Materiales

The v–i characteristic of a nonlinear resistor such as a diode or a transistor is often described by a curve on the i–v plane rather than by a mathematical relation. The v–i characteristic curve can be obtained by

Figure 2.38 Models of voltage/current amplifiers

using a curve tracer. To analyze circuits containing a nonlinear resistor, the load line analysis should be used. To grasp the concept of a load line, consider the graphical analysis of the circuit in Figure 2.39(a), which consists of a linear resistor R1, a nonlinear resistor R2, a DC voltage source Vs, and an AC voltage source of small amplitude v
. Let the v–i relationship of R2be denoted by v2ðiÞ and represented by the characteristic curve in Figure 2.39(b). A graphical method will be considered that yields the operating point (IQ, VQ), i.e. the pair of the current through and the voltage across R2for v
¼ 0. KVL can be applied around the mesh to write down the mesh equation as

R1iþ v2ðiÞ ¼ Vs ð2:49Þ

Since no specific mathematical expression of v2ðiÞ is given, no analytical method can be used to solve this equation, which is why a graphical method will be used. First, it may be considered to try plotting the graph for the LHS (left-hand side) of Equation (2.49) and finding its intersection with a horizontal line for the RHS (right-hand side), i.e. v¼ Vs, as depicted in Figure 2.39(b). Another way is to leave only the nonlinear term on the LHS and move the other terms into the RHS to rewrite the equation as

v2ðiÞ ¼ Vs R1i ð2:50Þ

and find the intersection, called the operating point and denoted by Q (quiescent point), of the graphs for both sides, as depicted in Figure 2.39(c). The straight line with the slope ofR1is called the load line.

This graphical method is better than the first one because it does not require a new curve to be plotted for v2ðiÞ þ R1i. That is why it is widely used to analyze nonlinear resistor circuits in the name of ‘load line analysis’. Let us take a look at the following example.

Note. All resistors appearing in this book except in this section are linear in the sense that their voltages are linearly proportional to their currents so that their voltage–current relationships are described by Ohm’s law (Equation (1.6a)) and, consequently, their v–i characteristics are described by straight lines passing through the origin with the slopes corresponding to their resistances on the i–v plane. However, they may have been modeled or approximated to be linear just for simplicity and convenience, because all physical resistors more or less exhibit some nonlinear characteristics. The problem is whether or not the modeling is valid in the range of practical operations so that it may yield the solution with sufficient accuracy to serve the objective of the analysis and design.

Note. A curve tracer is an instrument that displays the v–i characteristic curve of an electric element on a cathode ray tube (CRT) when the element is inserted into an appropriate receptacle (Reference [F-1].

Figure 2.39 Graphic analysis of a nonlinear resistor circuit

2.12 Load Line Analysis of Nonlinear Resistor Circuits 83

(Example 2.23) Small-Signal (AC) Analysis of Nonlinear Circuit

Consider the circuit in Figure 2.39(a), where a linear resistor R1and a nonlinear resistor R2in series are driven by a DC voltage source Vsin series with a small-amplitude AC voltage source producing the virtual voltage as

vsðtÞ ¼ Vsþ v
sin !t ðE2:23:1Þ

The voltage–current relationship, v2ði2Þ, of the nonlinear resistor R2is described by the characteristic curve in Figure 2.40.

As depicted in Figure 2.40, the upper/lower limits as well as the equilibrium value of the current i through the circuit can be obtained from the three operating points, i.e. the intersections (Q1, Q, and Q₂) of the characteristic curve with the following three load lines:

v¼ Vsþ v
 R1i ðE2:23:2aÞ

v¼ V_s R₁i ðE2:23:2bÞ

v¼ Vs v
 R1i ðE2:23:2cÞ

Although this approach gives the exact solution, no insight into the solution is gained from it. Instead, a rather approximate approach is taken, which consists of the following two steps:

1. Find the equilibrium (IQ, VQ) at the major operating point Q, which is the intersection of the characteristic curve with the DC load line (E2.23.2b).

2. Find the two approximate minor operating points Q⁰₁and Q⁰₂from the intersections of the tangent to the characteristic curve at Q with the two minor load lines (E2.23.2a) and (E2.23.2c).

Then the current will be obtained as

iðtÞ ¼ IQþ i
sin !t ðE2:23:3Þ

Figure 2.40 Variation of the voltage and current of a nonlinear resistor around the operating point

With the dynamic or small-signal or AC resistance r2ddefined to be the slope of the tangent to the characteristic curve at Q as

r2d¼dv2

Now find the analytical expressions of IQand i
in terms of Vsand v
. Referring to the encircled area around the operating point in Figure 2.40, i
can be expressed in terms of v
as

i
¼ QB ¼
^QQ

This corresponds to approximating the characteristic curve in the operation range by its tangent at the operating point. Noting that:

(a) the load line and the tangent to the characteristic curve at Q are at angles of (180 1) and 2to the positive i axis,

(b) the slope of the load line is tanð180 1Þ ¼  tan 1and must beR1, which is the proportion-ality coefficient in i of the load line equation (E2.23.2); tan 1¼ R1, and

(c) the slope of the tangent to the characteristic curve at Q is the dynamic resistance r2ddefined by (E2.23.4); tan 2¼ r2d,

Equation (E2.23.5) can be written as

i
¼ v

R1þ r2d

ðE2:23:6Þ

Now the static or DC resistance of the nonlinear resistor R2is defined to be the ratio of the voltage VQ

to the current IQat the operating point Q as R2s¼VQ

IQ

¼Vs R1IQ

IQ

ðE2:23:7Þ

so that the DC component of the current, IQ, can be written as I_Q¼ V_s

R1þ R2s

ðE2:23:8Þ

Finally, the above results are combined to write the current through and the voltage across the nonlinear resistor R2as follows:

iðtÞ ¼ IQþ i
sin !t¼ V_s

This result implies that the nonlinear resistor exhibits twofold resistance, i.e. the static resistance R2s

to a DC input and the dynamic resistance r2dto an AC input of small amplitude. That is why r2dis also called the (small-signal) AC resistance, while R2sis called the DC resistance.

[Remark 2.5] Operating Point and Static/Dynamic Resistances of a Nonlinear Resistor

1. For a nonlinear resistor R2connected with linear resistors in a circuit excited by a DC source and a small-amplitude AC source, its operating point Q¼ ðIQ; VQÞ is the intersection of its characteristic curve vðiÞ and the load line.

2.12 Load Line Analysis of Nonlinear Resistor Circuits 85

2. The v intercept of the load line (v¼ Vs R1i) is determined by the DC component (Vs) of the voltage source. The slope of the load line is determined by the equivalent resistance (R1) of the linear part seen from the pair of terminals of the nonlinear resistor. (See Problem 2.29.) 3. The static or DC resistance (R2s) is the ratio of the voltage VQto the current IQat the operating

point Q.

4. The dynamic or small-signal or AC resistance (r2d) is the slope of the tangent to the characteristic curve at Q.

5. Once R1, R2s, and r2dare obtained, the above formulas (2.51) and (2.52) can be used to find the voltage and current of the nonlinear resistor.

Note. The static or dynamic resistance are not used for linear resistors since they are identical.

Note. The relationship between the AC (small-signal) components of voltage across and current through the nonlinear resistor can be attributed to the Taylor series expansion of its VCR (voltage–current relationship) v2ðiÞ up to the first-order term around the operating point Q¼ ðIQ; VQÞ: