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11.5 Calculating Speed and Current PI Gains Based on Damping Factor

So far in this series, we have discussed how to distill the design of a cascaded speed controller from four PI coefficients down to two "system" parameters. One of those parameters is simply the bandwidth of the current controller. The other is the damping factor (δ). The damping factor represents the tradeoff between system stability and system bandwidth in a single number. Keep in mind that we are only considering loads which contain only torque and inertial components (that is, no torsional resonance or viscous damping). Let's move forward by taking a closer look at the damping factor in both the time and frequency domains.

Figure 12-8 illustrates the open-loop magnitude and phase response for a system where the current controller bandwidth is arbitrarily set to 100 Hz. For our purposes, it really doesn't matter what the current bandwidth is, as it only serves to provide a reference point on the frequency axis. However, the shape of the curves won't change, regardless of what the current bandwidth is. The damping factor is swept from 1.5 to 70 in 8 discrete steps to show how it affects system response. A value of 1.0 corresponds to the condition where the open-loop gain intercepts 0 dB right at the frequency of the current controller bandwidth. This results in pole/zero cancellation at this frequency with a phase margin of zero. It goes without saying that zero phase margin equals bad things for your system.

Figure 11-8 Speed Controller Open Loop Magnitude and Phase Response as a Function of δ

One of the goals with the damping factor equations is to achieve the maximum stability possible for a given bandwidth. This is seen on the open-loop phase plots which indicate the phase margin peaks to its maximum value right at the frequency where the open-loop gain plots cross 0 db. As the stability factor is increased, you eventually reach a point of diminishing returns as the signal phase shift approaches -90 degrees. However, the gain margin continues to improve at the expense of a much slower system response.

Figure 12-9 illustrates the closed loop magnitude response of the speed loop, again assuming a current controller bandwidth of 100 Hz. Just like the open-loop response, the actual current controller bandwidth is irrelevant in determining the shape of the curves and only serves to associate the curves with a specific frequency reference point.

Figure 11-9 Speed Controller Closed Loop Bandwidth as a Function of δ

The required frequency separation between the -3 dB cutoff point of the speed closed loop response and the current controller pole is clearly seen along the bottom of the graph for various values of the damping factor. As the damping factor approaches unity, the complex poles in the speed loop approach the required frequency sepdamped ringing. This is perhaps better visualized in Figure 12-10, which shows the normalized step response of the system for various values of the damping factor. Values below 2 are usually unacceptable due to the large amount of overshoot. At the other end of the scale, values much above 30 usually unacceptable due to the large amount of overshoot. At the other end of the scale, values much above 30 usually don't work either because of the extremely long rise and settling times, as seen in the step response curves. In-between these values is usually your design target window.

Figure 11-10 Step Response of Speed Controller as a Function of δ

So what should you do if you picked the lowest value you can tolerate for the damping factor, but you still aren't satisfied with the system response times? Your best recourse might be to increase the bandwidth of your current loop. But the problem with this is that it appears to be an iterative approach since you need to determine the current loop bandwidth first before you can determine what speed loop bandwidth is created from a given damping factor. However, we can take advantage of the fact that the frequency curves shown here can be normalized with respect to the current loop bandwidth irrespective of frequency. In other words, if your motor control system has a form similar to the one discussed in this section, irrespective of actual parameter values, you will get frequency curves (and transient step curves) that look like this, with only the frequency scaling (and time scaling) being different. So let's exploit this fact to develop a procedure which will minimize the iterative nature of the design process and allow us to set the current loop bandwidth as a function of the speed loop bandwidth:

Pick the frequency response (-3 dB cutoff frequency) you want for your speed loop (BW_s).
Using the shape of the curves in Figure 12-10, find the lowest value of damping factor that will produce a satisfactory response for your speed loop. We have found that it is OK to pick a damping factor with a little more overshoot than you prefer, since integrator clamping will remove most of it anyway. At this point, the scaling of the frequency and time axes is irrelevant.
Calculate the required current loop bandwidth to support the design requirements using the following formula (obtained by curve fit analysis):

Equation 61.

Where:

BW_c is the current controller bandwidth
= one of the current loop PI coefficients
L = the motor inductance
BW_s is the speed controller bandwidth
δ is the damping factor.

Proceed with calculating the four PI coefficients as discussed previously in this section.

EXAMPLE

An Anaheim Automation 24V permanent magnet synchronous motor has the following characteristics:

Rs = 0.4 ohms
Ls = 0.6 mH
Back-EMF = 0.0054 v-sec/radians (peak voltage phase to neutral, which also equals flux in Webers in the SI system)
Inertia = 2E-4 kg-m²
Rotor poles = 8

The desired speed bandwidth = 800 rad/sec, and we would like a damping factor (δ) of 4. Find the required current loop bandwidth to support the speed loop bandwidth, and then calculate the four PI coefficients.

SOLUTION

The required current bandwidth can be found directly from Equation 61:

Equation 62.

From Equation 62, we find

Equation 63.

Recall that

Equation 64.

Also recall that

Equation 65.

Finally, recall that

Equation 66.

and,

Equation 67.

The simulated speed transient step response for this example is shown in Figure 12-11 where the time axis is now scaled appropriately for this design example.

Figure 11-11 Simulated Step Response of Speed Controller Design from the Above Example

Our analysis so far has assumed that the only poles in the speed loop are the two at s = 0, and the one associated with the current controller. But what if other poles exist? For example, the speed feedback signal in many systems is often processed by a low-pass filter. So how does this affect our tuning procedure? This will be covered in the following section.