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Design >> Analog Design >> Physical significance of zero
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Message started by adesign on Jan 17^th, 2007, 9:38pm

Title: Physical significance of zero
Post by adesign on Jan 17^th, 2007, 9:38pm

Dear all,

Could you please explain the physical significance of a zero(both right half and left half) in a control system?

Regards,

Title: Re: Physical significance of zero
Post by bharat on Jan 19^th, 2007, 1:15am

It means; at that perticular frequency the output is zero even the input is present. To make the better undertanding, think of small signal model where the output current going to load and coming out of load is equal and finaaly current valus eis zero.
thanks
-Bharat

Title: Re: Physical significance of zero
Post by adesign on Jan 19^th, 2007, 1:42am

Thanks Bharat!

Here is my another query:

Consider the transfer function in s-domain

H(s) = 1+as

The zero frequency is given by

s = -1/a

==> jw = -1/a

At real frequency i.e. 1/a, the magnitude of H(s) is given by

|H(s)| = √2

In log magnitude term, it is:

= 3dB

This is not zero what you quoted.

Please comment.

Best regards,

Title: Re: Physical significance of zero
Post by Croaker on Jan 19^th, 2007, 5:14am

Bharat,
H(s) = 1 + as
H(-1/a) = 1 - 1 = 0

When you evaluate these s-domain functions, you have to fill in the actual s-value and then take the magnitude.

In the attached plot, there is a a zero at s = -4. i.e., H(s) = 1 + s/4

The point s=-4 does go to zero, but the slice of this 3d plot along the imag axis is what we look at for the frequency response, as just recently discussed in another post.

Hope this helps!

Title: Re: Physical significance of zero
Post by adesign on Jan 19^th, 2007, 5:34am

Does it mean that zero makes the transfer function to zero in s-domain whereas in real frequency domain the magnitude is 3dB?

When we plot a zero in Bode plot, there comes 3dB gain. Then where does that "zero" value come??

What'll be the effects of zero in time domain??

Title: Re: Physical significance of zero
Post by Croaker on Jan 19^th, 2007, 6:13am

That plot didn't get saved so well...

Here is the Matlab code (I found on the web) which you can modify to plot whatever. Modify the num and den vector...these are the transfer function co-efficients.

% Figure 3.2
%
% BM Mar 98

clear
clf
colordef(1,'black')

rx = -5:0.1:5;
ix = -5:0.1:5;
[rrx,iix] = meshgrid(rx,ix);
xx = rrx + j*iix;

num = [0.25 1];
den = [ 1 ];
z = zeros(size(xx));
nn = max(size(rx));
for jj = 1:nn
z(:,jj) = polyval(num,xx(:,jj))./polyval(den,xx(:,jj));
end
h = surf(rx,ix,20*log10(abs(z)));hold
plot3([-1 -1 ],[ -2 -2 ],[-30 10 ])
plot3([-1 -1 ],[ 2 2 ], [ -30 10 ])

xlabel('real')
ylabel('imag')
zlabel('magnitude (dB)')
view([50,40]);
fprintf(1,'Figure 3.2: press return to rotate\n')
pause

rotvec = 50:5:90;
ss = max(size(rotvec));
step = 30/(ss-1);
hvec = 40:-step:10;
for ii = 1:length(rotvec)
set(gca,'View',[rotvec(ii) hvec(ii)] )
drawnow
pause(2);
end

Title: Re: Physical significance of zero
Post by Croaker on Jan 19^th, 2007, 6:18am

adesign wrote on Jan 19^th, 2007, 5:34am:

Hopefully that plot helps. It's a surface with a hole (a zero) at s=-4. If there was a pole, it would look like someone stuck a pole under a blanket (the magnitude is going to infinity at that point)

You are correct. When looking at the s-plane you will see a zero magnitude value, but looking at the magnitude values along s = j*ω, you will see the 3 dB value as I mention below.

The zero is occurring exactly where the function says, at s = -1/a. But for frequency response you only look at the curve where s = j*ω. Now, along this axis you would see a 3 dB bump where s= 0+ j*ω= 0 + j*1/a. Note that s = -1/a + j * 0 is not the same thing. That's why you get a zero value at s = -1/a but a 3 dB bump at s = j*1/a.

Title: Re: Physical significance of zero
Post by mg777 on Jan 19^th, 2007, 10:55am

A zero represents differentiation of the input signal, while a pole integrates it. From a control theory perspective, a zero makes for a 'PD' controller which, like the proverbial hare, responds fast but with steady state error. OTOH a pole corresponds to a 'PI' controller that, like the tortoise, is slow to get there but does so with zero steady state error.

Here's a physical picture: say you're designing an elevator for a high rise building. You leverage a zero to get quick response but it won't do for the elevator to stop three feet short of the opening to a particular floor. Likewise you use poles for a slow but precise 'Driving Miss Daisy' ride that will align the door to the opening with millimetric precision (and leave passengers wishing they'd taken the stairs). The golden mean corresponds to a balance between haste and precision - it's called a PID controller and is non-trivial to design.

Non-linear circuits can be very different. A flip-flop regenerating its own input is like a snake chasing its tail. The unstable transient response takes off like a rocket, reaching steady state nirvana by saturating against one of the rails. If you design fighter aircraft you want 'em to whip around like that, but also not spin out of control or make Top Gun pass out. So you do an elaborate balance of linear and non-linear control. Even PID design is kid stuff compared to non-linear aircraft control amidst turbulence and countermeasures.

