## 3 Circuit Descriptions  
The CCCCTA is an active building block, which is useful in simplifying the circuit design. The CCCCTA device integrates a CCCII and an OTA into a versatile component to facilitate the implementation of analog signal-processing circuits. The CCCCTA does not need a resistor in practical applications, as its transconductance gain, \(g_{m}\), and parasitic resistance, \(R_{X}\), are electronic adjustable. Figure 2 shows the symbol of a multiple-output CCCCTA (MO-CCCCTA) [33]. The MO-CCCTA properties can be expressed as \(V_{X}=V_{Y}+I_{X}R_{X}\), \(I_{Z}=I_{Z\text{C1}}=I_{Z\text{C2}}=I_{X}\) and \(I_{O1}=I_{O2}=g_{m}V_{Z}\), where \(R_{X}\) is the parasitic resistance of the \(X\)-terminal, and \(g_{m}\) is the transconductance gain of the MO-CCCCTA. The bias currents, \(I_{S}\) and \(I_{B}\), of MO-CCCCTA can control the parasitic resistance, \(R_{X}\), and the transconductance gain, \(g_{m}\), respectively [33; 34]. Figure 3 shows the proposed schematic of the resistorless third-order quadrature oscillator, which consists of two MO-CCCCTAs and three grounded capacitors. The only grounded capacitors without a resistor structure in the proposed circuit are easily integrated into a single chip. In Figure 3, the transfer function, \(B/A\), realizes a current-mode lossy integrator, and the transfer functions, \(C/B\) and \(D/C\), realize two current-mode lossless integrators. Figure 1: Functional block diagram for realizing third-order quadrature oscillators. Figure 2: MO-CCCCTA symbol. Let \(D=A\), and the close-loop circuit analysis provides the following system characteristic equation. \[s^{3}C_{1}C_{2}C_{3}R_{X1}R_{X2}+s^{2}C_{2}C_{3}R_{X2}+sC_{2}+g_{m1}=0 \tag{8}\]  
and the CO and FO can be obtained as:  
\[\text{CO:}\;g_{m1}R_{X1}=\frac{C_{2}}{C_{1}} \tag{9}\]  
\[\text{FO:}\;\omega_{o}=\sqrt{\frac{g_{m1}}{R_{X2}C_{2}C_{3}}} \tag{10}\]  
As indicated by Equations (9) and (10), the CO can be adjusted by \(R_{X1}\) without disturbing the FO, and the FO can be controlled by \(R_{X2}\) without affecting the CO. In other words, the CO and FO are controlled independently by the bias currents, \(I_{S1}\) and \(I_{S2}\), respectively. As the parasitic resistances, \(R_{X1}\) and \(R_{X2}\), of the MO-CCCCTA can be electronically controlled by \(I_{S1}\) and \(I_{S2}\), respectively, both the CO and FO are independently and electronically controllable. In a steady state, the two voltage outputs and two current outputs in Figure 3 are:  
\[V_{o1}=-jk_{1}V_{o2},\;I_{o1}=jk_{2}I_{o2} \tag{11}\]  
where \(k_{1}=\omega_{o}C_{2}R_{X1}\), and \(k_{2}=\frac{\omega_{o}C_{2}}{g_{m1}}\). Equation (11) shows that the proposed circuit can obtain the quadrature voltage outputs and the quadrature current outputs simultaneously. The magnitude ratios of the quadrature voltage outputs and the quadrature current outputs are the function of the operating frequency. Thus, the value of \(R_{X2}\) will affect the oscillation frequency and the magnitude ratios of the generated quadrature signals. However, this problem can be solved by using \(C_{1}=C_{2}=C_{3}\), \(R_{X1}=R_{X2}\), and letting the product \(g_{m1}R_{X1}=1\). Subsequently, the amplitude of voltages and currents in Equation (11) are equal and ensure \(k_{1}=k_{2}=1\). The output current, \(I_{o3}\), can be given as follows:  
\[I_{o3}=g_{m2}V_{o1} \tag{12}\]  
where the \(g_{m2}\) of the second MO-CCCCTA can be independently tuned by the bias current, \(I_{B2}\). This means that the \(g_{m2}\)-value can be tuned by \(I_{B2}\), without disturbing the CO and FO. Thus, if a modulating signal is applied to \(I_{B2}\), then the AM/ASK signals can be obtained from \(I_{o3}\). From Equation (11), the magnitude of \(V_{o2}\) and \(V_{o1}\) or \(I_{o2}\) and \(I_{o1}\) is dependent on the factor \(k_{1}\) or \(k_{2}\). Therefore, the mechanism of amplitude limitation will depend on the factors \(k_{1}\) and \(k_{2}\)  
Figure 3: The proposed third-order quadrature sinusoidal oscillator.