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Stability Criteria of Type 3 PLL

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promach

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I suppose we can derive the stability criteria based on poles of open-loop transfer function of F(z) in expression (4.11). However, the pole analysis does not help in deriving the stability criteria at expression (4.23)

Could anyone tell me if I had missed anything ?

1.png

2.png

4.png
 

However, the pole analysis does not help in deriving the stability criteria at expression (4.23)

Equation 4.23 only does the parameterization of the values of the gains K1 and K2 that put the root of the equation over radius 1, which is the stability limit in the Z domain.
 

If you look at root of the denominator within equation (4.11), you will only find z=0 and z=1 as roots.

Or did I miss anything ?
 

Ok, but the criteria to determine instability is when root crosses the 'z=1' circle bound.
 
instability is when root crosses the 'z=1' circle bound.

Ok, then setting equation (4.12) to 0 gives us two solution which is not equation (4.23) at all.

k2=0 , OR k2=4*k3
 

Ok, then setting equation (4.12) to 0...

No, at this equation you set the Z variable at the left side to '1', which cancel the remaining constant '1' at the right side.
 

No, at this equation you set the Z variable at the left side to '1', which cancel the remaining constant '1' at the right side.

This still give the same solution: k2=0 , k2=4*k3

Please double check yourself
 

Likewise to you, I slipped on the same mistake: Equation 4.12 represents the expression for the Zero, not for the Pole. In order to obtain equation 4.23 you should have to agebraically develop equation 4.11 and then obtain the expression for the global denominator, since in this equation the result is presented as a sum of partial equations that, if expanded, would have different denominators. By the way, you omit part of the text (for example equation 4.22) but nevertheless, surely somewhere the author must have mentioned some criterion of stability analysis (eg Routh-Hurwitz), which should have generated the equations 4.24 for example.
 

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