**The complex number plane**

Recall the planes for resistance (R) and inductive reactance (XL ) from previous topics. This is the same as the upper-right quadrant of the complex number plane shown in above figure. 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.

#### No Negative Resistance

**The complex impedance plane, also called the resistance reactance (RX ) plane.**

There is no such thing, strictly speaking, as negative resistance. You 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 (or dynamic) conditions. But for most practical applications in the RX plane, the resistance value is always positive. You can remove the negative axis, along with the upperleft and lower-left quadrants, of the complex number plane, obtaining a half plane, as shown in above figure, and still get a complete set of coordinates for depicting complex impedances.

#### “Negative Inductors” and “Negative Capacitors”

Capacitive reactance, XC, is effectively an extension of inductive reactance, XL, into the realm of negatives. Capacitors act like “negative inductors.” It’s equally true to say that inductors act like “negative capacitors,” because the negative of a negative number is a positive number. Reactance can vary from extremely large negative values, through zero, to extremely large positive values.

#### Vector Representation of Impedance

Any impedance R + jX can be represented by a complex number of the form a + jb. Just let R = a and X = b. Now try to envision how the 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 XL gets larger, the vector grows longer. If R stays the same but XC gets larger negatively, the vector grows longer.

**Some points in the complex impedance plane, and their resistive and reactive components on the axes.**

Think of the point R + jX moving around in the RX plane, and imagine where the corresponding points on the axes lie. These points can be found by drawing dashed lines from the point R + jX to the R and X axes, so that the dashed lines intersect the axes at right angles. Some examples are shown in above figure.

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 twodimensional. To fully define impedance, you must render it on a two-dimensional coordinate system such as the RX plane. The resistance and the reactance can change independently of one another.

#### Absolute-Value Impedance

You’ll occasionally 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-Ω” speakers and “600-Ω” amplifier inputs. How, you ask, can manufacturers quote a single number for a quantity that is two-dimensional

and needs two numbers to be completely expressed?

That’s a good question, and there are two answers. First, figures like this refer to devices that have purely resistive impedances, also known as nonreactive impedances. Thus, the 8-Ω speaker really has a complex impedance of 8 + j 0, and the 600-Ω input circuit is designed to operate with a complex impedance at, or near, 600 + j 0. Second, you can talk about the length of the impedance vector (that is, the absolute value of the complex impedance), calling this a certain number of ohms. If you talk about impedance this way, however, you are being ambiguous. There can exist an infinite number of different vectors of any given length in the RX plane.

Sometimes, the uppercase italic 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 Ω” or “Z = 300 Ω nonreactive.” In this context, if no specific impedance is given, “Z = 8 Ω” can theoretically refer to 8 + j 0, 0 + j 8, 0 − j 8, or any other complex impedance point on a half circle consisting of all points 8 units from 0 + j 0. This is shown in following figure.

**Vectors representing an absolute-value impedance of 8 Ω.**