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Файл:Basics of electronics. Study aid
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With different resistors the inputs can have different weightings at the
R
output.
This idea can be extended to make a simple digital-to-analogue (D/A)
converter.
Once again, the negative sign preceding the formula indicates that the
output signal is opposite in polarity to input signals.
EXAMPLE
Calculate the output voltage of the summing amplifier circuit shown in
fig. 9.21 if R
Solution:
= R3 = R; R2 = R/2; R1 = R/4.
4
RRR
VVVVVVV
out
123 123
RR
R
42
42
.
There can be more than three inputs to an inverting adder.
Noninverting Adder
By feeding the output of an inverting adder into an inverter with a gain
of –1 the overall result is noninverting addition. The circuit of fig. 9.22
achieves the same result with a single op-amp.
In this circuit,
ViR
33
where
iii
312
because input bias current is
zero. Now, we get
Then
VR
Therefore
VV
1
i
1
VV V V
V
12
RR
12
VV
12
.
R
;
1
. If
3
VV
i
2
2
R
RR
123
3
R4
V
R5
-
R1
V
1
V
+
R2
V
2
R3
Fig. 9.22. Two inputs noninverting adder
.
2
then
V
out
12
2.VVV V
111
R
Now
VV
out
5
1
R
4
In general, we have
out
R
5
,
2
R
4
3
12
.
, so if
VVVV
RR RR
23 13
RRR RRR
123 213
12
RR RR
23 13
12
.
and
VVV
RR
VV V V
out
55
11
RRRRRRRR
44123213
There can be more than two inputs to a noninverting adder.
9.2.4. The Differential Amplifier
Figure 9.23 shows how the op-amp can be connected to operate as a
differential amplifier circuit. In this circuit application, the op-amp simply
uses its first internal amplifier stage, which (as mentioned earlier) is a diffamp. In this circuit, all four resistors are normally of the same value, and the
output voltage is equal to the difference between the two input voltages.
V
R1
1
R2
V
out
R1
V
2
R2
Fig. 9.23. Differential amplifier
R
In this circuit we have
VVV
out
In general, input impedances are different for the sources V
2
21
R
1
.
and V2.
1
9.2.5. The Integrator
Integrators are used in a variety of measurement and signal processing
applications. The basic op-amp integrator is shown in fig. 9.24. The
important principle of the circuit operation is , therefore the inverting
V
0
input potential equals 0 V.
112
R
C
V
out
Then
i
C
R
i
V
in
Fig. 9.24. Integrator circuit
V
in
i
R
and
ii
(because input bias current is zero)
C
11
VV idt Vdt
and we have
out C in
CRC
.
Figure 9.25 shows the integrator’s input/output waveforms, with the
peak-to-peak voltage of the output triangular wave that equals
V
2.
VV t
pp m
m
11
C
We can see that the positive input voltage gives a negative change in the
output. This is due to the use of the inverting circuit in the integrator.
V
in
V
m
t
1
t
2
0
t
V
out
V
m1
0
t
Fig. 9.25. Integrator’s input/output waveforms
113
Summing Integrator
Addition and integration can be combined by the summing integrator
of fig. 9.26.
In this circuit,
and
ii i
Therefore
i
1
12
R1
V
V
i1
1
R2
i2
2
i
Fig. 9.26. Two inputs summing integrator
V
R
1
,
1
V
2
i
2
, because inverting input voltage is zero ( 0V)
R
2
, because input bias current is zero.
11
V i dt i i dt
out
CC
12
C
V
,
out
where V
VVdtVdtV
out
is the initial value of V
0
RC RC
11
120
12
.
out
,
There can be more than two inputs to a summing integrator.
9.2.6. The Differentiator
The op-amp differentiator circuit (not to be confused with the previous
differential amplifier) is similar to the basic inverting amplifier except that R
is replaced by a capacitor, as shown in fig. 9.27(a).
The use of a capacitor in any circuit will develop problems as the
frequency of the input signal increases because the capacitive reactance is
inversely proportional to the frequency. This means that the reactance of the
input capacitor decreases for input signals that are higher in frequency.
