Points in the RL Plane

Inductive reactance can be plotted along a half line, just as can resistance. In a circuit containing both resistance and inductance, the characteristics become two-dimensional. You can orient the resistance and reactance half lines perpendicular to each other to make a quarter-plane coordinate system, as shown in following figure. Resistance is plotted horizontally, and inductive reactance is plotted vertically upward.

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The quarter plane for inductive reactance (XL ) and resistance (R). This is also known as the RL quarter-plane, or simply as the RL plane

In this scheme, resistance-inductance (RL) combinations form complex impedances. (The term impedance comes from the root impede, and fully describes how electrical components impede, or inhibit, the flow of ac. ) Each point on the RL plane corresponds to one unique complex impedance value. Conversely, each complex impedance value corresponds to one unique point on the RL plane.

You might ask, “What’s the little j doing in above figure?” For reasons that will be made clear in further topics, impedances on the RL plane are written in the form R + jXL, where R is the resistance in ohms, and XL is the inductive reactance in ohms. The little j is called a j operator and is a mathematical way of expressing the fact that reactance is denoted at right angles to resistance in compleximpedance
graphs.

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Four points in the RL plane.

If you have a pure resistance, say R = 5 Ω, then the complex impedance is 5 + j 0, and is at the point (5,0) on the RL plane. If you have a pure inductive reactance, such as XL = 3 Ω, then the complex impedance is 0 + j3, and is at the point (0,j3) on the RL plane. These points, and a couple of others, are shown in above figure.

In real life, all coils have some resistance, because no wire is a perfect conductor. All resistors have at least a tiny bit of inductive reactance, because they take up some physical space and they have wire leads. So there is really no such thing as a mathematically perfect pure resistance such as 5 + j 0, or a mathematically perfect pure reactance like 0 + j3. But sometimes you can get extremely close to theoretical ideals in real life.

Often, resistance and inductive reactance are both deliberately placed in a circuit. Then you get impedances values such as 2 + j3 or 4 + j1.5. These are shown in above figure. as points on the RL plane. Remember that values for XL are reactances, not actual inductances. Because of this, they vary with the frequency in an RL circuit. Changing the frequency has the effect of making complex impedance points move around in the RL plane. They move vertically, going upward as the ac frequency increases, and downward as the ac frequency decreases. If the ac frequency goes down to zero, the inductive reactance vanishes. Then XL = 0, we have pure dc, and the point is right on the resistance axis.