# Two-Terminal Electronic Circuits with Controllable Linear NDR Region and Their Applications
## 6 Applications
### Negative Differential Resistance Voltage-Controlled Oscillators  
Voltage-controlled oscillators are crucial elements of modern instrumentation, communication, navigation, and radar systems. We can classify VCOs as negative impedance (NI) and NDR oscillators. In the NI VCOs, the real part of the input impedance has a negative sign [51; 52]. This negative resistance compensates for the losses in the tank circuit. In the NDR VCOs, a negative resistance induced into the tank circuit neutralizes the tank circuit losses; this resistance is inversely proportional to the absolute value of \(R_{diff}\)[53]. The two-terminal circuits of Figures 3-5 can be the base of NDR VCOs. Figures 9-11 show the NDR VCOs based on MCCM, MWCM, and MIWCM. In the VCO circuits, instead of capacitors \(C_{1}\) and \(C_{2}\), two oppositely connected varactors, \(Var_{1}\) and \(Var_{2}\), are used. A voltage source \(V_{var}\) applies dc voltage to the cathodes of varactors. Resistor \(R_{0}\) is linked with the voltage source \(V_{var}\), preventing the parallel tank circuit's shunting. Figure 9: Voltage-controlled oscillator with multiple-output cascode current mirror. Figure 14: Current-voltage characteristics of the two-terminal NDR circuit with multiple-output improved Wilson current mirror; \(m=0\)—green curve, \(m=1\)—blue curve, \(m=2\)—red curve. Figure 12: Current-voltage characteristics of the two-terminal NDR circuit with multiple-output cascade current mirror; \(m=0\)—green curve, \(m=1\)—blue curve, \(m=2\)—red curve. Figure 13: Current-voltage characteristics of the two-terminal NDR circuit with multiple-output [PERSON] current mirror; \(m=0\)—green curve, \(m=1\)—blue curve, \(m=2\)—red curve. Since current \(I_{Z}\) is the same for all two-terminal NDR circuits, and current \(I_{Y}\) is different, the slope of the current-voltage characteristics in the NDR region is higher for the greater value of current \(I_{Y}\). Therefore, we can order the absolute negative resistance values of various two-terminal NDR circuits according to the following inequality:  
\[\left|R_{diff}^{C}\right|>\left|R_{diff}^{W}\right|>\left|R_{diff}^{ IW}\right|, \tag{25}\]  
where \(\left|R_{diff}^{C}\right|_{\prime}\) \(\left|R_{diff}^{W}\right|\), and \(\left|R_{diff}^{IW}\right|\) are, respectively, the absolute values of NDR in the two-terminal circuits with cascade, Wilson, and improved Wilson CM. Thus, inequalities (23) and (25) indicate that the oscillator with multiple-output improved Wilson CM has the most considerable self-excitation ability. In addition, the oscillator with multiple-output cascade CM has the least self-excitation ability. For comparison with the simulation results, we calculate the current \(I_{1}\) using (3), (5), and (7) with the following data of transistors MMBFU310 LT1 and BT92W: \(I_{DSS}=50\) mA, \(V_{P}=\) -3.5 V, \(V_{A}=11\) V, and \(V_{EB3}=0.6\) V. We use the same values of resistors \(R_{c}\) and \(R_{d}\) as in the simulation. Figures 15 and 16 show the calculated dependences of current-voltage characteristics in the NDR region for different two-terminal circuits. Tables 2 and 3 show the calculated currents \(I_{Y}\) and \(I_{Z}\) and the relative accuracy of the \(I_{Y}\) current calculation \(\Delta I_{Y}\) %, where \(\Delta I_{Y}\) % is given by  
\[\delta I_{Y}\%=\frac{I_{Y}(\text{calculated})-I_{Y}(\text{simulated})}{I_{Y}( \text{calculated})}\times 100\%. \tag{26}\]  
As shown in Table 3, the highest accuracy of current \(I_{Y}\) calculation when \(m=0\) belongs to Equations (5) and (7) for the two-terminal NDR circuit with an MWCM and MIWCM. The highest accuracy of the current \(I_{Y}\) calculation when \(m=1\), \(2\) has Equation (5) for the two-terminal NDR circuit with an MWCM. The worst accuracy of the current \(I_{Y}\) calculation when \(m=0\), \(1\), \(2\) has Equation (3) for the two-terminal NDR circuit with an MCCM. In general, we should note that the worst accuracy does not exceed \(12.3\%\), which indicates a sufficient engineering accuracy for calculating the current \(I_{Y}\).