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The RX plane
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Recall the planes for resistance (R) and inductive reactance (XL) from chapter 13. This is the same as the upper-right quadrant of the complex-number plane shown in Fig. 15-2. Similarly, the plane for resistance and capacitive reactance (XC) is the same as the lower-right quadrant of the complex number plane. Resistances are represented by nonnegative real numbers. Reactances, whether they are inductive (positive) or capacitive (negative), correspond to imaginary numbers.
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No negative resistance
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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.
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Reactance in general
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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
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270 Impedance and admittance
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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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Vector representation of impedance
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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
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15-5 The complex impedance RX plane.
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).
15-6 Some points in the RX plane, and their components on the R and X axes.
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.
272 Impedance and admittance
Absolute-value impedance
There will be times when you ll hear that the impedance of some device or component is a certain number of ohms. For example, in audio electronics, there are 8- speakers and 600- amplifier inputs. How can manufacturers quote a single number for a quantity that is two-dimensional, and needs two numbers to be completely expressed There are two answers to this. First, figures like this generally refer to devices that have purely resistive impedances. Thus, the 8- speaker really has a complex impedance of 8 j0, and the 600- input circuit is designed to operate with a complex impedance at, or near, 600 j0. Second, you can sometimes talk about the length of the impedance vector, calling this a certain number of ohms. If you talk about impedance this way, you are being ambiguous, because you can have an infinite number of different vectors of a given length in the RX plane. Sometimes, the capital letter Z is used in place of the word impedance in general discussions. This is what engineers mean when they say things like Z 50 or Z 300 nonreactive. Z 8 in this context, if no specific complex impedance is given, can refer to the complex value 8 j0, or 0 j8, or 0 j8, or any value on a half circle of points in the RX plane that are at distance 8 units away from 0 j0. This is shown in Fig. 15-7. There exist an infinite number of different complex impedances with Z 8 . Problems 15-1, 15-2, and 15-3 can be considered as problems in finding absolutevalue impedance from complex impedance numbers.
Problem 15-4
Name seven different complex impedances having an absolute value of Z 10. It s easy name three: 0 j10, 10 j0, and 0 j10. These are pure inductance, pure resistance, and pure capacitance, respectively. A right triangle can exist having sides in a ratio of 6:8:10 units. This is true because 62 82 = 102. (Check it and see!) Therefore, you might have 6 j8, 6 j8, 8 j6 and 8 j6, all complex impedances whose absolute value is 10 ohms. Obviously, the value Z 10 was chosen for this problem because such a whole-number right-triangle exists. It becomes quite a lot messier to do this problem (but by no means impossible) if Z 11 instead. If you re not specifically told what complex impedance is meant when a singlenumber ohmic figure is quoted, it s best to assume that the engineers are talking about nonreactive impedances. That means they are pure resistances, and that the imaginary, or reactive, factor is zero. Engineers will often speak of nonreactive impedances, or of complex impedance vectors, as low-Z or high-Z. For instance, a speaker might be called low-Z and a microphone high-Z.
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