Points in the RC Plane

Untitled
In a circuit containing resistance and capacitive reactance, the characteristics are two-dimensional in a way that is analogous to the situation with the RL plane from the previous topics. The resistance ray and the capacitive-reactance ray can be placed end to end at right angles to make a quarter plane called the RC plane (above figure). Resistance is plotted horizontally, with increasing values toward the right. Capacitive reactance is plotted downward, with increasingly negative values as you go down.

The combinations of R and XC in this RC plane form impedances. You’ll learn about impedance in greater detail in the next section. Each point on the RC plane corresponds to one and only one impedance. Conversely, each specific impedance coincides with one and only one point on the plane.

Any impedance that consists of a resistance R and a capacitive reactance XC can be written in the form R + jXC. Remember that XC is always negative or zero. Because of this, engineers will often write R − jXC instead.

If an impedance is a pure resistance R with no reactance, then the complex impedance is R − j0 (or R + j 0; it doesn’t matter if j is multiplied by 0!). If R = 3 Ω with no reactance, you get an impedance of 3 − j 0, which corresponds to the point (3,j 0) on the RC plane. If you have a pure capacitive reactance, say XC = −4 Ω, then the complex impedance is 0 − j 4, and this is at the point (0,−j 4) on the RC plane. Again, it’s important, for completeness, to write the “0” and not just the “−j 4.” The points for 3 − j 0 and 0 − j 4, and two others, are plotted on the RC plane in following figure In practical circuits, all capacitors have some leakage resistance. If the frequency goes to zero (pure dc), a tiny current always flows, because no capacitor has a perfect insulator between its plates. In addition to this, all resistors have a little capacitive reactance because they occupy a finite physical space. So there is no such thing as a mathematically perfect resistor, either. The points 3 − j0 and 0 − j4 represent an ideal resistor and an ideal capacitor, respectively—components that can be worked with in theory, but that you will never see in the real world.

Untitled

Sometimes, resistance and capacitive reactance are both placed in a circuit deliberately. Then you get impedances such as 2 − j 3 and 5 − j 5, both shown in above figure.

Remember that the values for XC are reactances, not the actual capacitances. If you raise or lower the frequency, the value of XC will change. A higher frequency causes XC to get smaller negatively (closer to zero). A lower frequency causes XC to get larger negatively (farther from zero, or lower down on the RC plane). If the frequency goes to zero, then the capacitive reactance drops off the bottom of the RC plane to negative infinity!