ECE 2A Lab #6
Lab 6
First-Order RC/RL
Circuits
Overview
This lab will explore the transient response of simple RC and RL circuits to understand the
role of the time constant. In the process we will introduce the based concepts of
differentiator, integrator, and frequency filtering circuits. We will then combine the RC time
constant material with an earlier comparator circuit from Lab #4 to create a simple timer.
Table of Contents
Background Information
Capacitors
Inductors
References
Pre-lab Preparation
Before Coming to the Lab
Required Equipment
Parts List
In-Lab Procedure
6.1
RC Circuits
RC Time Constant
The Circuit as an Integrator
The RC Circuit as a Low-Pass Filter
An RC Differentiator Circuit
6.2
RL Circuits
RL time constant
An RL differentiator/High-Pass Filter
6.3
A Simple RC Timing Circuit
2
2
3
4
5
5
5
5
5
5
5
6
6
7
7
7
7
8
1
© Bob York
2
First-Order RC/RL Circuits
Background Information
Capacitors
Capacitors are made from a layer of insulating material sandwiched between two conducting
sheets. The total capacitance generally scales with the materials and geometry according to
C
r A
(6.1)
d
where  r is an intrinsic property of the insulator called the “dielectric constant”, d is the
thickness of the insulator, and A is the net surface area of the conducting sheets. Large
capacitors therefore require a large sheet area and thin insulators with the maximum possible
dielectric constant.
Capacitors are usually classified according to the type of insulating material that is used.
“Film” capacitors use a thin film of material such as polypropylene, polystyrene, Teflon,
Mylar, etc. These films are then stacked together with metal electrodes (either a separate foil
or a thin layer of metal coating on the film). “Ceramic” capacitors use a high dielectricconstant material, generally barium titanate or some related ceramic material, which is mixed
in a thin paste between metal electrodes and then hardened together (“fired”) in a hightemperature furnace. “Electrolytic” capacitors use a thin oxide layer on one of the electrodes
(usually aluminum oxide on aluminum) as the insulator, with a liquid electrolyte used to
insure an intimate contact between the second electrode and the oxide. Figure 6-1 illustrates
some of the common capacitor types.
Film Capacitors
Ceramic
Electrolytics
• Low loss, low leakage
• Larger than ceramics, but
tighter tolerance and better
performance for a given price
• Small size, High voltage
• Tolerance, temp. dependence,
leakage, etc. vary strongly with
material; cost tradeoff
• Very high capacitance density
• ±20% tolerance
• Large size, poor loss and leakage
characteristics
Figure 6-1 – Comparison of commonly-available through-hole capacitor types.
It is difficult to control the capacitance values precisely during the manufacturing process,
so capacitors are specified for a certain “tolerance”. The statistics are essentially the same as
we discussed for resistors, and the same system of values applies for capacitors: E6 for
±20%, E12 for ±10%, etc. Electrolytic capacitors usually have a poor tolerance, typically
±20%, but ceramic and film capacitors can be obtained with tighter tolerances.
There are two very important practical issues with capacitors that you need to be aware of.
One is the voltage rating: This is the maximum voltage that the capacitor can withstand for
some period of time. If this voltage is exceeded the insulating material will fail and a large
current could flow, destroying the device. The second issue is polarity: some capacitors,
notably electrolytics and tantalum capacitors, are not reversible; they must be inserted in such
a way that any DC voltage will appear across its terminals in a specified polarity. So care
© Bob York
3
Background Information
must be taken when inserting them in a circuit—a polar capacitor mounted in the wrong way
will self destruct. In the case of electrolytics, this happens because one voltage polarity
promotes a chemical oxidation-reduction process
between the electrolyte and the insulating oxide
Value and voltage rating
printed on package
that slowly destroys the insulator. Electrolytics
have been known to erupt rather dramatically
Long-lead is “positive”
when biased improperly. For this reason, polar
capacitors usually have some marking on the
package to indicate the correct insertion; Figure
6-2 illustrates the markings for a typical
“-” sign on package
indicates “negative” lead
electrolytic capacitor. In a circuit schematic polar
capacitors are usually indicated by a “+” sign on Figure 6-2 – Symbol for a polar capacitor
one terminal as shown.
and associated markings for electrolytics.
There are several other non-idealities that
affect the behavior of capacitors in circuits. Unfortunately too many to discuss here in any
great detail! They suffer from leakage currents, vary with temperature and humidity, and
may have some problems at high frequencies. These issues will not be critical in ECE 2, but
they are important factors for engineers to understand and deal with in commercial designs.
In ECE 2 we will generally use inexpensive ceramic capacitors for small-values (<1μF)
and electrolytics for the larger values (>1μF). The actual capacitance value is marked in a
variety of different ways. For electrolytics the value is usually stamped clearly as shown in
Figure 6-2.
For ceramics and film
Third “multiplier” digit
capacitors a three-digit code is more
commonly used, illustrated in Figure 6-3.
10  104 pF  105 pF
This code gives the value in picofarads
( 1pF  1012 F ). Beware that this is not
 100nF
First two
the only capacitor code in use! Sometimes
“value” digits
 0.1 F
there is a 4-digit number which represents
the actual value in pF (i.e. no multiplier),
and other times a decimal value is shown
(like “.033”) which is the capacitance in Figure 6-3 – Three-digit code used for most of the
μF! Sometimes (but not always) there is a ceramic capacitors in ECE 2.
letter following the code that indicates the
tolerance, and sometimes (but not always) the voltage rating is shown. To further complicate
things, many modern devices are so tiny and the lettering so small and faint that it is often
hard to see the markings unless you have especially good eyesight (a magnifying glass is a
good investment for this line of work!). If the markings are confusing or ambiguous, out
advice is to go MEASURE the capacitance. In the ECE 2 lab there is an “LCR” meter that
will measure a wide range of capacitor and inductor values.
Inductors
Inductors are usually made by coiling wire around some cylindrical or toroidal (doughnutshaped) form. This winding form is often (but not always) a solid magnetic material such as
iron or iron-oxide (“ferrite”) called the “core”. The total inductance usually varies with the
material and geometry according to
L
r N 2 A

(6.2)
4
First-Order RC/RL Circuits
where r is an intrinsic property of the core material called the “magnetic constant”, N is
the number of turns of the wire, A is the average area of each turn, and  is the overall
length of the winding. Large inductors therefore require a lot of turns, a large turn area, and a
core with a large r .
In some respects inductors are very uncomplicated devices—after all, what could be
simpler than a coil of wire? But some complexity is hiding in the physics of the core material
which ultimately plays a significant role in the I-V characteristics of the device. Most
magnetic core materials lead to nonlinear behavior at certain current levels, and also can limit
the maximum operating frequency of the devices. But there are two even more basic
problems with inductors: 1) they are HUGE in comparison to other circuit components, and
2) they do not lend themselves easily to mass production and hence do not benefit from the
same economies of scale that other passive components enjoy. Because they are so big and
expensive they are used sparingly in modern microelectronics, mostly found today in
applications such as power-electronics (e.g. switch-mode power supplies, AC-to-DC and DCto-DC converters), high-frequency chokes, high-power filters, and certain radio circuits.
Solenoidal Inductors
Toroidal Inductors
• Inexpensive, low-profile
• Limited range of inductance values
• Poor confinement of magnetic fields
• Wide range of inductance and power handling
• Good confinement of fields
• Larger and more expensive than solenoidal
inductors
Figure 6-4 – Comparison of common inductors.
We remarked earlier that capacitors have an inconsistent marking system. Inductors are
even worse—most have no markings at all! In addition there is no consistent set of values
like the decade-based systems used with resistors and capacitors (E6, E12, E24, etc.); the
availability of certain values seems to be driven by demand and are very application-specific.
