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A Novel Multi-Feedback Differential Filter Instrumentation Amplifier for Βiosignals Acquisition Applications

A peer-reviewed version of this preprint was published in:
Electronics 2024, 14(1), 95. https://doi.org/10.3390/electronics14010095

Submitted:

17 November 2024

Posted:

18 November 2024

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Abstract
Efficient filtering in bio-signals acquisition is challenging. The resistance of the sources exhibits inter- and intra-subject variability or is unknown, thus using passive filters before the first amplification stage is problematic. Conversely, filtering after amplification does not effectively eliminate the amplified electrical noise, Main’s Interference, and the artifacts. In this context, the design and utilization of filters in the Analog Front End of biosensors, in conjunction with the first amplification stage, is not common but offers substantial advantages. In this study, the design of a novel Multi-feedback Differential Filter Instrumentation Amplifier (MFDFIA) is proposed. For addressing the aforementioned issues, the design and the equations governing the gain and bandwidth characteristics of the MFDFIA are presented and relevant topologies are explored. Even though MFDFIA has two op-amps in its first stage, due to its symmetric topology, the analysis can be conducted separately for the differential and common mode input signal with a simplified one op-amp equivalent circuit. Notably, MFDFIA’s CMRR is equal and depends only on the CMRR of the second stage. An exemplary application for EEG signal acquisition is provided, which has bandwidth 0.7 to 98.6 Hz, gain 50.3 db, power consumption 11.583 mW and 0.1-10 Hz noise 462.94 nVrms.
Keywords: 
Differential Instrumentation Amplifier
;  
Filter Instrumentation Amplifier
;  
Bio-signals Acquisition
;  
EEG
;  
Analog Front End
;  
CMRR
;  
Multi-feedback Differential Filter
;  
MFDFIA
;  
Sensors
Subject: 
Engineering  -   Electrical and Electronic Engineering

1. Introduction

1.1. The Medical Importance of EEG Technology

Electroencephalography (EEG) is a long-established method for monitoring the brain's electrical activity, primarily originating from pyramidal neurons in cortical layers IV and V [1,2]. Research is ongoing to detect subcortical electrophysiological activity using scalp EEG [3]. EEG is valued for its high temporal resolution, low cost, and non-invasive nature [1]. It typically involves scalp electrodes, but higher-quality signals can be obtained with intracranial EEG and Electrocorticography (ECoG), where electrodes are placed directly on the cortical surface [4]. EEG's strengths make it particularly useful for diagnosing epilepsy and other seizure disorders, as real-time electrical activity information is crucial [5]. It is also employed in diagnosing brain malignancies [6], CNS infections [7], sleep disorders [8], and other conditions. Beyond diagnostics, EEG is integral to Brain-Computer Interfaces (BCI), aiding those with movement and communication impairments and serving in research, entertainment, communication, and general well-being [9].
However, EEG has notable drawbacks, such as low spatial resolution [10], which only captures the synchronous activity of large neuron populations. Its low voltage (10-100 μV) and high susceptibility to noise, both physiological (e.g., EOG, ECG, pulse artifacts, EMG) and non-physiological (e.g., 50/60 Hz mains hum, electrode artifacts) [11], also pose challenges. Improving spatial resolution involves increasing electrode density and using source-localization algorithms [1]. Addressing noise and signal amplitude issues requires signal amplification and noise minimization to enhance the signal-to-noise ratio (SNR) [12]. The EEG signal comprises various neural oscillations, each with distinct frequencies and amplitudes. Filtering techniques, such as high-pass, low-pass, band-pass, and notch filters, are employed to isolate relevant frequencies and reject irrelevant noise [12]. These frequencies are categorized into bands corresponding to different brain states or activities. Traditionally, these bands included delta to beta waves (0.5-30 Hz) [11], but advancements in recording equipment and digital signal processing have expanded this range [13]. The frequency bands are:
  • Infra-slow oscillations (<0.5 Hz), observed in preterm neonates and non-REM sleep [14].
  • Delta waves (0.5-4 Hz), associated with deep sleep and found in infants and children [15].
  • Theta waves (4-7 Hz), linked to drowsiness and early sleep stages (N1, N2) [15].
  • Alpha waves (8-12 Hz), seen during quiet wakefulness, especially when eyes are closed [16].
  • Beta waves (13-30 Hz), present during active concentration and task completion [16].
  • Gamma waves (30-80 Hz), occurring in all brain states, prominent during alertness [17].
  • High-frequency oscillations (>80 Hz), including ripples (80-250 Hz) and fast ripples (>250 Hz), related to memory encoding and cognitive process synchronization [18].
To reduce noise, instrumentation amplifiers with common-mode rejection and high differential signal amplification are utilized [19]. This work aims to integrate these concepts into a single analog circuit, enhancing the EEG measurement chain.

