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4.4 Matching any two complex impedances - Smith Chart Matching 185 By "transforming" the reflection coefficient into an impedance we have reduced our level of abstraction however we have made things a bit more difficult graphically. That is because, whereas the magnitude of the reflection coefficient only varies between -1 and 1, thereby making it easy to represent it on a polar graph (section 3.6), the magnitude of a complex impedance may be just any value! Good old Phillip H. Smith of Bell Labs found a really clever way around this. First of all, he recognised that in an RF or Microwave Systems the characteristic impedances of the majority of the transmission lines are identical and equal to the internal impedance of the generator. He therefore thought to normalise each impedance in his circuit to the characteristic impedance of the RF system, Z 0 . Let us indicate the normalised impedances, and their respective resistances and reactances with lower case letters and un-normalised ones with upper case letters. For an impedance we define a normalised equivalent as To de-normalise our impedance all we need to do is multiply by so we haven't really added much complexity to our lives. Also note how an impedance equal to the characteristic impedance of the system, has a normalised value of 1. Next, and this is the very clever bit, he considered lines of constant normalised resistance (Figure 4.4-3) and lines of constant normalised reactance (Figure 4.4-4) and bent them in such a way as to fit in the unity radius polar plot of the reflection coefficient! This is shown in Figure 4.4-5. Figure 4.4-3 Constant normalised resistance circles Figure 4.4-4 Constant normalised reactance circles r=0 r=0.5 r=1 r=2 0 r r=0 r=0.5 r=1 r=2 0 +jx -jx x=0 x=0.5 x=-0.5 x=-1 x=1 x=0 x=0.5 x=-0.5 x=-1 x=1 Conquer Radio Frequency 185 www.cadence.com/go/awr

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