Therefore the input voltage applied to the op-amp and output voltage from
the op-amp increases with the frequency. Including an additional resistor
(Rin) in series with the input capacitor and additional capacitor (Cf) in parallel
with the feedback resistor, as shown in fig. 9.27(b), will decrease the highfrequency gain because gain will be a ratio of R/Rin.
114
C
f
R
i
R
V
in
C
V
R
in
i
V
out
in
C
V
out
(a) (b)
Fig. 9.27. The op-amp differentiator: (a) ideal and (b) improved differentiator circuit
dV
In this circuit,
Therefore
and if RC = 1 s then V
V
VV
Cin
and
iC
dV
ViRRC
out
is the differential of the input.
out
V
in
m
dt
in
.
dt
in
,
0
t
V
out
V
m
1
t
1
t
2
0
t
Fig. 9.28. Differentiator’s input/output waveforms
Figure 9.28 shows the differentiator’s input/output waveforms, with the
peak of the output square wave being equal to
V
m
VRC
m
1
We can see that the output maximum voltage V
maximum voltage V
if pulse duration t1 equals time constant = RC. It is
m
.
t
1
is equal to the input
m1
important not to forget about the polarity inversion in this circuit.
115
9.2.7. Active Filters
F
Z
Z
Passive filters are circuits that contain passive or non-amplifying
components (resistors, capacitors, and inductors) connected in such a way that
they pass certain frequencies while rejecting others. An active filter, on the
other hand, is a circuit that uses an amplifier with passive filter elements to
provide frequency paths with rejection characteristics. Active filters, like the opamp circuits shown in fig. 9.29, have several advantages over passive filters.
1. As the op-amp provides gain, the input signal passed to the output
does not attenuate, and therefore better response curves can be obtained.
2. The high input impedance and low output impedance of the op-amp
implies that the filter circuit does not interfere with the signal source or load.
3. Since active filters provide gain, resistors can be used instead of
inductors, and therefore active filters are less expensive.
Figure 9.29 illustrates how the op-amp can be connected to form four
basic active filter types.
Active High-Pass Filter
Figure 9.29(a) illustrates the simple op-amp circuit, the frequency
response, and relevant formulas for an
this inverting amplifier is dependent on the ratio of the feedback resistor R
to the input resistor R
. When capacitors are included in any circuit, the
in
active high-pass filter. The gain of
F
impedance (Z) must be considered instead of simply resistance, and the gain
is now equal to the ratio of the feedback impedance to the input impedance.
G
.
in
The input RC network offers a high impedance to low frequencies,
resulting in a low voltage gain. At high frequencies, the RC network has a
low impedance, causing a high voltage gain. The cutoff frequency for this
circuit can be calculated with the following formula when С
= C2.
1
1
f
c
2
.
RC
Active Low-Pass Filter
Figure 9.29(b) illustrates the op-amp circuit, frequency response curve,
and relevant formulas for an
active low-pass fitter. At low frequencies, the
capacitor reactance is high, and low-frequency signals will pass to the opamp input to be amplified and passed to the output. As the frequency
increases, the capacitive reactance of C
decreases; the greater part of the
1
signal will be shunted away from the op-amp and will not appear at the
116
output. The cutoff frequency for this circuit can be calculated with the
following formula when R
= R2.
1
R
2
V
C
in
C
1
R
2
1
V
out
f
c
2
V
.
RC
R
3
in
R
1
C
R
2
1
V
out
(a) (b)
1
R
C
2
R
3
C
1
1
R
C
1
1
R
C
2
4
V
in
R
2
V
out
V
R
in
R
2
3
V
out
(c) (d)
Fig. 9.29. Active filter circuits: (a) high-pass filter; (b) low-pass filter;
(c) bandpass filter; (d) band-stop filter
Active Bandpass Filter
Figure 9.29(c) illustrates how the op-amp can be connected in order to
form an active bandpass filter. At frequencies outside the band, V
is fed
out
back to the input and it is not attenuated. Therefore the input signal amplitude
is almost equal to the feedback signal amplitude. This results in almost
complete cancellation of the signal and therefore a very small output voltage.
On the other hand, for the frequencies within the band, a very small feedback
signal will appear at the negative input of the op-amp and will have a very
small degenerative effect. As a result, the change at the input of the op-amp
will be larger when the input signal frequencies are within this band, and the
output voltage will be also larger.
