Jim T's PLL Analysis

[Active8]

hi:

i was just wondering what the circuit for the phase detector in the
analysis is. i'd like to know how the phase detector gain was derived,
i.e.,

[Vp/(PI*Re)](ThetaRef - ThetaVCO)

http://www.analog-innovations.com/SED/PhaseLockedLoopAnalysis.pdf

TIA,
mike

 

[Russell Powell]

For LPLL's it's an analog mulitplier - probably a Gilbert mulitplier.
I haven't the means to quickly make you a postable schematic....



--
sincerely,

Russell Powell

Artisan Components
SR. FAE - U.S. Central
[email protected]
469-438-6589

 

[Jim Thompson]

hi:

i was just wondering what the circuit for the phase detector in the
analysis is. i'd like to know how the phase detector gain was derived,
i.e.,

[Vp/(PI*Re)](ThetaRef - ThetaVCO)

http://www.analog-innovations.com/SED/PhaseLockedLoopAnalysis.pdf

TIA,
mike

ANALOG phase detector, i.e. multiplier of sine waves.

Remember this dates to the mid '60's before Ron Treadway and I had
designed the edge-matcher.

Russell Powell was correct in his follow up.

...Jim Thompson

 

[Active8]

hi:

i was just wondering what the circuit for the phase detector in the
analysis is. i'd like to know how the phase detector gain was derived,
i.e.,

[Vp/(PI*Re)](ThetaRef - ThetaVCO)

http://www.analog-innovations.com/SED/PhaseLockedLoopAnalysis.pdf

TIA,
mike

ANALOG phase detector, i.e. multiplier of sine waves.

Remember this dates to the mid '60's before Ron Treadway and I had
designed the edge-matcher.

Russell Powell was correct in his follow up.

...Jim Thompson

ok. if figured that since it was an analog detector, you may have used a
multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

mike

 

[Kevin Aylward]

ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Active8]

ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

that doesn't explain the Vp/(PI.Re) part.

mike

 

[Kevin Aylward]

ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

that doesn't explain the Vp/(PI.Re) part.

mike

I haven't really looked at Jims paper, or what circuit preceded it, just
skimmed it right now, but the above is just a gain constant. Without
even looking at the details, one can use dimensional analysis and logic
to get the numbers. Its what Einstein used to get the gain constant for
T_uv=A.R_uv in his Gravitational field equations. e.g. What must the
output be for maximum phase? Since I don't know what his circuit is, I
would guess that there is a current given by Vp/Re. Presumably when
delta_theata = pi, its maximum output is Vp/Re.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Russell Powell]

One would have to use small signal analysis on the
circuit to determine the actual PD gain. Max. PD dc output voltage should
be reached when the phase difference betweeen the VCO output and
incoming reference is either 0 or PI - Meaning the incoming signal is
leaving "lock" range. It appears to me that the equation you mention should
be more like:
Vpd = Vo,max*cos(ThetaRef - ThetaVco), where Vo,max = Kd(PI/2),
where Kd is the PD gain.

Take caution, I know enough about PLL's to be dangerous.......



--
sincerely,

Russell Powell

Artisan Components
SR. FAE - U.S. Central
[email protected]
469-438-6589

Active8 wrote:
ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

that doesn't explain the Vp/(PI.Re) part.

mike

I haven't really looked at Jims paper, or what circuit preceded it, just
skimmed it right now, but the above is just a gain constant. Without
even looking at the details, one can use dimensional analysis and logic
to get the numbers. Its what Einstein used to get the gain constant for
T_uv=A.R_uv in his Gravitational field equations. e.g. What must the
output be for maximum phase? Since I don't know what his circuit is, I
would guess that there is a current given by Vp/Re. Presumably when
delta_theata = pi, its maximum output is Vp/Re.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Active8]

Active8 wrote:
ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

that doesn't explain the Vp/(PI.Re) part.

