Abstract: This study aimed to provide addition to the work by introducing the NAM equations as well as the pathological realizations of four important active building blocks. The four active building blocks are the Balanced Output Op Amp (BOOA), the Differential Difference Amplifier (DDA), the Differential Difference Operational Floating Amplifier (DDOFA) and the Differential Difference Operational Mirror Amplifier (DDOMA). The importance of the pathological realizations in the generation of alternative ideally equivalent realizations of an active building block is illustrated by an example on the BOOA.
INTRODUCTION
In the circuit synthesis method proposed by Haigh et al. (2005) and based on NAM expansion it is necessary to represent each passive and active circuit element by a NAM equation. The NAM stamp for representation of a nullor having a nullator connected between two nodes and the norator connected between two alternative nodes is derived by Haigh et al. (2005) based on a Voltage Controlled Current Source (VCCS) representation of the non-ideal nullor. The limit variables also called, the infinity variables is used by Haigh et al. (2006) to represent the nullor by a 2x2 NAM stamp. An alternative notation for the nullor elements which is defined as the bracket notation was also introduced by Haigh et al. (2005). Applications of the NAM model for the nullor using limit variables in circuit design was given in details by Haigh and Radmore (2006). Transformation method from symbolic transfer function to active-RC circuit by NAM expansion was demonstrated clearly by Haigh (2006). Recently the synthesis method based on NAM expansion using nullor elements (Haigh et al., 2005; Haigh, 2006) was extended to accommodate pathological mirror elements resulting in a generalized framework encompassing nullator, norator, pathological Voltage Mirror (VM) and pathological Current Mirror (CM) (Saad and Soliman, 2008b). Table 1 includes a summary of the NAM stamps using infinity parameters for the nullator-CM pair, VM-norator pair and VM-CM pair (Saad and Soliman, 2008b). The generalized NAM expansion method was used in the generation of gyrators (Saad and Soliman, 2008a; Awad and Soliman, 1999). NAM stamps and pathological representation of several types of active building blocks was given by Saad and Soliman (2010). Table 2 includes a summary of NAM stamps of the op amp and Current Op Amp (COA) (Soliman, 2009). Table 3 includes a summary of the NAM stamps of the current conveyor (CCII) and inverting current conveyor (ICCII) families (Saad and Soliman, 2008a; Soliman, 2009; Awad and Soliman, 1999). The generalized NAM expansion method was used in the generation of gyrators (Saad and Soliman, 2008a).
| Table 1: | NAM description of the four basic floating building blocks (Saad and Soliman, 2008a) |
Due to the importance of the NAM in the synthesis of active RC circuits; NAM stamps of BOOA, DDA, DDOFA and DDOMA are derived in this study and their pathological realizations are also given.
BALANCED OUTPUT OP AMP (BOOA)
The fully balanced integrator introduced by Banu and Tsividis (1983) is based on using the BOOA as the active building block together with two matched MOS transistors operating in the non-saturation region and two equal capacitors. This integrator provides cancellation of the even nonlinearities introduced by the MOS transistors. Full nonlinearity cancellation using the four MOS transistor cell operating in the non-saturation region was introduced by Czarnul (1986) using also the BOOA as the active building block. The BOOA was also used in realization of a continuous time CMOS balanced filter introduced by Banu and Tsividis (1985). The BOOA was also used in the realization of the universal op amp proposed by Ramírez-Angulo and Ledesma (2006). Also most recently the BOOA has been used in the design of high-speed low-power SC circuits (Amoroso et al., 2010).
