# An Electronically Adjustable Floating Memcapacitor Emulator Circuit Using CDBA
## 2 Proposed Memcapacitor Emulator Circuit  
Classical circuit elements such as resistors, capacitors and inductors are expressed by the relationships between voltage, current, charge and flux, which are defined as fundamental electrical quantities. In contrast to these relationships, the definition of the memcapacitor is given by a relation of the form \(F(\varphi,\sigma)=0\) where \(\varphi\) and \(\sigma\) are the time integrals in \((-\infty,t]\) of the voltage and charge, respectively. If this relation is expressible as a map \(\sigma\rightarrow\varphi\) (\(\varphi\rightarrow\sigma\)), the memcapacitor is said to be charge- (voltage-) controlled. By assuming that either map is differentiable, we can express the voltage in terms of the charge (or vice versa) in the form \(v=C_{M}^{-1}(\sigma)q\) (\(q=C_{M}(\varphi)v\)), where  
\[C_{M}^{-1}(\sigma)=\frac{v(t)}{q(t)}=\frac{d\varphi}{dt}\frac{dt}{d\sigma}= \frac{d\varphi}{d\sigma}, \tag{1}\]  
\[C_{M}(\varphi)=\frac{q(t)}{v(t)}=\frac{d\sigma}{dt}\frac{dt}{d\varphi}=\frac{ d\sigma}{d\varphi}. \tag{2}\]  
Here, \(C_{M}^{-1}(\sigma)\) is the inverse memcapacitance of the charge-controlled memcapacitor and \(C_{M}(\varphi)\) is the memcapacitance of the voltage-controlled memcapacitor. This means that the quantity \(C_{M}^{-1}(\sigma)\) (\(C_{M}(\varphi)\)) depends on the entire time history of the input variable. The memcapacitor therefore has a memory-dependent property and is an energy-storing electronic element [2,43]. Finally, in this work, we intend to design a charge-controlled memcapacitor emulator circuit that satisfies the following equation  
\[v(t)=C_{M}^{-1}\bigg{(}\int_{t_{0}}^{t}q(\tau)d\tau\bigg{)}q(t). \tag{3}\]  
However, (3) is inadequate to express any parasitic effects and has no internal state vector. Instead, it is more realistic to use the expression given by  
\[v(t)=(\alpha\pm\beta\sigma(t))q(t). \tag{4}\]  
Here, \(\alpha\) is the initial value of the memcapacitance and \(\beta\) represents the capacitance change corresponding to the charge flow in the memcapacitor, respectively [42,44]. In this section, a new memcapacitor emulator based on the CDBA is proposed. The CDBA, whose block diagram is shown in Figure 1, is implemented using two current feedback operational amplifiers, commercially available as AD844. The CDBA is an active element with four terminals that can be used in current and voltage mode analog signal processing circuits and filters. The \(p\) terminal of the CDBA is called the positive (non-inverting) input and the \(n\) terminal is called the negative (inverting) input. Both input terminals are virtually grounded, making them the preferred current inputs. The \(z\) terminal is a current output and the \(w\) terminal is a voltage output, which follows the voltage of the \(z\) terminal. The CDBA has low impedance inputs on the \(p\) and \(n\) terminals and a low impedance output on the \(w\) terminal. This allows it to be used as a cascadable active device [45]. The basic electrical descriptions of the terminals are expressed as follows  
\[i_{z}=i_{p}-i_{n},\;v_{p}=v_{n}=0,\;v_{w}=v_{z}. \tag{5}\]  
Among these, the CDBA is notable for offering a balanced trade-off between performance and simplicity. It provides a wide bandwidth in the range of 10-100 MHz, moderate input-referred noise (approximately 20-80 nV/ \(\sqrt{\text{Hz}}\)), and low to moderate power consumption (typically 200-600 \(\upmu\)W) [46; 47]. Thanks to its differential current input and buffered voltage output, the CDBA enables easy cascading in current-mode systems without significant loading effects or signal degradation. Although Operational Transconductance Amplifier (OTA) structures can achieve very low noise levels (as low as 10-50 nV/ \(\sqrt{\text{Hz}}\)) under optimized biasing conditions, they often suffer from limited linearity, narrow bandwidth (around 1-5 MHz), and pronounced sensitivity to process, temperature, and voltage variations [48], which can compromise reliability in practical designs.