[sdiy] 90-degree phase displacement network calculations
Bernard Arthur Hutchins, Jr
bah13 at cornell.edu
Mon Jan 25 04:28:49 CET 2021
FIVE POINTS
(1) (just an aside) A “digital” 90-Degree-PDN is called a “Hilbert Transformer”. In contrast to an implementation with analog all-pass (flat magnitude but imperfect phase difference) the HT has a perfect phase of 90-degrees phase difference but a non-flat magnitude response (trade-off):
http://electronotes.netfirms.com/EN199.pdf
EN Vol 20 No. 199 pp 10-12 Sept 2001
(2) Keep in mind that all PNDs have a TRANSIENT response because of the poles. They take time to reach steady-state; - the longer the cascades, and the higher the pole-Q, the longer the delay, and possible artifacts.
(3) Consider using “Inverse Chebyshev” to set the phases error instead of equiripple (Cauer). At a small expense in upper frequency error, (larger than equiripple), you should be able achieve near zero in the audio band ALL THE WAY TO ZERO frequency if that is your desire.
4) I have previously mentioned EN#168:.
http://electronotes.netfirms.com/EN168-90degreePDN.PDF
This approach allows a designer to build with second-order sections and do the entire design with even just ONE selected capacitor value, while cutting the number of op-amps in half, and avoiding extremes of RC products.
(5) There is time-honored concept called “Classical Sensitivity“ (CS). It is: the CS of a performance-parameter, X, to a variation of (tolerance of) a component, y, is:
SXy = (y/ X) [ (delta)X / (delta)y ]
= [ (delta)X/ X / (delta)y/ y ]
This can be estimated fairly easily by incrementing the relevant design equation a small amount, OR better by calculation via partial derivative since X(y,….) is generally available:
SXy = (y/X) [ partial(X)/(respect to y ) ]
For example, a pole frequency might be p=1/RC at first-order [SpR = -1], but p = 1/sqrt(R1R2C1C2) at second-order [SpR=-1/2] which says that a second-order approach is better. For more on tuning with Sensitivity see:
http://electronotes.netfirms.com/AN361.pdf
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