The symbolic representation of the BOOA is shown in Fig. 1a. In order to derive the NAM stamp of the BOOA assume that it is non-ideal and add the admittances Yc and Yd at the two output ports of the equivalent Voltage Controlled Voltage Source (VCVS) model as shown in Fig. 1b.
| Table 2: | NAM stamp of the op amp family (VOA and COA) |
| Fig. 1: | (a) Symbolic representation of the BOOA and (b) Equivalent VCVS model for the non-ideal BOOA |
| Table 3: | NAM stamp of the op amp family (VOA and COA) |
The NAM equation of the non-ideal BOOA is given by:
| (1) |
Setting Yc and Yd to their ideal values of infinity the NAM stamp of the ideal BOOA is obtained as:
| (2) |
It is desirable to see how the ideal BOOA equations are obtained from the above NAM stamp. From Eq. 1 the following voltage matrix equation can be obtained by dividing third row by Yc and fourth row by Yd thus:
| (3) |
Setting Yc and Yd to their ideal values of infinity the above equation simplifies to:
| (4) |
The output voltage Vc is obtained from the first equation in the above matrix as:
| (5a) |
Similarly the output voltage Vd is obtained from the second equation in the above matrix as:
| (5b) |
Equation 5a and b represent the BOOA where Av is voltage gain which is very high and frequency dependent and is theoretically infinity.
DIFFERENTIAL DIFFERENCE AMPLIFIER (DDA)
The DDA was introduced by Sackinger and Guggenbiihl (1987) as a new analog building block. New CMOS realizations of the DDA and several new circuit applications were given by Zarabadi et al. (1992) and Huang et al. (1993). The symbolic representation of the DDA is shown in Fig. 2a. In order to derive the NAM stamp of the DDA assumes that it is non-ideal and has finite output admittance Yo. Figure 2b represents the VCVS model of the non-ideal DDA, which can be represented by the following NAM equation:
| (6) |
| Fig. 2: | (a) Symbolic representation of the DDA and (b) Equivalent VCVS model for the non-ideal DDA |
From the above Equation the Y matrix of the DDA in the non-ideal case is given by:
| (7) |
Setting Yo to its ideal value of infinity the NAM stamp of the ideal DDA is obtained as:
| (8) |
It is of interest to verify that the ideal DDA equation can be derived from the above NAM equations. The last row in the NAM Eq. 6 can be written in the form of the following voltage equation:
| (9) |
This voltage matrix equation can be written as:
| (10) |
In the ideal case Yo equal to infinity and Eq. 9 reduces to:
| (11) |
The output voltage Vo is obtained as follows:
| (12) |
In the ideal case Av equal to infinity and the above equation simplifies to:
| (13) |
The above equation represents the well-known equation defining the DDA in the ideal case with infinity gain.
DIFFERENTIAL DIFFERENCE OPERATIONAL FLOATING AMPLIFIER (DDOFA)
The DDOFA was introduced by (Mahmoud and Soliman, 1998) as a new analog building block. New CMOS realization and several new analog circuit applications were also introduced by Mahmoud and Soliman (1998). The symbol of the DDOFA is shown in Fig. 3a where Gm in the ideal case is infinity. Figure 3b represents the VCCS model of the DDOFA from which the following NAM equation is obtained:
| (14) |
The above NAM equation can also be written as follows:
| (15) |
In the ideal case the NAM stamp of the DDOFA becomes:
| (16) |
It is of interest to verify that the ideal DDOFA equation can be derived from the above NAM equations, dividing both sides of Eq. 14 by Gmi thus:
| (17) |
| Fig. 3: | (a) Symbolic representation of the DDOFA and (b) Equivalent VCCS model for the DDOFA |
In the ideal case the transconductance Gmi approaches infinity thus:
| (18) |
Each of the above two rows gives the same equation which represents the ideal DDOFA and given by:
| (19) |
DIFFERENTIAL DIFFERENCE OPERATIONAL MIRROR AMPLIFIER (DDOMA)
The DDOMA was introduced by (Soltan and Soliman, 2009) as a new analog building block. New CMOS realization and analog circuit applications were also introduced by Soltan and Soliman (2009). The symbol of the DDOMA is shown in Fig. 4a where Gm in the ideal case is infinity. Figure 4b represents a four VCCS model of the DDOMA from which the NAM equation is obtained:
| (20) |
| Fig. 4: | (a) Symbolic representation of the DDOMA and (b) Equivalent VCCS model for the DDOMA |
The above NAM equation can also be written as follows:
| (21) |
From Eq. 20 therefore:
| (22) |
In the ideal case the transconductance Gmi approaches infinity thus:
| (23) |
Each of the above two rows gives the same equation which represents the ideal DDOFA and given by:
| (24) |
The derived four new NAM stamps are very useful in computing small signal characteristics of analog circuits using Nodal Admittance (NA) analysis methods using CAD tools (Tlelo-Cuautle et al., 2010a).
