Recent Advances in Circuits, Systems, Signal Processing and Communications
Operational amplifiers for measurement purposes: non-inverting
amplifier integral and derivative
JAROSLAV LOKVENC, RENE DRTINA, JOSEF SEDIVY
Department of Technical subjects and Department of Informatics
Faculty of Education and Faculty of Science, University of Hradec Kralove
Rokitanskeho 62, 500 03 Hradec Kralove
CZECH REPUBLIC
jaroslav.lokvenc@uhk.cz, rene.drtina@uhk.cz, josef.sedivy@uhk.cz, http://ktp.katedry.cz
Abstract: This paper presents new possibilities for involvement frequency dependent circuits with operational
amplifiers that are designed for use in measurement, low frequency and DC applications. Provide basic
background, principles and equations for these circuits. This involvement should be used everywhere where it
is necessary to process and edit analog electrical signal. The article presents and explains the involvement of
various operational amplifiers explained the uses of operational amplifiers, allowing for the integral and
derivative of the input voltage. Input voltage can be used and an optional large range of frequencies. All is used
with single selectable passive impedance. The aim is to calculate and suggest such involvement, which processed do not invert phase voltage.
Key-Words: operational amplifier, differential amplifier, amplifier integration, derivative amplifier, frequency
dependence, measurement technique
1 Introduction
Capture and processing of analog signals whose
level is a function of frequency, usually requires the
use of frequency-dependent circuits. They often use
active 4-poles that implement the integration or
derivation of the processed signal. Steady and largely standardized operational amplifiers involved in
the function of integration and derivation amplifiers
are described in numerous publications [1]. Figure 1
shows a typical diagram of the circuit realizing the
function.
U 2 = ∫ U 1 dt
Fig.2 Scheme of typical derivative amplifier
Both circuits use a combination of resistor, capacitor
and inverting operational amplifier circuit. This
does not mean that these circuits can´t be resolved
any other way, in the simplest schematic solutions
with minimal number of components used. Research
work in the field of measurement technology, new
materials and power electronics require new approaches in the design of analog sensors and analog
signal processing. Department of Electrical Workers
laboratory technical subjects Faculty of Education,
University of Hradec Králové for many years engaged in the development of innovative circuit with
operational amplifiers and their applications in sensing amplifiers for high-voltage measurements in
the low-frequency technology, management and
control circuits. Despite the ongoing digitization is
for basic analog signal processing circuits
irreplaceable.
Fig.1 Scheme of typical integration amplifier
Next figure 2 shows typical circuit diagram then,
realizing the function:
U2 =
ISBN: 978-960-474-359-9
dU 1
dt
157
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In the literature [2] is for with one-input inverting
operational amplifier with a resistor R0 derived for
inverting transfer equation (1) that the condition
R1 = R2 = R leads to a new shape inverting transfer
Ain.
2 Non-inverting amplifier with an
impedance of integration in the virtual
zero
Involvement of operational amplifiers with unbalanced differential input and a resistor in a virtual zero,
as in [2] can be modified to the next version, which
allows an easy way to obtain the integral in the large
input voltage range of selectable frequencies using
single selectable passive impedance. It retains the
advantages of simple universal circumferential
arrangement of other elements of the amplifier. In
the original diagram (Fig. 3) instead of resistor R0
locally series impedance circuit used in combination
with RC or RL type.
Ain = −
(1)
1
1
⋅
R0
2 1
+
A0 2 ⋅ R0 + R
(2)
When connected in parallel both inputs are then the
sum of transmission Ain and An. Assuming that the
voltage gain operational amplifier load A0 is greater
than 106 and the amplification circuit is chosen in
the range of 10-3 to 103, can be members of the equations (1) and (2), includes the A0 put equal to zero
and adding the following simplified receive transmissions resulting Ac shaped transmission
Fig.3 Diagram unbalanced differential amplifiers
with input [1]
The involvement of the amplifier (Fig. 4) is
designed to integrate sinusoidal frequencies from
several Hz and above depending on the type of
operational amplifier can be operational frequencies
up to tens of MHz. The circuit can be, for example
in conjunction with a suitable current sensor type
dI/dt is used for measuring alternating currents in a
large current transmitters and frequency range,
practically the entire current range radio frequency
of 100 kHz to 100 MHz.