1.2. Problem Statement and Contributions of this Manuscript

Two aspects of great importance in EEG analog circuit design are signal amplification and filtering of unwanted signal components (e.g. noise, electromagnetic interference, artifacts). The typical design incorporates the amplification stage (usually by using an instrumentation amplifier, which also rejects some of the noise), before the filtering stage. This is done because the process of filtering also attenuates the signal, adding noise which would then be amplified along with the rest. By placing the filtering stage after the amplification stage, the issue is that the unnecessary signal components (e.g. 50 Hz mains hum) are also amplified. Thus, this technique is not a panacea, because the filters must handle the extra amplified added noise, and in general cannot cancel all the effects of noise amplification.
As an alternative, the initial filtering can be placed before the first amplification stage. In the particular case of biosignal acquisition this is deemed problematic, for the reason that the filters are usually in contact with the signal source that has variable impedance. The impedance of the signal source, which consists of the biological source plus the measurement electrodes, exhibits inter- and intra-subject variability or is unknown. Moreover, the way electrodes are placed appropriately on the measurement spot, the quality of the placement, the gel application and its durability in case of wet electrodes and the relative movement between the electrodes and the measured spots are factors that also affect the impedance of the measured system. This phenomenon makes the design of passive filters imprecise because their characteristics and filtering frequency are affected in real time by their contact with the measurement source.
Our design, by integrating a band-pass filter into the instrumentation amplifier, aims to accomplish simultaneous filtering and amplification, to mitigate the limitations of either aforementioned configuration and minimize the noise that remains in the signal.

3. Design of Novel Multi-Feedback Differential Filter Instrumentation Amplifier Topology

In this study, the design of a novel differential instrumentation amplifier combined with a multi-feedback filter is proposed. The present design combines the stage of filtering and amplification. The novelty of this paper lies in solving the drawbacks of having the first filtering stage right before or after the first amplification stage. In case of passive filters, the disadvantages of placing a filter before the amplification is that the resistance of the source exhibits inter- and intra-subject variability or unknown and especially in biological signal acquisition settings usually it is not stable. Thus, the characteristics of the passive filter will change and it cannot be placed on the input, if precision is desirable.
The disadvantages of placing a filter after the amplification is that frequency ranges that can cause noise are amplified, and filtering cannot cancel the totality of the presence of the amplified noise. Thus, a novel approach of tackling the above problems is utilizing a multi-feedback filter. This is a type of electronic filter that uses multiple feedback elements for signal processing. A differential filter is a type of filter that is based on the principles of differential amplification. It is typically used to isolate, amplify, or filter the signal that differs between two inputs or between an input and a reference. In this novel design the fusion of the above type of filters is proposed to create a novel Multi-feedback Differential Filter Instrumentation Amplifier.

3.1. A Novel Construction of a Multi-Feedback Differential Instrumentation Amplifier Filter Topology (MFDFIA)

The implementation of this novel MFDFIA is achieved with the incorporation of a Multifeedback Filter. In particular, two symmetrical filters are constructed, one for each input, that produce the differential output of the first stage of the Instrumentation Amplifier. The symmetry is apparent with respect to the middle point between the impedances Z as can be seen in Figure 3.1.1. In order to get the largest rejection of the common signal, an absolute symmetry in the circuit is necessary, which depends upon the component values and their tolerances. This mandate is achieved with precision resistors and type I capacitors (C0G/NP0). The prices of this specific type of capacitor set a limit to the capacity that can be chosen to less than μF.
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Figure 3.1.1. The circuit of the MFDFIA.
Figure 3.1.1. The circuit of the MFDFIA.
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  • The gain of MFDFIA:
V A O U T − V B O U T V A − V B = 1 + R 2 Z ∗ C 2 Z s + 1 C 2 Z ∗ 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2
For further details on how those equations were constructed refer to Appendix A. Notice that component numbers have already been replaced by the corresponding lower index numbers due to symmetry and equal values.
From the transfer function it appears that we have a circuit which always has at least gain equal to 1 (0dB) and it will never act like a real filter, due to the constant unity in the transfer function (non-inverting amplifier topology that give rise to a unity constant in the transfer function), which dominates in high frequencies. However, the rest of the transfer function presents a form which acts like a filter with the following characteristics:
  • A total DC gain R 2 Z  
  • 1st degree High Pass filter C 2 Z s + 1 C 2 Z  
  • and a 2nd degree Low Pass filter 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2
The above analysis identifies the behavior of the circuit in terms of the difference of the output voltages to the difference of the input voltages. However, it is not clear what happens to the common and differential signal.

