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ELEC 350L
Electronics I Laboratory
Fall 2005
Lab #5: Relaxation Oscillator (Astable Multivibrator)
Introduction
Many electronic devices and systems require a source of pulses, often called a “clock signal,” for
proper operation. One example is the clock circuit found in all computers that controls the
timing of the digital signals within the CPU. Another is the signal generator on your lab bench.
The waveform most often required in a large number of applications is the square wave.
Sometimes a nonsymmetrical square wave, also called a rectangular wave, is required. There are
many ways to generate square waves, but if a high-quality signal source is not required, one of
the easiest and least expensive approaches is to use an op-amp. In this lab experiment you will
build a type of square wave generator circuit known as a relaxation oscillator (also called an
astable multivibrator).
Theory of Operation
A common square wave generator circuit based on an op-amp is shown in Figure 1. One of the
first things you might notice about this circuit is that it has no input port other than the
connections to the power supply. Rather than serving as processors of signals (like amplifiers
and filters, which have inputs and outputs), oscillators instead are sources of signals (which have
only outputs). Note that there are resistors (R2 and R3) connected from the output port of the opamp to both its inverting and noninverting terminals. The design is unusual in that it
incorporates both positive and negative feedback. The positive feedback dominates, primarily
because the capacitor voltage cannot change instantaneously, whereas the voltage at the junction
of the voltage divider formed by R1 and R2 can. Thus, a virtual short cannot be assumed to exist
between the two input terminals. We will see that the circuit behaves cyclically, switching
regularly between two unstable output states (hence the name astable multivibrator;
multivibrator is another name for oscillator).
R3
C
+
vc
−
+15 V
−
vo
+
−15 V
R2
R1
Figure 1. Relaxation oscillator.
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Because of the positive feedback, the output of the op-amp will be near either the positive power
supply voltage VPOS (+ 15 V in Figure 1) or the negative supply voltage VNEG (–15 V). The
reference voltage at the noninverting input will be VPOS or VNEG, where

R2
.
R1  R2
For example, if R1 and R2 are equal in value, the reference voltage will be 0.5VPOS. Thus, the
state of the output (vo = VPOS or VNEG) will depend upon the value of the capacitor voltage vc
(which is the same as v−) compared to the voltage at the junction of the voltage divider (v+). The
capacitor charges through resistor R3 with a time constant of R3C because no current flows into
the inverting input of the op-amp. The capacitor voltage vc would eventually reach VPOS or VNEG
if the circuit did not react to the change in vc before then.
Consider what happens when vo is saturated at VPOS. The voltage at the noninverting input (v+) is
VPOS. The output voltage charges the capacitor through resistor R3. The capacitor voltage
becomes more positive, rising toward the value VPOS. However, the capacitor voltage never rises
that far, because the output voltage changes abruptly to VNEG when vc just goes over the value
VPOS. In response, the current through R3 changes direction, and vc drops as the capacitor
charges toward the new negative output voltage VNEG. (Voltage vc starts at the positive value
VPOS, drops through zero, and then grows more negative.) But once vc reaches the value VNEG,
the output voltage changes state again back to VPOS. The capacitor voltage rises again, becoming
less negative and then more positive, and the cycle repeats. The output voltage cycles between
VPOS and VNEG as long as power is applied to the circuit. The capacitor alternately charges,
discharges, and charges again with the time constant R3C.
Figure 2 shows a plot of the output voltage vo and the capacitor voltage vc vs. time. The figure
illustrates why the circuit is sometimes called a relaxation oscillator; the capacitor voltage tries to
“relax” to a steady-state value after every transition in the output voltage. The circuit is also
V
POS
V
POS
0
t
V
NEG
V
NEG
T/2
T
Figure 2. Output voltage vo (solid line) and capacitor voltage vc (dashed line)
produced by the relaxation oscillator. The capacitor voltage does not actually go
above VPOS or below VNEG. The dashed lines are extended to illustrate the
exponential behavior of vc.
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called an astable multivibrator because the output continuously alternates between two states,
each of which is initially stable but then becomes unstable when the capacitor voltage
approaches the reference voltage v+. The term multivibrator is just another name for an
oscillator, specifically one that generates pulses.
The period T of the square wave is controlled by the values of R3, C, and . A formula for T can
be derived from the equation that gives the voltage across the capacitor. When the output
voltage is at the value VPOS, the time-varying capacitor voltage is given by
v c (t )  VPOS  VNEG  VPOS  e  t / R3C ,
where t is measured from the instant the output voltage made its most recent transition from VNEG
to VPOS. Note that at t = 0 in this case the capacitor voltage is VNEG, the value of vc that causes
the output to switch from VNEG to VPOS. As t   , vc would reach the value VPOS; however, it
never gets there before the output changes state again. During the other half-cycle, when the
output voltage is VNEG, the capacitor voltage is
v c (t )  VNEG  ( VPOS  VNEG ) e  t / R3C ,
where again t measures the time that has passed since the most recent output transition. If we
make the assumption that VNEG = –VPOS (that is, that the power supply voltages are equal in
magnitude and opposite in sign), then the equations for vc become