Ac =
R
2 ⋅ R0
(3)
From equation (3) shows that a single resistor R0 is
easy to change the overall transmission amplifiers, if
we choose appropriately the same size of the other
resistors in the circuit. Adhering to itself instead of
resistor R0 serial combination of R0, L (while the coil
resistance can easily be considered as part of the
total resistance R0), then equation (3) is modified in
the shape:
Ac =
Fig.4 Diagram of involvement of amplifier with
impedance in virtual zero
R

jωL 

2 R0 ⋅ 1 +

R
0


(4)
Lower limit frequency f0 of the integration amplifier
f0 =
ISBN: 978-960-474-359-9
R0
2
+1+
2 ⋅ R0 + R
A0
The differential operational amplifier can be used
for its non-inverting input, which is not included in
the resistor R0, derived in the same condition R1 = R2
= R, the transfer An non-inverting input of this, the
pattern shape
An =
3 Analysis of the
amplifier integration
1
transmission
158
R0
2π ⋅ L
(5)
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Since approximately ten times the lower limit
frequency f0 is a circuit according to Figure 4
integrator with transfer
Ac =
1
2L
jω
R
(6)
On the contrary, reduce the spread of the integrator
for frequencies f < f0 at a constant value given by
equation (3), for the stability of participation desirable and necessary.
4 An example of the integration
amplifier and connection properties
After the theoretical solution was the involvement
of the integration amplifier with an impedance of
the virtual zero (as shown in Figure 5) Multisim
simulation program was tested with an ideal operational amplifier for band integration from 100 kHz
to 100 MHz (with an operational amplifier LM 107
operates to 10 MHz). Where used for the simulation
program parameters defined circuits. Following
parameters were verified on a real connection
diagram. For the operational amplifier μA725 and
the value of R0 = 220 Ω and L = 15.7 H is the
integration of the appropriate band from 50 Hz to 10
kHz.
DC level shift output of non-ideal operational amplifier (μA725) by about 5 mV arises only for a maximum transfer order of 100 for low frequencies f < f0
outside the area of integration and can eliminate the
external offset compensation circuit, for example in
[3]. It does not matter if the intended use of additional small circuit phase shifts obtained integral stress
and emphasis is placed on the correct amplitude, can
be used to separate the interfering DC component
obtained capacitor voltage.
Fig.5 Specifications of operational amplifier μA725
Verification measurements showed real involvement
with the concurrence of the calculated and simulated
values in the tolerance to 5% for operational
amplifier MAA725 (μA725) using standard-based
components
5 Non-inverting amplifier with a
shunt impedance of the virtual zero
Involvement of operational amplifiers with unbalanced differential input and a resistor in a virtual zero,
as in [1] can be modified into the next version,
which allows an easy way to obtain the derivative in
a large input voltage range of selectable frequencies
using single selectable passive impedance. Involvement retains advantages of simple peripheral universal amplifier arrangement of other elements, but
instead of resistor R0 is the involvement of locally
applied impedance type C (capacitor). Similar scheme was published in [2] as a non-inverting differentiator, especially for slowly varying DC voltage. For
some types of operational amplifiers, but it was
prone to oscillation. Therefore, there is the involvement of elected such treatment, which specifically
limits the upper frequency limit differentiator, eliminating the above-mentioned instability[3].
Connecting the amplifier (Fig. 7) is designed for
sinusoidal frequency derivative from a few Hz
amount and type of operating amplifier may be
functional up to frequencies of tens of kHz. It is
involved in [4]. It expanded further in the second
impedance inverting input of operational amplifier.