117
Active Band-Stop Filter
Figure 9.29(d) illustrates how the op-amp can be connected to form an
active band-stop filter, also known as a band-reject or notch filter. The basic
operation of this circuit is opposite to that of the previously discussed
bandpass filter. At frequencies outside the band, the feedback signal will be
heavily attenuated, and therefore the degenerative effect will be small and the
output voltage large. On the other hand, at frequencies within the band, the
feedback signal will not be greatly attenuated, and therefore the degenerative
effect will be large and the output voltage small.
9.3. Signal Generators
A signal generator is a circuit (or device) which produces simple
repetitive waveforms. Such devices contain an electronic oscillator, a circuit
that is capable of creating a repetitive waveform. Modern devices such as
digital-to-analogue converter may use digital signal processing to synthesize
waveforms. The most common waveform is a sine wave, but sawtooth, step
(pulse), square, and triangular waveform oscillators are commonly available
as arbitrary waveform generators (AWGs). If the oscillator operates above
the audio frequency range (>20 kHz), the generator will often include some
sort of modulation function such as amplitude modulation (AM), frequency
modulation (FM), or phase modulation (PM).
Signal (or Function) generators are typically used in simple electronics
repair and design; where they are used to test a circuit. A device such as an
oscilloscope is then used to measure the circuit’s output. Function generators
vary in the number of outputs, frequency range, frequency accuracy and
stability, and several other parameters.
9.3.1. Sinusoidal Oscillator
A more common oscillator phase shift network is shown in fig. 9.30.
R1
C1
R2
C2
V1
Fig. 9.30. Wien network
118
V2
f
R
R
R
V
V
0
2
2
RC
12
1
RC
21
, then
.
RCC
1212
1
.
1
1
0
2fRC
and
V
V
1
2
1
.
3
For this network,
we have
At f
0
CCC
If
12
and
RR
12
Op-Amp Wien Bridge Oscillator
1
One way of obtaining the gain required for a Wien-bridge oscillator is to
use the op-amp circuit as shown in fig. 9.31.
In this case we have the simplified Wien network with equal values of
resistance (R) and capacitance (C) so to compensate for the loss of the phase
shift network the amplifier must have a noninverting gain of 3. Now the
amplifier input will be the Wien network output (V
) which is connected to
2
the noninverting input of the op-amp. To provide a gain of 3
R
2
13.
R
1
2
When
R
21
, the amplifier gain is 3 and oscillation occurs at
1
frequency
0
2fRC
.
R1 R2
V
out
C
R RC
Fig. 9.31. Op-amp Wien-bridge oscillator
If the gain is less than 3, the circuit does not oscillate. If it is greater than
3 then the amplitude of the oscillations builds up until nonlinearity reduces
the gain, but this causes flattening of waveform peaks (fig. 9.32). In other
words, the waveform is sinusoidal, i. e. harmonic distortion is added.
119
Sine Wave
V
out
V
V
sat
0
t
Fig. 9.32. Output Signal Distortions
The circuit shown in fig. 9.31 oscillated at f0, however the distortion is
noticeable. The solution is to introduce a nonlinear component into the
negative feedback network that determines the gain. A suitable nonlinear
component for this purpose is a thermistor or a lamp.
9.3.2. Rectangular Waveform Generators
For the rectangular waveform generator, op-amps will have significant
differential input voltages so that their outputs will be at one of the supply
rail voltages.
Transistors of op-amp output stage are operated at cutoff or in
saturation.
Timings are determined by charge and discharge of capacitors in RC
circuits.
In some cases discrete components have advantages over ICs.
Rectangular Waveform Definitions
A rectangular waveform is one which switches between two voltage
levels V
and V2 (fig. 9.33).
1
The voltage levels may be of the same or opposite polarity; in many
cases one of them is 0 V. For many purposes it is necessary for the transition
time from V
to V2 and the return from V2 to V1 to be as short as possible.
1
V
out
V
2
t
1
t
2
0
V
1
t
T
Fig. 9.33. Rectangular waveform
The rise time of a pulse is defined as the time it takes the leading edge
to rise from 0.1 to 0.9 of its amplitude, as illustrated in fig. 9.34. The
time
is defined as the time it takes the leading edge to fall from 0.9 to 0.1 of
its amplitude.
120
fall
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