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Positive Feedback and Oscillators Physics 3330
Experiment #5
Fall 2005
Experiment #5
5.1
Fall 2005
Positive Feedback and Oscillators


Purpose

In this experiment we will study how spontaneous oscillations may be caused by positive
feedback. You will construct an active LC filter, add positive feedback to make it oscillate, and
then remove the negative feedback to make a Schmitt trigger.

Introduction
By far the most common problem with op-amp circuits, and amplifiers in general, is unwanted
spontaneous oscillations caused by positive feedback. Just as negative feedback reduces the gain
of an amplifier, positive feedback can increase the gain, even to the point where the amplifier
may produce an output with no input. Unwanted positive feedback is usually due to stray
capacitive or inductive couplings, couplings through power supply lines, or poor feedback loop
design. An understanding of the causes of spontaneous oscillation is essential for debugging
circuits.

On the other hand, positive feedback has its uses. Essentially all signal sources contain
oscillators that use positive feedback. Examples include the quartz crystal oscillators used in
computers, wrist watches, and electronic keyboards, traditional LC oscillator circuits like the
Colpitts oscillator and the Wien bridge, and lasers. Positive feedback is also useful in trigger and
logic circuits that must determine when a signal has crossed a threshold, even in the presence of
noise.

In this experiment we will try to understand quantitatively how positive feedback can cause
oscillations in an active LC filter, and how much feedback is necessary before spontaneous
oscillations occur. We will also construct a Schmitt trigger to see how positive feedback can be
used to detect thresholds.

Readings
The readings in H&H and Bugg are optional for this experiment.
1.
Horowitz and Hill, Section 5.12 to 5.19. If you are designing a circuit and want to
include an oscillator, look here for advice. Amplifier stability is discussed in Sections
4.33-4.34.
2.
Bugg discusses the theory of spontaneous oscillations in Chapter 19. You may want to
read Section 19.4 on the Nyquist diagram after you read the theory section below. Experiment #5
5.2
Fall 2005
Theory

LC ACTIVE BANDPASS FILTER
The circuit for the active LC filter is shown
in Figures 5.1 and 5.2. Recall from the
theory section of Experiment #4 that the gain
of an inverting amplifier is G = R
F
/R when
the open loop gain is large. The basic idea of
this filter is to replace R
F
with a resonant
circuit whose impedance becomes very large
at the resonant frequency. Then there will be
a sharp peak in the gain at the resonant
frequency. If we replace R
F
with the
impedance Z
F
shown in Fig. 5.2 and do a fair bit of
algebra, we can show that
G( )
= Z
F
R =
Z
0
R
i
0
+ 1<i>Q
2 02
+ i
0
Q + 1
,

where we have defined the resonant frequency 0
,
the characteristic impedance Z
0
, and the Q: 0
= 1<i>LC, Z
0
= L
C, Q=
Z
0
r .

With quite a bit more work you can show that the peak in the magnitude of G occurs at the
frequency peak
= 0
1
+ 2
Q
2
1
Q
2
,
which is very close to 0
when Q is large. The gain at the peak is
G( peak
)
= Q Z
0
R .
This last formula is only approximate, but corrections to the magnitude are smaller by a factor of
1/Q
2
and thus not usually important. There is quick way to get this final formula if you already
know that a parallel tank circuit has an impedance of Z
F
( 0
) = Q
2
·r at resonance, and that peak
is very close to
0
. Then we get G( 0
) = Z
F
( 0
)/R = Q
2
·r/R, which is the same result.


Figure 5.1 Active Bandpass Filter
+ Vin
Vout
ZF
R
A

C
L
Figure 5.2 Parallel Resonant Circuit
r
Z F Experiment #5
5.3
Fall 2005

GAIN EQUATION WITH POSITIVE FEEDBACK
To turn our band-pass filter into an
oscillator, we add positive feedback as
shown in Fig. 5.3. This circuit now has
both positive and negative feedback paths,
so we need a more general gain equation
than we have derived before. The two
divider ratios are defined as before
B
+
=
R
2
R
1
+ R
2
, B =
R
R
+ Z
F
.
The voltage V
out
at the output is related to
the voltages V
+
and V at the op-amp
inputs by
V
out
= A V
+
V (
)
.

