Download Closed Loop Gain of an Operational Amplifier

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Closed Loop Gain of an Operational Amplifier
Nisha Lad
Abstract
Operational Amplifiers (Op-Amp) are incredibly useful integrated circuits which are used for signal processing.
The gain in amplification of a signal by an Op-Amp is highly important in electromagnetics, such as in loudspeakers and antennas. An experiment was conducted to investigate the effects of how the closed loop gain of
an Op-Amp circuit varies with frequency for different feedback values. It was found that at high values of
feedback, the bandwidth was larger compared with a low feedback value where the equivalent open loop gain
was obtained.
I.
Introduction
Electronic circuits can be classified into two types:
analogue and digital. Digital circuits are used in
computers where two signal levels are needed,
whereas in analogue circuits, such as the Op-Amp,
electrical signals can vary continuously.
IA.
Where the closed loop differential input signal is [V in
- Vout] [1]. In this case the desirable effect is
achieved, as G(f) decreases with increasing frequency
which in turn increases the differential input signal
[Vin - Vout] such that Vout, and hence the gain remain
constant over a large range of frequencies, known as
the bandwidth.
Theory
The purpose of an Op-Amp is to amplify a weak
signal resulting from the difference in voltage
between two inputs, known as the non-inverting (Vn)
and inverting (Vi) inputs, giving rise to its name
‘differential amplifier’. The Op-Amp requires power
to amplify the small input-signal (Vin) and hence the
maximum possible output-signal (Vout) obtained, as a
function of frequency, is described by Equation 1.
Where G(f) is the equivalent ‘open-loop’ gain when
there is no feedback and [Vn-Vi] is the differential
voltage [1].
In practice the ‘open-loop’ mode is not used due to
the fact excessively high gains are obtained at low
frequencies across a short bandwidth. The desirable
effect is generally a smaller gain that is constant over
a wide range of frequencies. This is achieved by
‘feeding back’ a fraction of Vout into the inverting
input of the Op-Amp, shown in Figure 1, so that its
input depends upon its output and thus results in the
formation of a closed loop circuit. The ratio between
the input potential and output potential is described
by equation 2 and equation 3 describes the
relationship between Vout and the gain if a fraction of
Vout, , is fed into the inverting input Vi, shown in
Figure 1.
Fig. 1 Circuit diagram of the non-inverting amplifier
(triangle) used in the investigation. The circuit consists of a
potential divider resistor network used to feedback a
fraction, , of Vout into the inverting input (Vi) resulting in
differential amplification. [1]
Manipulating (3) gives the closed loop gain described
by equation 4. Consequently, at lower frequencies
where G(f) is much greater than unity, the gain is
approximated
by
equation
5.
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Where Glf indicates the low frequency gain of the
closed loop circuit and is independent of frequency
[1]. The closed loop gain, Glf, will remain constant
with increasing frequency so long as G(f)>>Glf. This
relationship breaks down when the open loop gain,
G(f), reduces until G(f)Glf. Through manipulation of
the potential divider relation given by equation 6, the
low frequency gain is therefore determined via
equation 7.
II.
Prior to starting the experiment, calibration of the
Cathode Ray Oscilloscope (CRO) was performed for
both Channels 1 (input) and 2 (output) in order to
ensure inaccuracy in readings, and hence any
systematic
errors
resulting
from
skewed
measurements, were not introduced. It was ensured
that the variation knobs were not rotated after
calibration, as this would also introduce systematic
errors in measurements, and that the oscilloscope
inputs were set to measure DC signals to compensate
for any DC offset.
Experimental Method
An experiment was conducted to investigate how the
closed loop gain of the Op-Amp circuit varies with
frequency for the low frequency gain, Glf, values of
≃10, 100 and 1000. This was achieved by changing
the configuration of resistors R1 and R2 (shown in
Figure 2) and using (7), hence the fraction of Vout, ,
which was fed back into the inverting input (Vi) was
varied, whilst a small oscillating signal was fed into
the non-inverting input (Vn). The Vout and Vin were
then measured over a large range of frequencies and
the corresponding gains were calculated. The circuit
used in this investigation is shown in Figure 2 and the
Op-Amp in Figure 3.