3.2. Simplified Analysis of the Multi-Feedback Differential Filter Instrumentation Amplifier

Common Signal Gain

Studying the behavior of the circuit for the common signal, it can be stated that for V a = V b , due to circuit symmetry, the corresponding nodes share the same voltage. More specifically, the voltages at the inputs of the operational amplifiers are equal:
V c o m = V a = V A M P 1 + = V A M P 1 − = V b = V A M P 2 + = V A M P 2 −
Also due to symmetry and equal inputs we have
V X A = V X B
Since nodes XA and XB of the circuit in Figure 3.1.1 share the same voltages the current IX equals zero, so the Z branch is an open circuit.
This fact allows the removal of the branch between nodes XA and XB, so the circuit simplifies as seen in following figure:
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Figure 3.2.1. The equivalent circuit of the MFDFIA for common signal.
Figure 3.2.1. The equivalent circuit of the MFDFIA for common signal.
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The common signal transfer function is the following:
V c o m _ o u t V c = 1 + s C 2 R 2 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2
For further details on how those equations were constructed, refer to Appendix B.
From the transfer function, excluding unity constant, it appears that we have:
  • No total DC gain
  • 1st degree High Pass filter s C 2 R 2
  • and a 2nd degree Low Pass filter 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2

Differential Signal Gain

Studying the behavior of the circuit for the differential signal, it can be stated that for V a = − V b , due to circuit symmetry, the corresponding nodes have the opposite voltage. More specifically, the voltages at the inputs of the one operational amplifier are opposite from the inputs of the other operational amplifier.
V d i f f = V a = V A M P 1 + = V A M P 1 − = − V b = − V A M P 2 + = − V A M P 2 −
So due to symmetry we have V X A = − V X B .
Since nodes XA and XB of the circuit in Figure 3.1.1, have opposite voltages and the current flows through two equal resistors (Z1,Z11), it is concluded that the voltage will be zero in the middle point (node X).
This fact allows the "cutting" of the aforementioned circuit at this node so the circuit simplifies as seen in following figure:
V d i f f _ o u t V a = 1 + R 2 Z ∗ C 2 Z s + 1 C 2 Z ∗ 1 C 1 C 2 R 1 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 1 C 2 R 1 R 2
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Figure 3.2.2. The equivalent circuit of the MFDFIA for differential signal.
Figure 3.2.2. The equivalent circuit of the MFDFIA for differential signal.
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For further details on how those equations were constructed. refer to Appendix C.
From the transfer function, excluding the unity constant, it appears that we have:
  • A total DC gain R 2 Z  
  • 1st degree High Pass filter C 2 Z s + 1 C 2 Z
  • and a 2nd degree Low Pass filter   1 C 1 C 2 R 1 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 1 C 2 R 1 R 2

3.3. Conclusions and Examples of the Simplified Analysis

It is noticed that any branch such as the Z’s branch, from the region of the negative input of one operation amplifier to the region of the negative input of the other operational amplifier, does not affect the common signal, regardless of the complexity of the circuit.
As for the common signal, it is also noticed that connecting any component such as the capacitor C2 to ground, creates a current flow from the output to GND and therefore a voltage drop across node X, thus forcing the output Vcom_out to rise in order to compensate this voltage drop, so there will be common signal amplification.
The gain of the differential signal depends on the Z resistor and the capacitor connected to the ground C2, which are in parallel connection.
The circuits proposed in this work are simulated using the LTSPICE software. The input signals in the simulation were chosen with the rationale that the circuit is designed to expect input signals from a few mV to a few hundred mV. The resistance in the signal generation part of the circuit is noiseless so the noise can be added specifically when it is relevant to the simulation. It must be also stated that when acquiring the signal, the circuit in principle cannot differentiate the signal that comes from inside the brain from the noise induced electrodes.
The circuit works as a proof of concept and thus generally, it is not mandatory to follow the exact values of the circuit components. The proposed values were chosen having in mind the incorporation of the circuit in a potential EEG AFE signal acquisition system.