These voltages are themselves determined by the divider networks
V = V
in
+ B V
out
V
in
(
)
,
V
+
= B
+
V
out
.

Combining all three relations yields the gain equation

G
= V
out
V
in
= A 1 B (
)
1
A B
+
B (
)
.

In the absence of positive feedback (B
+
= 0, pot wiper at bottom) this reduces to the gain
equation for the inverting amplifier discussed in Experiment #4.

OSCILLATION THRESHOLD
If we increase B
+
by moving the pot wiper up, the denominator in the gain equation will
decrease and the gain will increase, resulting in a sharper and sharper peak in the filter response
near 0
. The system will oscillate when the gain becomes infinite, since infinite gain implies
that there is an output even with zero input. The gain will be infinite when the denominator in
the gain equation is zero. If the value of the loop gain A is very large at the resonant frequency
then, to a good approximation, the denominator will be zero when B+ = B, or when


Figure 5.3 Positive Feedback LC Oscillator
+ Vin
Vout
ZF
R
A
R 1
R 2
10 k 10-turn
F
Z
R
R
R
R
R
+
=
+
2
1
2 Experiment #5
5.4
Fall 2005

If the oscillation occurs near 0
we can replace Z
F
with Z
F
(
0
) = Q
2
·r = Z
02
/r. With this
substitution we find that the oscillation threshold occurs at
.
2
0
2
1
2
Z
rR
rR
R
R
R
B
threshold
threshold
+
=
+
=
+


This derivation is correct, but it leaves several questions unanswered. What does it really mean
to have infinite gain? What does the system do if we increase B+ past the threshold for
oscillations? How do we analyze a system for stability when we dont know what frequency it
will oscillate at? See the Appendix to this experiment for a more complete treatment of stability
that addresses these questions. Experiment #5
5.5
Fall 2005
Problems
1.
Design an active bandpass filter with a resonant frequency of 16 kHz, a Q of 10, and a
closed loop gain of one at the peak of the resonance. Choose suitable component values
for the parallel LC circuit shown in Figures 5.1 and 5.2, using the inductor that you made
in Experiment #3. Use the value of the inductance that you measured earlier. The series
resistor shown in Figure 5.2 will have two contributions, one from the losses in your
inductor, and one from an actual resistor that you must choose to get the correct Q. If
you do not know what the loss of your inductor is, assume it is zero for now.
2.
To make an LC oscillator you will add positive feedback as shown in Figure 5.3. Predict
the value of the divider ratio B
+
where spontaneous oscillations will just begin. B
+
is
defined as
B
+
=
R
2
R
1
+ R
2
.
Also predict the oscillation
frequency.

3.
The circuit shown in Figure 5.4 is
called a Schmitt trigger. It has only
positive feedback. To figure out what it
does, suppose a 1 kHz, 2 V p-p sine
wave is connected to V
in
, and try to
draw the waveforms for V
in
, V+, and
V
out
, all on the same time axis.
Suppose that the op-amp saturation levels are ± 13 V, and that the divider is set so V
+
=
0.5 V when V
out
= + 13 V. (You should really use a comparator chip rather than an op-
amp for circuits like this. See H&H Section 4.23.)

Figure 5.4 Schmitt Trigger
+ V
in
V
out
A
R
1
R
2
10 k 10-turn
V
+ Experiment #5
5.6
Fall 2005
Experiment

1.
Build and test an active LC bandpass filter following the design worked out in Problem 1
above. Measure the resonant frequency, closed loop gain on resonance, and Q. If the
circuit does not perform as expected, figure out why and adjust the circuit until you meet
the design goals. Use your final measurements to refine your values for the circuit
parameters. Make sure you correct your value for r so that it includes the inductor losses
and corresponds to the observed Q.


2.
Convert the bandpass filter into an oscillator by adding positive feedback, as described in
the theory section. Observe the output as a function of input for various values of the
positive feedback divider ratio B
+
. Then ground the input (V
in
= 0) and measure the
threshold value of the divider ratio at which spontaneous oscillation begins. Measure the
oscillation frequency near threshold. Compare your observations with theory.


3.
Remove the negative feedback from your circuit to create a Schmitt trigger. Compare the
observed waveforms with those predicted in Problem 3.

Appendix: St