base of 400 µ sec div . Adjust the trigger level and horizontal position so that the
trace approximates the appearance of Figure 2.
3. Use the instructions “Tektronix TDS 3012 – Storing a waveform to diskette,”
attached, and store the transient waveform to a 3 1 2 ” diskette. Use the file
format appropriate for use with Matlab, because Matlab functions will be used in
the data reduction.
4. Save your components in a safe place so that the same components can be used
for next week’s laboratory. Next week’s laboratory will use the results you obtain
this week, and the identical components must be used in that lab to obtain a good
result. Also, note the layout of your circuit and make a sketch of it so that you
can reproduce it for next week’s lab.

IV. Data reduction and report

The final step is to plot the data you have down-loaded from the oscilloscope and to
perform a least-squares fit to this data. This will be done by using the Matlab
functions “pltdata,” “pltwavform,” “rlctranfit” and “decsin” which your instructor
will provide to you on a floppy diskette. Please return the diskette next class period,
after you have copied these functions and the data.

Given an estimated “starting” set of four waveform parameters, and the actual data
from the measured waveform, the Matlab function “rlctranfit” will return a set of four
“actual” wave parameters representing the best least-squares fit to the data. The first
three estimated parameters passed to rlctranfit, with their values as calculated in
equations (12 and (13 are:

V A Ro (2   d )
A =                         = 0.539 volts
Ro + R g + R
= 1275

The last estimated parameter is the time delay before the step is applied to the
waveform. It is taken from the oscilloscope trace and is set to be as near
400 × 10 6 sec as possible. To summarize, the estimated parameters are passed to the
“rlctranfit” function as a Matlab row vector of the form:

estparm = [A         d   t d ].

The estimated or starting parameters should be the best estimate possible for best
results. The numbers given above are probably adequate, though you may want to
use actual measured values in your estimate.
1. Copy the Matlab functions “pltdata,” “pltwavform,” “rlctranfit” and “decsin” into

2. Use the instructions “Tektronix TDS 3012 – Storing a waveform to diskette,” to
variable produced in your workspace will be a matrix with two columns. The first
column will be the time samples of the waveform and the second column will be
the corresponding voltage samples. The variable will have the same name as the
file which produced it, something like “TEK00000.”

3. Plot the down-loaded waveform by typing, at the Matlab command prompt2:

pltdata(TEK00000)

4. Create the estimated parameter data by typing at the command prompt:

estparm = [0.539 1275 6955 400e-6]

5. Invoke the “least squares fit” function “rlctranfit” by typing, at the command
prompt:

[tdat, fitpars] = rlctranfit(estparm,TEK00000);

(This is an example. The name of your waveform data matrix may have a name
different from ‘TEK00000’.)

Type “fitpars” at the Matlab command prompt to see the “fitted” parameters. The
“fitpars” vector gives wave parameters corresponding to the actual data, that is, it
will give values [A         d  t d ] for the actual data. These values may be used
in the equations (12 to find measured values for 0 and Q0 .
Record the “fitted” values for [A         d  t d ] and calculate values for     0

and Q0 for the actual circuit. Include these calculations in your report.

6. At the Matlab command prompt, type

hold

to hold the plot for multiple plotting.

7. Plot a waveform computed from the “fitted” values on the same graph as the data
by using the function ‘pltwavform.’ At the command prompt, type

pltwavform(tdat,fitpars)

2
Of course, hit “return” after each command.
8. Add x and y axis labeling and a graph title by typing, at the command prompt:

xlabel(‘Time in Seconds’)
ylabel(‘RLC Transient Voltage’)
title(‘Transient Response of an RLC Series Circuit to a Step of Voltage’)

9. On the graph, select “edit” then “Copy Figure” and paste the resulting graph in

10. Compare the measured values for 0 and Q0 with values for these quantities
computed from nominal component values and measured circuit resistances. List
and discuss possible reasons for differences between these values.

11. Record the values of 0 and Q0 and the measured resistance values for use in
preparing the pre-laboratory for next week’s session.
Tektronix TDS 3012
Storing a waveform to diskette:
With waveform displayed on the oscilloscope,

1. Insert a floppy in the disk drive
2. From the “set of 6 brown buttons” at the upper right of the control panel, press the
“Save/Recall” button.
3. From the menu in the lower screen margin, press the button to select “Save
Waveform Chx” (‘x’ is the number of the channel which was most recently
selected with the ‘Ch 1’ or ‘Ch 2’ buttons.)
4. Press the button beside the floppy disk icon in the right screen margin menu.
5. Select the format from the menu in the right screen margin.
a) Select ‘internal file format’ if the stored waveform is to be displayed on
the oscilloscope later.
b) Select ‘Spreadsheet File Format’ if the stored waveform is to be loaded
c) Select ‘Mathcad File Format’ if the stored waveform is to be loaded into
6. Select ‘Save Channel x to Selected File’ from the right screen margin menu. (It
may take several seconds to save the waveform.)

To Recall to Display on Scope (Internal Format Files; “.ISF suffix”)
1. Insert a floppy in the disk drive
2. From the “set of 6 brown buttons” at the upper right of the control panel, press the
“Save/Recall” button.
3. Select “Recall Waveform” from the lower screen margin menu.
4. Select “From File” from the right screen margin menu.
5. Select a file by turning the control knob just to the right of the “select” and
“coarse” buttons in the upper left corner of the control panel.
6. Select a “ref” location (ref1, ref2, etc.) in which to store the waveform from the
7. The waveform will be displayed...

1. Spread sheet files are saved in Microsoft Excel format
2. To load into Matlab, type ‘uiimport’ at the Matlab command line (note two i’s).
3. A user interface will be displayed. Browse to file you wish to load.
4. Use ‘comma’ as column separator.
5. File will be imported to Matlab as a two column array. Column ‘1’ is the time
base; column ‘2’ is the data.
6. Type ‘who’ at the Matlab command line to see the variable you have loaded. It
will have the same name as the file you imported.

```
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