Recent Researches in Circuits and Systems

Unusual involvement of 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 [email protected], [email protected], [email protected], 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: 978-1-61804-108-1

dU 1 dt

125

Recent Researches in Circuits and Systems

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: 978-1-61804-108-1

R0 2 +1+ A0 2 ⋅ R0 + R

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

126

R0

2π ⋅ L

(5)

Recent Researches in Circuits and Systems

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

ISBN: 978-1-61804-108-1

127

Recent Researches in Circuits and Systems

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

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)

A0 z + =

(3)

R 2Z1

Ani =

(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 R 2 R2 A0 R0

(12)

Z 2 (2 Z 1 + R )

(13)

Z 1 (2 Z 2 + R )

Z 2 (2Z 1 + R )

Z 1 (2Z 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: 978-1-61804-108-1

1 + A0 (Z 1 R ) + 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

1 Z1 R

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 =

(11)

(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 =

128

1 jωC1

(16)

Recent Researches in Circuits and Systems

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: 978-1-61804-108-1

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

129

Recent Researches in Circuits and Systems

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 [9]. 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 [8]. 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 subfrequency 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 larger dimensions, such as heavy electrical engineering, where one pole of the capacitor is often connected to the ground. Such involvement can also use

ISBN: 978-1-61804-108-1

References: [1] CASIER, H. et al. Analog circuit design. Springer, 2008. ISBN 978-1-4020-8262-7. [2] LOKVENC, J. - DRTINA, R. Feedback linear ohmmeter. In Scienece and Education Nitra. 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. Banská Bystrica. UMB. 2004. ISBN 80-8083-040-1. [4] BIOLKOVA, V. et al. State-Space Averaging (SSA) Revisited: On the Accuracy of SSABased Line-To-Output Frequency Responses of Switched DC-DC Converters in WSEAS Transactions on Circuits and Systems, volume 9, issue 2, 2010, E-ISSN: 2224-266X [5] MILLAN-GIRALDO, M. SANCHEZ, J.S. A Comparative Study of Simple Online Learning Strategies for Streaming Data. In: WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Issue 10, Volume 7, 2008. Print ISSN: 11092734, E-ISSN: 2224-266X [6] ERDEI, Z. DICSO, A-L. NEAMT, L. CHIVER, O. Symbolic Equation for Linear Analog Electrical Circuits using Matlab. In: WSEAS Transactions on Circuits and Systems, volume 9, issue 7, 2010, E-ISSN: 2224-266X [7] LONG, K. Intercept of Frequency Agility Signal using Coding Nyquist Folding Receiver in WSEAS Transactions on Signal Processing volume 9, issue 2, 2010, E ISSN 2224-3488. [8] DONDON M. CIFUENTES, M. TSENOV, G. MLADENOV, V. A practical modeling for the design of a sigma delta class D power switching amplifier and its pedagogical application. In: Proceedings of the 15th WSEAS International Conference on Circuits 2011, Published by WSEAS Press. ISBN: 978-1-61804-017-6 [9] LOKVENC J. A non-inverting derivator. Tesla electronics 5, 1972, č.1, s.26-27.

130

Unusual involvement of 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 [email protected], [email protected], [email protected], 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: 978-1-61804-108-1

dU 1 dt

125

Recent Researches in Circuits and Systems

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: 978-1-61804-108-1

R0 2 +1+ A0 2 ⋅ R0 + R

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

126

R0

2π ⋅ L

(5)

Recent Researches in Circuits and Systems

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

ISBN: 978-1-61804-108-1

127

Recent Researches in Circuits and Systems

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

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)

A0 z + =

(3)

R 2Z1

Ani =

(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 R 2 R2 A0 R0

(12)

Z 2 (2 Z 1 + R )

(13)

Z 1 (2 Z 2 + R )

Z 2 (2Z 1 + R )

Z 1 (2Z 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: 978-1-61804-108-1

1 + A0 (Z 1 R ) + 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

1 Z1 R

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 =

(11)

(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 =

128

1 jωC1

(16)

Recent Researches in Circuits and Systems

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: 978-1-61804-108-1

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

129

Recent Researches in Circuits and Systems

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 [9]. 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 [8]. 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 subfrequency 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 larger dimensions, such as heavy electrical engineering, where one pole of the capacitor is often connected to the ground. Such involvement can also use

ISBN: 978-1-61804-108-1

References: [1] CASIER, H. et al. Analog circuit design. Springer, 2008. ISBN 978-1-4020-8262-7. [2] LOKVENC, J. - DRTINA, R. Feedback linear ohmmeter. In Scienece and Education Nitra. 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. Banská Bystrica. UMB. 2004. ISBN 80-8083-040-1. [4] BIOLKOVA, V. et al. State-Space Averaging (SSA) Revisited: On the Accuracy of SSABased Line-To-Output Frequency Responses of Switched DC-DC Converters in WSEAS Transactions on Circuits and Systems, volume 9, issue 2, 2010, E-ISSN: 2224-266X [5] MILLAN-GIRALDO, M. SANCHEZ, J.S. A Comparative Study of Simple Online Learning Strategies for Streaming Data. In: WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Issue 10, Volume 7, 2008. Print ISSN: 11092734, E-ISSN: 2224-266X [6] ERDEI, Z. DICSO, A-L. NEAMT, L. CHIVER, O. Symbolic Equation for Linear Analog Electrical Circuits using Matlab. In: WSEAS Transactions on Circuits and Systems, volume 9, issue 7, 2010, E-ISSN: 2224-266X [7] LONG, K. Intercept of Frequency Agility Signal using Coding Nyquist Folding Receiver in WSEAS Transactions on Signal Processing volume 9, issue 2, 2010, E ISSN 2224-3488. [8] DONDON M. CIFUENTES, M. TSENOV, G. MLADENOV, V. A practical modeling for the design of a sigma delta class D power switching amplifier and its pedagogical application. In: Proceedings of the 15th WSEAS International Conference on Circuits 2011, Published by WSEAS Press. ISBN: 978-1-61804-017-6 [9] LOKVENC J. A non-inverting derivator. Tesla electronics 5, 1972, č.1, s.26-27.

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