Example of a Low-Pass Differential Filter

The AC Analysis of the filter was performed using the following SPICE model:
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Figure 3.3.1. The circuit of the MFDFIA with Va, Vb, simulating the differential signal and Vc simulating the common signal.
Figure 3.3.1. The circuit of the MFDFIA with Va, Vb, simulating the differential signal and Vc simulating the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3.3.2. Τhe gain of the common and differential signal as a function of frequency.
Figure 3.3.2. Τhe gain of the common and differential signal as a function of frequency.
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It is noticed that the differential signal is amplified as expected from the chosen values of the resistors and the circuit presents a form like a low-pass filter, while for the common signal the circuit presents the form of a high-Q filter with no general amplification.

Proof of Equivalence with the Simplified Circuits for Differential and Common Signal

The AC Analysis of the simplified circuits for differential and common signal was performed using the following SPICE model:
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Figure 3.3.3. The simplified circuits of the MFDFIA for the differential signal and the common signal.
Figure 3.3.3. The simplified circuits of the MFDFIA for the differential signal and the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3.3.4. Τhe gain of the simplified circuits for the common and the differential signal as a function of frequency.
Figure 3.3.4. Τhe gain of the simplified circuits for the common and the differential signal as a function of frequency.
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It is noticed that the results of the AC analysis for the gain of the differential and the common signal in Figure 3.3.2 are the same with the results in Figure 3.3.4.

Example of a Band-Pass Differential Filter

By replacing the resistor Z with RC, a band-pass differential filter is created, as the capacitors cut the DC current and create a zero solution in the nominator of the fraction in the transfer function. So, the circuit will act like a band pass filter. The transfer function becomes a third order polynomial making difficult to specify the contribution of the existence of this zero in the filters behavior. However, it will be near 1/RC. The AC Analysis of the filter was performed using the following circuit:
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Figure 3.3.5. The circuit of the Band-Pass Differential Filter with Va, Vb, simulating the differential signal and Vc simulating the common signal.
Figure 3.3.5. The circuit of the Band-Pass Differential Filter with Va, Vb, simulating the differential signal and Vc simulating the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3. 3.6. Τhe gain of the common and differential signal as a function of frequency.
Figure 3. 3.6. Τhe gain of the common and differential signal as a function of frequency.
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One of the most noteworthy elements of this design is that the common and the differential input can be amplified in different frequency ranges.
It is also observed that the Z resistor affects only the differential signal and not the common signal which makes the differential amplifier useful in a variety of application in a diverse number of fields, i.e. biomedicine, geophysics, telecommunications, defense, etc.. Thus, the proposed approach allows the design of complex circuits (filters of higher degree) affecting only the differential signal.

Elimination of the Common Signal Gain

From the simplified analysis of the differential input signal, the “point” between the Z resistors in the circuit has zero voltage. This “point” is a “virtual ground”. If the branches which were connected to the real ground, are connected to the “virtual ground”, the current that flows through these branches will be zero, only in the case of common input signal. So the circuit for the common signal is essentially a buffer and the gain of the circuit of the common signal is eliminated. However, the circuit for the differential signal normally “sees” the filter as if it had the branches connected to the real ground. This fact is clear from the simplified analysis.
The AC Analysis of the filter was performed using the following circuit:
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Figure 3.3.7. The circuit with Va, Vb, simulating the differential signal and Vc simulating the common signal.
Figure 3.3.7. The circuit with Va, Vb, simulating the differential signal and Vc simulating the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3 3.8. Τhe gain of the common and differential signal as a function of frequency.
Figure 3 3.8. Τhe gain of the common and differential signal as a function of frequency.
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As is seen in the AC analysis, by connecting the capacitors in the middle of the Z branch, on the one hand the common-mode signal is indeed eliminated, on the other hand the differential signal’s amplification is not affected and the topology still produces a ~31db gain to it.