mike

I haven't really looked at Jims paper, or what circuit preceded it, just
skimmed it right now, but the above is just a gain constant. Without
even looking at the details, one can use dimensional analysis and logic
to get the numbers. Its what Einstein used to get the gain constant for
T_uv=A.R_uv in his Gravitational field equations. e.g. What must the
output be for maximum phase? Since I don't know what his circuit is, I
would guess that there is a current given by Vp/Re. Presumably when
delta_theata = pi, its maximum output is Vp/Re.

you might be right. but if delta_theta were 3PI/2, that's greater than
Vp/Re.

mike

 

[Jim Thompson]

Active8 wrote:
ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

that doesn't explain the Vp/(PI.Re) part.

mike

I haven't really looked at Jims paper, or what circuit preceded it, just
skimmed it right now, but the above is just a gain constant. Without
even looking at the details, one can use dimensional analysis and logic
to get the numbers. Its what Einstein used to get the gain constant for
T_uv=A.R_uv in his Gravitational field equations. e.g. What must the
output be for maximum phase? Since I don't know what his circuit is, I
would guess that there is a current given by Vp/Re. Presumably when
delta_theata = pi, its maximum output is Vp/Re.

you might be right. but if delta_theta were 3PI/2, that's greater than
Vp/Re.

mike

Vp/Re has the dimensions of *current*... delta_theta/pi(radians) is
*dimensionless* ;-)

...Jim Thompson

 

[Active8]

Vp/Re has the dimensions of *current*... delta_theta/pi(radians) is
*dimensionless* ;-)

...Jim Thompson

i'd laugh, but that wasn't funny.

obviously if i can write the laplace notation for a circuit and follow
the rest of your PLL analysis (nice little paper, BTW), i can see that
Vp/Re is current ( you even drew an arrow!). the original question was
how you arrived at Kp = Vp/PI.Re... i guess it doesn't matter since
these days, i'm looking at specs with

Kp = 500uA/rad

and such with no other info and a circuit designed for me that i'm
supposed to assume is right. then, with analog devices, you just use
their EDA tool.

that takes all the fun out of it.

mike

 

[Active8]

i'd laugh, but that wasn't funny.

obviously if i can write the laplace notation for a circuit and follow
the rest of your PLL analysis (nice little paper, BTW), i can see that
Vp/Re is current ( you even drew an arrow!). the original question was
how you arrived at Kp = Vp/PI.Re... i guess it doesn't matter since
these days, i'm looking at specs with

Kp = 500uA/rad

and such with no other info and a circuit designed for me that i'm
supposed to assume is right. then, with analog devices, you just use
their EDA tool.

that takes all the fun out of it.

mike

even if i take the Vp/Re part on blind faith, the PI term had me
wondering. i'm going to do the trig thing... sin(a).sin(a+b), integrate
and divide by the interval... PI?

mike

 

[Active8]

One would have to use small signal analysis on the
circuit to determine the actual PD gain.

hi Russell:

the "circuit" is a current source [Vp/(PI.Re)](deltaTheta) followed by a
loop filter driving a VCO - Kv/s. it's all pretty straight forward
except the Kp part. Since Jim said the PD was a multiplier, i can only
guess at a transistor with emitter Re operating non-linearly as a
multiplying mixer (as opposed to an audio mixer).

Max. PD dc output voltage

PDs put out pulses, or in this case, with a multiplier, you'd get sum
and difference. perhaps, by "DC out" you mean the average, yes?

should
be reached when the phase difference betweeen the VCO output and
incoming reference is either 0 or PI - Meaning the incoming signal is
leaving "lock" range.

that's a reasonable guess (but 0 rad?). it's the only way i see the PI
term in the denominator happening.

