# Two-Terminal Electronic Circuits with Controllable Linear NDR Region and Their Applications
## 4 Modeling the Drain Current of Transistor \(T_{0}\)  
As we can see from (3), (5), and (7), to calculate the total current \(I_{1}\) in the NDR region, we should know the drain current \(I_{D}\) of the transistor \(T_{0}\). The modeling of current \(I_{D}\) depends on the type of transistor \(T_{0}\).  
If transistor \(T_{0}\) is a JFET, the Shockley equation well represents the drain current in the saturation region:  
\[I_{D}=\frac{I_{DSS}}{V_{P}^{2}}\left(V_{P}-V_{GS}\right)^{2}, \tag{8}\]  
where \(I_{DSS}\) is the drain current of transistor \(T_{0}\) at zero bias, \(V_{P}\) is the negative pinch-off voltage, and \(V_{GS}\) is the gate-source voltage.  
The voltage between gate and source of transistor \(T_{0}\) is negative and consists of two parts: the voltage due to voltage divider \(R_{c}\), \(R_{d}\):  
\[\frac{R_{c}V_{1,2}}{R_{c}+R_{d}}, \tag{9}\]  
and the voltage drop across resistor \(R_{c}\) because of the current through this resistor, i.e.,  
\[\frac{I_{D}R_{d}}{R_{c}+R_{d}}\,R_{c}. \tag{10}\]  
Combining (9) and (10) gives  
\[V_{GS}=-\frac{R_{c}V_{1,2}}{R_{c}+R_{d}}-I_{D}(R_{c}\|R_{d}). \tag{11}\]  
Substituting (11) to (8) and providing some mathematical manipulations, we obtain the following quadratic equation for determining the value of the current \(I_{D}\):  
\[(R_{c}\|R_{d})^{2}I_{D}^{2}+\left[2\left(V_{P}+\frac{V_{1,2}R_{c}}{R_{c}+R_{d} }\right)(R_{c}\|R_{d})-\frac{V_{P}^{2}}{I_{DSS}}\right]I_{D}+\left(V_{P}+ \frac{V_{1,2}R_{c}}{R_{c}+R_{d}}\right)^{2}=0. \tag{12}\]  
If transistor \(T_{0}\) is a MESFET, we can model the current \(I_{D}\) by one of the nonlinear large-signal models [42; 43; 44]. For example, the popular Curtice model in the saturation region is as follows [42]:  
\[I_{D}=b(V_{GS}-V_{TH})^{2}(1+\lambda V_{DS})\tanh(\alpha V_{DS}), \tag{13}\]  
where \(b\) is the transconductance, \(\lambda\) is the channel length modulation coefficient, \(\alpha\) is the coefficient of the hyperbolic tangent function, \(V_{DS}\) is the drain-source voltage, and \(V_{TH}\) is the threshold voltage.  
The voltage \(V_{DS}\) we find by applying [PERSON]'s voltage law to the circuit of Figure 1.  
\[V_{1,2}-(V_{1,2}-V_{D})-V_{DS}+V_{GS}=0, \tag{14}\]  
where \(V_{D}\) is the voltage at the drain of transistor \(T_{0}\).  
Substituting \(V_{GS}\) from (11) into (14), we obtain  
\[V_{DS}=\frac{R_{d}V_{1,2}}{R_{c}+R_{d}}-I_{D}(R_{c}\|R_{d})-(V_{1,2}-V_{D}). \tag{15}\]  
It is evident that, for all circuits in Figures 3-5, the voltage \((V_{1,2}-V_{D})\) is equal to  
\[V_{1,2}-V_{D}=2V_{EB}, \tag{16}\]  
where \(V_{EB}\) is the emitter-base voltage of BJT used in the current mirror.  
Substituting (11) and (15) into (13), we obtain the following nonlinear equation for determining the value of the current \(I_{D}\):  
\[\begin{split}\bigg{[}(R_{c}\|R_{d})^{2}I_{D}^{2}+2\Big{(}V_{TH}+ \frac{V_{12}R_{c}}{R_{c}+R_{d}}\Big{)}(R_{c}\|R_{d})I_{D}+\Big{(}V_{TH}+\frac{V _{12}R_{c}}{R_{c}+R_{d}}\Big{)}^{2}\bigg{]}\times\\ \Big{\{}\lambda\Big{[}-(R_{c}\|R_{d})I_{D}+\frac{V_{12}R_{d}}{R_ {c}+R_{d}}-(V_{1,2}-V_{D})\Big{]}+1\Big{\}}\times\\ \tanh\Big{\{}\alpha\Big{[}-(R_{c}\|R_{d})I_{D}+\frac{V_{12}R_{d} }{R_{c}+R_{d}}-(V_{1,2}-V_{D})\Big{]}\Big{\}}-I_{D}/b=0.\end{split} \tag{17}\]  
Solving Equation (17), we can find the value of \(I_{D}\) for the selected two-terminal NDR circuit. Then, by substitution of \(I_{D}\) into (3), (5), or (7), we can calculate the total current \(I_{1}\).  
The large-signal modeling of HEMT and PHEMT is quite similar to MESFET modeling [45].