3.4. Design Optimization of the Multi-Feedback Differential Filter in the Case of EEG

In the case of EEG, our goal is to amplify the differential input signal between 1-30 Hz without amplifying the common signal. Since our filter has to be broadband, we do not need a high Q filter. In order to achieve this, we remove the capacitors attached to ground. By removing the capacitors, the filter acts like a 1st order low pass filter, with a pole created from the feedback resistor and the feedback capacitor.
In order to achieve a bandpass filter, we replace Z with a series RC, which produces a zero near 1Hz. With these modifications we can design a bandpass filter with slope of almost 20 db/decade, which is equal to the slope of the previously analyzed filter in the case of low Q.
We notice again that the unity constant in the transfer function limits the gain of the filter to 0dB and not lower than this, so in high frequencies the transfer function is dominated by the unit. Because of these alterations, the resistor at the negative input of the operation amplifier is not necessary. This derived improved model version can be seen in the next figure.
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Figure 3.4.1. The circuit of the optimized filter for EEG.
Figure 3.4.1. The circuit of the optimized filter for EEG.
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The AC Analysis of the filter was performed using the following SPICE model:
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Figure 3.4.2. The circuit of the optimized MFDFIA with Va, Vb, simulating the differential signal and Vc simulating the common signal.
Figure 3.4.2. The circuit of the optimized MFDFIA with Va, Vb, simulating the differential signal and Vc simulating the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3.4.3. Τhe graphical representation of the gain of the common and differential signal as a function of frequency.
Figure 3.4.3. Τhe graphical representation of the gain of the common and differential signal as a function of frequency.
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Due to the unit in the transfer function of the circuit, which does not allow the drop of the filter’s gain at high frequencies below 0db, it is necessary to add a low pass filter (at least first degree) before placing the A/D converter, as it is seen in the following figure:
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Figure 3.4.4. The circuit of the optimized MFDFIA, including a low pass filter, with Va, Vb, simulating the differential signal and Vc simulating the common signal.
Figure 3.4.4. The circuit of the optimized MFDFIA, including a low pass filter, with Va, Vb, simulating the differential signal and Vc simulating the common signal.
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The output of the AC analysis is seen in the following figure:
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Figure 3.4.5. Τhe graphical representation of the gain of the common and differential signal as a function of frequency.
Figure 3.4.5. Τhe graphical representation of the gain of the common and differential signal as a function of frequency.
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4. Discussion

The designed novel MFDFIA constitutes an effective solution of the problems stated in the introduction of this work, which are especially true in biosignal acquisition, where maintaining signal integrity is crucial (pre- and post- amplification filtering problems). As shown in section 3 of this work, this novel circuit design can mitigate the aforementioned problems.
By integrating a multi-feedback filter directly into the instrumentation amplifier, the MFDFIA circuit enables simultaneous amplification and filtering of biosignals. A significant advantage of the MFDFIA is its high input impedance, achieved by utilizing the positive input of the amplifiers as the signal input which diminishes issues associated with electrode-skin impedance variability, as the filtering characteristics are practically independent of the source impedance.
This results in an input resistance that is orders of magnitude greater than the resistance of the electrodes, ensuring minimal loading effect on the signal source and preserving the integrity of the biosignals. This characteristic makes it compatible with both wet and dry electrodes which is particularly beneficial in applications like EEG, where electrode-skin impedance is usually unknown.
The symmetric topology simplifies analysis by allowing separate examination of differential and common-mode signals using as the equivalent electronic model only a single-op-amp circuit. By simplifying the model, it is obvious that many complex filters can be created. As a general rule, any branch connected to ground will increase the gain of the common signal.
The use of virtual ground leads to the decoupling of the filter operation in terms of the common and differential signal. This results in the common signal being neither amplified nor filtered, in which case the circuit acts as a buffer. In the case of the differential component of the signal the circuit works normally as a filter.
Using as second stage a differential ADC 24-bit, the MFDFIA can exhibit a high CMRR that depends solely on the second stage of the amplifier. Since the first stage (MFDFIA) behaves as a buffer with respect to the common mode signal, the CMMR is practically dependent on preserving the symmetry of the components, which is a function of their quality and particularly tolerance levels of the components used. This enhances its ability to suppress unwanted common-mode signals such as mains interference and artifacts prevalent in biosignal measurements. The satisfactory noise performance, depending on the components choice of the engineer, can results in a high signal-to-noise ratio (SNR), essential for accurate detection and acquisition of low-amplitude biosignals, meeting the stringent requirements of EEG systems and other bio-amplifier circuits.
On the other hand, a feature of the MFDFIA is its dual-bandwidth capability, as shown in Section 3.3 and 3.4, allowing it to amplify and filter signals within two distinct bandwidth ranges. This flexibility enables the amplifier to function as both a broadband amplifier and a low-pass filter, depending on application requirements. Such adaptability extends its utility to other applications. As an example, in telecommunications the engineers could use the common mode signal to transmit the clock and the differential signal to transmit the information and the MFDFIA can be used to decouple the clock from the information on the receiver’s side.

