Servo terminology differs

closed loop transfer function

Figure 2: CLTF curves are often used to illustrate the bandwidth of a system.

open loop transfer function

The open-loop transfer function (OLTF) represents all frequency-dependent blocks that make up the servo loop, namely controls, drives, devices, and sensors. The OLTF for some systems can be obtained with the servo loop open, but in many cases it must be derived from the model or obtained with the loop closed using the test points in Figure 1 . is found using the formula CLTF/(1-CLTF), where CLTF is the complex representation of the closed-loop transfer function. Referring to Figure 1, CLTF can be measured by using a random or swept sine drive system at In2 and measuring Out2/In2. OLTF can be found by driving In2 and measuring Out2/Out3. The open-loop and closed-loop transfer functions are shown in Figure 2.

loop gain

Servo gain is attributed to each component in the servo loop. The gain may be only single, or it may be much lower or many orders of magnitude higher. There are many types of gain in servo loops. Some gains are frequency dependent and some gains are constant at all frequencies. The most noteworthy servo gains are control gain, drive gain, motor gain, device gain, and sensor gain. Each servo gain is multiplied to create the overall loop gain. Changing any one component will affect closed loop stability. Some gain examples are component-specific:

  • For encoders, gain is counts per micron.

  • For capacitive probes, the units are volts/micron.

  • For rotating motors, the motor gain is Newton meters per ampere.

  • For current drivers it is Amps/Volts.


Servo bandwidth describes the maximum frequency at which the control system exerts a beneficial effect on the control system. Electrical engineers working with low-pass filters such as Butterworth filters, elliptical filters, or Chebyshev filters often quote the -3 dB amplitude frequency as the bandwidth of the filter. The equivalent value for a servo system is the -3 dB point of the closed loop transfer function. However, this definition is not universal. Many control engineers refer to 0 dB of the open-loop or loop transfer function as the bandwidth of the control system.

Be clear about what you mean. Are you citing the -3 dB point in closed loop or the 0 dB point in open loop? Ask others what they mean when they use the word so there is no chance of miscommunication leading to the control system being over-engineered or not performing as expected.

In control textbooks, bandwidth (BW) is described as a representation of the reaction time of a system. BW = 0.35 * TR, where TR is the rise time, is the most commonly cited formula, but also the most commonly misused formula when describing closed-loop mechanical systems. This formula applies to first-order systems; closed-loop motion control systems behave differently than first-order systems. Using this formula can definitely lead people in the wrong direction.

There are several variations of the bandwidth formula that take into account the system damping ratio zeta. The Zeta of a closed-loop system is related to the phase margin of the system.

Almost all formulas that equate bandwidth and response time for second-order systems are based on unsaturated linear systems. If you compare the response time of actual step response data to the calculation through the bandwidth formula, they are rarely equal; the most likely cause is voltage or current saturation. The voltage or current driving the system is limited; and if the step value selected in combination with the proportional gain and integral gain is too large, voltage or current saturation will occur, slowing down the response of the system compared to the predicted response time of the formula .

Bandwidth and loop gain

Bandwidth is often mistakenly used to represent a control system’s ability to track, reject, or attenuate errors. The most common misconception is that the higher the bandwidth, the better the performance. For tracking, attenuating, or suppressing interference, it is loop gain, not bandwidth, that matters. Loop gain is frequency dependent. Two systems can have the same bandwidth but completely different low-frequency gains. Both systems have the same mathematically calculated response time, but one is better at tracking or suppressing low-frequency interference than the other.

Related to this concept is understanding the limitations of interference suppression or tracking as the frequency approaches the bandwidth frequency. If we use the 0 dB crossover in OLTF as our bandwidth frequency, this means that the loop gain before the bandwidth is higher than 0 dB, but also very small, eg only a few dB.

Loop gain is what drives noise rejection, and if the loop gain is lower, the noise immunity or tracking capability will also be lower. This means that one cannot expect the same performance at frequencies before the bandwidth as compared to frequencies more than twice below the bandwidth frequency. This is a key concept to understand when writing and responding to control system requirements.

Another common misconception is that systems with large mass or inertia cannot have high servo bandwidth or cannot move and stabilize at high speeds. This statement is simply not true. The mass or inertia of the system is just another gain in the cycle. The more mass or inertia there is in the system, the lower the gain of the device. To compensate for low device gain, the control gain can be increased to achieve maximum bandwidth, provided stability rules are met. However, in mass servo systems, the limiting factor is the power available to move the load or suppress interference. This is where it gets confusing. Device lacks current or voltage.

Motors with larger motor constants heat less, and the higher available current or voltage will eliminate saturation. The limiting factor in large mass or inertial systems is not servo bandwidth; it is servo bandwidth. Rather, it is the components chosen to drive the mass – the motor and drive electronics.

Loop Gain and Bandwidth Example

How to use increased low frequency (integrator) gain to improve current loop performance? Figure 3 shows a simplified diagram of the working principle of a linear motor.

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