Band Stop Filter with a Synthetic Inductor with Series Resistance and a Real Operational Amplifier

This paper deals with application of simple synthetic inductor with series resistance (non-ideal gyrator). Those inductors are seldom used. However, if we use this inductor in the arm of bridge, the circuit is able to achieve high performance. In this way we can get band stop filter with good performance. In the paper we solve influence of real OPA properties. Theoretical considerations are verified by means of simulations (MicroCap) and experiments. Based on theoretical considerations, simulations and experiments are finally determined by criteria that must meet real operational amplifier to make the circuit performed well.


Introduction
In essence, the properties of an inductor can be simulated by a simple circuit with an amplifier called a synthetic inductor or gyrator.This basic circuit is presented in Fig. 1.The amplifier can be realized with vacuum tubes, transistors and integrated amplifying structures.This circuit is widely used for designing a band-stop and band pass filters.Although some authors (e.g.[2] and [3]) states: "The drawback of the circuit is that the quality factor is poor".
Nevertheless, this circuit (with an operational amplifier) is used to design the band-stop filter since 1971 [4], as shown in Fig. 2. The series resistance presented in Fig. 2, which is generally undesirable, is t functionally utilized in his circuit.This circuit has been described in literature [5], [6], [7] and [8] several times.However, it is necessary to describe the influence of the real synthetic inductor on the properties of the band-stop filter in detail (see Fig. 2).

Prescott Inductor
Figure 1 shows the Prescott inductor (non-ideal gyrator).To find out the input impedance, current ÎA and voltage ÛX must be calculated by using superposition theorem.So we can come up with expressions: where p = jω, and input impedance If Ĝ = 1, the expression for input impedance is described as: Expression from Eq. ( 4) has been mentioned commonly in literature.The input impedance appears as a resistor R S and inductor L connected in series.Where series resistor is defined as and inductor If amplifier (follower) is realized with an operational amplifier, it follows that where Â is the gain of the operational amplifier without feedback.Assuming that operational amplifier is well compensated and its gain is sufficiently described by the frequency of the first order model (e.g.[9] and [10]), we obtain: where ω 1 is the dominant (first) pole of transmission, ω T is the extrapolated transition frequency (ω T = A 0 ω 1 ), and A 0 is the gain (DC) for ω < ω 1 .
Further, we assume that that ω > ω 1 and normal operating mode of an operational amplifier while the normal values for ω 1 range from 2π • 5 rad•s −1 to 2π • 50 rad•s −1 .Then, the approximate relationship is valid when: and and by applying this formula to the Eq. ( 3), we get the following expression for input impedance Real series R SR resistance under these conditions is now frequency dependent and determined by the relation as well as real inductor 3: Equivalent circuit of the circuit in Fig. 2.

Band Stop Filter with Synthetic Inductor
Fig. 3 shows equivalent circuit of the circuit in Fig. 2.
If we make R a = 2R, we can determine an expression relating Ûin and Ûout (so that we can find the gain) by the following expression: If the operational amplifier op-amp2 is ideal (ω T → ∞), then R SR = 2R and L R = CR 2 .At resonance (ideal state): Therefore A quality factor Q is determined as follows For frequency ω 0 is input of operational amplifier OP-AMP1 connected to a "balanced bridge R 1 − R 1 ; 2R − 2R" and ideally Û (ω 0 ) → 0.
By modifying Eq. (18) and Eq. ( 19) we obtain the relationships suitable for design of the ideal circuit: Simply, we select R and then use the above relationships.The bandwidth B is determined by elemental expression When varying ω 0 by means of C S , B remains constant, whereas Q varies.
A more complex situation occurs when op-amp OP-AMP2 is non-ideal (and ω 0 > ω 1 ), then By solving this equation, we get Resulting expression for series resistor R SR at frequency ω r using Eq. ( 24) is determined as: This means that the operational amplifier OP-AMP1 (for R a = 2R) at frequency ω r is not balanced and transmission is different from zero.From Eq. ( 16), it is evident that Substituting for R SR from Eq. (25) into Eq.( 26) we get after modifications (for R a = 2R -see Fig. 3): 4.