So the only good recourse for an experimentalist is to measure the inductance directly with an
LCR meter. But note: because the properties of the core material depend strongly on the
amount of current in the windings, most inductors are designed to have a certain inductance
at some specified DC current level, and it is not uncommon for inductances to vary
significantly around this nominal specification at other current levels.
References
■ http://en.wikipedia.org/wiki/Types_of_capacitor
■ http://en.wikipedia.org/wiki/Ceramic_capacitor
■ http://en.wikipedia.org/wiki/Electrolytic_capacitor
■ http://en.wikipedia.org/wiki/Inductor
© Bob York
Pre-lab Preparation
5
Pre-lab Preparation
Before Coming to the Lab
□
Read through the details of the lab experiment to familiarize yourself with the
components and testing sequence.
□
One person from each lab group should obtain a parts kit from the ECE Shop.
Required Equipment
■ Provided in lab: Bench power supply, Function Generator, Oscilloscope, Decade Box
■ Student equipment: Solderless breadboard, and jumper wire kit
Parts List
Qty
1
2
1
2
Description
10mH 19DCR power inductor
PC mount tactile switch
0.01uF 50V ceramic capacitor, radial lead
100uF 25V electrolytic capacitor, radial lead
We will also use the 741 op-amp, LEDs, and selected resistors from earlier labs
In-Lab Procedure
6.1 RC Circuits
RC Time Constant
□
Build the RC circuit shown in Figure 6-5, using the decade box for the resistor, R, and the
0.01 μF capacitor in your kit. Note that the 50 Ω resistor is just shown for reference—it
represents the internal resistance of the
Input
Output
function generator. You can verify the
(Ch1)
(Ch2)
50Ω
R
capacitor value using the LCR impedance
meter in the lab. Adjust the output of the
Vg
function generator to a 1 kHz square wave,
C
Vin
Vout
with an amplitude of 1V (2 volts peak-toFunction
0.01μF
peak). Be sure that the DC offset switch
Generator
on the function generator is off (pushed in)
and that the vertical scope inputs are DC
coupled (not AC coupled). It will be Figure 6-5 – A simple first-order “RC” circuit.
convenient to view the input and output
signals simultaneously on the oscilloscope screen, which is accomplished by the
connections shown below. Set the vertical positions so that the 0 volt (ground) level is
the center of the screen for both channels:
□
Starting with R=100 Ω, observe and record the input and output waveforms. They should
be similar. Now increase the resistance to 5 kΩ and record the resulting waveforms in
your notebook. Physically the series resistance limits the flow of current so the voltage
6
First-Order RC/RL Circuits
across the capacitor rises more slowly as the resistance is increased. Expand the time to
zoom in on one transition, and use the cursors on the oscilloscope to estimate the time
constant,  , from the waveform; remember,  is time it takes for the voltage change by
1  e1  63% of the total step voltage (final value minus initial value). Compare with the
theoretical prediction based on the RC product.
Another figure of merit commonly used in such cases is the rise-time, tr , and fall-time, t f ,
defined as the time it takes for the signal to progress from 10% to 90% of the total step size
(final value minus initial value). The rise time can be expressed as
tr   ln
□
0.9
 2.2 RC
0.1
(6.3)
Use the cursors again to estimate the risetime/falltime and compare with the prediction of
(6.3). For this measurement it may help to adjust the vertical sensitivity so that the
waveform fills the screen.
The Circuit as an Integrator
□
Now increase the resistor to 20 kΩ. The time constant is now longer than the duration of
the square pulse so the output waveform does not quite reach the peak value of the input
waveform. Increase the resistance again to 100 kΩ and record the resulting waveform in
your notebook. You will probably have to adjust the vertical scale since the peak
amplitude will continue to shrink. Can you understand what is happening? The output is
now approximately a triangular wave, which is essentially integral of the input signal.
Verify that the circuit is acting as an integrator by changing the input waveform to
triangular and sine waves, making appropriate plots of both in your notebook.
The important point here is that there are two time-periods involved: the RC time constant  ,
and the time-period T of the square wave excitation. For   T we see the full RC charging
transient after every transition, but for   T we only see the initial linear part of the
charging transient, and hence the circuit functions as an integrator.