5. Conclusions

To recapitulate, the novel MFDFIA effectively overcomes the limitations of traditional biosignal acquisition circuits by integrating filtering and amplification into a single stage. This design addresses challenges posed by variable electrode impedance and efficiently suppresses common-mode interference, enhancing the quality of acquired biosignals. Compared to existing designs, the MFDFIA offers a comprehensive solution by combining amplification and filtering in a single stage, reducing component count and system complexity. Its robust performance in gain, bandwidth, noise, and power consumption positions it as a valuable component in biosignal acquisition systems, such as the EEG, and other applications requiring reliable and high-quality signal amplification and filtering, such as applications requiring sensitive sensing.
Simulation results confirm that the amplifier achieves the necessary bandwidth and gain for EEG applications while maintaining low power consumption and low noise levels. Overall, the MFDFIA offers a robust and versatile solution for improving signal integrity in biosignal acquisition systems.

Author Contributions

Conceptualization, A.D.; methodology, A.D., D.P.G. and K.P.; software, A.D., D.P.G. and K.P.; validation, A.D., D.P.G., I.S., E.A. and K.P.; formal analysis, A.D. and D.P.G.; investigation, A.D., D.P.G., I.S. and E.A.; resources, A.D., E.A. and K.P.; data curation, A.D., D.P.G. and I.S.; writing—original draft preparation, A.D., D.P.G. and I.S.; writing—review and editing, A.D., D.P.G., I.S., E.A. and K.P.; visualization, A.D. and D.P.G.; supervision, E.A. and K.P.; project administration, A.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in the study are included in the article.

