In the majority of the cases, analog compensators for PSUs, take the form of an inverting op-amp which is usually internal to the controller IC, plus some external capacitors and resistors. Of course the capacitors and resistors determine the position of the poles and zeros based on the transfer function of the op-amp circuit. Building on the material covered in the previous articles, we are now in an ideal position to cover this topic. One thing to remember about PSU compensators is that there is nothing scary about them. We are simply dealing with an inverting op-amp with some impedances, just like what we studied in our very first few lectures at university. Consider the generic inverting op-amp in Figure 1. Please register to download the complete version of this resource We know from university lectures that the transfer function of this circuits is: Please register to download the complete version of this resource Where Z1 and Z2 are usually a combination of capacitors and resistors. With this simple circuit we can construct almost all of the compensators that are used in analog power supplies. Type I Compensator If I now replace Z2 with a capacitor (C = 10nF) and Z1 with a resistor, (R = 1.6kΩ) I will have created myself a simple Type I compensator. Furthermore, by selecting the values of the capacitor and the resistor I can control the position of any poles and zeros that this compensator has. Please note that for simplicity of teaching we are not considering the dc reference source usually connected to the non-inverting input and the biasing resistor usually connected between the inverting input and ground. These two elements only affect the dcanalysis and do not impact the ac transfer function of our compensator; in other words they have no impact on the position of our poles and zeros My transfer function therefore becomes: Please register to download the complete version of this resource My gain at a certain frequency f will be: Please register to download the complete version of this resource And, after taking into account the inverting action of the op-amp, the phase will be: Please register to download the complete version of this resource Looking at the above equations there some very important observations to point out:
Type I compensators are relatively rare in high performance power supplies and we have included it here for teaching purposes and to explain the concept of the position of the pole at origin. Type II and Type III compensators are by far more common and the vast majority of power supplies are stabilised with only just these two circuits. Let us go through them. Type II Compensator This is one of the most popular compensators as it is almost always used in current mode power supplies. We will discuss current mode control in a later article. The circuit, including a simple buck power stage is given in figure 2. Please register to download the complete version of this resource The transfer function is for this compensator is: Please register to download the complete version of this resource Referring back to our previous lecture notes, we can immediately identify that we have 1 pole at origin (ωp0), a second pole (ωp2) and a zero (ωz1). Please note that the equations are in rad/s and as always the component values determine the position of these poles and zeros: Please register to download the complete version of this resource It is easy to see that by appropriate selection of the component values we can place our poles and zeros in such a way to shape our Bode plots so that we meet the stability criteria as discussed in earlier articles. You can see from the transfer function that we have one zero. This of course means that we can force a maximum phase boost of 90o. But sometimes 90o of phase boost is just not enough; a case in point being almost all our voltage mode power supplies. We therefore need another compensator that allows more phase boost; this is our Type III. Type III Compensator The circuit for our Type III compensators is given in figure 3 Please register to download the complete version of this resource With an addition of just an extra resistor and capacitor (R3 and C2) we can convert our Type II compensator into a type III and get the extra phase boost that we need. The transfer function is given below: Please register to download the complete version of this resource We can see now that again we have our desirable pole at origin to remove our steady state offset, but this time we have an extra pole (ωp3) and more importantly and extra zero (ωz2) compared to our Type II compensator. The presence of 2 zeros of course suggest that we can have 180o of phase boost compared to our Type II’s 90o. The equations determining the positions of our poles and zeros are: Please register to download the complete version of this resource How Well Does the Transfer Function Match with Reality The good news is that with just these two transfer functions we can pretty much stabilise most of the analog power supplies in the world. We will learn how to place the poles and zeros and step-by-step real life design in future articles. For now let us experiment with the values of resistors and capacitors to place our poles and zeros and then measure a real compensator to see if it matches with our theory. We will use Biricha Digital’s automated power supply design software to first design a Type III compensator, then we will build and measure with a Bode 100 network analyser. You can see from figure 4 that Biricha WDS has placed the poles and zeros of the compensator and calculated the component values. The nearest preferred values of the components are given in the right hand column of figure 4. Inserting the component values in the equations of the Type III compensator above should give exactly the same poles and zeros as calculated by WDS and shown in the figure. Please note that the equations given earlier are in rad/s, but in WDS they are in Hz. For now we are not interested in the pole zero placement strategy; we will discuss detailed design in a later article. For now we are just making sure that our equations are correct and the simulated frequency response fits well with the real life one. Please register to download the complete version of this resource The comparison between the simulated Bode plot and the real measurement of a real compensator with the component values of figure 4 is shown in figure 5. You can see that we have almost a perfect match and therefore our equations are correct and the component values accurately place the poles and zeros of the compensator exactly where we expect them to be. All we need to do now is to devise a strategy for placing our poles and zeros such we meet our specifications and the stability criteria. We will do this, in detail in the next few articles. Please register to download the complete version of this resource Concluding Remarks
In this article, we discussed compensators, their poles and zeros, transfer functions and clarified what we mean by the phrase “placing the pole at origin at a certain frequency”. We showed, once again, that our mathematical presentations of our circuits match almost perfectly with real measurements.Furthermore, we detailed all the necessary equations relating the values of capacitors and resistors to the position of our compensators’ poles and zeros. We are now ready to start designing our compensators to stabilise our power supplies. In the next few articles we will discuss voltage mode and current mode control strategies and go through the compensator design procedures in a step-by-step manner. Things to Try 1 – Download a copy of Biricha WDS PSU Design software 2 – Attend one of our Analog Power Supply Design workshops 3 – Visit OMICRON Lab website for more information about Bode 100 Bibliography [1] Biricha Digital’s Analog Power Supply Design Workshop Manual [2] OMICORN Lab website Comments are closed.
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