#### What is in this article?:

- Sizing a motion control system
- How many KVs are there?

Multiple calculations for closed-loop electrohydraulic servo systems are presented, including a detailed explanation of the VCCM equation.

**How many ***K*_{V} s are there?

*K*

_{V} I’ve been discussing *K*_{V}*K*_{V}

Therefore, when tested in accordance with ISO 10770-1, two lands are connected in series (both have the same flow), and their individual pressure drops add up to the supply pressure — plumbing and other losses notwithstanding. With the valve shifted fully open in one of its two directions, it has some published rated flow, provided the pressure drop is set to the flow rating pressure. The value calculated for *K*_{V} depends on the value used for the flow rating pressure.

### Three scenarios

Three outcomes are possible for the pressure value used in the *K _{V}* formula:

1. If it is the total, (supply pressure as dictated by ISO), then the resulting coefficient is called *K _{V}*,Total. It is the combined effect of two individual metering lands in series.

2. If it is the measured differential pressure from the

*P*port to the

*A*port, then the resulting value is called the powered land coefficient, or

*K*. It is the effect of only metering land — the one connecting the

_{VPL}*P*port to the

*A*port.

3. If it is the measured differential pressure from the

*B*port to the

*T*port, then the resulting value is called the return land coefficient, or

*K*. It is the effect of only that land internally connecting the

_{VRL}*B*port to the

*T*port.

Conditions two and three are then repeated with the valve shifted in the opposite direction, and the result is two more values for* K _{V}*. Note that there are powered and return lands for both directions of spool shift, so there needs to be a designation that distinguishes them:

• *K _{V PL, PA}* is the powered land coefficient when the spool is shifted to flow

*P*to

*A*,

•

*K*is the return land coefficient for flow from

_{V RL, PA}*P*to

*A*,

•

*K*is the powered land coefficient for flow from

_{V PL, PB}*P*to

*B*, and

•

*K*is the return land coefficient when shifted to flow

_{V RL, PB}*P*to

*B*.

In the symmetrical valve, all four individual lands have the same value; only the total coefficient is different. But the most important thing is that all the coefficients can be determined from published data, and they can be used to determine flows and pressures at any and all operating conditions.

### The VCCM equation

We can develop a method to simultaneously select the optimum components for a given servo-proportional application using the valve coefficient concept and Bernoulli’s principle. Figure 1 depicts the starting point for the analysis.

There are four different valve lands, and they control the power delivered to the two ends of a cylinder. Shifting the spool to extend the cylinder (*P* to *A* and *B* to *T*) opens *K _{VPL, PA}* and

*K*, while

_{VRL, BT}*K*and

_{V PL, PB}*K*are in overlap — that is, they are essentially shut off.

_{V RL, AT}The mathematical procedure involves solving for the cylinder pressures on both sides of the piston in relation to the supply pressure, valve coefficients, valve ratio, and cylinder ratio. The math also takes into account the load variables, speed of propulsion, and load force. The result is the VCCM equation, which is solved to size the components for optimal operation. The pressure drop across the valve is implicit in the equation:

*f _{L} = P_{S} A_{PE} –v^{2}(A_{PE}^{3} *÷

*K*÷

_{VPL}^{2}) (1+ρ_{v}^{2}*ρ*

_{c}^{2}) where *f _{L}* is the load force that must be overcome,

*P*is supply pressure,

_{S}*A*is the cylinder size,

_{PE}*v*is the speed of cylinder propulsion,

*K*is the degree to which the valve is open,

_{VPL}*ρ*is the symmetry of the valve, and

_{v}*ρ*is the cylinder area ratio (cap side of piston area/rod side area).

_{c}Solutions to the equation can be used for analyzing the system to ensure that the operating envelope is adequate for the application. The operating envelope is constructed from the VCCM equation with the valve fully open and the valve coefficient at its maximum value. The operating envelope describes the force and velocity available for a given system.

The VCCM equation is the basis for all sizing procedures. If the valve, cylinder, and supply pressures are not properly sized, results can be less than the required degree of control or poor system performance. The worst case is that the application will not meet its design goals.

Pressure metering characteristics define the stopping conditions, including pressures at a given load and the stopping position within the VCCM equation. More specifically, the valve ratio, ρv, and the cylinder ratio, *ρ _{c}*, from the VCCM equation are defined as:

*ρ _{v} = K_{VPL} *÷

*K*÷

_{VRL}and ρ_{c}= A_{PE}*A*

_{RE}.Note that two different values are used for the valve ratio: one for extending the cylinder, and one for retracting it. Similarly, two different values are used for the cylinder ratio, depending on whether it is extending or retracting. The value for extending is greater than one, and the value for retracting is less than one.

### **Closing the loop**

The system in Figure 2 illustrates the positional servomechanism, which is the ultimate subject of this discussion. The cylinder in Figure 2 transmits some load force, *f _{L}*, while its piston is connected to a position sensing transducer with a transfer characteristic of

*H*, the feedback signal. The value of

*H*is a voltage that goes into an error calculator, where it is subtracted from the command voltage,

*C*. The error signal,

*E*is fed into the servo/proportional amplifier to shift the valve.

As long as the error signal is not zero, the valve will shift, causing the cylinder to move the load to a point where command and feedback signals are equal. In this state, the error is zero, the valve current goes to zero, the spool centers, and the load and cylinder stop. In principle, that is how it works. In reality, it is a bit more complicated.