DESIGN AND PRACTICE OF SIMPLE FIRST-ORDER ALL-PASS FILTERS USING COMMERCIALLY AVAILABLE IC AND THEIR APPLICATIONS

Abstract First­order all­pass filter circuits, both non­inverting and inverting, could be the focus of this article, which could include the design and implementation of first-order all-pass filter circuits. Using a standard integrated circuit (IC): AD830, as well as a single resistor and a single capacitor, the proposed first-order all-pass filters could well be built. The AD830 is an integrated circuit (IC) manufactured by Analog Devices Corporation that is available for purchase. The pole frequency and phase response of the proposed all­pass filters could well be directly modified by attuning the resistor in the circuit. Aside from that, the output voltage has a low impedance, making it appropriate for use in voltage­mode circuits. In addition, the proposed first­order all­pass filter is used to design the multiphase sinusoidal oscillator, which serves as an example of an application wherein the oscillation condition can be adjusted without impacting the frequency. The gain and phase responses of the proposed all­pass filters, as well as their phase response adjustment, time­domain response, and total harmonic distortion of signals, are all shown via computer simulation using the PSPICE software, as well as their experimental results. For the proposed circuits, a statistical analysis is coupled with a Monte Carlo simulation to estimate the performance of the circuits. In accordance with the results of this study, a theoretical design suitable for developing a worksheet for teaching and learning in electrical and electronic engineering laboratories has already been developed.

A summary and comparison of the previous publications and researches in the literatu res  can be reported in Table 1.
The purpose of this article is to present a design idea for all-pass filters based on the commercially available IC:AD830. Both the noninverting and inverting firstorder allpass filters have low output impedance as well as have no need for the constraint matching condition of active or passive elements. A noninverting allpass filter is applied to a multiphase oscillator as a sample.
To confirm the performance of the first-order all-pass filter and a multiphase oscillator circuit is shown with PSPICE simulation and experimental results according to the theories.

Materials and methods 1. The details of AD830
The AD830 is a difference amplifier in an 8pin package produced for commercially available by Analog Devices Corporation [30]. Fig. 1 shows the electrical symbol, and pin configuration of AD830. The input voltages (pin 1, 2, 3 and 4) of the AD830 are highimpedances and the output voltage (pin 7) is lowimpedance. The supply voltage of AD830 can be operated from ±5 V to ±15 V. The following mathematical function can be used to describe the electrical characteristics of the AD830: (1) A o ≌ ∞ is the voltage gain in the open loop. Fig. 2 demonstrates how easily the AD830 may be set up to create the difference between three signals, V Y1 , V Y2 , and V Y3 , in which the applied differential signal is precisely replicated at the output. The voltage output of the circuit in Fig. 2 can be written as: (2) Сontinuation of the Table 1 Original Research Article: full paper (2022), «EUREKA: Physics and Engineering» Number 3 Engineering

Concept voltage-mode first-order all-pass filters 1. Non-inverting first-order all-pass filters
The method for synthesizing allpass filters as presented in Fig. 3 is a conceptual design of the noninverting firstorder allpass filter (APF+) on a circuit block diagram that consists of the first-order high-pass filter, the amplifier at defined k 1 = 2, and the summing. The circuit of the proposed noninverting APF is schematically designed as in Fig. 4, a and Fig. 4, b.  The proposed noninverting APF is shown in Fig. 4, where a uses a single capacitor, a single resistor, and two AD830s, where the first AD830 acts as an amplifier of the input signal (k 1 ) at high-input-impedance without resistors. The second AD830 connects a capacitor and a resistor as a firstorder highpass filter and sums the input signal, with the output voltage of the circuit being lowimpedance. Additionally, the proposed noninverting APF is shown in Fig. 4, Engineering where b is simply designed with a single AD830, a single capacitor, and a single resistor. Fig. 4, b, shows the AD830 connected to a capacitor and a resistor to create a high-pass filter. It can be seen that the AD830 in Fig. 4, b can function as both an amplifier and a summing amplifier. Furthermore, the voltage output port of the circuit is lowimpedance, so it can be conveniently cascaded or connected to other stages or circuits. The electrical properties of AD830 in equation (2) of the circuits in Fig. 4, a, b can be analyzed and presented by the voltage transfer function of the proposed APF circuit as follows: The pole frequency and voltage gain of the proposed noninverting APF are the same that can be analyzed in equations (4), (5), respectively.
The voltage gain of the proposed noninverting APF can be analyzed and written as: According to equation (6), the input and output voltages are equal. From equations (4), (5), the adjustment of the pole frequency and the phase response can be controlled via R 1 and C 1 . Fig. 5 shows the design concept of the inverting firstorder allpass filter (APF-) that consists of a first-order low-pass filter, an amplifier defined as k 2 = 2, and summing. From this concept, the circuit in Fig. 6, a is then designed with two AD830s, a single capacitor, and a single resistor. The first AD830 is an amplifier of the input signal (k 2 ). The second AD830 connects a capacitor and a resistor to be the low-pass filter and sums the signal from pin Y 2 . This design is for highinputimpedance and lowoutputimpedance. The inverting APF in Fig. 6, b is designed with only a single AD830, a single capacitor, and a single resistor. The output port of the inverting APF is low impedance. This design can reduce the amount of equipment required from the circuit in Fig. 6, a.