Maximum Input Voltage
The results of the experiments showed that the amplitude of the input voltage is limited by the output of OP-AMP2.If the voltage ÛSI is limited, we cannot use a linear model, which has always been used.
Using the assumption that the operational amplifier is ideal, we get: By analysing Eq. ( 28) it was found that the maximum allowable amplitude for the input signal is where U CC− , U CC+ define the value of the supply voltage, Q is the quality factor (Eq. ( 19)) [6].

Real Quality Factor of Band Stop Filter
The above considerations are concerning the exact resonant frequency ω r .However, by substituting L R and R SR in Eq. ( 16) for transfer function we obtain relationship from which it is not possible to determine the bandwidth B (change of 3 dB), see [11].Simulation results from [11] are briefly summarized in Tab. 1.
The results in Tab. 1 shows how the ratio Q r /Q is dependent on an expression ω T /(Qω 0 ), see below.From data in Tab. 1 it is evident that for the real values of 1.42 20 1.22 50 1.10 100 1.07

Verification of Theoretical Considerations
To verify theoretically derived relationship, a number of simulations and measurements under various conditions has been carried out.It turned out that it is necessary to modify the obtained relations into an appropriate (normalized) form.For this reason, the product of ω T RC s was modified as follows: Then It is apparent that the ratio ω T /(Qω 0 ) is important.For ω T /(Q • ω 0 ) → ∞, we obtain the ideal value 1 or zero.
Simulations were performed in a Microcap software.For the simulations, we used resistors R 1 = 10 kΩ, R = 5 kΩ, the capacity C s ranged from 80 pF to 80 nF, the capacity C ranged from 128 nF to 1280 nF; the transition frequency f T of operational amplifiers ranged from 50 kHz to 10 MHz (see Fig. 2).The frequency of the op amp second pole has always been shifted, thus was higher than the frequency f T .It turned out that the properties of the operational amplifier op-amp1 practically do not affect (ω r /ω 0 ) and T min , if (ω T /ω 0 ) > 10.This obviously affects the overall transfer function since the decrease of transmission op-amp1 by 3 dB, under these conditions, is at a frequency ω T /2.Equation ( 16) can be adjusted into Table 2 summarizes the results obtained experimentally in comparison with the ideal values calculated using Eq.(32) and Eq.(34).Values ω T /(Qω 0 ) were obtained by different combinations of ω T , Q, and ω 0 .It was confirmed that the result of the product is decisive in comparison with the individual expressions in the normalized form.
Tab. 2: Results of simulation (Ra = 2R) and calculations for the circuit of Fig. 2 according to the ratio ω T /(Q • ω 0 ).Although Eq. ( 33) is explicitly verified by the relationship Eq. (34), we have verified simulations for several values of R SR /(2R) and these results are shown in Tab. 3. Simulations were carried out under similar conditions as in previous case (see Tab. 3).The difference lies in the fact that the second pole of the response Tab.3: Results of simulation (Ra = 2R) and calculations for the circuit of Fig. 2 according to the ratio ω T /(Q • ω 0 ).Tab. 4: Results of the simulations [11] and calculations according to equations Eq. ( 32) and Eq. ( 34 curve of the operational amplifier op-amp2 was fixed to 2 MHz. It is obvious that the position of the second operational amplifier pole (op-amp2, f 2 ) substantially affects primarily the value of R SR , especially when transmission on f 2 "emerges over 0 dB (here for f T > 1 MHz) as shown in Fig. 4. The ratio f r /f 0 remained practically unchanged.
In [11], the simulations were carried out for three different real operational amplifiers, see Fig. 5.
Ideally, the desired frequency f 0 = 100.658kHz and Q = 1.58.The results of the simulations and calculations are summarized in Tab. 4.
The simulations show that the derived relationships describe the behaviour of the circuit at the retention frequency precisely.Derived relations were verified by measurements.The circuit of the band stop filter was realized according to Fig. 2.This circuit was consisted of the following components: resistor values R 1 = 15 kΩ (twice), R a = 2•15 kΩ, R = 15 kΩ (twice) -resistors with an accuracy of one percent, the capacitance value C = 1 µF, operational amplifiers op-amp1 and op-amp2 -both µA741, power supply ± 12 V.The amplitude of the input voltage is 1.41 V. Assume f T = 1 MHz.The results of measurement and calculation for different values of C S are summarized in Tab. 5 and Tab. 6.For the given f T /(Qf 0 ), the measured values (f r /f 0 ) are determined only accuracy of passive elements".The same statement applies to the minimum transmission T min .If the resistance tolerance is 1 %, the actual resistance value R a can be, for example, up to 30 kΩ + 300 Ω and thus the actual value of the resistor R S = 2R may be up to 30 kΩ − 300 Ω.
