Absolutevalue Impedance
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 “8ohm” speakers and “600ohm” amplifier inputs. How can manufacturers quote a single number for a quantity that is twodimensional 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 “8ohm” speaker really has a complexnumber impedance of 8 + j 0, and the “600ohm” input circuit is designed to operate with a complexnumber 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 halfplane.
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 halfplane whose length is 8 units. This is shown in Fig. 1512. There can exist an infinite number of different complex impedances with Z = 8 ohms in a purely technical sense.
Fig. 1512 . Vectors representing an absolutevalue impedance of 8 ohms.
Absolutevalue Impedance Practice Problem
Problem
Name seven different complex impedances having an absolute value of Z = 10 ohms.
Solution
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 ageold Pythagorean theorem!) Therefore, you can have 6 + j 8, 6 − j 8, 8 + j 6, and 8 − j 6. These are all complexnumber impedances whose absolute values are 10 ohms.
If you’re not specifically told which particular complexnumber impedance is meant when a singlenumber 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 “lowZ” or “highZ.” There is no formal dividing line between the realms of low and high impedance; it depends to some extent on the application. Sometimes a reactancefree 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 McGrawHill. 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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