## 2 Description of the Proposed Method  
The proposed approach involves the use of complementary and inverse transformations. One can obtain a complementary voltage-mode network by interchanging the input and ground of a circuit [17; 18], as shown in Figure 1. The resulting transfer function (TRANSF) T\({}_{2}\) of the obtained circuit would be the complement (COMP) of the original voltage transfer function T\({}_{1}\); _i.e._, T\({}_{2}\) = 1 - T\({}_{1}\). According to inverse transformation by interchanging the output norator and the input voltage source of a circuit (with its transfer function T\({}_{3}\) = Y/U), the inverse transfer function (INV) can be obtained as T\({}_{3}\)' = 1/T\({}_{3}\) = Y/U'), as shown in Figure 2[12]. The inverse network can also be obtained by interchanging the output pathological current mirror and the input source of a circuit [19]. To obtain a sinusoidal oscillator from the biquadratic band pass filter, the combination of circuit transformations can be exploited, as the stages shown in Tables 1 and 2. Figure 1: The complementary transformation for voltage-mode a network. (**a**) Original network; (**b**) complementary network of (**a**). To describe the synthesis method clearly, a biquadratic voltage-mode band pass filter with its transfer function given by Equation (1) is considered. The pole frequency \(\omega_{o}\) and bandwidth (BW) of the filter can be computed as \(\omega_{o}=\sqrt{c}\) and \(\mathrm{BW}=\omega_{o}/\mathrm{Q}=b\), where \(\mathrm{Q}\) is the quality factor of the filter. The coefficient \(b\) and \(c\) are always greater than zero for a stable system. \[T\big{(}s\big{)}=\frac{V_{0}}{V_{in}}=\frac{as}{s^{2}+bs+c} \tag{1}\]  
Applying the proposed procedure in Tables 1 or 2 to this filter, we obtain its new transfer function as Equation (2). \[T^{\prime}\big{(}s\big{)}=\frac{V_{0}}{V_{in}}=-\frac{as}{s^{2}+\big{(}b-a \big{)}s+c} \tag{2}\]  
Namely, the new filter with \(\omega_{o}=\sqrt{c}\) and \(\omega_{o}/\mathrm{Q}=(b-a)\) is derived. For a positive-feedback system in Figure 3 with a stable amplifier stage \(A(s)\) and a frequency-selective network \(B(s)\) connected in a positive-feedback loop, the closed-loop gain \(A(s)\) can be expressed by: \begin{table}
\begin{tabular}{c c c c c} \hline
**Procedure** & **Stage (0)** & \begin{tabular}{c} **Stage (1):** \\ **COMP of (0)** \\ \end{tabular} & \begin{tabular}{c} **Stage (2):** \\ **INV of (1)** \\ \end{tabular} & \
\begin{tabular}{c} **Stage (3):** \\ **COMP of (2)** \\ \end{tabular} \\ \hline TRANSF & T & 1-T & 1-T &
\begin{tabular}{c} T \\ T-1 \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: Another combination of complementary-inverse transformation for a voltage-mode network. Figure 2: The inverse transformation of a circuit. (**a**) Original network; (**b**) inverse network of (**a**). \begin{table}
\begin{tabular}{c c c c c} \hline
\begin{tabular}{c} T \\ T-1 \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: The combination of complementary-inverse transformation (TRANSF) for a voltage-mode network. COMP, complement; INV, inverse transfer function. \[A_{f}\left(s\right)=\frac{X_{0}}{X_{in}}=\frac{A(s)}{1-A(s)B(s)} \tag{3}\]  
If at a specific pole frequency \(\omega_{o}\) the loop gain \(A(s)B(s)\) is equal to unity, \(A\)/ will be infinite. That is, at this frequency, the system defined as an oscillator will have a finite output for zero input signal. Thus, the condition for the feedback loop of Figure 3 to provide sinusoidal oscillations of frequency \(\omega_{o}\) is:  
\[1-A(j\omega_{o})B(j\omega_{o})=0 \tag{4}\]  
This is known as the Barkhausen criterion. Therefore, a linear oscillator can be synthesized by zeroing the input source of the filter [18]. The obtained characteristic equation of the oscillator can be expressed as Equation (5). The oscillation condition is given by Equation (6) and oscillation frequency can be expressed by \(\omega_{o}=\sqrt{c}\).