RLC Impedance Help (page 2)
We’ve seen how inductive and capacitive reactance can be represented along a line perpendicular to resistance. In this section we’ll put all three of these quantities— R, X L , and X C —together, forming a complete working definition of impedance .
The Rx Half-plane
Recall the quarter-planes for resistance R and inductive reactance X L from the preceding sections. This is the same as the upper-right quadrant of the complex-number plane. Similarly, the quarter-plane for resistance R and capacitive reactance X C 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.
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 source of dc, such as a battery, can be treated as a negative resistance; in other cases, a device can behave as if its resistance were negative under certain changing conditions. Generally, however, in the RX (resistance-reactance) half-plane , the resistance value is nonnegative (Fig. 15-11).
Fig. 15-11 . The RX impedance half-plane showing some points for specific complex-number impedances.
Reactance In General
Now you should get a better idea of why capacitive reactance X C is considered negative. In a sense, it is an extension of inductive reactance X L into the realm of negatives, in a way that generally cannot occur with resistance. Capacitors act like “negative inductors.” Interesting things happen when capacitors and inductors are combined.
Reactance can vary from extremely large negative values, through zero, to extremely large positive values. Engineers and physicists always quantify reactances as imaginary numbers. In the mathematical model of impedance, capacitances and inductances manifest themselves perpendicularly to resistance. Thus ac reactance occupies a different and independent dimension from dc resistance. The general symbol for reactance is X ; this encompasses both inductive reactance X L and capacitive reactance X C .
Vector Representation Of Impedance
Any impedance Z can be represented by a complex number R + jX , where R can be any nonnegative real number and X can be any real number. Such numbers can be plotted as points in the RX half-plane or as vectors with their end points at the origin (0 + j 0). Such vectors are called impedance vectors .
Imagine how an impedance vector changes as either R or X or both are varied. If X remains constant, an increase in R causes the vector to get longer. If R remains constant and X L gets larger, the vector also grows longer. If R stays the same as X C gets larger (negatively), the vector grows longer again. Think of point representing R + jX moving around in the plane, and imagine where the corresponding points on the resistance and reactance axes lie. These points can be found by drawing straight lines from the point R + jX to the R and X axes so that the lines intersect the axes at right angles. This is shown in Fig. 15-11 for several different points.
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 and the corresponding vector from 0 + j 0 to R + jX in various scenarios. This is how impedance changes as the resistance and reactance in a circuit are varied.
Sometimes you’ll read or hear that the “impedance” of some device or component is a certain number of ohms. For example, in audio electronics, there are “8-ohm” speakers and “600-ohm” 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-ohm” speaker really has a complex-number impedance of 8 + j 0, and the “600-ohm” input circuit is designed to operate with a complex-number impedance at or near 600 + j 0. Second, engineers sometimes talk about the length of the impedance vector, calling this a certain number of “ohms.” If you talk about “impedance” in this way, then theoretically you are being ambiguous because you can have an infinite number of different vectors of a given length in the RX half-plane.
The expression “ Z = 8 ohms,” if no specific complex impedance is given, can refer to the complex vectors 8 + j 0, 0 + j 8, 0 − j 8, or any vector in the RX half-plane whose length is 8 units. This is shown in Fig. 15-12. There can exist an infinite number of different complex impedances with Z = 8 ohms in a purely technical sense.
Fig. 15-12 . Vectors representing an absolute-value impedance of 8 ohms.
Absolute-value Impedance Practice Problem
Name seven different complex impedances having an absolute value of Z = 10 ohms.
It’s easy to name three: 0 + j 10, 10 + j 0, and 0 − j 10. 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 6 2 + 8 2 = 10 2 . (It’s the age-old Pythagorean theorem!) Therefore, you can have 6 + j 8, 6 − j 8, 8 + j 6, and 8 − j 6. These are all complex-number impedances whose absolute values are 10 ohms.
If you’re not specifically told which particular complex-number impedance is meant when a single-number ohmic figure is quoted, it’s best to assume that the engineers are talking about nonreactive impedances . This means that they are pure resistances and that the imaginary, or reactive, factors are zero. Engineers often will speak of nonreactive impedances as “low-Z” or “high-Z.” There is no formal dividing line between the realms of low and high impedance; it depends to some extent on the application. Sometimes a reactance-free impedance is called a pure resistance or purely resistive impedance .
Purely resistive impedances are desirable in various electrical and electronic circuits. Entire volumes have been devoted to the subject of impedance in engineering applications. Insofar as basic physics is concerned, we have gone far enough here. A somewhat more detailed yet still introductory treatment of this subject can be found in Teach Yourself Electricity and Electronics , published by McGraw-Hill. Beyond that, college textbooks in electrical, electronics, and telecommunications engineering are recommended.
Practice problems of these concepts can be found at: Alternating Current Practice Test
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