6/06
Electrical Signal Response
ELECTRICAL SIGNAL RESPONSE
About this lab
The response of passive electric circuits to pulse or sinusoidal
excitation is governed by basic physical properties of the circuit elements, and the
corresponding forms of magnetic and electric energy storage and of resistive dissipation.
Thus, they cannot be circumvented. Understanding the time response of the various
element combinations allows design planning.
References: Physics: Serway&Beichner, 6th ed v.2; Chapter 33 (Saunders, 2000)
Physics: Cutnell & Johnson, 6th ed., Sections 20.13; 23.1-23.4(Wiley, 2001)
BRING YOUR TEXTBOOK TO LAB
Apparatus
> Circuit board with mounted R, L, C components and banana jack sockets. Alligator
clip lead for easily disconnecting small resistor from circuit.
> Labtec speakers with Mic and earphone sockets, and signal amplitude (volume)
control.
> Interface board between circuit block and computer sound card (via speaker), carrying
circuit drive signal and (two) circuit read signals, and connecting grounds.
> PC computer with FFTScope (scope and FFT signal analysis modes).
> Digital multimeter for accurate determination of external circuit resistors.
Introduction
R (resistance), L (inductance) and C (capacitance) are the
fundamental passive components of linear electric circuits. They implement the basic
electrical phenomena:
Dissipative energy loss from electric particle collisions:
(Resistance),
Electrostatic energy storage in the form of electric field:
(Capacitance),
Magnetic energy storage in the form of magnetic field:
(Inductance).
In pairs, their circuit influence can be expressed in terms of certain combinations of
values, known as time constants:
RC:
R*C exponential charging/discharging time constant τ RC
LR:
L/R exponential current increase/decrease time constant τ LR
LC:
1/sqrt(L*C) natural oscillation angular period (in sec/radian)
τ LC .
These time constants arise directly out of Kirchkoff's circuit (differential) equations, or by
differentiation with respect to time of the energy equivalent, which again gives
Kirchkoff's equations. Essential physical content of these equations is:
a) Energy conservation (loop voltages sum to zero if there is no dissipation, or if an EMF
makes up resistive power losses), and
b) Charge conservation ('current in' = 'current out' at nodes; the current in a single loop
circuit is the same everywhere (with added definition of displacement current in
capacitors).
The major signals of interest are digital pulses (the basis of low-error communication,
made possible by the advent of fast solid-state switching devices) and sine waves (the
basis of efficient, narrow band communication, long possible because of the resonant
tuning properties of circuit elements combinations such as those we will study).
The responses of element combinations to pulse and to sine signals are related, not
independent, as we will see.
Physical L's and C's may not be pure types; they may also have some dissipative
character, expressed as internal resistance R L or R C . Then, four quantities (L and RL, C
and R C) would be required to characterize completely a physical inductor or capacitor.
Before proceeding , study some examples of response vs. time (Oscilloscope) or vs.
frequency (FFT) , using the FFTScope program.
Figure 1
RC response to square drive pulse @ 100 Hz. Sampling time interval =
0.1 second. Scope mode (mono).
R ext = 2660 ohms.
Figure 2
RL response to 1:100 drive pulse @ 10 Hz. Sampling time interval = 0.1
second. Scope mode (mono)
Rext = 10 ohms (in parallel w/ 2660 ohms). Ignore initial negative spike; fit positive
decay portion. Read current-proportional voltage (CR) across R ext.
Figure 3
FFT mode. Series RLC frequency response curve to all-sines (white
noise) excitation.
R external ~ 51 Ω. The pulse response time curve is conjugate.
Figure 4
Series RLC amplitude and phase resonance.
White-noise excitation. FFT mode w/averaging:
R ext = 10 ohms. Resonance Left/Right ratio about ¼ indicates internal Rint (from
L&C) is around 30 ohms. L/R = VR ext/Vdrive . Phase is between drive voltage and
response current.
Figure 5
Series RLC time response to pulse excitation (1:100 pulse ).
R external ~ 51 Ω. Scope mode. The frequency response curve is conjugate.
Figure 6
Parallel RLC amplitude and phase resonance.
White-noise excitation.
FFT mode w/averaging
R ext = 2660 ohms. L/R ratio does not become zero at resonance (“tank” impedance
does not become infinite) because of R internal of L and C. Resonance ratio indicates
tank impedance at resonance is only around 10,000 ohms. Phase is between drive
voltage and circuit response current.
Read current-proportional voltage V across R ext . i = VR ext / R ext.
Experiment Your qualitative and quantitative assignments are:
1. To study all the various responses of your circuit to pulse and sine signal excitation,
including the parallel resonance,
2. To determine as accurately as you may the circuit parameters L and C. You may
also attempt to determine the lossy equivalent resistances RC and RL , if you wish.
(One strategy in finding the lossy internal resistances RC and RL is to attempt a fit with
total resistance (Rext+Rinternal), with the numerical value of Rext inserted. Perhaps easier
is just to fit with Rtotal and subtract Rext to determine Rinternal.)
