Points in the RC plane

Chapter Points in the RC plane

Teach Yourself Electricity and Electronics Third Edition Book
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Teach Yourself Electricity and Electronics Third Edition Book

  • In this problem, you need to put the numbers in the formula and solve for the un-known C. Begin with the equation 1001/(6.2810.0C)Dividing through by 100, you get:11/(62810.0C)Multiply each side of this by C, and you obtain:C1/(62810.0)This can be solved easily enough. Divide out C1/6280 on your calculator, andyou’ll get C0.000159. Because the frequency is given in megahertz, this capacitancecomes out in microfarads, so that C 0.000159µF. You might rather say that this is 159pf (remember that 1 pF 0.000001µF).Admittedly, the arithmetic for dealing with capacitive reactance is a little messierthan that for inductive reactance. This is the case for two reasons. First, you have towork with reciprocals, and therefore the numbers sometimes get awkward. Second, youhave to watch those negative signs. It’s easy to leave them out. But they’re importantwhen looking at reactances in the coordinate plane, because the minus sign tells youthat the reactance is capacitive, rather than inductive.Points in the RC planeCapacitive reactance can be plotted along a half line, or ray, just as can inductive reac-tance. In fact, capacitive and inductive reactance, considered as one, form a whole linethat is made of two half lines stuck together and pointing in opposite directions. Thepoint where they join is the zero-reactance point. This was shown back in Fig. 14-1.In a circuit containing resistance and capacitive reactance, the characteristics aretwo-dimensional, in a way that is analogous to the situation with the RL plane from theprevious chapter. The resistance ray and the capacitive-reactance ray can be placedend to end at right angles to make a quarter plane called the RC plane (Fig. 14-5). Re-sistance is plotted horizontally, with increasing values toward the right. Capacitive re-actance is plotted downwards, with increasingly negative values as you go down.The combinations of R and XC in this RC plane form impedances. You’ll learn aboutimpedance in greater detail in the next chapter. Each point on the RC plane corre-sponds to one and only one impedance. Conversely, each specific impedance coincideswith one and only one point on the plane.Impedances that contain resistance and capacitance are written in the form RjXC. Remember that XC is never positive, that is, it is always negative or zero. Becauseof this, engineers will often write RjXC, dropping the minus sign from XC and re-placing addition with subtraction in the complex rendition of impedance.If the resistance is pure, say R3Ω, then the complex impedance is 3 j0 and this cor-responds to the point (3,0) on the RC plane. You might at this point suspect that 3 j0 is thesame as 3 j0 , and that you really need not even write the “j0” part at all. In theory, both ofthese notions are indeed correct. But writing the “j0” part indicates that you are open to thePoints in the RC plane251