Inverting first-order all-pass filters
The voltage transfer function of the proposed inverting APF circuit in Fig. 6, a, b is written as follows: From equation (7), the pole frequency, voltage gain, and phase response of the proposed inverting APF can be given as: and From equation (10), the input and output voltages are equal. The pole frequency and phase response can be modified with equations (8), (9), respectively, by adjusting R 1 and C 1 . The nonideal of the voltage in practice is partly caused by parasitic components. They can be found at the input ports and output port of the AD830. The high-impedance ports are the Y 1 , Y 2 , Y 3 , and Y 4 ports. These parasitic elements of the ports have resistors and capacitors connected to the ground as well as the output port X and they are in a series of resistors with lowparasitic values. Fig. 7 shows the details of the parasitic elements. It can be described as follows.

3. Non-ideal study of non-inverting APF
The parasitic elements of AD830 are included in the characteristic equation of the proposed noninverting APF in Fig. 4, b. Thus, the new features when R 1 >>R X were studied and analyzed as follows: The pole frequency and phase response of noninverting APF circuits are modified to (12), (13), respectively. where

4. Non-ideal study of inverting APF
The non-ideal study of AD830 are incorporated into the characteristic equation of the proposed inverting APF in Fig. 6, b. As a result, the new transfer functions at R 1 >>R X were analyzed as follows: From equation (14), the pole frequency and phase response of inverting APF circuits are modified as and Parasitic elements have an effect on the performance of the proposed noninverting APF and inverting APF. The impact of the pole frequency and phase response is caused by equations (12), (13), (15), and (16), which can be resolved by slightly adjusting the resistance. The multiphase sinusoidal oscillator (MSO) is an example of the application of the proposed APF. The MSO is designed by cascading three proposed noninverting APFs and positioning them in a feedback loop to show an application example in Fig. 8. The output voltage at the lowimpedance of the MSO is at nodes V o1 , V o2 , and V o3 . It can be cascaded or connected to other stages or circuits without the need for a voltage buffer. The system loop gain of the proposed MSO can be written as follows: LG The phase of the system loop gain can be expressed in equation (18).
Equation (17) can be used when n is an odd number. The output signals have a phase of 360/n, which is in accordance with equation (18). An oscillator circuit can generate sinusoidal signals if the oscillation requirement is achieved, as in: Original Research Article: full paper (2022), «EUREKA: Physics and Engineering» Number 3 Engineering As can be observed in equations (19), (20), by adjusting K, the oscillation condition can be easily changed without affecting the frequency. Also, the frequency of oscillation can be adjusted by C 1 and R 1 .