□
In the above steps we adjusted the value of the resistance to observe the integrator
behavior. Alternatively we could keep the resistor fixed and change the frequency
instead. At approximately what frequency will the circuit behave as an integrator for a
resistor value of 10kΩ? Verify your answer experimentally using a square-wave
excitation.
The RC Circuit as a Low-Pass Filter
Our focus so-far has been on the transient response of the RC circuit due to a step function
excitation (the leading and falling edges of the square-wave). But for sinusoidal (AC)
excitation the output will also be sinusoidal, so it makes more sense to focus on how the
amplitude or phase changes as a function of frequency.
Switch the function generator to a sinusoid with a 1V amplitude (2V peak-to-peak) and
zero DC offset. Starting with a frequency of 100 Hz, record the output amplitude versus
frequency up to 100 kHz, using R=5 kΩ. Since we are covering several decades of
frequency it is best to choose the frequencies appropriate to a log scale like this: 100 Hz,
200 Hz, 500 Hz, 1 kHz, 2 kHz, 5 kHz, etc. Take data quickly; this step should not last
long. In your lab report you will plot your results on a log scale in frequency.
In the context of sinusoidal signals the simple RC circuit of Figure 6-5 is called a “low-pass
filter”, because low frequency signals will pass through but high frequency signals get
□
© Bob York
7
RL Circuits
strongly attenuated. There are many uses for filters in electrical engineering; a simple and
familiar example is the cross-over network in an audio speaker system or surround-sound
system, which directs low frequency audio to the big “sub-woofers”.
An RC Differentiator Circuit
By simply switching the placement of the
capacitor and resistor as shown in Figure 6-6
the operation of the circuit can be changed
significantly.
□
C
50Ω
Vg
0.01μF
Vout
Vin
Build this circuit, and drive it with a 1 kHz
R
Function
square wave using a 5 kΩ resistor. Sketch
Generator
the resulting waveforms in your notebook.
Can you see that the circuit is now acting as
a differentiator? Prove this to yourself by
Figure 6-6 – RC (high-pass) circuit.
changing the excitation to triangular and
sine waves. Then using the square wave signal again, zoom in on one of the transitions
by adjusting the time base. Can you identify any characteristic property of the network
from the shape of this signal?
6.2 RL Circuits
The RL circuit is very similar to the RC circuit in terms of the mathematical form of the
response, despite very different physical mechanisms involved. Therefore we will not
examine this case quite as thoroughly:
RL time constant
□
Build the circuit of Figure 6-7 using a 10mH
inductor and a 100 Ω resistor. As before,
drive the circuit with a 1 kHz square wave
and estimate the time constant. Compare
with the theoretical prediction. Can you
reconcile the data with the theory? Try to
identify extra sources of resistance in the
circuit. Sketch the waveforms in your
notebook and compare with the earlier RC
circuits for similarities.
L
50Ω
10mH
Vg
Vin
R
Vout
Function
Generator
Figure 6-7 – RL (low-pass) circuit.
An RL differentiator/High-Pass Filter
□
□
Now switch the order of the inductor and
resistor to construct the circuit of Figure 6-8,
this time using a 1 kΩ resistor, and verify
the action of this circuit as you did earlier
for the RC differentiator circuit.
This circuit also functions as a high-pass
filter, because only high frequency signals
go through without attenuation. Show that
this is true by taking data over frequency for
50Ω
R
Vg
Vin
Function
Generator
L
10mH
Figure 6-8 – RL (high-pass) circuit.
Vout
8
First-Order RC/RL Circuits
a sine-wave excitation as you did earlier in connection with the RC low-pass filter. In
your lab report you will make a careful plot of this data..
6.3 A Simple RC Timing Circuit
Time constant circuits arise in a number of different contexts in electrical engineering. One
particular application is a simple timer function where the RC charging time is exploited to
create a known time-delay.