M.G.Rajan
www.eecalc.com

Title: Re: Physical significance of zero
Post by Croaker on Jan 19^th, 2007, 11:26am

M.G.Rajan, nice intuitive and insightful post! It's often not too hard to get the answers right in a class but hard to really 'get it' without immersing yourself in the area for a while.

Could you comment on a right-hand zero vs. a left-hand zero? The RHZ has a phase response like a left-hand pole but a regular zero magnitude response.

Title: Re: Physical significance of zero
Post by mg777 on Jan 19^th, 2007, 2:25pm

Croaker, you are right about the phase difference between right and left half zeros. The RHP zero is kinda funny - for a step input it tries to cancel the proportional term with a negative going impulse. That means the rise time will increase, so it actually behaves like a pole!? This intuition is borne out mathematically, since for small s

1/(1+αs) ≈ 1 - αs

One place we naturally come across a RHP zero is the (g_m - sC_μ) term in the numerator of the collector/drain voltage gain transfer functions derived from the hybrid-Pi small signal model of transistors. You can derive it using the following circuit for a common-emitter amplifier (Spice netlist in lieu of a diagram).

vs 1 0 ac
cpi 1 0 {small or zero} *doesn't matter since it's charged neat by an ideal voltage source
cmu 1 2 cmu
g 2 0 1 0 gm
rl 2 0 rl

The physical significance of this zero is that even with the base/gate driven with zero source or contact resistance (R_s, r_bb etc = 0) the displacement current through the drain-gate capacitance acts like an inertia. Kind of trying to cause an 'fT effect' if you will.

M.G.Rajan
www.eecalc.com

Title: Re: Physical significance of zero
Post by mg777 on Jan 19^th, 2007, 2:31pm

Following up my last post, I just realized that (g_m - sC_μ) can also be viewed as a transit time delay of C_μ/ g_m. This is probably more intuitive from the viewpoint of the Ramo-Shockley theorem.

M.G.Rajan
www.eecalc.com

Title: Re: Physical significance of zero
Post by Croaker on Jan 19^th, 2007, 4:17pm

mg777 wrote on Jan 19^th, 2007, 2:31pm:

Are you sure about that? The transit time is related to the diffusion capacitance of a BJT. Cb = gm * τ...maybe you are confusing Cb with Cu.

Unless you're relating the change in base charge over the change in base-collector voltage to transit time in a way I don't see.

I don't understand what you mean by an inertia and fT effect.

In my mind, the main problem with that zero in the CS amp is that it makes the phase margin worse if not dealt with somehow.

Title: Re: Physical significance of zero
Post by mg777 on Jan 20^th, 2007, 5:35am

You're right, C_μ/gm isn't a physical transit time, thanks for catching my mistake. I had meant collector transit time, but even that doesn't make sense since C_μ will decrease with increasing collector depletion layer thickess while that should increase the transit time. The collector transit time τ_c makes the α of a BJT vary as

α = α₀ sinc(θ) exp(-jθ)

where θ = ωτ_c/2 and sinc(θ) = sin(θ) / θ. My mistake was to attribute the exp(-jθ) ≈ 1-θ to an RHP. Close, but no cigar.

M.G.Rajan
www.eecalc.com

Title: Re: Physical significance of zero
Post by adesign on Jan 21^st, 2007, 11:22pm

mg777 wrote on Jan 19^th, 2007, 2:25pm:

The RHP zero is kinda funny - for a step input it tries to cancel the proportional term with a negative going impulse. That means the rise time will increase, so it actually behaves like a pole!?

Rajan,

Could you please elaborate more into this?

Regards,

Title: Re: Physical significance of zero
Post by mg777 on Jan 22^nd, 2007, 7:39am

Consider (1 - τs) operating on a unit step, where τ is positive i.e; the transfer function has an RHP zero. In time domain that's

(1 - τ d/dt) u(t) = u(t) - τ⋅δ(t)

If you don't like the impulse singularity you can replace u(t) with a smooth function nice(t) whose derivative is a blip(t). In that case the output will be

nice(t) - τ.blip(t)

The idea is still the same - the RHP zero tries to oppose, to first order, the proportional response of the system. In effect trying to mimic a LHP pole!

Of course, an LHP zero (1 + τs) will boost the signal i.e; the blip will add to the input rather than subtract from it. To first order an LHP zero would correspond to an RHP pole i.e; an 'unstable' system.

M.G.Rajan
www.eecalc.com

Title: Re: Physical significance of zero
Post by adesign on Jan 23^rd, 2007, 1:06am

Thank you Rajan, Bharat and Croaker for the detailed discussion!!

Best Regards,

Title: Re: Physical significance of zero
Post by avlsi on Jan 23^rd, 2007, 3:01am

Mr. Rajan,

I have a got a doubt. I have been taught all these days that RHP zero makes system unstable and not LHP zero.

Please clarify me on this.

I liked your practical explanation regaring poles and zeroes.

Title: Re: Physical significance of zero
Post by Croaker on Jan 23^rd, 2007, 7:44am

A RHZ has the same phase effect as a LHP. So yes, it could degrade phase margin and make the system unstable.

Title: Re: Physical significance of zero
Post by bharat on Jun 11^th, 2007, 10:04pm

Thanks for your splendid explaination.
Still I want to ask;
Does this mean that differentiator response time is less as compared to integrator response time for the same unit step?

mg777 wrote on Jan 19^th, 2007, 10:55am: