Phase shift help (how's your maths?!)

[HankMcSpank]

I’m trying to work out the phase shift for 330Hz through an all pass filter which has a value of 68K for R1 & 10nf for C1.....

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Now formulas aren’t a strongpoint of mine, but apparently, this one will provide the phase shift for any given frequency through the above circuit ….




(source = http://www.maxim-ic.com/appnotes.cfm/an_pk/559


Where where w is the frequencies in rad/s.

I want to determine the phase shift for 330Hz, so I used this online calculator to get that converted to radians per sec

http://www.sciencelab.com/data/conve...nversion.shtml

therefore w (radians per second) for 330Hz is 2,073.451149

R1 = 68,000
C1 = 0.00000001


Ok, don't laugh, but I tried to use that formula above....

Top part of the equation (within the brackets)….

2 x 2,073.451149 totalling 4146.902298

divided by 68,000 (ie R1 @68K) x 0.00000001 (C1 @10nF) totalling 0.00068

Therefore 4146.902298 divided by 0.00068 = 6098385.7323529411764705882352941 (this is the top number within the brackets)



Bottom part of the equation (again within the brackets)….

4299199.667289420201 (ie ‘w’ or 2,073.451149 squared)

Minus….

1 divided by 0.00068 = 1470.5882352941176470588235294118

Which then must be squared = 2162629.7577854671280276816608997

Therefore we end up with 4299199.667289420201 - 2162629.7577854671280276816608997, resulting in....

2136569.909503953072972318339101 (this is the bottom number within the brackets)

Therefore to get the number within brackets, we must divide the top number by the bottom number…

6098385.7323529411764705882352941 / 2136569.909503953072972318339101

Therefore the number in brackets is 2.8542879431306799315135558591161

Now a scientific calculator tells ne that the tan of the above number is 0.049857973507496406210421036060919 degrees.

now tan -1 means I must shift the decimal point one place to the right?

So, am I right in thinking that the phase shift @330Hz through the above all pass is just a mere 0.5 degrees?

[richie00boy]

Excellent working there, just a small mistake at the very last hurdle. Tan -1 means inverse tan, you should have the function on your scientific calculator, usually shift or 2nd then press Tan.

So you need to find the inverse tan of 2.8543 which is approx 70.7 degrees.

[jcx]

some of us are just lazy - let (Free!) software do the math



the inset is SciLab console calc for those who need better than the LtSpice cursor resolution

[HankMcSpank]

Thanks richie00boy - I knew that if there was likely to be a mistake, it'd be in the last part ...as I've not used tan, cos, etc since leaving school about 30 years ago (& thanks for not laughing!). On the standard XP 'calculator ...in scientific mode, do you know how to do an inverse tan on it?

That said, in the light of jcx's post above - it looks like either the formula in that Maxim sheet is wrong...or my interpretation of it is!


jcx ...I'm all for letting the software take the strain - & I note that it's got to a different outcome than my calculations - 331Hz through that circuit has a 109 degree lag ...I'd love to know why! (I'm assuming there's is right!). I'll try & work out how to get access to what you've got there - it looks funky (though the thought of having to knock up the whole circuit just to establish a phase shift isn't that appealing - I guess once it's done once though....)

[richie00boy]

Simply tick the Inv checkbox on the second row of buttons below the display.

Sometimes simulated circuits show different results because op-amps have their own phase shifts as well, but it's usually just a very small difference. Something else is amiss.

[HankMcSpank]

Originally Posted by richie00boy

Simply tick the Inv checkbox on the second row of buttons below the display.

Many thanks!

Originally Posted by richie00boy

Sometimes simulated circuits show different results because op-amps have their own phase shifts as well, but it's usually just a very small difference. Something else is amiss.

That'll be me!

When I get a mo, I'll try again, but this time rather than covert to radians per sec, maybe I'll just give it a go with Frequency, extract from that datasheet....




"where w is the frequencies in rad/s, or 2×pi×f, when f is in Hertz. "

[HankMcSpank]

Ok, I used Excel (using Radians per sec again)...

Input Values….
Resitance =68,000
Capacitance =0.00000001
R1C1 =0.00068000
w (Radians per sec) =2,073.451149
w² =4,299,199.66728942

Top Line of formula
2w = 4,146.90229800
R1C1 = 0.00068
Result (2w / R1C1) = 6,098,385.73235294

Bottom Part of formula (sub brackets)
1 / R1C1 = 1,470.58823529

Bottom Part of formula
w² minus 1 / R1C1 = 4,297,729.07905413
[w² minus 1 / R1C1]² = 18,470,475,236,947.4

Inverse Tangent of 18,470,475,236,947.4 =89.99900000 (let's call it 90 degrees!)

So a different figure again!

I am uncertain which order to do the maths on the lower part of the equation....

do I square 1/R1C1 then subtract that result from w²

*or*

do subtract 1/R1C1 from w² & then square that result?

[richie00boy]

The first one. If it was all to be squared there would be brackets around the whole bottom part with the square to the right of it. Although very occasionally you do get poorly written formulas.

[HankMcSpank]

Ok, richie00boy ....we were right with our first calculation!

The phase shift at 330Hz for an all pass filter circuit with the values of 68K & 10nf is +70.69213237 degrees

How do I know?

Well, I know the centre frequency for that circuit with those resistance & capacitance values - it's approx 235Hz (where this will yield a 90 deg phase shift). Here's the formula for the centre frequency...

1 / 2xPi x C1R1

So then I created a spreadsheet to work out what the same circuit's phase shift would be for 330Hz....it came out at +70.69213237 degrees.

Then, in my Excel spreadsheet's cell - where I enter the frequency as an input - I changed it to 235Hz (the centre frequency of the circuit) ....& the spreadsheet result changed to 90 degrees (ie the centre freq phase shift) ....bingo!

Alas, I'm unable to upload the spreadsheet, as this forum doesn't allow Excel Spreadhseet uploads. But if anyone wants a copy of the spreadsheet drop me a PM with your email address.

[jcx]

109.307... degrees is the "correct" answer

the sim's -109 comes from the software's phase unwrapping algorithm

a simple calculator style atan function is going to return a value between +/- pi/4 rad == +/-90 degrees while the "true" value of the phase shift may have an additional +/-n*180 degree phase

at "0" freq the cap is open and the circuit has a +1 gain == 0 phase shift

at high freq the cap is a short and the circuit is an inverter with -1 gain == 180 (or -180) degree phase shift

so the phase shift smoothly varies from 0 to either of +/-180 which means it goes beyond the +/-90 range of the elementary atan function

I believe the sign ambiguity enters with +/-180 degrees being "equal" so the software chooses one - probably using the magnitude response slope - which would be correct for a minimum phase circuit - which this is not


if you sim the circuit with a .tran you can see the phase shift is clearly greater than 90 degrees at 330 Hz, a frequency sweep should show the phase smoothly varying as I described above