when vo  VPOS


when vo  VNEG .
v c (t )  VPOS 1    1 e  t / R3C
and
vc (t )  VNEG 1    1e  t / R3C
Note from Figure 2 that the output voltage spends equal amounts of time in the positive state and
in the negative state. We can therefore use either equation for the capacitor voltage to find the
period T of the square wave. We’ll use the first one (valid when the output is positive).
The output voltage remains positive until the capacitor voltage rises to VPOS. At that instant in
time the output switches to VNEG, and the capacitor voltage begins to drop again. Thus,
vc (0.5T )  VPOS because the output voltage changes state after only half a period. Substituting
this “end condition” (as opposed to an “initial condition”) into the expression for the timevarying capacitor voltage with t = 0.5T yields


v c (0.5T )  VPOS  VPOS 1    1 e 0.5T / R3C .
In the right-hand equality the VPOS factors cancel out, leaving
  1    1 e 0.5T / R C ,
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or
1 
 e  0.5T / R3C .
1 
3
Taking the natural logarithm of both sides leads to the formula for the period of the square wave,
1  
T  2 R3C ln 
 ,
1  
which can also be written
1  
T  2 R3C ln 
 .
1  
We would have obtained the same result if we had started with the equation for the capacitor
voltage for the case when the output voltage is negative.
Quite often the resistors R1 and R2 are chosen to be equal, which sets  = 0.5 and results in a
simplified equation for the period given by
T  2.2 R3C for   0.5 ,
because
 1  0.5 
2 ln 
  2.2 .
 1  0.5 
Experimental Procedure

Assemble a relaxation oscillator like the one shown in Figure 1. Use power supply voltages
of ±15 V, and make the resistors in the positive feedback network (R1 and R2) equal in value.
Choose reasonable values for R1 and R2 based upon your experience with op-amps and the
requirement to limit the output current of the op-amp.

Add the circuit shown in Figure 3 to the output of the op-amp. The devices D1 and D2 are
called light-emitting diodes (LEDs). Like regular diodes, LEDs conduct in one direction
only. Since the LEDs shown in Figure 3 are oriented in opposite directions, one LED should
light when the output of the op-amp is positive, and the other should light when the output is
negative. The LEDs used in this lab glow well when they draw around 20 mA. Interestingly,
an LED has a steady 1-V drop across it when it conducts current. Assuming that the opamp’s output voltage saturates at around 14 V, and knowing that the drop across an LED is
1 V when it is on, find an appropriate value for the LED current-limiting resistor R4.
R4
to vo
D1
D2
Figure 3. Light-emitting diodes used to indicate output voltage polarity.
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
Using the parts available to you, select reasonable values for C and R3 to produce a square
wave with a period of approximately 0.5 sec. This period is long enough to allow you to see
the two LEDs blinking alternately when you apply power to the circuit. Assemble the
circuit, including the output indicator shown in Figure 3, and apply power to it.

Devise a method to estimate the period (or frequency) of the square wave without using the
oscilloscope or any other piece of bench-top test equipment, and use this method to verify
that your circuit is working properly. Be sure to record the details of your measurement
method in your notebook.

Increase the oscillation frequency by changing the capacitor value to 0.001 F. Calculate the
new frequency, and verify your calculation by displaying the output waveform on the
oscilloscope. Measure the length of one period of the square wave, and make sure it
corresponds to the frequency you measured.

Now display the capacitor voltage and the output voltage together using the same scale,
making sure that both waveforms have the same 0-V reference. Sketch or print out the
display, and measure the oscillation frequency again. Give a valid explanation for the change
in the operation of the circuit. (Hint: The coaxial oscilloscope leads are in parallel with the
capacitor.)

Decrease the time/division setting of the oscilloscope so that the slope of the output voltage
transition from positive to negative (or negative to positive) is exaggerated on the display.
Calculate the time rate of change of the voltage during the transition. How does this result
relate to the slew rate specification of the op-amp?
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