Fig. 3 Schematic of the 8-pin 741 Op Amp integrated
circuit as an electrical component used in the investigation.
The triangle represents the amplifier. Pins 4 and 7
correspond to the connection of the power supply, 2 and 3
correspond to the feedback resistor network and 6 to the
oscilloscope [2].
Initially, the circuit was connected to set up a
feedback loop as shown in Figure 2, with a resistor
configuration of R1=1MΩ and R2=10kΩ, resulting in
Glf ≃ 100 by using (7). The oscilloscope was adjusted
to read an appropriate sine wave signal for both Vin
and Vout. The frequency was varied from the signal
generator from 10Hz to 1MHz and measurements
were taken from channels 1 and 2 to obtain Vin and
Vout respectively. Subsequently, the gains were
calculated using (2), thus a logarithmic graph was
plotted of log(gain) as a function of log(f(Hz)), as
shown in Figure 4.
In order to reduce the proportion of random
uncertainties, signals displayed on the oscilloscope
were adjusted to be as sharp and as faint as possible.
Furthermore, channels 1 and 2 were both triggered in
order to ensure repeated signals displayed were in
phase for ease of measurement. This procedure was
repeated for resistor configurations of R1=1MΩ and
R2=1kΩ, resulting in a Glf ≃ 1000 and finally
Fig. 2 Schematic diagram of the 741 Operational Amplifier
circuit board and resistor network connected to a
synthesised function generator (ISO-Tech 1Hz-1MHz) and
oscilloscope (ISO-Tech ISR622 20MHz) used in the
investigation. Yellow arrows represent an example
configuration of resistors R1 and R2 via yellow leads to
allow a fraction  to be fed back [2].
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R1=100kΩ and R2=10kΩ, resulting in a Glf ≃ 10.
This enabled a different fraction of Vout, , to be fed
back and the resulting gains were calculated.
III.
remained constant over a much greater bandwidth, as
seen in Figure 4.
The input voltage Vin remained constant at (10.0±0.5)
mV over the three resistor configurations, as
expected.
Results and Analysis
It was found that for a larger feedback, and hence a
smaller Glf value, the larger the bandwidth over which
the gain remained constant. After a certain threshold
value, which depended on the value of feedback, the
gain decreased for higher frequencies. This is shown
in Figure 4. Additionally, it was also found that the
results obtained coincided with the theory, in which
the smaller feedback value, and hence larger the Glf,
very high gains were obtained which remained
constant over a short bandwidth compared to larger
bandwidths for lower Glf’s.
For example, for Glf ≃ 1000, the equivalent open-loop
gain was obtained in which a large gain remained
constant over a short bandwidth from 10-500Hz at a
value of 950.00±68.97, which then decreased with
increasing frequency, shown in Table 1. The
uncertainty in the synthesized function generator was
40 parts per million [3], hence the uncertainties in
f(Hz) were negligible compared with the values of
f(Hz), for this reason they are not shown in Tables 1,2
and 3.
f(Hz)
Vout (±0.05V)
Gain
10
1.00
100.00±7.07
5000
1.00
100.00±7.07
7500
0.84
84.00±6.53
Table. 2 Results obtained from a feedback of Glf ≃ 100.