It appears to me that the equation you mention should
be more like:
Vpd = Vo,max*cos(ThetaRef - ThetaVco), where Vo,max = Kd(PI/2),
where Kd is the PD gain.

see http://www.analog-innovations.com/SED/PhaseLockedLoopAnalysis.pdf

***
Jim goes on to analyze the circuit using what i gave as Kp as Kp only
it's not a voltage, but a current source. that (not just Jim) had me a
bit confused. i see Pds with charge pumps in PLL chips and ASICs with
the circuits designed around them and no info whatsoever beyond that.

i had to assume that Kp.Kf.Kv can be

volts/rad * volts/volt * rads/sec.volt = rads

*OR* for a current source PD...

Amps/rad * Volts/Amp * rads/sec.volt = rads since a current pumped into
the filter will charge it up. that's the way i attacked the Micrel
MICRF501's PLL.

i was going to ask if it's OK to treat it that way, so while i'm here,
is it?

the spec gives the PD gain as 500uA/rad. i treated that as Kp = 500e-6.
then i used 500e-6(delta_Theta) just like Jim did to find the trsansfer
function:

Theta_out/Theta_ref
***

Take caution, I know enough about PLL's to be dangerous.......

do spill your guts, then

BTW, you said:

For LPLL's it's an analog mulitplier - probably a Gilbert mulitplier.
I haven't the means to quickly make you a postable schematic....

i remember a Gilbert cell from somewhere. vague memory. kinda remember a
ring diode mixer topology. or maybe FETs like in an active DBM. then
we're back to "what the hell is Re?"

as for schems, the better ascii art you find here (not counting this)...

+----->|-
| D2
-
^ D1
|

is created by Andy´s ASCII-Circuit v1.22.310103 Beta www.tech-chat.de

there's your tip for the day

thanks,
mike

--
sincerely,

Russell Powell

Artisan Components
SR. FAE - U.S. Central
[email protected]
469-438-6589

Active8 wrote:
ok. if figured that since it was an analog detector, you may have
used a multiplier. so how do you get

[Vp/(PI*Re)](ThetaRef - ThetaVCO) amps/rad

from it? i can't quite see that coming from a mult output.

Use trig and Expand y=sin(x+a).sin(x)

that doesn't explain the Vp/(PI.Re) part.

mike

I haven't really looked at Jims paper, or what circuit preceded it, just
skimmed it right now, but the above is just a gain constant. Without
even looking at the details, one can use dimensional analysis and logic
to get the numbers. Its what Einstein used to get the gain constant for
T_uv=A.R_uv in his Gravitational field equations. e.g. What must the
output be for maximum phase? Since I don't know what his circuit is, I
would guess that there is a current given by Vp/Re. Presumably when
delta_theata = pi, its maximum output is Vp/Re.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Jim Thompson]

i'd laugh, but that wasn't funny.

obviously if i can write the laplace notation for a circuit and follow
the rest of your PLL analysis (nice little paper, BTW), i can see that
Vp/Re is current ( you even drew an arrow!). the original question was
how you arrived at Kp = Vp/PI.Re... i guess it doesn't matter since
these days, i'm looking at specs with

Kp = 500uA/rad

and such with no other info and a circuit designed for me that i'm
supposed to assume is right. then, with analog devices, you just use
their EDA tool.

that takes all the fun out of it.

mike

I was just letting the thread run for awhile, for amusement value.

Now I'll have to try and remember what the hell I was doing nearly 40
years ago ;-)

...Jim Thompson

 

[Ken Taylor]

I was just letting the thread run for awhile, for amusement value.

Now I'll have to try and remember what the hell I was doing nearly 40
years ago ;-)

...Jim Thompson

Heck, I have trouble remembering what I was doing 40 *minutes* ago!!

Ken

 

[Active8]

I was just letting the thread run for awhile, for amusement value.

i can accomodate you. i found something so amusing i wish my scanner ran
on OSs older than win98, so i could post the side by side for
comparison. haven't got around to connecting it to the Linux box yet.
i'd have to load Sane and i'm anything *but* sane right now.

Now I'll have to try and remember what the hell I was doing nearly 40
years ago ;-)

don't feel bad, i'm having to look up crap i haven't touched in years.