+5V
The circuit in Figure 6-9
+5V
510Ω
combines a very simple RC
R
circuit (inside the dashed box)
Yellow
10kΩ
with the op-amp comparator
circuit that we examined in Lab
LM741
100μF
#4. A simple adjustable voltage
divider circuit is used to set the
C
Switch
Vref
Red
reference
voltage
for
the
comparator, Vref . The circuit
-5V 510Ω
functions as follows: with the
switch open the capacitor is
+5V
charged to +5V, and since this is
1kΩ
10kΩ
potentiometer
always  Vref the output will stay
in its high state (near +5V), Figure 6-9 – A simple LED timer circuit.
lighting the red LED. When the
switch is pressed the capacitor is quickly discharged to 0V and the output goes low, switching
on the yellow LED. The capacitor then begins to charge up again through resistor R. The
output stays low until the voltage across the capacitor reaches Vref , at which point the red
LED is switched back on. So the yellow light is on for a short period of time after the button
is pressed, and this time period is set by the RC time-constant and Vref .
wiper
+
□
Start building this circuit by first mounting the 741 op-amp and establishing the proper
bias. Refer to the data sheet for the pinout, and use ±5V for the power supplies.
□
Next, add the 10k trimpot and
1k resistor, then add the LEDs
and 510 Ω bias resistors. The
resistors and LEDs are leftover
parts from earlier labs.
Lastly add the RC timing
network and the pushbutton
“tactile” switch. Note: this is
our first circuit using a large
electrolytic capacitor, so be
sure you are familiar with the Figure 6-10 – Typical pinout for a tactile pushbutton switch.
relevant material from the
background section; electrolytics must be inserted in the correct polarity!!
The tactile switch we are using is called a “momentary on” switch, meaning that it is
normally off until someone presses the button. A typical pinout is shown in Figure 6-10, but
you should use your ohmmeter to make a simple continuity test to verify the correct pin
connections.
□
© Bob York
9
A Simple RC Timing Circuit
□
Now turn on the power-supplies and adjust the potentiometer to 5(1  e 1 )  3.16 V . This
sets the threshold voltage such that the time period is exactly one time constant. After the
power-supply has been on for a few seconds the red LED should be on, and the yellow
LED should be off.
□
Now press the button and release it. Use your watch to monitor how long the yellow
LED stays lit, and compare this with the theoretical prediction based on   RC .
□
Can you calculate the Vref that would be needed to keep the yellow LED alight for two
time constants? Adjust the potentiometer accordingly and verify your design. Include
the details of the calculation in your lab report.
□
Now design the timer so that the yellow LED stays lit for 10 seconds. You will need to
use a different charging resistor R. Calculate the necessary R and Vref and verify your
design experimentally. Include the details of the calculation in your lab report.
Congratulations!
You have now completed Lab 6
As noted in the previous lab: keep all your leftover electrical components!
Amplitude, Volts
Notes on the Report
In your report, you should (wherever possible) compare the theory and experiment, and
discuss why the circuits behave as they do for different combinations of resistance (time
constant) and frequency. For example, in the case of the RC circuit with a square-wave
excitation the output waveform
10
closely resembles the input
waveform when   T ; this
1
occurs because the charging or
discharging
transients
are
0.1
completed well before the next
pulse transition occurs. Be sure
0.01
to include any necessary
mathematics to support you’re
0.001
answers.
If you use a result
0.0001
from your textbook, reference
1
10
100
1000
10000
100000
the relevant equation number
Frequency, Hz
and/or page.
Programs
like
Matlab,
Figure 6-11 – Example of a log-log (Bode) plot in Excel.
Mathematica, etc. can and
should be used wherever possible. For the output voltage-versus frequency plots you should
use a log-log scale. Figure 6-11 shows a log-log plot done in Excel (the file for this plot is on
the course web site, as well as a brief excel tutorial).
The time constant you found in connection with the RL Time Constant section should
have differed noticeably from what you would predict based on the resistance shown in the
schematic. Be sure to explain a possible source of this difference.