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 nonideal 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 activeRC 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 nullatorCM pair, VMnorator pair and
VMCM 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).
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 nonsaturation 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 nonsaturation 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
RamírezAngulo and Ledesma (2006). Also most
recently the BOOA has been used in the design of highspeed lowpower 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 nonideal and add the admittances Y_{c} and Y_{d} 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 nonideal BOOA 
Table 3: 
NAM stamp of the op amp family (VOA and COA) 

The NAM equation of the nonideal BOOA is given by:
Setting Y_{c} and Y_{d} to their ideal values of infinity the NAM stamp of the ideal BOOA is obtained as:
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 Y_{c} and fourth row by Y_{d}
thus:
Setting Y_{c} and Y_{d} to their ideal values of infinity the above equation simplifies to:
The output voltage V_{c} is obtained from the first equation in the above matrix as:
Similarly the output voltage V_{d} is obtained from the second equation in the above matrix as:
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 nonideal and has finite
output admittance Y_{o}. Figure 2b represents the
VCVS model of the nonideal DDA, which can be represented by the following NAM
equation:

Fig. 2: 
(a) Symbolic representation of the DDA and (b) Equivalent
VCVS model for the nonideal DDA 
From the above Equation the Y matrix of the DDA in the nonideal case is given by:
Setting Y_{o} to its ideal value of infinity the NAM stamp of the ideal DDA is obtained as:
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:
This voltage matrix equation can be written as:
In the ideal case Y_{o }equal to infinity and Eq. 9 reduces to:
The output voltage V_{o} is obtained as follows:
In the ideal case A_{v }equal to infinity and the above equation simplifies to:
The above equation represents the wellknown 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 G_{m} in the ideal case is infinity. Figure 3b
represents the VCCS model of the DDOFA from which the following NAM equation
is obtained:
The above NAM equation can also be written as follows:
In the ideal case the NAM stamp of the DDOFA becomes:
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 G_{mi} thus:

Fig. 3: 
(a) Symbolic representation of the DDOFA and (b) Equivalent
VCCS model for the DDOFA 
In the ideal case the transconductance G_{mi} approaches infinity thus:
Each of the above two rows gives the same equation which represents the ideal DDOFA and given by:
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 G_{m}
in the ideal case is infinity. Figure 4b represents a four
VCCS model of the DDOMA from which the NAM equation is obtained:

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:
From Eq. 20 therefore:
In the ideal case the transconductance G_{mi} approaches infinity thus:
Each of the above two rows gives the same equation which represents the ideal DDOFA and given by:
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
(TleloCuautle et al., 2010a).
Most recently the derivation of NAM stamps of the Operational Transresistance
Amplifier (OTRA) and the Current Operational Amplifier (COA) have been reported
by SanchezLopez 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; GarciaOrtega et al., 2007;
Masmoudi et al., 2005; TleloCuautle
et al., 2010b).