Pole-at-zero

Figure 7-5 The Integrator (pole-at-zero) Operational Amplifier...

Note that the integrator has a single-pole — at zero frequency . Therefore, we will often refer to it as the pole-at-zero stage or section of the compensation network. This pole is more commonly called the pole at the origin or the dominant pole. 

The basic reason why we will always strive to introduce this pole-at-zero is that without it we would have very limited dc gain. The integrator is the simplest way to try and get as high a dc gain as possible. 

Note that in general, the transfer function of a pole-at-zero function such as this will always have the following form... 

Let us try to connect the dots now. Both the first- and second-order filters we have discussed gave us poles. That is because they both had s in the denominators of their transfer functions — if s takes on specific values, it can force the denominator to become zero, and the transfer function then becomes infinite, and we get a pole by definition. The values of 5 at which the denominator becomes zero are the resonant (or break) frequencies, that is, the locations of the poles. For example, a hypothetical transfer function 1/s will give us a pole at zero frequency (the pole-at-zero we talked about earlier). 

If we were using only an op-amp integrator (no LC-pole cancellation), the open-loop gain will then fall with a slope of —2 instead of —3. But that is not enough. However, if we introduce just one zero (besides the pole-at-zero and the ESR zero), we can get the —1 intersection at crossover, as we are seeking. 

There are two poles pi and p2 (besides the pole-at-zero pO ), and two zeros, zl and z2 provided by this compensation. Note that several of the components involved play a... 

Type 1 compensation provides only a pole-at-zero, and in fact can only work with current mode control (that too with the ESR zero below crossover). Note that it is just a simple integrator. 

Note The way we have separated the terms of the transconductance op-amp, the pole-at-zero (fpO H3) seems to be dependent only on Cl (no resistance term). However, we could have also clubbed the voltage divider section HI along with H3 (since these are simply cascaded blocks, in no particular order). Then the pole-at-zero would have appeared differently (and also included a resistance term). However, whichever way we proceed, the final result, that is, H, will remain unchanged. In other words, HI, H2, and H3 are just intermediate mathematical constructs in calculating H (with no obvious physical meaning of their own necessarily). That is why the actual pole-at-zero frequency of the entire feedback block is designated as fpO, not fpO H3. 

This is the heat capacity of a one-dimensional oscillator according to Einstein. The heat capacity deviates at low temperatures. It is not possible to expand into a Taylor series around T 0. In other words, the function has a pole at zero, which emerges as an essential singular point. A more accurate formula is due to Debye, n... 

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