But, if external resistance Rext is >> internal resistance RC and RL,, the shape of the
response curve will not be very sensitive to the internal resistances when fitting the
curve, so thought will be necessary as to what data is fit for this purpose.
Also, if the uncertainty in value of Rext is comparable to or greater than the internal
resistances, their determination will again be quite difficult, so an independent
determination of Rext is desirable, rather than relying on the nominal value.)
Procedure
Setup
An intermediate connection block distributes signals between the circuit
block and the computer, via mike and line sockets of one of the Labtec speakers, which is
connected to the “Line In” port of the PC.
Rext
G
Figure 7
L
C
3
2
1
Circuit block
Four brass blocks with banana sockets and wing nuts clamp tightly the three circuit
elements. You may loosen a wing nut, but do not lift the brass blocks – the original L
and C circuit elements will remain undisturbed. Only R ext (initially ~ 10 Ω in parallel
with 2600 Ω) may be altered (increased) by disconnecting the alligator clip connected to
one end of the small (~ 10 ohm) resistor.
There are five wires from the intermediary connection board, in two sections: drive and
read. Two ground wires (drive gnd and read gnd) always remain plugged in to block G.
The read left wire (rl) always remains plugged into block 3. The remaining two wires
move together: they are the drive signal (ds) and the read right (rr).
Two read wires are needed to send signals back to the computer for analysis: drive
amplitude (the read right lead) and the circuit response amplitude (the read left lead). The
read left circuit response signal reads voltage across the the external circuit resistor Rext.
It is therefore proportional to the circuit current.
The computer FFTScope program can analyze the two signals with respect to time (scope
mode), or with respect to frequency (FFT mode). These are complementary. Computer
frequency analysis is limited to one half the range of the computer sound card employed,
which is about 48 kHz.
FFTScope display in scope mode can be either mono (circuit response signal) or stereo
(both circuit response and drive signals). In FFT mode, the mono display shows the net
amplitudes (sine + cosine) of the various response signal frequency components, and in
stereo mode, both the frequency amplitudes and the phase (relative sine and cosine
amplitudes of a given frequency response component)
(The circuit grounding arrangement produces a 180 degree inversion of the phases, which
can easily be corrected in analysis by a program such as Graphical Analysis.)
Read signals are between the SR position and G. CR (“Current Read”) is really a voltage
signal; because it is across a resistor, it has the phase of the circuit current and the
magnitude of the current could be calculated as CR/R ext.
> For (R ext L) connection, drive signal and read right go to block 1, and blocks 2 and 3
are shorted with a banana jumper (C removed from circuit).
> For (R ext C) connection, drive signal and read right go to block 2, leaving L dangling.
Use Large R ext.
> For series (R ext LC) connection, drive signal and read right go to block 1.
> For parallel (R ext LC) (tank) connection, drive signal and read right go to block 2, and
blocks 1 and 3 are shorted with banana jumper.
You may measure Rext with a multimeter, for accuracy. It is best to remove momentarily
the two G wires, to eliminate any outside connections.
It is understood that L and C in the circuit block figure stand for the combinations [ L +
any internal lossy resistance of the inductor ] and similarly for the capacitor.
Data and analysis
Study the response Figures above. Take data for the various
cases and verify the theory by fitting theoretical forms (exponential or damped sine for
pulse excitation, or phase and amplitude resonant shapes for sine excitation, as
appropriate).
First check the fit values of R, L and C against nominal values. Then you may try to
determine the internal resistive loss parameters RL and RC, if you wish.
Your data may copied and pasted (Ctrl-C) into Graphical Analysis for fitting with
appropriate theoretical form, in which the circuit parameters are fitting parameters,
explicitly or implicitly. You may need to modify a standard form (e.g., damped sine vs.
time) or specify a non-standard form (e.g., resonant phase or amplitude vs. f or ω.
FFTScope outputs frequency. And before fitting, for scope waveform data, you will need
to change from scope time (which may start, say, at 29.2967 milliseconds) to “Good
Time”, by defining “Good Time” as (Scope Time – first scope time entry numerical
value). Then plot vs. “Good Time”.
If you are modifying a sine to incorporate exponential damping, take care that the
exponential multiplies only the sine, not the vertical offset parameter. Also, note that the
damping time constant for RLC “ringing” after pulse excitation is (2L/R), not (L/R) as for
the simple RL circuit (see Serway and Beichner, 5th, Eqs. 32.31, 32.7, 32.10; also Figure
32.21 p 1031-1033).
In copying scope mode decay curves from FFTScope, avoid any kinks. Use 1:100 pulse
for the RL and RLC circuits, as illustrated – this restricts to frequencies less than 1:100 of
the sampling rate, which is no problem. (The 1:100 pulse maintains the set repetition
freqeuncy, but replaces equal up and down levels by a much shorter up and a much longer
down level - much more of an impulsive drive.) The inductor will generate “backEMF's” (Faraday's law, Lenz' Law), large at times of rapid change. In copying the RL
time curve for decay fitting, avoid the negative spike and any overshoot; choose an
exponential-looking portion. Use small Rext to observe series RLC circuit, large for
parallel. With white noise excitation, TURN FFT AVERAGING ON.
Use square pulses for the RC.