The gain obtained remained constant from 10-5000Hz and
then began to decrease with increasing frequency up to
1MHz [2].
f(Hz)
Vout (±0.01V)
Gain
10
0.11
11.00±0.74
75000
0.11
11.00±0.74
100000
0.10
10.00±0.51
Table. 3 Results obtained from a feedback of Glf ≃ 10. The
gain obtained remained constant from 10Hz to 100kHz and
then began to decrease with increasing frequency up to
1MHz [2].
f(Hz)
Vout (±0.50V)
Gain
10
9.50
950.00±68.90
500
9.50
950.00±68.90
1000
5.00
500.00±55.90
Table. 1 Results obtained from a feedback of Glf ≃ 1000,
where distortion in signals became prominent; hence the
uncertainties were larger than propagated values. The gain
obtained remained constant from 10-500Hz and then began
to decrease with increasing frequency up to 1MHz [2].
Fig. 4 Plot of log(gain) as a function of log(Frequency(Hz))
from the data obtained for low frequency gain values of G lf
≃10, 100 and 1000. [2].
The results acquired from Glf ≃ 1000 were similar for
the resistor configuration for Glf ≃ 100, where the
gain obtained remained constant over a larger
bandwidth of 10-5000Hz at a value of 100.00±7.07. It
then decreased with increasing frequency shown in
Table 2. A similar behaviour was also obtained for Glf
≃ 10, where the gain remained constant at a much
lower value of 11.00±0.74, although the bandwidth
was much greater for frequencies of 10Hz-100kHz.
This data obtained is shown in Table 3. Therefore, it
was observed that for each decreasing Glf, the gain
obtained was much smaller than the previous and
From Figure 4, it was found that the rate at which the
gain decreased were not the same for all Glf values.
This did not support the theory given by (2), and may
be a result of signals being susceptible to noise at
much greater frequencies (≃ 105Hz) as the Op-Amp
becomes unstable. Consequently, uncertainties were
adjusted to be greater than those propagated at higher
frequencies to compensate for signals becoming more
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distorted, due to the fact difficulty in taking precise
measurements became apparent and obtaining a
reliable noise-free output was harder than predicted.
Subsequently, suitable improvements include taking
repeat readings, which would improve the reliability
of the investigation and reduce the proportion of
random errors in results. Additionally, taking more
measurements at smaller intervals of frequency would
make the curves more defined, particularly about the
turning point. This would also increase the accuracy
of the investigation and make anomalous data points
easily detectable.
In order to reduce the amount of noise obtained,
hence acquire less distortion and precise
measurements at greater frequencies, a power supply
with a greater voltage could be used as this would
produce higher amplitude signals.
IV.
V.
References
[1] Dr P. Bartlett, “Experiment NX 741, Operational
Amplifiers”, 13/01/2014, (UCL Department of
Physics and Astronomy, Teaching Laboratory 1)
[2] Nisha Lad, Lab Book 2, Experiment 1,
“Operational Amplifiers” (UCL Department of
Physics and Astronomy, Teaching Laboratory 1)
[3] “Synthesized Function Generator User Manual,
82RS-21200M01”, pp 11, (UCL Department of
Physics and Astronomy, Teaching Laboratory 1)
Conclusions
From the results obtained for each Glf, the gain, G,
was found to be constant over a certain range of
frequencies, known as the bandwidth, which then
steadily decreased for increasing frequencies and
eventually reaching zero. According to Figure 4, the
bandwidth is shorter for a higher Glf and is larger for a
low Glf.
This is in agreement with the theory due to the fact
that the amplitude gain would be at its maximum so
long as G(f) >> Glf where G(f) is the equivalent openloop gain when is no feedback. Hence, when Glf is
very large (1000), there is a small bandwidth over
which G(f) remains constant and conversely if Glf is
small (10), there is a much greater bandwidth at
which G(f) remains constant.
The results obtained are in accordance with what was
expected, excluding the noise, and were comparable
with colleagues’ results.
Possible further work includes investigating other
resistor configurations, in determining whether they
follow the behaviour as obtained in Figure 4.
Additionally, the Op-Amp can be investigated further
with other electrical components, such as how it can
be used to amplify the potential difference between
two LDRs, which in turn allows current to flow to a
motor.
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