# Universal Voltage Conveyor and its Novel Dual-Output Fully-Cascadable VM APF Application
## 2 Circuit Description  
The circuit symbol of the UVC is shown in Figure 1a. Using standard notation, relations between its individual terminals can be described by the following hybrid matrix:  
\[\left[\begin{array}{c}i_{\rm X}\\ v_{\rm YP}\\ v_{\rm YN}\\ i_{\rm W}\\ v_{\rm ZP}\\ v_{\rm ZN}\end{array}\right]=\left[\begin{array}{cccccc}Y_{\rm X}&\alpha_{1} (s)&-\alpha_{2}(s)&0&0&0\\ 0&Z_{\rm YP}&0&\delta_{1}(s)&0&0\\ 0&0&Z_{\rm YN}&\delta_{2}(s)&0&0\\ 0&0&0&Y_{\rm W}&0&0\\ \gamma_{1}(s)&0&0&0&Z_{\rm ZP}&0\\ -\gamma_{2}(s)&0&0&0&0&Z_{\rm ZN}\end{array}\right]\left[\begin{array}{c}v_{ \rm X}\\ i_{\rm YP}\\ i_{\rm YN}\\ v_{\rm W}\\ i_{\rm ZP}\\ i_{\rm ZN}\end{array}\right], \tag{1}\]  
where \(Y_{\rm X}=sC_{\rm X}+1/R_{\rm X}\), \(Y_{\rm W}=sC_{\rm W}+1/R_{\rm W}\) are parasitic admittances and \(Z_{k}=R_{k}\) (\(k=\)YP, YN, ZP, ZN) are the parasitic resistances at relevant terminals of the UVC, respectively, discussed in detail for the readily available UVC-N1C 0520 chip in [22; 23; 24]. Parameters \(\alpha_{j}(s)\), \(\delta_{j}(s)\), and \(\gamma_{j}(s)\) are, respectively, frequency-dependent non-ideal current and voltage gains for \(j=\{1,2\}\). Ideally, these parameters are equal to unity. Using a single-pole model, they can be defined as:  
\[\alpha_{j}(s)=\frac{\alpha_{\rm oj}}{1+\tau_{\alpha_{j}}s},\ \ \ \ \ \delta_{j}(s)=\frac{\delta_{\rm oj}}{1+\tau_{\delta_{j}}s},\ \ \ \ \ \gamma_{j}(s)=\frac{\gamma_{\rm oj}}{1+\tau_{\gamma_{j}}s}. \tag{2}\]  
Here, \(\alpha_{\rm oj}\) is DC current, and \(\delta_{\rm oj}\) and \(\gamma_{\rm oj}\) are DC voltage gains of the UVC, respectively. The bandwidths \(1/\tau_{\alpha_{j}}\), \(1/\tau_{\delta_{j}}\), and \(1/\tau_{\gamma_{j}}\) depend on the fabrication of active devices, and in current technologies on the order of a few gigapad/s are ideally equal to infinity. Hence, at low and medium frequencies--i.e., \(f\ll(1/(2\pi))\times\min\{1/\tau_{\delta_{j}},1/\tau_{\delta_{j}},1/\tau_{ \gamma_{j}}\}\)--Equation (2) becomes:  
\[\alpha_{j}(s)\cong\alpha_{\rm oj}=1+\varepsilon_{\alpha_{\rm ij}},\ \ \ \ \ \delta_{j}(s)\cong\delta_{\rm oj}=1+\varepsilon_{\delta_{\rm Vj}},\ \ \ \ \ \gamma_{j}(s)\cong\gamma_{\rm oj}=1+\varepsilon_{\gamma_{\rm oj}}, \tag{3}\]  
whereas \(\varepsilon_{\alpha_{\rm ij}}\), \(\varepsilon_{\delta_{\rm Vj}}\), and \(\varepsilon_{\gamma_{\rm oj}}\) are current and voltage tracking errors, respectively, and satisfy the inequalities \(|\varepsilon_{\alpha_{\rm ij}}|\ll 1\), \(|\varepsilon_{\delta_{\rm Vj}}|\ll 1\), and \(|\varepsilon_{\gamma_{\rm oj}}|\ll 1\).  
Considering an ideal UVC and assuming \(R_{1}=R_{2}=R\) and \(C_{1}=C_{2}=C\) for the proposed dual-output first-order VM APF shown in Figure 1b, routine analysis yields ideal voltage transfer functions (TFs) in the following forms:  
\[T_{1}(s)=\frac{V_{\rm o1}}{V_{\rm in}}=\frac{sCR-1}{sCR+1},\ \ \ \ \ T_{2}(s)=\frac{V_{\rm o2}}{V_{\rm in}}=-\frac{sCR-1}{sCR+1}, \tag{4}\]  
while the phase responses of the TFs (4) are calculated as:  
\[\varphi_{1}(\omega)=180^{\circ}-2\mathrm{arctg}(\omega CR),\ \ \ \ \ \varphi_{2}(\omega)=-2\mathrm{arctg}(\omega CR), \tag{5}\]  
Figure 1: (a) Circuit symbol of UVC; (b) proposed new voltage-mode first-order all-pass filter.  
which indicates that the proposed VM APF can simultaneously provide phase shifting between \(\pi\) (at \(\omega=0\)) to 0 (at \(\omega=\infty\)) and 0 (at \(\omega=0\)) to \(-\pi\) (at \(\omega=\infty\)) at output terminals \(V_{01}\) and \(V_{02}\), respectively. Finally, both the zero and pole frequencies are equal, and can be found as \(\omega_{\rm z}=\omega_{\rm p}=1/(CR)\). Therefore, their sensitivities to passive elements are \(S_{C,R}^{\omega_{\rm z,p}}=-1\); i.e., are not higher than unity in magnitude.