By substituting these values into Eq.( 26) [R S → 30 kΩ −300 Ω; 2R → 30 kΩ +300 Ω] we obtain: This corresponds to transmission (at f r ) −40 dB.The value of T min is determined by accuracy of resistors not by properties of the operational amplifier op-amp2.The non-optimal op-amp (described f T /Qf 0 ) then increases the value of T min , but the trend remains, i.e. the lower the ratio f T /(Qf 0 ), the higher the T min .
Tab. 5: Calculated values for the circuit of Fig. 2 under the conditions described in the text.
C S Calculation Eq. ( 18) Eq. ( 19) -Eq.(32) Eq. ( 34) According to [5], when setting the minimum transition T M IN (at f 0 ) it is possible to set the resistance value R a individually for each set frequency f 0 (f r ) so that the transition T min is less than −40 dB.Practically useful circuit connection is achieved by changing the resistance R a by a serial combination of trimmer and fixed resistance (see Fig. 2).Our experiments used a combination of 27 kΩ and a 6.8 kΩ resistors.Again, it was always possible to set a transition T min less than −45 dB without changing the frequency.This setting compensates the actual operational amplifier properties and the final tolerance of the resistors at the same time (of course, this problem is not the case with simulations since the resistor values are set precisely).
We must pay attention to the required value of Q = 7.91 (see C s = 1 nF in Tab. 5).Detailed measurements (for the conditions in Tab. 5) are shown in Tab.7 for the amplitude of the input signal U IN A = 1.41 V (i.e.effective value 1 V).In addition to the output voltage Û0 , the voltage at the output op-amp2 ( ÛSI ), which is defined by Eq. ( 28), is measured.At the required value of Q = 7.91, a significant resonance maximum is achieved, resulting in a limitation of the voltage at the op-amp2 output (at ±12 V supply).The circuit is "non linear" for U IN = 1.41 V and this leads to degradation of its properties.
If the power supply voltage is increased to 15 V, the ÛSI / ÛIN ratio at 310 Hz is equal to 7.167 (17.11 dB) and at 321 Hz at 7.50 (17.50 dB) with the amplitude U SI A = 10.58V.The quality factor Q was returned to 7.8, the value T min is −28.6 dB (at 322 Hz), while the influence of inaccurate (real) resistors is still applied here.We will achieve the same effect by lowering the amplitude U IN A below 1.2 V.The maximal input voltage is determined by substituting Q in Eq. ( 29): For comparison, of a circuit with a lower quality factor (Q = 3.40) was measured (see in Tab. 5 -C S = 5.4 nF), again for ±12 V supply, R a = 2R; The circuit will still work in linear mode, see Tab. 8.
In both cases, harmonic distortion was observed for frequencies higher than 40 kHz (amplitude of 1.41 V).This is a slew rate distortion where the power frequency f p is defined by the following relationship (see for instance [10]): where SR is the slew rate (V•s −1 ), U A is the amplitude of signal.
For operational amplifier µA741, a typical slew rate SR = 0.5 V•µs −1 = 5 • 10 5 V•s −1 is given.Assuming that U A = 1.41 V, the power frequency f p becomes f p = 5 • 10 5 2π • 1.41 = 56.44 kHz. (41) Therefore it can be assumed that the actual value of the slew rate is lower.If the threshold frequency is From Eq. (34), the desired [f T /f 0 Q] for T min can be determined (at R a = 2R): It is obvious that for practical use it is advisable to provide f T /f 0 Q greater than 200.

Conclusion
Although circuits of this type (such as the one showed in Fig. 2) have already been investigated, the article brings new results that allow to design a band stop filter according to real properties.From Sec. 7. it is clear that the circuit will be suitable for relatively low f 0 values, for example in medical applications

Fig. 4 :
Fig. 4: Quantitative representation of the transmission of the operational amplifier at a fixed position of the second pole f 2 .