Research Article  Open Access
SFG Synthesis of General HighOrder AllPass and AllPole Current Transfer Functions Using CFTAs
Abstract
An approach of using the signal flow graph (SFG) technique to synthesize general highorder allpass and allpole current transfer functions with current follower transconductance amplifiers (CFTAs) and grounded capacitors has been presented. For general nthorder systems, the realized allpass structure contains at most n + 1 CFTAs and n grounded capacitors, while the allpole lowpass circuit requires only n CFTAs and n grounded capacitors. The resulting circuits obtained from the synthesis procedure are resistorless structures and especially suitable for integration. They also exhibit lowinput and highoutput impedances and also convenient electronic controllability through the value of the CFTA. Simulation results using real transistor model parameters ALA400 are also included to confirm the theory.
1. Introduction and Motivation
In 2008, the conception of the current follower transconductance amplifier (CFTA) has been introduced [1]. The CFTA device is slightly modified from the conventional current differencing transconductance amplifier (CDTA) [2] by replacing the current differencing unit with a current follower. Conceptually, the CFTA element is a combination of the current follower and the multioutput operational transconductance amplifier. Consequently, several structures for realizing currentmode active filters using CFTAs were developed [3–7]. Interesting circuit realizations of general thorder currentmode allpass filters can be found in [8–15]. However, the works in [8–13] do not include electronic controllability. Moreover, filter structures in [8–12, 15] needed external passive resistors. Although the electronically tunable filters were reported in [14, 15], the input terminals of these structures are not in lowimpedance level. The configuration of [15] also employs an external passive resistor for CDTAbased circuit realization. In addition, the realization of general highorder allpole lowpass current transfer function using current conveyors was introduced in [16]. The circuit still requires a large number of active and passive components, at most current conveyors, n resistors and n capacitors.
In this paper, the signal flow graph (SFG) procedure is applied to synthesize the highorder, allpass, and allpole current transfer functions using CFTAs as active elements together with grounded capacitors as passive elements. The approach is based on drawing a signal flow graph directly from the given transfer function and then obtaining, from the graph, the active filter involving CFTAs. The design procedure shows that the resulting structures are canonical in the number of active components, CFTAs for realizing nthorder allpass circuit and CFTAs for realizing thorder allpole circuit. Only n grounded capacitors are used as passive components for all circuit realizations, making them suitable for integration point of view. As desired, the realized circuits have lowinput and highoutput impedance properties, which especially enable cascading. The circuits also have low sensitivity characteristics, and exhibit electronic controllability of important filter coefficients via transconductance gains () of CFTAs. To demonstrate the proposed approach, the thirdorder currentmode allpass filter and Butterworth allpole lowpass filter were designed and simulated using PSPICE program.
2. Basic Concept of the CFTA
The symbolic representation of the CFTA and its behavior model are shown in Figure 1. Assuming the standard notation, the terminal defining relations of this device can be characterized by the following [3–6]: where is the transconductance gain of the CFTA and is an external impedance connected to the terminal. The CFTA consists essentially of the current follower at the input part and the multioutput transconductance amplifier at the output part. According to (1) and Figure 1, the terminal forms the current input terminal at ground potential () and the output current at the terminal () follows the current () through the terminal. The voltage drop at the terminal () is then converted to a current at the terminal () by a parameter. In general, the value is adjustable over several decades by a supplied bias current/voltage, which lends electronic controllability to design circuit parameters.
(a) Schematic symbol
(b) Behavioral model
The possible implementation of the bipolar technologybased CFTA is shown in Figure 2 [6, 7]. It is mainly composed of a current follower circuit constructed by transistors and a multipleoutput transconductance amplifier . In this case, the transconductance gain () of the CFTA is directly proportional to the external bias current , which is approximately equal to and mV at 27°C.
3. SFG Synthesis of General HighOrder AllPass Current Transfer Functions
3.1. Realization Procedure
The general form of an thorder allpass current transfer function can be expressed by the following expression:
Equation (3) can be represented by the SFG as shown in Figure 3. It is clearly seen that the graph consists of two basic operations, which are multioutput current follower and current lossless integrator, as redrawn in Figures 4(a) and 4(c), respectively. Using the current and voltage relations of the CFTA given in (1), we easily find that these two subgraphs can be realized using CFTA by the subcircuits as shown in Figures 4(b) and 4(d), respectively. For the CFTAbased realization, it can readily obtain the CFTAC circuit by interconnecting the corresponding subcircuits of Figures 4(b) and 4(d) according to the overall signal flow graph representation of Figure 3. Therefore, the CFTAC circuit realizing any thorder allpass current transfer function can be shown in Figure 5. For this realization, it has to be noted that, for general nthorder filter function, the proposed filter configuration requires CFTAs as active elements and capacitors as passive elements. Also note that the circuit realization uses only grounded capacitors that are suitable for the integrated circuit implementation point of view and also exhibits lowinput impedance and highoutput impedance terminals that are desirable for cascading in currentmode operation [17].
(a)
(b)
(c)
(d)
From the CFTAbased circuit realization, the design equations can be obtained through comparing the SFG representation of Figure 3 with Figure 5. The results are summarized as follows: It should be noted from the above expressions that the coefficients () of the realized function can be tuned electronically by adjusting the value of the CFTA.
