Seminar: Moderne Methoden der analogen Schaltungstechnik

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Seminar Moderne Methoden der analogen
Schaltungstechnik

• Teil II A Root-locus technique
• Eugenio Di Gioia

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Root-locus technique

• Trajectory of the poles and zeros of the
  feedback-amplifier H(s) on the s-plane when the
  low-frequency loop-gain ßADC varies.
• For every loop-gain the position of poles and
  zeros must be calculated analytically or
  simulated
• More information about the amplifier performance
  than in frequency-domain techniques can be
  obtained
• Drawback high computational effort

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Example three-pole transfer function
TF of the amplifier with three real coincident
poles
Overall gain of the feedback amplifier, ß is
resistive
The poles of the feedback amplifier are given by
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Root-locus
Real negative pole
Complex-conjugate poles
The poles are shifted on the s-plane depending on
ßADC For ßADC0 (no feedback) s1s2s3sp1
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Root-locus
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Root-locus

• The root-locus shows that if the loop-gain
  becomes larger than 8, the amplifier will be
  unstable
• This happens because two of the three poles enter
  the RHP
• Compensation can shift the three open-loop poles
  further on the left, allowing larger values of
  ßADC to be used

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Construction of the root-locus

• Analytical calculation is complicated (an
  equation of n-th order must be solved)

Overall Gain with feedback
Assuming
and
We obtain
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Construction of the root-locus

• The root-locus is determined by solving the
  denominator of H(s) for every value of the loop
  gain ßDCADCT

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Root-locus

• T 0 Open-Loop
• The roots are given by the poles of the
  amplifier A and the feedback network ß
• Simplifying

The poles of H(s) are the poles of A(s) in this
case
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Root-locus

• T ? 8
• The roots of the equation are given by the zeros
  of A and ß
• The poles of H(s) are given by the zeros of ß in
  this case
• Conclusion by varying T, the poles of H(s) move
  from the poles of ßA to the zeros of ßA
• If the zeros of ßA are less than the poles, the
  poles move toward infinity

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Rules for Root-Locus construction

• The branches of the root-locus start at the poles
  of ßA for T0 and terminate at the zeros of ßA
  for T?8
• If ßA has all zeros in the LHP the locus is on
  the real axis if there is an odd number of zeros
  and poles to the right
• Segments of the locus that are on the real axis
  between two poles must branch out from the real
  axis

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Use of Rule 2 3
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2
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Rules for Root-Locus construction

• The locus is symmetrical with respect to the real
  axis
• Branches of the locus leave the real axis at
  right angles
• When branches break away from the real axis, they
  do that at the point where the vector sum of
  reciprocals of distances to the poles equals the
  vector sum of reciprocals of distances to the
  zeros
• If ßA has all zeros in the LHP, branches go to
  infinity with angles of (2n-1)p/(NP-NZ),
  n0,1,,NP-NZ-1
• Branch asymptotes intersect the real axis at

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Examples of root-loci