Results and discussion
A simulation of the proposed first-order all-pass filters used sample circuits as in Fig. 4, b; 6, b by verifying the circuit performance and theoretical validity using the PSPICE program. To simulate, the AD830 macromodel was adopted. Both proposed APFs used passive elements with values of R 1 = 1 kΩ and C 1 = 1 nF. The supply voltage chosen for use was ±5 V. Fig. 9, a, b show the gain and phase responses of the noninverting and inverting APFs, respectively, when compared to a theoretical model. The simulation results represent the phase response from 180° to 0° of the noninverting APF and the phase response from 0° to 180° of the inverting APF. The voltage gain of both APFs was approximately 0 dB. The pole frequencies of the noninverting APF and the inverting APF were about 157 kHz and 156 kHz, respectively. When calculated using equations (4), (8), it deviated from the theoretical figure by about 1.35 % and 1.97 %. Thus, it will be seen that the simulation results were greater compared to the theoretical analysis.
The results in Fig. 10, a, b followed adjustments in the phase response of the noninverting APF in Fig. 4, b and the inverting APF in Fig. 6, b which adjusted the value of R 1 in both circuits to 250 Ω, 500 Ω, 1 kΩ, and 2 kΩ, respectively. The phase responses at 90° of the noninverting APF were transformed to 79.04 kHz, 157.73 kHz, 314.25 kHz, and 623.93 kHz, respectively, with the inverting APF transformed to 79.43, 159.56, 316.22, and 630.54 kHz, respectively. The results agreed with the theoretical analysis when equations (4), (8) were compared.   The Monte Carlo (MC) method was utilized to examine the tolerance error of the passive device since this has an effect on the proposed APF. The MC was set to a Gaussian distribution for 100 samples, and the tolerance errors of the resistor and the capacitor were 1 % and 10 %. The MC analysis of both the proposed APFs is illustrated in Fig. 11, a for the gain response of the noninverting APF, in Fig. 11, b for the gain response of the inverting APF, Fig. 11, c for the phase response of the noninverting APF, and Fig. 11, d for the phase response of the inverting APF. The histogram of the gain and phase at frequency 159.15 kHz of both APFs is illustrated in Fig. 12, and Table 2. The noninverting APF has a mean gain response of -0.0052 dB, a median of -0.0052 dB, and a standard deviation of 0.000087. The noninverting APF has a mean phase response of 89.62°, a median of 88.53°, and a standard deviation of 5.84. The inverting APF has a mean gain response of -0.0048 dB, a median of -0.0048 dB, and a standard deviation of 0.000092. The inverting APF has a mean phase response of -90.57°, a median of -90.62°, and a standard deviation of 5.36.  The total harmonic distortion (THD) of the output noninverting APF and inverting APF in simulation is shown in Fig. 13 when sweeping the input signal from 0.1 V p-p to 2 V p-p . The percentage THD of both APFs was a minimum of 0.01 % and a maximum of 0.30 %.
The simulation results of the proposed multiphase sinusoidal oscillator were obtained using the circuit in Fig. 8 by configuring the supply voltage of ±5 V, the passive elements as R 1 = 470 Ω and C 1 = 0.47 nF. The MSO was able to generate sinusoidal signals when K ≥ 1, R G1 = 10 kΩ and R G2 = 100 Ω were configured to conform to equation (19). Fig. 14, a shows the simulation of the initial state signals, and Fig. 14, b shows the steadystate signals of the sinusoidal output waveform. The output spectrum of multiphase signals in Fig. 15 shows an output frequency of about 400.0 kHz, which deviated from 415.97 kHz by about 3.83 % from equation (20), and the total Engineering harmonic distortion (THD) of V o1 , V o2 , and V o3 were about 3.41 %, 1.58 %, and 2.90 %, respectively. The simulation results were analyzed using the MC because the tolerance error of the passive device affected the MSO. The MC was set to a Gaussian distribution for 100 samples, and the tolerance errors of each resistor and capacitor were 1 % and 10 %, respectively. Fig. 16, a, b demonstrate the spread of the frequency domain and histograms of the output (V o1 ). The mean, median, and standard deviation of the frequency oscillation were 410.87 kHz, 407.63 kHz, and 24.48 kHz, respectively. To verify the circuit performance and theoretical validity of the proposed firstorder allpass filter circuit by the experimental noninverting APF in Fig. 4, b and the inverting APF in Fig. 6, b. The experiment was conducted by using the commercially available IC: AD830. The proposed APF in Fig. 4, b; 6, b was connected to a supply voltage of ±5 V by the Siglent SPD3303C power supply. The passive elements are chosen as R 1 = 1 kΩ and C 1 = 1 nF. The performance of the circuit was measured using a Keysight DSOX3024T oscilloscope. The frequency response analyzer was set at 200 mV p-p amplitude and sweep frequency from 1 kHz to 10 MHz.
