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There is no such thing, strictly speaking, as negative resistance. That is to say, one cannot have anything better than a perfect conductor. In some cases, a supply of direct current, such as a battery, can be treated as a negative resistance; in other cases, you can have a device that acts as if its resistance were negative under certain changing conditions. But generally, in the RX (resistance-reactance) plane, the resistance value is always positive. This means that you can remove the negative axis, along with the upper-left and lower-left quadrants, of the complex-number plane, obtaining a half plane as shown in Fig. 15-5.

Now you should get a better idea of why capacitive reactance, XC, is considered negative. In a sense, it is an extension of inductive reactance, XL , into the realm of negatives, in a

D 1 2 3 4 5 6 7 8 9 0

way that cannot generally occur with resistance. Capacitors act like negative inductors. Interesting things happen when capacitors and inductors are combined, which is discussed in the next couple of chapters. Reactance can vary from extremely large negative values, through zero, to extremely large positive values. Engineers and physicists always consider reactance to be imaginary. In the mathematical model of impedance, capacitances and inductances manifest themselves perpendicularly to resistance. The general symbol for reactance is X; this encompasses both inductive reactance (XL) and capacitive reactance (XC).

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of each year, stocks are divided into 10 portfolios based on their BE/ME ratios. Portfolio 1 consists of the stocks with the lowest BE/ME ratios, portfolio 2, 3, 4 . . . consist of groups of stocks with increasing BE/ME ratios, with portfolio 10 consisting of the stocks with the highest BE/ME ratios. The monthly returns for each of the 10 portfolios are measured over 12 months. The following July, each stock s BE/ME ratio is recomputed and stocks are reassigned to the ten portfolios based on their new BE/ME ratios ranging from lowest to highest. Stocks may move from one portfolio to another. The returns for the portfolios are again measured over the ensuing twelve months. The return calculation and sorting process occurs again and again over the 26-year period, and an average return of each of the 10 portfolios is computed. Fama and French further subdivide portfolios 1 and 10 into two portfolios each, resulting in a total of twelve portfolios. Table 2-3 and Exhibit 2-4 present summaries of their findings:

source. The CM8880 requires input/output control lines to animate the touch-tone decoder via the microprocessor. These 3 data-bit pins are used to select the modes of the CM8880: the chip select (CS) pin, the read/write (R/W) pin, and the resistor select (RS0) pin. Table 14-3 depicts all the combinations of the three control pins. The CM8880 is active only when CS equals 0. The R/W bit determines the data direction: 1 read and 0 write. The RS bit determines whether the transaction involves data (DTMF tones) or internal functions, i.e., 1 instructions/status and 0 data. Before you can use the CM8880, you have

Any impedance R jX can be represented by a complex number of the form a jb. Just let R a and X b. It should be easy to visualize, now, how the impedance vector changes as either R or X, or both, are varied. If X remains constant, an increase in R will cause the vector to

Vector representation of impedance 271 get longer. If R remains constant and XL gets larger, the vector will also grow longer. If R stays the same as XC gets larger (negatively), the vector will grow longer yet again. Think of the point a jb, or R jX, moving around in the plane, and imagine where the corresponding points on the axes lie. These points are found by drawing dotted lines from the point R jX to the R and X axes, so that the lines intersect the axes at right angles (Fig. 15-6).

0 0 0 0 1 1 1 1

Now think of the points for R and X moving toward the right and left, or up and down, on their axes. Imagine what happens to the point R jX in various scenarios. This is how impedance changes as the resistance and reactance in a circuit are varied. Resistance is one-dimensional. Reactance is also one-dimensional. But impedance is two-dimensional. To fully define impedance, you must render it on a half plane, specifying the resistance and the reactance, which are independent.

TABLE 2-3 1992 Fama and French Stock Return Study Portfolios Based on Ascending Book Equity/Market Equity Ratios

0 0 1 1 0 0 1 1

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