Fig.6 Simulation integration scheme non-inverting
amplifier with impedance in the virtual zero
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Ain = −1
(10)
To transfer from the input terminal connection to the
non-inverting input of op amp transfer are loaded Ad
shaped divider:
Ad =
Fig.7 Diagram of derivative amplifier with
impedance in the virtual zero
6 Analysis of the
amplifier derivation
A0 z + =
Involvement in figure 7 can be used in a simpler
case without impedance Z2. You can then transfer to
calculate the total involvement of the Ac operational
amplifier used equation (3)
(3)
Ani =
R
2Z 1
(7)
R
C1
2
(8)
Ac =
Thus executed differentiator only works well with
an ideal operational amplifier. With a true operational amplifier, which has effectively limited the
upper frequency band limit frequency derivative,
but it is usually prone to instability in the form of
transient or permanent parasitic oscillations [5]. To
derive the overall transfer Ac full involvement
applies to the first operational amplifier transfer Ain
conducted from the input terminals of the inverting
input connection (equation (2) in [3])
Ain = −
1
1
1 R1 R1
1 R1
+
⋅
+
+
⋅
A0 A0 R2 R2 A0 R0
(12)
1
+
A0 (Z 1 R ) + R
Z 2 (2 Z 1 + R )
(13)
Z 1 (2 Z 2 + R )
Z 2 (2 Z 1 + R )
Z 1 (2 Z 2 + R )
−1
(14)
Which can be adjusted to obtain the

Z 
R 1 − 1 
Z2 

Ac =
Z
2Z1 + R 1
Z2
(15)
If you put in this equation Z2 → ∞ (impedance not
use), it goes equations (15) to form Ac = R/2Z1,
which is identical with equation (7). To the extent
that is in place Z1 and Z2 capacity reactance (C1 ≥
100·C2) when
(9)
When equal R1 = R2 = R and gain operational amplifier load A0 = 106 and order more leads equation (9)
the outcome:
ISBN: 978-960-474-359-9
1
Z1 R
If both inputs of operational amplifier connected
across the circuit elements on a common input
terminal, the added equation (10) and (13) in the
resulting transfer of Ac
We choose the impedance Z1 reactance capacitor C1,
and then we get
Ac = jω
(11)
Here again we neglect 1/A0 member. Total noninverting transmission Ani is then obtained as the
product of equations (11) and (12), which leads after
adjustment for the final shape
After substitution we obtain R0 = Z1 equation
Ac =
(Z 2 R ) + R
The symbol ║ is parallel combination impedance,
and transmission of non-inverting A0z+ input
operational amplifier at the output in the form
transmission
R
Ac =
2 ⋅ R0
Z2 R
Z1 =
160
1
jωC1
(16)
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Z2 =
1
jωC 2
(17)
Then after substituting (16) and (17) into (15) and
get after adjusting for the Ac transmission amplifier
derivative equation
R
(C1 − C 2 )
2
Ac =
R
1 + jω C 2
2
jω
(18)
Fig.8 Simulation diagram derivative amplifier with
an ideal operational amplifier
Transfer Ac has the value 1 at the frequency f0
f0 =
1
πRC1
(19)
And to the frequency fh
fh =
1
πRC 2
(20)
A circuit derives, but it is advisable not to elect the
highest derived frequencies higher than 0,1·fh, to
avoid creating too large a negative phase error from
the derived phase +90° tension. It is also desirable to
choose the value of fh (20) at least 10 times lower
than the marginal operating frequency operational
amplifier that occurred phase conditions suitable for
the creation of own oscillations. In connection with
the real circuit operational amplifier (Fig. 9) also
reduced the differential input resistance operational
amplifier parallel resistor Rd connected between the
inputs of operational amplifier. This is achieved
even higher resistance against oscillations involving
mutual negative feedback inputs [6].
Fig.9 Simulation diagram derivative amplifier with
real operational amplifier LM 741
Verification measurements on real involvement with
MAA741 operational amplifier (LM741) showed
concurrence with the calculated and simulated
values within 10% for the frequency band from 40
to 120 Hz when using standard-based components.
Using fast operational amplifiers required to ensure
stability
according
to
the
manufacturer's
instructions. For use in measuring circuits for
electricity, we have successfully used operationally
reliable precision bipolar instrumentation amplifier
Operational MAA725 (μAA725) with accurate
compensation,
according
to
manufacturer's
recommendations.
7 Example of a derivation and
properties of the amplifier connection
After the theoretical solution was the involvement
of a derivation impedance amplifier with a virtual
zero (as shown in Figure 7) simulation program was
tested with an ideal operational amplifier for derivations range 1 Hz to 1 kHz, then subsequently with
the LM741 operational amplifier with internal. The
operational amplifier type LM741 and the values of
C1 = 100 nF, C2 = 1nF (Fig. 9) is appropriate band
1 Hz to 100 Hz integration, while reducing the input
of the differential resistance of the resistor Rd as an
effective measure against parasitic resonances).
ISBN: 978-960-474-359-9
8 Conclusion
Proposed involvement of operational amplifiers has
been developed especially for the measurement
technology and signal processing for current and
voltage sensors in the electricity and industrial
drives. Primarily, therefore, expected operating
frequency of 50/60 Hz with the functionality of up
to 5 kHz used in life-drives. The main benefits and
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Recent Advances in Circuits, Systems, Signal Processing and Communications
larger dimensions, such as heavy electrical engineering, where one pole of the capacitor is often
connected to the ground. Such involvement can also
use circuits constructed from discrete components.
These circuits are usually designed as a customer for
a specific use and usually are made of selected
components with minimum tolerances [8]. These
circuits can be reasonably expected to significantly
lower variance parameters, and thus significantly
reduce the negative effects.
advantages of our proposed engagement and the
derivation of the integration amplifier lies in the fact
that, unlike the conventional circuit (Fig. 1, Fig 2)
are not frequency-dependent elements in the direct
signal path. Parts ensures integration of derivative or
passing the signal is always one pole connected to
signal ground. This limited their impact on the
possible introduction of spurious signal via
capacitive or inductive coupling to the main signal
path.
For the temperature dependence of the integration amplifier offset voltage are the conclusions
drawn in [7]. There are, however, set the value of
transmission to the DC input voltage up to 100, and
an integrator is used exclusively for processing AC
signal, which can be from the DC component, if a
defect in other downstream signal processing circuit, separate. The advantage of the circuit is that
the processed does not invert phase voltage and
enables saving inverter voltage. A significant advantage is fact that the internal resistance of the coil
used may count resistor R0 and used as the involvement of an "ideal" inductance L. Temperature
dependence of the offset voltage of the derivation-it
involved the same amplifier as the classical differential amplifier. For DC input signal transfer is
determined by the involvement of only the transmission signal summation for a particular type of amplifier and is usually several orders of magnitude
below the processed signals. Differentiator is the
same quality and process sub-frequency AC signals.
The advantage of the circuit is that the processed
does not invert phase voltage inverter allows you to
save a further operational amplifier. A significant
advantage is the fact that large leakage resistance of
capacitors do not affect the transmission characteristics and involvement of the relatively low values of
the surrounding resistors can be used in connection
space is considered ideal. The circuit can also be
used for example in conjunction with a suitable
frequency switchable decimal sinusoidal generator
for directly showing the linear meter capacity with a
wide range of measured values from 1 μF to 10 pF
in easily achievable frequency range 10 Hz to 1
MHz. The advantage in this case is the fact that one
pole of the measured capacitor (C1) is grounded.
This is especially useful for measuring capacitors
ISBN: 978-960-474-359-9
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[1] CASIER, H. et al. Analog circuit design.
Springer, 2008. ISBN 978-1-4020-8262-7.
[2] LOKVENC, J. - DRTINA, R. Feedback linear
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UKF. 2003. ISBN 80-8050-624-8.
[3] LOKVENC, J. DRTINA, R. Feedback linear
ohmmeter with the measuring range 1mΩ to
5 Ω. In Technical Education. s.297-301. Banská
Bystrica. UMB. 2004. ISBN 80-8083-040-1.
[4] BIOLKOVA, V. et al. State-Space Averaging
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[5] MILLAN-GIRALDO, M. SANCHEZ, J.S. A
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TRANSACTIONS on CIRCUITS and SYSTEMS
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[6] ERDEI, Z. DICSO, A-L. NEAMT, L.
CHIVER, O. Symbolic Equation for Linear
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volume 9, issue 7, 2010, E-ISSN: 2224-266X
[7] LONG, K. Intercept of Frequency Agility
Signal using Coding Nyquist Folding Receiver
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[8] LOKVENC J. A non-inverting derivator. Tesla
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