3.2. Nonideal Analysis
In this subsection, the effect of the nonidealities of the CFTA on the circuit performance is discussed. In case of the nonideal characteristic condition, the port relations of the CFTA given in (1) can be rewritten as follows: where is the nonideal current gain from f to z terminals, and is transconductance inaccuracy factor from to terminals of the CFTA, respectively. Reanalyzing the realized structure in Figure 5 with (5) yields the following nonideal current transfer function: where In (7), and (for ) represent the parameters and of the ithCFTA, respectively. Practically, these transfer gains differ from unity by tracking errors of the CFTA. More specifically, denotes the parasitic resistance at the terminal of the ()thCFTA, which is ideally equal to infinity.
3.3. Design Example
To demonstrate the proposed design procedure, the thirdorder allpass transfer function is considered. Generally, the current transfer function of the normalized thirdorder allpass filter is defined as follow: It can be easily shown that the above transfer function can be represented by the SFG shown in Figure 6(a), and the corresponding circuit realization of this graph is thus shown in Figure 6(b). In this example, the design equations of the circuit are found as follow: Thus, the normalized component values are obtained as , and . Routine circuit analysis of Figure 6(b) yields the current transfer function as follows: The active and passive sensitivities of various parameters for the designed circuit of Figure 6(b) are analyzed and the results are given in Table 1. It is clearly seen that all the sensitivities are low and within unity in magnitude.

(a) SFG representation
(b) CFTA realization
4. SFG Synthesis of General HighOrder AllPole Current Transfer Functions
Considering the SFG of Figure 3 if all the forward parts are removed, we obtain the suitable SFG as redrawn in Figure 7(a). In this case, the corresponding current transfer function is expressed in the following form: It is apparent that this equation is now in the form of general thorder allpole lowpass current transfer function. Since the SFG representation of this function is slightly modified from Figure 3, then the CFTAbased circuit realization can easily be obtained by using the same realization procedure presented in Figure 4. The resulting circuit is shown in Figure 7(b). It is clearly seen that the circuit includes at most CFTAs and grounded capacitors. Furthermore, the design equations of the circuit in Figure 7(b) are also the same as given in (4).
(a) SFG representation
(b) CFTA realization
As an example of the useful application of the given second structure, let us consider the thirdorder allpole lowpass Butterworth function. In this case, the current transfer function has the canonical form: The SFG representing (12) can be drawn in Figure 8(a). Referring to Figure 4, the circuit realization of this graph is shown in Figure 8(b) and has the following: Thus, the normalized component values are obtained as , , , and A/V. Routine circuit analysis of Figure 8(b) yields the current transfer function as follows: From (14), the coefficient sensitivities to active and passive components ( and ) are 1 or 0. Also, the active and passive sensitivities of the natural angular frequency and quality factor ( and ) are calculated and found as 1 or 0. Thus, all the sensitivities are small.
(a) SFG representation
(b) CFTA realization
5. Computer Simulations and Results
To verify the theoretical analysis, the proposed design procedure given above has been simulated with PSPICE simulation program. To implement the CFTA active device in simulations, the bipolar technology structure depicted in Figure 2 has been employed using transistor model parameters PR100N (PNP) and NP100N (NPN) of the bipolar arrays ALA400. The DC supply voltages and bias currents were set as and , respectively.
As an example, the illustrative currentmode thirdorder allpass filter of Figure 6(b) was designed with rad/sec. For this purpose, the denormalized component values were chosen as nF, mA/V ( A), mA/V ( A), mA/V ( A), and mA/V ( A). The simulated frequency responses comparing with the theoretical values are shown in Figure 9. We see that the phase response is found to vary with frequency from 0° to −540°, with a value of −270° at the pole frequency kHz. From the results, the simulated was found to be 154 kHz, which is in close proximity to the ideal value. Next, applying a sinusoidal current input signal of 20 A peak at 159 kHz to the circuit, the input and output waveforms are given in Figure 10. This results in 4.86 s time delay at the output current corresponding to −278° phase shift. In Figure 11, the frequency characteristics of the CFTAbased currentmode allpass filter in Figure 5 for and 2 are also depicted. It can be observed that the simulation results agree very well with the ideal ones.
(a) Gain responses
(b) Phase responses
For the CFTAbased thirdorder allpole lowpass Butterworth circuit realization of Figure 8(b), the circuit with the pole frequency of kHz was designed with the following settings: nF, mA/V, mA/V and mA/V. The simulated results and ideal values of the gain responses are depicted in Figure 12.
6. Conclusions
This study presents a signal flow graph (SFG) approach to synthesize the general, highorder, allpass, and allpole lowpass current transfer functions by an active circuit. By using the SFG representation, any nthorder allpass current transfer function can be realized by employing CFTAs and grounded capacitors, and thorder allpole lowpass current transfer function can also be realized employing CFTAs and grounded capacitors. It has been shown that the design procedure given here is resistorless structure, electronic tuning property, and suitable for integration. Furthermore, the circuit has low sensitivity. PSPICE simulation results which agree very well with the theoretical analysis are also included.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang (KMITL). The author also expresses gratitude to Mr. Kritsakon Intawichai for his performing simulations in initial stage.
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Copyright
Copyright © 2014 Worapong Tangsrirat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.