Two-terminal resistive elements

Two-terminal, resistive elements are two-terminal network elements whose behavior can be described with an instantaneous \(v-i\) relation. Hence, the shape of their \(v-i\) plot does not depend on the rate of change of \(the\)voltage or the current across the resistive element. Passive, resistive, two-terminal elements have their \(v-i\) characteristic pass through the origin: they do not carry any current when shorted. \sidenote [][-2.5cm]{Or, alternatively, the voltage across them equals zero if they are left open,} Fig. 59 shows the schematic symbol of a two-terminal, passive, nonlinear, resistive element with associated signs for the voltage across and the current through it.

Fig. 59 Schematic symbol of a two-terminal, non-linear resistor with reference directions for voltage and current.

Voltage-controlled and current-controlled notation

There are two representation methods for the \(v-i\) characteristic of these elements: the voltage-controlled and the current-controlled representation. Two-terminal, resistive elements of which the branch current is uniquely defined by the branch voltage, are called voltage-controlled. Two-terminal, resistive elements of which the branch voltage is uniquely defined by the branch current, are called current-controlled elements.

Monotonously increasing or decreasing \(v-i\) relations can be expressed in both voltage-controlled and current-controlled notation.

Examples of voltage-controlled and current-controlled, nonlinear resistors with a non-monotonic relation between voltage and current are shown in Fig. 60.

Fig. 60 A. \(i(v)\) function of a voltage-controlled, nonlinear resistor B. \(v(i)\) function of a current-controlled, nonlinear resistor.

A voltage-controlled representation of a \(v-i\) characteristic is shown in Fig. 60A. A voltage-controlled \(v-i\) relation is written in the form

\[I=I\left( V\right) .\]

In words: the branch current is uniquely defined by the branch voltage.

An example of a current-controlled representation of the characteristic of a nonlinear, two-terminal, resistive element is shown in Fig. 60B. We speak of a current-controlled \(v-i\) relation if the branch voltage is uniquely defined by the branch current:

\[V=V(I).\]

Resistive two-terminal elements

Some examples of nonlinear, resistive elements and their characteristics are shown in Fig. 61. The \(v-i\) relations of these elements are given table Table 5.

Fig. 61 Examples of two-terminal, resistive elements: A. Linear resistor B. Independent current source C. Independent voltage source D. Ideal diode.

Table 5 Examples of two-terminal nonlinear resistive elements

element	parameters	\textbf{current-controlled }	\textbf{voltage controlled }
linear resistor	R	\(V(I)=I\)R	\(I\left( V\right) =\frac{V}{\text{R}}\)
voltage source	V	\(V(I)=\ \)V	\(I\left( V\right) =\) undefined
current source	I	\(V(I)=\) undefined	\(I\left( V\right) =\ \)I
ideal diode	IS	\(V(I)=\frac{kT}{q}\ln\left( 1+\frac{I}{\text{IS}}\right) \)	\(I\left( V\right) =\ \)IS \(\left( \exp\left( \frac{qV}{kT}\right) -1\right) \)
		valid for \(I>-\) IS

Complementary devices

Let us consider two current-controlled, nonlinear, resistive elements of which the behavior is described by their respective \(v-i\) relations \(V(I)\)\ and \(V_{c}(I)\). These elements are said to be complementary if:

\[V_{c}(I)=-V(-I).\]

Fig. 62 \(v-i\) Characteristics of a non-linear, two-terminal resistor and its complementary device.

Similarly, two nonlinear, resistive elements with their respective voltage-controlled notations \(I(V)\) and \(I_{c}(V)\) are said to be complementary if:

\[I_{c}\left( V\right) =-I\left( -V\right) .\]

A graphical interpretation of this relation is shown in Fig. 62. It illustrates that the \(v-i\) relation of a complementary, two-terminal element is obtained from is normal version by rotating it over \(180\) degrees. Note that a two-terminal, resistive element is always complementary to its reversely connected element.

Operating point

The quiescent operating point, or shortly the operating point of a device is a point on the \(v-i\) curve of the device at which it is operating in the absence of a signal. At a later stage, we will show that if the (small-signal) resistance of a two-terminal resistive element in its operating point is negative, the element exhibits amplifying capabilities. As a matter of fact, we will see that the amplifying capabilities of active devices strongly depend on the device’s operating point. Selection and design of proper operating conditions of active devices is an important activity of analog circuit design engineers.

Fixing the operating point

Fig. 63 A. \(v-i\) characteristic of a PN diode B. Definition of the operating point and the application of bias sources. C. \(v-i\) characteristic of the biased diode.

The operating point of a two-terminal device can be altered by placing a voltage source in series with the element and a current source in parallel with it. This process is called biasing, and the added voltage and current sources are called bias sources.

Fig. 63 illustrates the process of biasing a diode. Fig. 63A shows the unbiased diode connected to a signal voltage source V1 with source resistance \(R_{s}\). If the signal voltage \(v\) equals zero, the diode operates in the origin of its \(v-i\) characteristic. Hence, the quiescent operating point is \((0,0)\). If the signal voltage deviates from zero, a signal current \(i\) will flow through the source.

Let us now change the operating point of the diode, while maintaining the quiescent operating conditions of the signal source. In other words: we still want no current through V1 if \(v=0\).

