Loop Filters

I have been working my way through the loop filter types, so far:

• Low Pass
• 1st order PID
• Passive Lead-Lag
• Active Lead-Lag

No problem with the first three:

Low Pass

PID

Lead-Lag

The Active Lead Lag Filter

Here are the analog domain basic loop filters. The problem filter is shown at the bottom of the image:

(source: http://www.globalspec.com/reference/38390/203279/2-6-second-order-phase-locked-loops)

I have plotted the S domain versus the the modelled Z domain and the Z domain is good up until about Nyquist, so no problems:

The recursive equation was:

• LP0 = (69 * PD0 + 10 * PD1 - 59 * PD2 - 0 * LP1 + 64 * LP2) \ 64

The issue is that this filter cannot be used directly. The DC gain is +46dB and as the structure is that of an integrator and the PD is 0 to K, the filter just blows up:

This can be fixed by using a differential PD (i.e. -K/2 to +K/2) or by using a filter like the second filter below:

(source: elcrost.com/wp-content/uploads/2016/11/pll-circuit-page-2-rf-circuits-next-gr-phase-locked-loop-fm-demodulator-part3_pll_i-pdf-tutorial-zambezi.jpg)

After substituting the PD for a differential PD (i.e. -K/2 to +K/2) I get something that basically works:

(Note that the vertical scale has changed compared to the other PLL filters)

The analog equivalent is:

K	4000		=KpKv
R1	5600	R
R2	5600	R
C	0.000000100	F
T1	0.00056
T2	0.00056
WN	2673		=sqrt(K/T1)
D	0.75		=T2/2*sqrt(K/T1)

The main advantage of the last filter is greater range of control of damping factor and natural frequency.

Choosing a Loop Filter (The Shoot Off)

Now that I have all the filters working which one to use going forward? Active Lead-Lag or PID?

Active Lead-Lag Loop

Inputs:

• K (KpKv)
• R1
• R2
• C

Intermediate:

• T1 (=R1C)
• T2 (=R2C)
• WN (=sqrt(K/T1))
• D (=T2/2*sqrt(K/T1))

Code (after calculating Z-transform coefficients):

• PD2 = PD1
• PD1 = PD0
• PD0 = PD * K - K \ 2
• LP2 = LP1
• LP1 = LP0
• LP0 = (B0 * PD0 + B1 * PD1 + B2 * PD2 - A1 * LP1 - A2 * LP2) \ D
• LP = LP0

PID Loop (First Order)

Inputs:

• WN (natural frequency)
• D (damping factor)
• SR (sample rate)

Intermediate:

• T1 = 1 / WN / WN
• T2 = 2 * D / WN

Output:

• Kp = Proportional factor
• Kp = (1 + 1 / Tan(1 / (2 * T2 * SR))) / (2 * T1 * SR)
• Kp ~ (1 + 2 * T2 * SR) / (2 * T1 * SR)
• Ki = Integral factor
• Ki = 1 / (T1 * SR)

Code:

• PD1 = PD0
• PD0 = PD
• LP = Kp * PD0 + (Ki + Kp) * PD1 - LP

Conclusion (The Shoot Off Results)

It looks rather like a "no-brainer" to go with PID from a computational point of view (all other things appearing to be equal). Funny how the common wisdom of PID in the discrete domain is confirmed!

AlanX