Most recently the derivation of NAM stamps of the Operational Trans-resistance Amplifier (OTRA) and the Current Operational Amplifier (COA) have been reported by Sanchez-Lopez et al. (2010).
PATHOLOGICAL REALIZATIONS
The second part of this paper includes the pathological realizations of the four active building blocks considered in this paper. The pathological realizations are very useful in the generation of alternative ideally equivalent realizations of an active building block (Carlin, 1964; Cabeza and Carlosena, 1993).
| Fig. 5: | (a) Pathological realization I of the BOOA and (b) Pathological realization II of the BOOA, (c) Pathological realization III of the BOOA |
Three equivalent alternative pathological realizations of the BOOA are given next. The first pathological realization of the BOOA employs one nullator, one VM and two norators is shown in Fig. 5a. The second pathological realization of the BOOA employs three nullators, three norators and two grounded resistors and is shown in Fig. 5b. This realization is based on replacing the VM by its nullor equivalent circuit given by Saad and Soliman (2010) and Awad and Soliman (1999). The third pathological realization of the BOOA employs three nullators, three norators and two floating resistors and is shown in Fig. 5c. This realization is based on replacing the VM by its nullor equivalent circuit given by (Saad and Soliman, 2010; Awad and Soliman, 1999).Three alternative ideally equivalent realizations of the BOOA are shown in Fig. 6 and are obtained from Fig. 5. The circuit derived in Fig. 6a uses an OA and an inverting CCII-. The circuit derived in Fig. 6b uses a nullor, two OAs and two grounded resistors. The circuit derived in Fig. 6c uses a nullor, two OAs and two virtually grounded resistors.
The pathological realization of the DDA employs one nullator; four VM and five norator four of them are dummy and shown in Fig. 7.
| Fig. 6: | (a) Realization I of the BOOA based on Fig. 5a, (b) Realization II of the BOOA based on Fig. 5b and (c) Realization III of the BOOA based on Fig. 5c |
| Fig. 7: | Pathological realization of the DDA |
The pathological realization of the DDOFA employs one nullator; four VM and five norator four of them are dummy and shown in Fig. 8.
| Fig. 8: | Pathological realization of the DDOFA |
| Fig. 9: | Pathological realization of the DDOMA |
The pathological realization of the DDOMA employs one nullator, four VM, one CM and four dummy norator and shown in Fig. 9. Alternative pathological realizations of the DDA, DDOFA and DDOMA can be derived by replacing the VMs by equivalent nullator norator realization and are not included to limit the length of this letter.
It is important to point out that for a physical realization of the circuit the total number of nullator plus the number of VM must equal to the number of norator plus the number of CM as explained by Awad and Soliman (1999).
CONCLUSIONS
The NAM stamp of the BOOA is derived from a two VCVS model assuming finite output impedance at both output terminals. Pathological representation of the BOOA using a nullator, VM and two norator is also given. The NAM stamp of the DDA is derived from a VCVS model assuming finite output impedance. Pathological representation of the DDA using a nullator, four VM and five norator four of them are added to achieve equal number of nullator plus VM and number of norator is also given. The NAM stamp of the DDOFA is derived from a two VCCS model. Pathological representation of the DDOFA using a nullator, four VM and five norator is also given. Finally the NAM stamp of the DDOMA is derived from a four VCCS model. Pathological representation of the DDOMA using a nullator, four VM, one CM and four dummy norators is also given. The NAM equations and the pathological realizations will be useful in the design automation of analogue integrated circuits (Amiri et al., 2008; Chong et al., 2007; Garcia-Ortega et al., 2007; Masmoudi et al., 2005; Tlelo-Cuautle et al., 2010b).