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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 chapter 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 (Fig 14-5) 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 X C in this RC plane form impedances You ll learn about impedance in greater detail in the next chapter 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 X C can be written in the form R + jX C Remember that X C is always negative or zero Because of this, engineers will often write R jX C instead If an impedance is a pure resistance R with no reactance, then the complex impedance is R j 0 (or R + j 0; it doesn t matter if j is multiplied by 0!) If R = 3 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 X C = 4 , 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 Fig 14-6 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.

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p r , pi ^ )

^ hr(pr - pi)

capacitive reactance (XC ) and resistance (R ) This is also known as the RC quarter-plane, or simply as the RC plane

Figure 815: Notation and geometry for analysis of rover to base separation Therefore, Rr Ro = hr (pr po ) + (cos( ) 1) ho po pi 2 Ro 2 (8128)

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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 Sometimes, resistance and capacitive reactance are both placed in a circuit deliberately Then you get impedances such as 2 j3 and 5 j5, both shown in Fig 14-6 Remember that the values for X C are reactances, not the actual capacitances If you raise or lower the frequency, the value of X C will change A higher frequency causes X C to get smaller negatively (closer to zero) A lower frequency causes X C 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!

Using the approximation sin( ) (accurate to third order in ) we have | | pr po ho (po pi ) = d Ro

Using this information, eqn (8128) reduces to Rr Ro hr (pr po ) + d2 2Ro (8129)

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Recall from the last chapter that RL impedances can be represented as vectors The same is true for RC impedances In Fig 14-6, four different complex impedance points are shown Each point is represented by a certain distance to the right of the origin (0,j 0), and a certain displacement downward The first of these is the resistance, R, and the second is the capacitive reactance, X C The complex RC impedance is a two-dimensional quantity Impedance points in the RC plane can be rendered as vectors, just as they can in the RL plane Then the points become rays, each with a certain length and direction The magnitude and direction for a vector, and the coordinates for the point, both uniquely define the same complex impedance The length of the vector is the distance of the point from the origin, and the direction is the angle measured clockwise from the resistance (R ) line, and specified in negative degrees The equivalent vectors, for the points in Fig 14-6, are shown in Fig 14-7

In the transition from eqn (8118) to eqn (8119) this error term is incorporated into the measurement error The error can be substantial for d > 1km

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