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In the circuit below, \$\tau \$ represents a time delay. We are using a first-order Pade approximation for this time delay. This means that \$\ e^{-s\tau} = (1-s\tau)/(1+s\tau)\$.

Below is the circuit. Does the feedback term, \$\ B = R + (1-s\tau)/(1+s\tau)\$? What exactly is the loop gain?

Op-amp with time delay

Hector
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  • You draw electrical components in your schematic, but the block with \$ \tau \$ is unfamiliar to me (is it just a gain?). Could you please explain? Next, although it *seems* obvious what input and what output is, I'd still draw and label input and output in the schematic as well. – Huisman Dec 12 '19 at 06:35
  • @Huisman \$\ \tau\$ is a time delay, represented as \$ e^{-s\tau} \$. As for the inputs and outputs, those we have to determine they are not given to us. The problem comes only with this schematic. – Hector Dec 12 '19 at 06:39
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    Where is the problem? You have two blocks (inverting integrator and delay unit) which form a closed loop. The transfer functions for both blocks are given....so it should not be a problem to find the transfer function for the open loop.... – LvW Dec 12 '19 at 08:34
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    Should that be: \$e^{-s\tau}= \frac{e^{-s\tau /2}}{e^{s\tau /2}}\rightarrow \frac{1-s\tau /2}{1+s\tau /2}\$? – Chu Dec 12 '19 at 17:40

1 Answers1

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Below is the open loop circuit without feedback.

Open loop ckt. w/o feedback

We know that since this is an ideal op-amp, \$V_+ = V_- = 0V \$. When we do KCL at the inverting terminal, we get \$|V_{out}/V_{in}| = 1/(sCR)\$.

Now we know this is the open loop voltage gain, or \$A\$. When we add feedback, our feedback term is \$\tau = e^{-s\tau}\$. The first order approximation for this is \$(1-s\tau/2)/(1+s\tau/2) \$.

op-amp with time delay

The feedback term represents \$B \therefore AB=[(1/(sCR)*(1-s\tau/2)/(1+s\tau/2))]\$. AB is the loop gain.

Hector
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    This answer is - of course - correct. However, only under the assumption that the hardware realization of the delay block (resp. the corresponding approximation) is decoupled from the integrator. – LvW Dec 15 '19 at 09:56