Multi-terminal resistive elements

Most active electronic devices are three-terminal elements. The behavior of a multi-terminal resistive element with \(n\) terminals can be described by \(n-1\) relations that have all \(n-1\)\ branch voltages and \(n-1\) branch currents as arguments. The general form of such a set of equations is

(22)\[f_{1...n-1}\left( V_{1}...V_{n-1},I_{1}...I_{n-1}\right) =0. \label{ex-general-multiterrm-R}\]

Complementary multi-terminal elements

Let us consider two \(n\)-terminal resistive elements. One is described by a set of multi-variate functions \(f_{1...n-1}\left( V_{1}...V_{n-1},I_{1} ...I_{n-1}\right) \)\ and the other by \(g_{1...n-1}\left( V_{1} ...V_{n-1},I_{1}...I_{n-1}\right) \). These two n-terminal elements are complementary if

(23)\[f_{1...n}\left( V_{1}...V_{n-1},I_{1}...I_{n-1}\right) =-g_{1...n-1}\left( -V_{1}...-V_{n-1},-I_{1}...-I_{n-1}\right) . \label{ex-compl-multiterm-R}\]

NMOS and PMOS transistors are complementary if the above is true. Similar holds for NPN and PNP bipolar transistors.

Resistive two-ports

Nonlinear resistive two-ports can be represented by a pair of nonlinear functions:

\[f_{1,2}\left( V_{i},V_{o},I_{i},I_{o}\right) =0.\]

The arguments of the functions are the port voltages \(V_{i}\) and \(V_{o}\) and the port currents \(I_{i}\) and \(I_{o}\). By selecting two independent variables and two dependent variables out of the four variables, we obtain six different representation methods. These representation methods have been listed in table Table 6.

Table 6 Six representation methods for nonlinear resistive two-ports

Voltage-controlled representation:	\(I_{i}=I_{i}\left( V_{i};V_{o}\right) \)	\(I_{o}=I_{o}\left( V_{i};V_{o}\right) \)
Current-controlled representation:	\(V_{i}=V_{i}\left( I_{i};I_{o}\right) \)	\(V_{o}=V_{o}\left( I_{i};I_{o}\right) \)
Hybrid 1 representation:	\(I_{i}=I_{i}\left( V_{i};I_{o}\right) \)	\(V_{o}=V_{o}\left( V_{i};I_{o}\right) \)
Hybrid 2 representation:	\(V_{i}=V_{i}\left( I_{i};V_{o}\right) \)	\(I_{o}=I_{o}\left( I_{i};V_{o}\right) \)
Transmission 1:	\(I_{o}=I_{o}\left( V_{i};I_{i}\right) \)	\(V_{o} =V_{o}\left( V_{i};I_{i}\right) \)
Transmission 2:	\(V_{i}=V_{i}\left( v_{o};I_{o}\right) \)	\(I_{i} =I_{i}\left( V_{o};I_{o}\right) \)

Complementary two-ports

Let us consider two resistive two-ports that are described by their respective multi-variate functions \(f_{1,2}\left( V_{i},V_{o},I_{i},I_{o}\right) \)\ and \(g_{1,2}\left( V_{i},V_{o},I_{i},I_{o}\right) \). These two-ports are complementary if

\[f_{1,2}\left( V_{i},V_{o},I_{i},I_{o}\right) =-g_{1,2}\left( -V_{i} ,-V_{o},-I_{i},-I_{o}\right) .\]

Operating point

Nonlinear resistive two-ports may, under specific operating conditions, exhibit amplifying capabilities. Similar as with two-terminal devices, such operating conditions may be established by adding bias sources. Biasing of a port requires the insertion of a voltage sources in series with the port and a current sources in parallel with the port. This is illustrated in Fig. 66. Because the output port quantities depend on the input port quantities, two of those bias sources may be selected by design, while the remaining two follow from the ones selected and the device equations. The two selected by design will be referred to as independent bias sources and the two remaining as dependent bias sources.

Fig. 66 Biased non-linear two-port.

Let us consider a nonlinear two-port described by two voltage controlled relations:

\[\begin{split}I_{i} & =I_{i}\left( V_{i},V_{o}\right) ,\\ I_{o} & =I_{o}\left( V_{i},V_{o}\right) .\end{split}\]

After biasing the input port in the operating point \((V_{iQ},I_{iQ})\) and the output port in the operating point \((V_{oQ},I_{oQ})\), we obtain the modified two-port equations for deviations from the operating point as

\[\begin{split}i_{i} & =I_{iQ}\left( v_{i},v_{o}\right) ,\\ i_{o} & =I_{oQ}\left( v_{i},v_{o}\right) ,\end{split}\]

where \(v_{i},i_{i}\) and \(v_{o},i_{o}\) represent the voltage and current excursions from the quiescent operating point of the input port and the output port, respectively. The modified functions \(I_{iQ}\left( v_{i},v_{o}\right) \) and \(I_{oQ}\left( v_{i},v_{o}\right) \) can be obtained from the original two-port functions \(I_{i}\left( V_{i},V_{o}\right) \) and \(I_{o}\left( V_{i},V_{o}\right) ,\) and the operating points \((V_{iQ},I_{iQ})\) and \((V_{oQ},I_{oQ})\) of the input port and the output port, respectively:

\[\begin{split}I_{iQ}\left( v_{i},v_{o}\right) & =I_{i}\left( V_{iQ}+v_{i},V_{oQ} +v_{o}\right) ,\\ I_{oQ}\left( v_{i},v_{o}\right) & =I_{o}\left( V_{iQ}+v_{i},V_{oQ} +v_{o}\right) .\end{split}\]

Such description methods can also be given for other two-port representation methods.

Linearization and available power gain

For small excursions from the operating point a linearized two-port model \ can be used. In amplifier design, the anti-causal transmission-1 matrix representation will often be used (see Chapter Modeling and specification of amplifiers and Chapter Modeling and specification of amplifiers).

The transmission-1 matrix parameters can be obtained from \(v-i\) relations of the biased device as

\[\begin{split}A & =\left. \frac{\partial V_{iQ}\left( v_{o},i_{o}\right) }{\partial v_{o}}\right\vert _{i_{o}=0},\\ B & =-\left. \frac{\partial V_{iQ}\left( v_{o},i_{o}\right) }{\partial i_{o}}\right\vert _{v_{o}=0},\\ C & =\left. \frac{\partial I_{iQ}\left( v_{o},i_{o}\right) }{\partial v_{o}}\right\vert _{i_{o}=0},\\ D & =-\left. \frac{\partial I_{iQ}\left( v_{o},i_{o}\right) }{\partial i_{o}}\right\vert _{v_{o}=0}.\end{split}\]

Because the direction of the positive output current of the two-port differ from the positive current direction in the elements, minus signs appear in the expressions for \(B\) and \(D\).

The available power gain of a two-port can be expressed in the transmission-1 matrix parameters and the source impedance; see expressions ((11)), ((13)) and ((14) ). A biased, nonlinear, multi-terminal device can have an available power gain that exceeds unity, a property that will be exploited in amplifiers.