SPRUHJ1I January 2013 – October 2021 TMS320F2802-Q1 , TMS320F28026-Q1 , TMS320F28026F , TMS320F28027-Q1 , TMS320F28027F , TMS320F28027F-Q1 , TMS320F28052-Q1 , TMS320F28052F , TMS320F28052F-Q1 , TMS320F28052M , TMS320F28052M-Q1 , TMS320F28054-Q1 , TMS320F28054F , TMS320F28054F-Q1 , TMS320F28054M , TMS320F28054M-Q1 , TMS320F2806-Q1 , TMS320F28062-Q1 , TMS320F28062F , TMS320F28062F-Q1 , TMS320F28068F , TMS320F28068M , TMS320F28069-Q1 , TMS320F28069F , TMS320F28069F-Q1 , TMS320F28069M , TMS320F28069M-Q1
At the end of last section, we discussed the possibility of using a single parameter that could help tune the speed PI loop in a motor control system. To develop this parameter, let's review the open-loop transfer function for the entire speed loop:
Where:
Assuming that the pole at occurs at a higher frequency than the zero at , and that the unity gain frequency occurs somewhere in-between these two frequencies, we should end up with a Bode plot that looks something like Figure 12-7.
The reason the shape of this curve is so important is because the phase shift at the 0 dB frequency determines the stability of the system. In order to achieve maximum phase margin (phase shift: 180°) for a given separation of the pole and zero frequencies, the 0 dB frequency should occur exactly half way in-between these frequencies on a logarithmic scale. In other words,
and,
Combining Equation 50 and Equation 51 we can establish that:
From Equation 49, we see that ωpole and ωzero are already defined in terms of the PI coefficients. Therefore,
Where "δ" we will define as the damping factor. The larger δ is, the further apart the zero corner frequency and the current loop pole will be. And the further apart they are, the phase margin is allowed to peak to a higher value in-between these frequencies. This improves stability at the expense of speed loop bandwidth. If δ = 1, then the zero corner frequency and the current loop pole are equal, which results in pole/zero cancellation and the system will be unstable. Theoretically, any value of δ > 1 is stable since phase margin > 0. However, values of δ close to 1 result in severely underdamped performance.
We will talk more about δ later, but for now, let's turn our attention towards finding the last remaining coefficient: . From Equation 50 we see that the open-loop transfer function of the speed loop from Equation 49 will be unity gain (0 dB) at a frequency equal to the zero inflection point frequency multiplied by δ. In other words,
By performing the indicated substitution for "s" in Equation 54 and solving, we obtain:
Finally, we can solve for :
At this point, let's step back and try to see the forest for the trees. We have just designed a cascaded speed controller for a motor which contains two separate PI controllers: one for the inner current loop and one for the outer speed loop. In order to get pole/zero cancellation in the current loop, we chose as follows:
sets the bandwidth of the current controller:
Once we have defined the parameters for the inner loop current controller, we select a value for the damping factor (δ) which allows you to precisely quantify the tradeoff between speed loop stability and bandwidth. Then it is a simple matter to calculate and :
The benefit of this approach is that instead of trying to empirically tune four PI coefficients which have seemingly little correlation to system performance, you just need to define two meaningful system parameters: the bandwidth of the current controller and the damping coefficient of the speed loop. Once these are selected, the four PI coefficients are calculated automatically.
The current controller bandwidth is certainly a meaningful system parameter, but in speed controlled systems, it is usually the bandwidth of the speed controller that we would like to specify first, and then set the current controller bandwidth based on that. In the next section, let's take a closer look at the damping factor, and we will come up with a way to set the current loop bandwidth based on the desired speed loop bandwidth.