Use large Rext (c. 3000 ohms) for parallel resonant circuit (because a parallel LC
combination has a high impedance at resonance), small R for series (because the converse
is true). Measure R external with meter and record.
Use Graphical Analysis for data record and curve fit. Add additional (manual) data
columns, pages and graphs as desired.
Copy your GA fit for later reference: Go up to a Student folder, then to 230 folder. Save
As (descriptive, e.g., RC initials). Put enough information into the graph title that you
can figure out later what you did.
Strategies Pulse excitation. See previous comments. A good fit to an RL time
constant τ R L would give you the product [L/(R ext + R L)], one equation for the two
unknowns L and R L. (Note that a GA data fit of form exp(-Kt) would give K, inverse to
τ, which enters as exp(-t /τ) ). Another equation is needed, which might be obtained by
altering R ext to R' ext and obtaining a new τ' = [L/(R' ext + RL)]. Further such would over
determine L and R L .
τ 's obtained with small R ext are sensitive to internal R's, whereas R ext >> RL determines
L immediately, up to order ( RL/ R ext ) (i.e., τ is insensitive to RL for large Rext), with
similar considerations for determination of RC .
The white noise FFT's show quickly the resonant frequencies f 0 = ( ω 0 /2 π ), in both
amplitude and phase. (Which do you think is more sensitive?)
But there is also a complementary strategy – use of sine excitation, where theory gives the
voltage ratio CR/SR, (read this ratio out in FFTScope), and also the phase between
applied voltage Signal Feed (SF) and Current Response (CR). (Remember that current
read CR is really a voltage read across R ext.) Changing the function generator frequency
is quick – no setup alteration. Spanning the frequency range from mainly resistive
impedance to mainly reactive impedance might make some sense.
An expression for current amplitude vs. frequency in the series RLC circuit, e.g. Serway
5th, Eq. 33.22, p 1053, would suffice also for the RC and RL cases (setting XL or XC to
zero). Eq. 33.25 similarly can give the phase vs. frequency. (The frequency appears
implicitly in these expressions, since XL = ω L and XC = 1/ω C). (The Pythagorean
expressions reflect the 90o phase relation between reactive and resistive voltages; the
relative – sign between XL and XC reflects the 180o relative phase in their voltage
variations.)
With I max equal to our CR/R ext and ∆V max our SR, rewriting Eq. 33.23 in our
nomenclature gives the voltage divider expression for the RL combination
CR/VR = R ext /sqrt[(R ext + R L )^2 + (2 π f L)^2]
(where the known numerical value of R ext would be used, instead of the symbol), with
equivalent expressions for the RC and RLC combinations. Note that the separate
amplitudes CR and SR are given on demand by FFTScope (Osc) (View menu), as is the
frequency f. Inserting several pairs f vs. the ratio CR/SR (you would have to do your own
division, or could input separately to GA the CR and SR amplitudes, with GA filling a
calculated ratio column) and fitting to the above form (for RL) should give the two
unknown fit parameters L and R L .
Thus, taking CR/SR amplitude ratios for several frequencies, inserting by hand into GA
and fitting to an expression such as the above (for RL) should yield numerical values and
errors for the fit parameters L and RL. Note that only amplitude ratios enter – we don't
need to fit an absolute amplitude scale factor. But, be sure not to overdrive – check the
drive sine for flat-topping. Adjust, if necessary, with the speaker volume control.
The RL fit expression for relative phase of current to signal voltage (i.e., of CR to SR)
(Serway, Eq. 33.25, p 1053, with XC set to zero) would be
φ = phase = arctan(X L / (R ext + R L ) = arctan[ωL/(R ext + R L )]
where, again, the known numerical value of R ext would be entered.
To obtain phase in Osc mode, you would view both sines (stereo) and calculate phase
from relative shift. This might not be so accurate. Alternatively, in FFT mono mode,
read the peak frequency. Then switch to stereo, find the same frequency on the phase
curve and read.
Report
Try various methods of determining the four parameters, using series
circuits. (The parallel resonant circuit is more complicated, though straightforward,
because it contains two loops, vs. one for the simpler series resonant circuit.)
Keep records of methods and results (with fit parameter errors, if given), and with
running comments.
Finally, start a clean page and prepare a neat and organized report, with your best values
for C, RC, L and RL. List the various methods used and other results (mention any
approximations). Say why you think one method might be better than another.
Submit the final, organized report plus any other material your instructor specifies, such
as fit print outs.
Appendix
Referring to Serway, Physics v2. 6 th ed.,
Current amplitude vs. sine frequency (e.g..with white noise drive) in a series circuit has
the form:
i=
1
 R ( X
2
2
L
2
2
- XC )
X L =  L and
XC =
with proportionality to drive voltage, where
1
C
and the phase variation with frequency is given by
arctan (
X L− X C
.
R
The current amplitude variation with frequency for RC or RL circuits can be obtained by
setting one or the other reactance to zero.
For a parallel resonance ("tank") circuit, XL and XC (including any associated internal
resistances) are in parallel. The impedance Z for the parallel circuit can be calculated
using the standard form for resistors in parallel.