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
In the last section we explained how to calculate the P and I coefficients (actually the and coefficients in a series structure) for a current loop controller for a motor. We saw that could be used to eliminate the zero in the closed-loop system response, resulting in a system having only one real pole (that is, well behaved and stable). sets the bandwidth of the closed-loop system response.
In this section let's back out and take a look at the speed control loop, which contains another PI controller. Can designing the speed loop be just as simple? Do the coefficient values perform the same system functions they did with the current controller?
It turns out that closing the speed loop is a little more complicated than closing the current loop. Also, to properly design the speed loop, we need to know more system parameters than we did for the current loop. This can be seen in Figure 12-6 which shows all of the components that comprise a cascaded speed control loop. By "cascaded," we mean a control system that consists of an outer loop with one or more inner loops. It bears mentioning again that we are only considering the case of a load with a single lumped sum inertia which is tightly coupled to the motor shaft and no viscous damping.
Let's start with the current control loop since this is where we left off in our last section. Assuming we design the current loop as discussed in my previous section, the closed-loop transfer function is:
Where is the error multiplier term in the current regulator's PI structure.
is not visible to the outside world since it is set to cause pole/zero cancellation within the current controller's transfer function. To avoid confusing the coefficients of the speed controller with those of the current controller, we will call the speed controller's coefficients and as shown in Figure 12-6. In the series form of the PI controller, is the error multiplier term (), and is the integrator multiplier term (). We can use the same equation we did in the last section to define the transfer function of the speed PI controller:
The transfer function from motor current to motor torque will vary as a function of what type of motor you are using. For a Permanent Magnet Synchronous Motor under Field Oriented Control, the transfer function between q-axis current and motor torque is:
Where:
P = the number of rotor poles
λr = the rotor flux (which is also equal to the back-EMF constant (Ke) in SI units)
For an AC Induction machine, the transfer function between q-axis current and motor torque would be:
Where:
P = the number of stator poles
Lm = the magnetizing inductance
Lr = the rotor inductance
Id = the component of current that is lined up with the rotor flux
For now, let's assume we are using a Permanent Magnet Synchronous Motor.
Finally, the load transfer function from motor torque to load speed is:
Where:
J = the inertia of the motor plus the load
Multiplying all these terms together results in the composite open-loop transfer function:
Let's combine all the motor and load parameters at the end of this equation into a single constant K:
Simplifying, we get:
From inspection of Equation 48, we can determine the following characteristics of the speed controller's open-loop transfer function:
In order for stable operation, the pole at must be higher in frequency than the zero at . Other than that, there is an infinite number of combinations of and which could be used to yield different system responses, depending on whether you want higher bandwidth or better stability. There is a procedure to define a single parameter which is proportional to system stability and inversely proportional to bandwidth, which can be used to set both and automatically to yield the maximum phase margin for the selected bandwidth. We will cover the details of this procedure in the next section.