Acknowledgments

Despoina-Polyxeni Georgiou and Ioannis Stamelos would like to thank the Biomedical Engineering student group "MedHub Athens" from the School of Medicine, National and Kapodistrian University of Athens, of which they are members, for inspiring and supporting their participation in this work.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Kirchhoff current laws (KCL) for circuit in Figure 3.1.1 are:
I X = I C 12 + I R 11 + I R 12
I X = − I C 2 − I R 1 − I R 2
Kirchhoff voltage laws (KVL) for circuit in Figure 3.1.1 are:
V X A − V X B = 2 Z I X
V X A − V A O U T = I R 2 R 2
V X B − V B O U T = I R 12 R 12
Kirchhoff current laws (KCL) for circuit in Figure 3.1.1 are:
V X A − V A R 1 = s C 1 V A − V A O U T ⇒ V X A − V A = s C 1 R 1 V A − V A O U T
V X B − V B R 11 = s C 11 V B − V B O U T ⇒ V X B − V B = s C 11 R 11 V B − V B O U T
Ohm’s Laws in Figure 3.1.1 are:
I C 2 = s C 2 V X A
I C 12 = s C 12 V X B
I R 1 = s C 1 V A − V A O U T
I R 11 = s C 11 V B − V B O U T
From equations (A.4) and (A.5) it can be derived that:
V A O U T − V B O U T = V X A − V X B − I R 2 R 2 + I R 12 R 12 R 2 = R 12
V A O U T − V B O U T = ( V X A − V X B ) + R 2 ( I R 12 − I R 2 )
I R 12 − I R 2 = V A O U T − V B O U T R 2 − V X A − V X B R 2
From equations (A.6) and (A.7) it can be derived that:
( V X A − V A ) − ( V X B − V B ) = s C 1 R 1 V A − V A O U T − s C 11 R 11 V B − V B O U T
C 1 = C 11 , R 1 = R 11 ⇒
( V X A − V X B ) − ( V A − V B ) = s C 1 R 1 ( V A − V B ) − s C 1 R 1 ( V A O U T − V B O U T ) ⇒
( V X A − V X B ) = ( s C 1 R 1 + 1 ) ( V A − V B ) − s C 1 R 1 V A O U T − V B O U T # A . 13
From equations (A.8) and (A.9) it can be derived that:
I C 12 − I C 2 = s C 12 V X B − s C 2 V X A C 12 = C 2 ⇒ I C 12 − I C 2 = − s C 2 V X A − X X B
From equations (A.10) and (A.11) it can be derived that:
I R 11 − I R 1 = s C 11 ( V B − V B O U T ) − s C 1 ( V A − V A O U T ) C 1 = C 11 ⇒
I R 11 − I R 1 = s C 1 V A O U T − V B O U T − s C 1 V A − V B
From equations (A.1) and (A.2) it can be derived that:
2 I X = I C 12 − I C 2 + ( I R 11 − I R 1 ) + I R 12 − I R 2
From equations (A.3), (A.12), (A.14), (A.15) and (A.16) it can be derived that:
V X A − V X B Z = − s C 2 V X A − X X B + s C 1 V A O U T − V B O U T − s C 1 V A − V B + V A O U T − V B O U T R 2 − V X A − V X B R 2
⇒ 1 Z + s C 2 + 1 R 2 V X A − X X B = s C 1 + 1 R 2 V A O U T − V B O U T − s C 1 V A − V B
From equations (A.17) and (A.13) it can be derived that:
1 Z + s C 2 + 1 R 2 ( s C 1 R 1 + 1 ) ( V A − V B ) − 1 Z + s C 2 + 1 R 2 s C 1 R 1 ( V A O U T − V B O U T ) = s C 1 + 1 R 2 V A O U T − V B O U T − s C 1 V A − V B
⇒ 1 Z + s C 2 + 1 R 2 s C 1 R 1 + 1 + s C 1 V A − V B =
= [ 1 Z + s C 2 + 1 R 2 s C 1 R 1 + s C 1 + 1 R 2 ] V A O U T − V B O U T
⇒ V A O U T − V B O U T V A − V B = s 1 Z C 1 R 1 + s 2 C 2 C 1 R 1 + s C 1 R 1 1 R 2 + 1 Z + s C 2 + 1 R 2 + s C 1 s 1 Z C 1 R 1 + s 2 C 2 C 1 R 1 + s C 1 R 1 1 R 2 + s C 1 + 1 R 2
∗ R 6 ⇒ V A O U T − V B O U T V A − V B = s 2 C 2 C 1 R 1 R 2 + s 1 Z C 1 R 1 R 2 + C 1 R 1 + C 2 R 2 + C 1 R 2 + R 2 Z + 1 s 2 C 2 C 1 R 1 R 2 + s 1 Z C 1 R 1 R 2 + C 1 R 1 + C 1 R 2 + 1
⇒ V A O U T − V B O U T V A − V B = 1 + s C 2 R 2 + R 2 Z s 2 C 2 C 1 R 1 R 2 + s C 1 1 Z R 1 R 2 + R 1 + R 2 + 1
⇒ V A O U T − V B O U T V A − V B = 1 + 1 C 2 C 1 R 1 R 2 ∗ s C 2 R 2 + R 2 Z s 2 + s C 1 1 Z R 1 R 2 + R 1 + R 2 C 2 C 1 R 1 R 2 + 1 C 2 C 1 R 1 R 2
V A O U T − V B O U T V A − V B = 1 + 1 C 1 R 1 C 2 Z R 2 R 2 ∗ s C 2 Z + 1 ∗ 1 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2 #
V A O U T − V B O U T V A − V B = 1 + R 2 Z ∗ C 2 Z s + 1 C 2 Z ∗ 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2 #
V A O U T − V B O U T V A − V B = 1 + R 2 Z ∗ C 2 Z s + 1 C 2 Z ∗ 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2

