#### What is in this article?:

- The importance of frequency response
- Some background
- Limitations to closed-loop bandwidth

## Limitations to closed-loop bandwidth

The maximum closed-loop bandwidth (frequency response) must be less by an amount called the separation ratio, which always has a value less than one. In mathematical terms:

*f _{max}* <

*p*× (lesser of

_{s}*f*or

_{v}*f*), where

_{n}*f _{max}* is the maximum system closed bandwidth

*p _{s}* is the separation ratio, and

*f _{v}* and

*f*are the valve and hydromechanical resonant frequencies in Hz.

_{n}When HMRF dominates (<* f _{v}*), the separation ratio is controlled entirely by the damping ratio of the hydromechanical system:

*p _{s}* = 2

*Z*

_{n}where *Z _{n}* is the damping ratio, a measure of the tendency for an oscillation to subside.

Two conditions contribute to *damping* — internal leakage from one side of the actuator to the other (whether from within the actuator or the control valve) and friction (whether from the actuator or its load). Because manufacturers strive to reduce internal leakage and friction, it should come as no surprise that the degree of damping in most hydromechanical systems can be very low. In fact, when the load can be moved with negligible friction (as when supported by a recirculating linear ball bearing), the damping ratio may be as low as 0.03 or 0.05. Admittedly, the system friction and the damping ratio are the most elusive quantities to evaluate in a system. Nonetheless, they, along with the frequencies, absolutely govern the performance limits of the system.

### Sample calculations

Consider an example to demonstrate this discussion. Suppose that a system's hydromechanical resonant frequency has been calculated and found to be 18 Hz. Further suppose that its servovalve has a 90° phase lag frequency of 65 Hz, and we estimate the hydromechanical damping ratio, due to both friction and internal valve leakage, to be about 0.05. We can calculate the maximum possible closed-loop system bandwidth:

*f _{max}* <

*p*x (lesser of

_{s}*f*or

_{v}*f*)

_{n}*f _{max}* < 2 x 0.05 x 18

*f _{max}* < 1.8 Hz

The maximum closed-loop bandwidth, *f _{max}*, is only 1.8 Hz, which is only a tenth of the HMRF! At startup time, we increase the system bandwidth by increasing the servo amplifier gain. If we increase the gain until we have 1.8 Hz of bandwidth, and then attempt further increases, the servo loop will break into sustained oscillation, rendering it worthless. The gain must be decreased to re-establish stability.

System bandwidth is important because a direct inverse relationship exists between it and positioning accuracy, or, more correctly, the positioning error and the following error. It has been shown that:

∆*x _{p}* = (∆

*I*×

_{T}*G*)/(2 ’

_{sp}*f*).

_{sys}Where ∆*x _{p}* is the expected steady state positioning error (in.)

∆*I _{T}* is the total expected valve current variation (amperes) caused by eight known external disturbances,

*G _{sp}* is the speed gain [(in.x

*A*) / sec] at the highest expected speed and load, and

*f _{sys}* is the actual closed-loop system bandwidth (Hz).

The output position is never where we want it to be — only close. There are eight known disturbances in the electrohydraulic positional servo mechanism that cause imperfect positioning:

• valve temperature changes

• supply pressure variations

• tank port pressure variations

• breakaway friction

• load variations

• valve hysteresis

• valve threshold, and

• valve dead zone.

All of these must be resolved into an equivalent valve current, then added together to yield the total expected valve current, delta *I _{T}*. In general, evaluating these eight "error contributors" for a given system is more than a trivial process. Experience tells us, however, that for zero-lapped valves with "typical" servovalve performance, delta

*I*is about 2% or 3% of rated valve current. If the valve is proportional and has substantial overlap, then we usually use the overlap only and ignore the other seven contributors.

_{T}Technically, speed gain must be calculated - using the characteristics of the selected control valve - for the worst-case loading condition. If the designer uses good engineering practice for selecting the control valve (if the valve is selected to provide maximum power transfer at the worst-case load-and-speed combination), then speed gain *G _{sp}* will equal approximately the target design actuator speed divided by about 2/3 of valve rated current. Armed with this information, we can now estimate the expected system "accuracy."

Suppose the system we are designing must propel a load at 21 in./sec under worst-case conditions using a servovalve with essentially zero overlap. The numerator of the equation at the upper-left portion of this page can be evaluated first:

∆*I _{T}* ×

*G*= (0.02 ×

_{sp}*I*x 21) ÷ (0.67 ×

_{R}*I*), where

_{R}*I _{R}* is the rated valve current, which cancels out of the equation.

Now, *∆I _{T} G_{sp}* = (0.02 × 21) × 0.67

∆*I _{T}G _{sp}* = 0.63 in./sec.

If we assume that the servoloop has been tuned to the maximum allowed before instability occurs, then *f _{sys}* is set to

*f*, so the error can be estimated:

_{max}∆*x _{p}* = ∆I

_{T}G

_{sp}÷ 2 ’ f

_{sys}.

But because the numerator has already been evaluated:

∆*x _{p}* = 0.63 ÷ (2 ’ × 1.8),

∆*x _{p}* = ±0.055 in.

We could expect, then, that the long-term positioning capability of this system would be about 0.055 in.

*Jack L. Johnson is a contributing editor for Hydraulics & Pneumatics and president of IDAS Engineering, Milwaukee.*

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