...Jim Thompson

expanding sin(theta).sin(theta + deltaTheta), integrating over 0 to PI
and dividing by PI didn't do it.

but since you want amusement, i did come up with something else.

where you started with

Vp.Vf.(delta_Theta).Kv/s and rearranged to get

theta_vco/theta_ref... your xfer function,

i'm not sure why you did that at all. using the delta_Theta term, that
is.

not being able to pull the cobwebs off the brain, i just took

G(s)
theta_vco(s) = ------------- . theta_ref(s) and plugged in
1 + G(s)H(s)

Vp
Kp = -----
PI.Re

Kf = what you had *after* i verified that

K(s+a)
G(s)H(s)= ------ ; i.e. Type II system
s^2

indicated a filter of the form

s + a RzCs + 1
Kf = ----- = --------
s Cs

Kp.Kf.Kv
so G(s) = --------
s

and H(s) = 1 ... there's something missing from your method... the

feedback function, not that it matters when it's 1.

and i got the same answer for theta_vco/theta_ref as you did.

how the hell those answers work out the same when i used the classic
closed loop xfer function and you used the open loop transfer function
(?) multiplied by delta_theta is beyond me. or does the inclusion of

(theta_ref - theta_vco) in your xfer function somehow *effectively*
close the loop? i can almost see that.

if you want to see the whole work, i'll show ya. somehow ;-) i can see
how the math works. in line 2 of your work, you've rearanged the eq so
that you end up with 1 + G(s)H(s) on the left side. it's *emerged*, so
to speak. it wouldn't have happened if you hadn't included

(theta_ref - theta_vco) in the whole mess.

was that amusing enough? did you spray some WD-40 on your time machine?
careful, 40 years is a long time... your flux capacitor may be dry
add booze?

Best Regards,
mike

 

[gwhite]

even if i take the Vp/Re part on blind faith, the PI term had me
wondering. i'm going to do the trig thing... sin(a).sin(a+b), integrate
and divide by the interval... PI?

The pi is from the Fourier expansion of a square wave. One of the detector
input signals is assumed to switch the transistors, diodes, or whatever
("square"). The other input to the detector is a sinusoid. The theta_r -
theta_v term (or whatever the designation) is the approximation of sin(theta_r -
theta_v) for "small angles." That is, lock is assumed and the phase error is
near zero.

 

[Jim Thompson]

The pi is from the Fourier expansion of a square wave. One of the detector
input signals is assumed to switch the transistors, diodes, or whatever
("square"). The other input to the detector is a sinusoid. The theta_r -
theta_v term (or whatever the designation) is the approximation of sin(theta_r -
theta_v) for "small angles." That is, lock is assumed and the phase error is
near zero.

That sounds correct. I did this eons ago, but your description would
fit: square wave VCO but analog input... IIRC this was for a TACAN
aircraft guidance system... it's all starting to come back ;-)

...Jim Thompson

 

[Active8]

that sounds like fun. thanks.

That sounds correct. I did this eons ago, but your description would
fit: square wave VCO but analog input... IIRC this was for a TACAN
aircraft guidance system... it's all starting to come back ;-)

...Jim Thompson

ah ha! that would explain the frequency. would you like some wodka?
cigarrette? you cannot lie to us. we know who you are. now tell us, who
are you? ;-)

 

[Jim Thompson]

that sounds like fun. thanks.
ah ha! that would explain the frequency. would you like some wodka?
cigarrette? you cannot lie to us. we know who you are. now tell us, who
are you? ;-)

I keep a ½ gallon of "Wodka" in the freezer... I make a mean Sake
martini (4 parts vodka to 1 part Sake) goes perfectly with Sashimi ;-)

"Cigarrette"... never cared for them, smoked a pipe for 36 years...
quit smoking when we moved into the new house (war department declared
it a no-smoking zone ;-)

Had a heart attack 4 years later... quitting smoking will do that to
you ;-)

...Jim Thompson