Appendix B

Kirchhoff current laws (KCL) for circuit in Figure 3.2.2 are:
I R 1 + I C 2 + I R 2 + I X = 0
Ohm’s laws for circuit in Figure 3.2.2 are:
I R 1 = V X − V a R 1
I R 1 = s C 1 V a − V d i f f _ o u t
I C 2 = s C 2 V X
I R 2 = V X − V d i f f _ o u t R 2
I X = V X Z
From equations (C.2) and (C.3) it can be derived that:
V X − V a R 1 = s C 1 V a − V d i f f _ o u t
⇒ V X = V a s C 1 R 1 + 1 − s C 1 R 1 V d i f f _ o u t
From equations (C.1),(C.3),(C.4),(C.5),(C.6):it can be derived that:
s C 1 V a − s C 1 V d i f f _ o u t + s C 2 V X + V X R 2 − V d i f f _ o u t R 2 + V X Z = 0
⇒ s C 1 V a + V X s C 2 + 1 R 2 + 1 Z − V d i f f o u t s C 1 + 1 R 2 = 0 # C . 8
From equations (C.7),(C.8) it can be derived that:
s C 1 V a + V a s C 1 R 1 + 1 s C 2 + 1 R 2 + 1 Z − s C 2 + 1 R 2 + 1 Z s C 1 R 1 V d i f f _ o u t − V d i f f _ o u t s C 1 + 1 R 2 = 0
⇒ V a s C 1 + s 2 C 1 C 2 R 1 + s C 1 R 1 1 R 2 + s C 1 R 1 1 Z + s C 2 + 1 R 2 + 1 Z
= V d i f f _ o u t s 2 C 1 C 2 R 1 + s C 1 R 1 1 R 2 + s C 1 R 1 1 Z + s C 1 + 1 R 2
⇒ V d i f f _ o u t V a = s 2 C 1 C 2 R 1 R 2 + s C 1 R 2 + C 1 R 1 + C 1 R 1 R 2 Z + C 2 R 2 + R 2 Z + 1 s 2 C 1 C 2 R 1 R 2 + s ( C 1 R 1 + C 1 R 1 R 3 Z + C 1 R 2 ) + 1
⇒ V d i f f _ o u t V a = 1 + s C 2 R 2 + R 2 Z s 2 C 1 C 2 R 1 R 2 + s ( C 1 R 1 + C 1 R 1 R 2 Z + C 1 R 2 ) + 1
⇒ V d i f f _ o u t V a = 1 + 1 C 1 C 2 R 1 R 2 ∗ s C 2 R 2 + R 2 Z s 2 + s C 1 R 1 + C 1 R 1 R 2 Z + C 1 R 2 C 1 C 2 R 1 R 2 + 1 C 1 C 2 R 1 R 2
⇒ V d i f f _ o u t V a = 1 + C 2 R 2 s + 1 Z C 2 ∗ 1 C 1 C 2 R 1 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 1 C 2 R 1 R 2
⇒ V d i f f _ o u t V a = 1 + R 2 Z ∗ C 2 Z s + 1 C 2 Z ∗ 1 C 1 C 2 R 1 R 2 s 2 + s 1 C 2 1 Z + 1 R 2 + 1 R 1 + 1 C 1 C 2 R 1 R 2

Appendix C

Kirchhoff current laws (KCL) for circuit in Figure 3.2.1 are:
I R 1 + I R 2 + I C 2 = 0
Ohm’s laws for circuit in Figure 3.2.1 are:
I R 1 = V X − V c R 1
I R 1 = s C 1 V c − V c o m _ o u t
I R 2 = V X − V c o m _ o u t R 2
I C 2 = s C 2 V X
From equations (B.2),(B.3) it can be derived that:
V X − V c R 1 = s C 1 V c − V c o m _ o u t
⇒ V X = V c s C 1 R 1 + 1 − s C 1 R 1 V c o m _ o u t
From equations (B.1),(B.3),(B.4),(B.5) it can be derived that:
s C 1 V c − s C 1 V c o m _ o u t + s C 2 V X + V X R 2 − V c o m _ o u t R 2 = 0
⇒ s C 1 V c + V X s C 2 + 1 R 2 − V c o m _ o u t s C 1 + 1 R 2 = 0
From equations (B.6),(B.7) it can be derived that:
s C 1 V c + V c s C 1 R 1 + 1 s C 2 + 1 R 2 − s C 2 + 1 R 2 s C 1 R 1 V c o m _ o u t − V c o m _ o u t s C 1 + 1 R 2 = 0
⇒ V c s C 1 + s 2 C 1 C 2 R 1 + s C 1 R 1 1 R 2 + s C 2 + 1 R 2
= V c o m _ o u t s 2 C 1 C 2 R 1 + s C 1 R 1 1 R 2 + s C 1 + 1 R 2
⇒ V c o m _ o u t V c = s 2 C 1 C 2 R 1 R 2 + s C 1 R 2 + C 1 R 1 + C 2 R 2 + 1 s 2 C 1 C 2 R 1 R 2 + s ( C 1 R 1 + C 1 R 2 ) + 1
⇒ V c o m _ o u t V c = 1 + s C 2 R 2 s 2 C 1 C 2 R 1 R 2 + s ( C 1 R 1 + C 1 R 2 ) + 1
⇒ V c o m _ o u t V c = 1 + 1 C 1 R 1 ∗ s s 2 + s C 1 R 1 + C 1 R 2 C 1 C 2 R 1 R 2 + 1 C 1 C 2 R 1 R 2
V c o m _ o u t V c = 1 + s C 2 R 2 1 C 1 R 1 C 2 R 2 s 2 + s 1 C 2 1 R 2 + 1 R 1 + 1 C 2 C 1 R 1 R 2

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