Log-z Plane Root Locus Plots

Instead of making root locus plots in the z plane, it is sometimes convenient to make them in the log-z plane. In the z plane, the ordinate is the imaginary part of z and the abscissa is the real part of z. In the log-z plane, the ordinate is the 

There are two effects of using this new coordinate system  

The limit of stability becomes the imaginary axis in the log-z plane and the region of stability is the left half of the log-z plane. This is analogous to the situation in the continuous s plane. 

The first effect is obvious from the definition of the logarithm of z as given in Eq. (19.44). Inside the unit circle, the magnitude of z is less than 1. Therefore the In I z I is negative. On the unit circle, In z = 0. So the unit circle in the z plane maps into the left half of the log-z plane. 

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Figure 19.7h,c compare z-plane and log-z-plane root locus plots for first-order and second-order systems. For the first-order system, the single root moves to minus infinity in the log-z plane as z goes to zero. Then the root comes back... 

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