The gain and phase responses of the noninverting APF and the inverting APF are shown in Fig. 17, a, b, respectively. It was found that the gain responses were about 0 dB for all frequencies of both APFs, which is consistent with the theoretical analysis results in equations (6) and (10). The phase responses of noninverting APF varied from about 180° to -17°. The pole frequency was about 158.5 kHz with a 90° phase shift. The phase responses of the inverting APF ranged from approximately 0° to -195°. The pole frequency was about 154.88 kHz with a -90° phase shift. These results are consistent with equation (5) of the noninverting APF and equation (9) of the inverting APF.   Fig. 18, b shows the gain and phase response of experimental results from inverting APF when the input has a frequency of 158.55 kHz and an amplitude of about 200 mV p-p . The results correspond with the following equations (9), (10). The phase relationship between waveforms V in and V out of the noninverting APF is about 90.06° at the frequency Engineering of 158.4 kHz, and the phase relationship between waveforms V in and V out of the inverting APF is about 90° at the frequency of 158.55 kHz. Both results are shown in Fig. 19, a, b, respectively, which agree with the theoretical analysis in phase response equation (5) of the noninverting APF and phase response equation (9) of the inverting APF.
The results in Fig. 20 demonstrate the adjustment in the phase response of noninverting APF in Fig. 4, b by adjusting the value of R 1 to 500 Ω, 1 kΩ, and 2 kΩ, respectively. It is evident that the phase responses at 90° of the frequency transformed to 77.62 kHz, 158.48 kHz, and 309.02 kHz. The results of the phase responses agree with the theoretical analysis in equation (5).
The proposed multiphase sinusoidal oscillator in Fig. 8 was set with a voltage supply at ±5 V. Moreover, passive components were defined using C 1 = 0.47 nF and R 1 = 470 Ω. When the oscillation condition was set as R G1 = 10 kΩ and R G2 = 100 Ω to define K = 1.01, which corresponds to the conditions in equation (19), then the circuit generated sinusoidal signals.   Fig. 21 shows the results of the multiphase sinusoidal waveforms. The oscillation frequency was 402.25 kHz, which inaccurate was about 3.29 % of the theoretical value of the equation (20). Additionally, the phase relationship between V o1 , V o2 , and V o3 output voltages was 119.20°, 120.90°, and 120.10°, respectively. This error may occur as a consequence of passive element to lerance issues. The tolerance errors of the MSO were about 3 % for three passive resistors and 30 % for three passive capacitors. However, the resulting frequency and phase of the MSO are still consistent with the theoretical.
The example relationship between the waveforms of the MSO in Fig. 22 is about 120° at output V o1 and V o2 when measured by using the XY-mode which the result agrees with equation (18).  The THD of the output voltage MSO (V o1 ) in Fig. 23, a is approximately 0.240 %, and the amplitude of the other harmonic is 56.8 dB greater than the first harmonic. The THD of the output voltage MSO (V o2 ) is about 0.351 %. As shown in Fig. 23, b,  Engineering harmonic is 53.2 dB, while the THD of the output voltage MSO (V o3 ) in Fig. 23, c is about 0.283 % and the magnitude of the first to other harmonic is 54.0 dB, respectively. It can be seen that a commercially available integrated circuit used in the proposed APFs makes lab tests simpler and cheaper. The pole frequency and phase response of noninverting and inverting APFs are in line with the theoretical analysis and MC analysis as presented in Table 2. It is interesting that the proposed APFs are appropriately used in electrical and electronic engineering study. However, the proposed APFs are still limited because the AD830 can operate at a maximum frequency of about 10 MHz.

Conclusions
In the proposed noninverting and inverting APF creation concepts, each kind of APF can be constructed with a commercial IC AD830, a single capacitor, and a single resistor. The APFs can adjust the phase response by adjusting the resistance. Additionally, the voltage output ports of both first-order APFs are low-impedance, so they can be cascaded or connected to other stages or circuits. The multiphase sinusoidal oscillator was used to prove the proposed noninverting APF. The simulation and experimental results confirmed the performance of the circuit. Also, the results agree with the theoretical analysis. The experiment results can be described as an example that the noninverting and inverting APF had a pole frequency of about 158.54 kHz and 158.55 kHz, respectively, which is an error of 0.383 % and 0.377 %, respectively. The adjustment of phase responses of 90° was experimentally demonstrated by changing resistors by 500 Ω, 1 kΩ, and 2 kΩ. After changing these, the pole frequencies were 77.62 kHz, 158.48 kHz, and 309.02 kHz, respectively. In applying APF, the multiphase sinusoidal oscillator can be generated an oscillation frequency of 402. 25  Engineering with an error of 3.29 %. From these, the APFs then are appropriate for developing a worksheet for teaching and learning in electronic laboratories.