Fig. 63B shows the way in which this can be achieved. The operating point of the diode has been changed to \((V_{Q},I_{Q})\) through the addition of a bias voltage \(V_{Q}\) in series with the diode and a bias current \(I_{Q}\) in parallel with the diode. The bias voltage source V2 provides the bias voltage \(V_{Q}\) and the bias current source I1 provides the bias current \(I_{Q}\). In the quiescent state, the bias current \(I_{Q}\) causes a voltage \(V_{Q}\) across the diode. The bias voltage source V2 compensates for this voltage, such that the quiescent operating conditions of the signal source have not changed. Please notice that this way of biasing makes the operating point insensitive for variations in \(R_{s}\). In fact, any change in a passive, resistive termination of the biased diode will not alter its operating point.

Dependent and independent bias sources

Either the bias voltage, or the bias current can be selected by design, while the other follows from the one selected and the \(V-I\) characteristic of the device. If the current \(I_{Q}\) is selected by design, the value of the voltage source should equal the voltage across the diode, with the current \(I_{Q}\) flowing through it, hence: \(V_{Q}=V(I_{Q})\). If the voltage \(V_{Q}\) is selected by design, the value of the current source equals \(I_{Q}=I(V_{Q})\).

Biasing errors

Since the real-world \(v-i\) characteristic of an element to be biased is usually not fully known and temperature-dependent, biasing errors will be inevitable and error reduction may be required to improve the biasing accuracy and temperature stability. At a later stage we will present techniques for the reduction of biasing errors.

Biased device \(v-i\) characteristic

The \(v-i\) characteristic of the biased device can be found from the \(V(I)\) function of the unbiased device and the operating point \((V_{Q},I_{Q})\).

The voltage-to-current transfer of the biased device can be written in terms a modified nonlinear transfer \(I_{Q}(v)\):

\[i=I_{Q}\left( v\right) ,\]

where \(v\) and \(i\) are the voltage and the current excursion from the operating point, respectively.

The function \(I_{Q}(v)\) can be obtained from the original nonlinear device characteristic \(I(V)\) as

\[I_{Q}\left( v\right) =I\left( V_{Q}+v\right) .\]

Similarly, but now in current-controlled notation, we may write

\[v=V_{Q}\left( i\right) ,\]

where

\[V_{Q}\left( i\right) =V\left( I_{Q}+i\right) .\]

The modified functions \(I_{Q}\left( v\right) \) and \(V_{Q}\left( i\right) \) now pass through the origin, while the quiescent operating point of the nonlinear device itself is \((V_{Q},I_{Q})\).

Linearization and available power gain

For small excursions from the operating point the \(v-i\) relation can be linearized. The small-signal conductance and resistance in the operating point in voltage-controlled or current-controlled notation, are:

\[\begin{split}i & =g_{Q}v;\qquad g_{Q}=\left. \frac{I_{Q}\left( v\right) }{dv}\right\vert _{v=0},\\ v & =r_{Q}i;\qquad r_{Q}=\left. \frac{V_{Q}\left( i\right) }{di}\right\vert _{i=0},\end{split}\]

respectively.

We will now study the conditions under which nonlinear resistive two-terminal elements can provide an available power gain larger than unity. As discussed earlier, an available power gain that exceeds unity is a distinguishing property of amplifiers.

Fig. 64 A. Resistive load connected to a voltage source with source resistance \(R_s\). B. A nonlinear resistor is placed between the source and the load. C. The nonlinear resistor is biased in an operating point \((v_Q, i_Q)\). In this operating point its small-signal equivalent resistance equals \(r_Q\).

Let us hereto consider a signal source that consists of a voltage source with a nonzero source resistance, as shown in Fig. 64A. The source voltage equals \(V_{s}\) and the source resistance equals \(R_{s}\). The source is connected to a load that has a resistance \(R_{\ell}\).

The available power of the source equals

\[P_{\operatorname*{av}}=\frac{V^{2}}{4R_{s}}.\]

Let us now place a biased two-terminal nonlinear device \(R\left( v,i\right) \) between the source and the load as shown in Fig. 64B. The device is biased in an operating point \(Q\) and its small-signal equivalent resistance in this operating point is assumed \(r_{Q}\). The maximum power will be delivered to a load with a resistance \(R=R_{s}+r_{Q}.\) The available power \(P_{\operatorname*{av} }^{\prime}\) of the series connection of the source and the biased nonlinear device equals

\[P_{\operatorname*{av}}^{\prime}=\frac{V^{2}}{4\left( R_{s}+r_{Q}\right) }.\]

The available power gain \(G_{P}\) of the two-terminal device \(R\) is obtained as the ratio of \(P_{\operatorname*{av}}^{\prime}\) and \(P_{\operatorname*{av}}\):

\[G_{P}=\frac{P_{\operatorname*{av}}^{\prime}}{P_{\operatorname*{av}}} =\frac{R_{s}}{R_{s}+r_{Q}}.\]

Fig. 65 Typical \(v-i\) characteristic of a tunnel diode.

It appears that the available power gain will exceed unity if the small-signal resistance \(r_{Q}\) of the two-terminal device in the operating point \(Q\) has a negative value. The tunnel diode is an example of a two-terminal device that has a negative small-signal resistance in a certain operating region. The \(v-i\) characteristic of a tunnel diode is shown in Fig. 65.