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Ideally, positive-displacement pumps generate flow that is independent of outlet pressure. The lower the internal leakage, the less the pump’s flow reacts to changing outlet pressures. In fact, a textbook positive-displacement pump would have *no* internal leakage and exhibit *no* reaction to changes in outlet pressure.

This is the essence of ideal positive displacement: the pump behaves as a positive flow source with the pressure being a consequence of some impeding load or resistance placed in the output flow path. If the output flow path is blocked, outlet pressure can rise to a destructive level, or the torque becomes so high that it stalls the prime mover. Either way, the consequences can be dire. For this reason, all positive-displacement hydraulic pumps must be accompanied by pressure-limiting devices, usually an external relief valve.

In contrast, the electrical generator is a positive voltage source. The interpretation here is that with the voltage-pressure analogy, the pump and the generator are not analogous. Were there to be a perfect analogy, then the electrical generator would behave as a positive current source, making it analogous to the hydraulic pump. Either that, or the hydraulic pump would be a positive pressure source, making it perfectly analogous to the generator. The fundamental hydraulic, or “natural” source, is a positive flow generator and the fundamental, or “natural” generator, is a positive voltage source.

This discussion raises the question, “Are there positive pressure sources in the hydraulic real and positive current sources in the electrical realm?” The answer in both cases is an emphatic yes. However, the two are constructed from their more natural, respective counterparts.

Consider, first, the hydraulic realm. The practical manifestation of the positive pressure source is the pressure-compensated pump. The conventional pump, starting with either a vane or piston construction, each with variable displacement, is configured to sense its own outlet pressure. When the pressure reaches a settable value, the sensing circuits use the pressure to apply force to the displacement changing mechanism so that the displacement reduces with rising pressure. The result is an automatically implemented pressure regulator that uses internal feedback to make the adjustments.

**Pressure-Flow Relationships**

Figure 11 shows the pressure-flow characteristics of a typical pressure-compensated pump and its two distinct regions of operation. The displacement at low pressure is a maximum value, being held there by a spring or spring equivalent. As such, it operates as a simple fixed-displacement pump. It is labeled *Positive flow region* in the figure. At zero pressure, the *ideal flow* is delivered to the pump’s outlet port.

The internal leakage depends on the outlet pressure, because leakage is caused by internal clearances and pressure. The higher the pressure, the higher the internal leakage. This phenomenon accounts, then, for the fact that pressure increases result in lower flow being delivered to the outlet port when there’s rising pressure.

Two simple functions describe a hydraulic pump’s performance. The relationship between output flow and pressure, when operating in the positive flow region and using practical engineering units of measure, is approximated with:

Q = D/60 âœ• N – 1/R_{L} âœ• P (5)

where:

Q is the output flow in volume units per second, depending on the unit’s choice for D.

D is the displacement of the pump in volume units per revolution of the shaft, cm^{3}/rev, or in.^{3}/rev.

N is the shaft rotational speed in rpm.

P is the outlet pressure in any desirable units, i.e., bar, Mpa, psi.

R_{L} is the internal, laminar leakage resistance, i.e., the reciprocal of the slope of the Positive Flow Region curve of Fig. 11 and units must be consistent with those of Q and P, eg, cm3/second/Mpa.

For the mechanical parts of the circuit, relating pressure and input torque can be expressed as:

T_{in} = D/2π + R_{FW} âœ• N (6)

where:

T_{in} is the input torque needed to turn the shaft “against the output pressure,” in newton-meter or in-lb.

R_{FW} is the “friction and windage resistance loss coefficient” in newton-meter/rpm or in-lb/rpm.

If R_{L} becomes infinite and R_{FW} becomes zero, or negligibly small, there are no losses in the machine. That means that the overall efficiency would be100%. Alas, it is not. Nonetheless, there is ample reason to have a perfect machine in the pantheon of mathematical models.

The internal leakage resistance (R_{L}) and internal friction resistance (R_{FW}) are used to explain why the ideal machine is impossible. The internal moving parts must slide past one another and be lubricated and pressure-balanced. Therefore, consequential leakage paths exist. Friction and windage are also inevitable in any machine with moving parts. Thus, they are said to be parasitic, because if we build the machine, these loss elements come along whether we want them or not. The only possibility is to limit them, because they cannot be eliminated.

**The Norton Equivalence Circuit**

Equation 5 is mechanized perfectly with the simplified hydraulic analytical schematic of Fig. 12. Those familiar with electrical technology should recognize this circuit as the Norton equivalent circuit. It has a flow source with internal parallel bypass resistance; the remaining flow makes its way to the output world where it can be felt and measured. Everything within the *Pump boundary* is a mathematical artifice, but representative of the real physical phenomena.

The *I* within the pump symbol indicates that it represents an ideal flow generator. Its symbol is necessary in the hydraulic realm because the standard ISO symbols are interpreted in their practical, industrial sense. That is, the pump symbol represents the real pump, including all of its imperfections, nuances, and idiosyncrasies. Explaining all of those imperfections requires a more nuanced set of symbols; thus, the notation for an ideal pump as well for the motor.

Laminar flow in hydraulic fluids is directly proportional to pressure, but it depends on the fluid properties, most notably the viscosity. As such, laminar flow is perfectly analogous to Ohm’s law. Laminar flow also depends on the geometry of the flow path. Laminar flow paths are characterized by long flow paths, small cross-sectional flow areas, or high viscosity. As a result, several maxims come into play: the longer the path, the higher the resistance; the smaller the flow area, the higher the resistance; and the higher the viscosity, the higher the resistance.

A Norton equivalent circuit contains an ideal current generator (positive current source) in parallel with the Norton resistance, which is an internal parallel flow path analogous to the pump’s internal leakage. It is important to realize that the current generator is *not* a natural kind of generator. It must be constructed of other elements, such as transistors, operational amplifiers, and the voltage generator.

Interestingly, one of the first positive-current sources is named after its creator, Brad Howland of MIT. The Howland current pump has obvious similarities to a positive-displacement pump, at least in performance characteristics. But it is *not* a pump with pistons. Instead, it is an electronic circuit designed to provide a controlled, positive current to some external load.

Figure 13 contains a generic Norton equivalent circuit, and even the most casual observer must be struck by the similarities to Fig. 12. The Norton model of an electrical circuit explains why output current decreases concurrently with an increase in outlet voltage. The culprit is the internal bypass resistance, which takes away current from the ideal Norton generator. It same explanation applies to the hydraulic pump, where internal bypass leakage takes fluid flow away from the outlet port.

**Dealing with Ideals**

The ideal flow pump in Fig. 12 is referred to as a *dependent* flow generator, so called because the amount of flow that it produces depends on a variable or action that does not appear in the figure. The other variable is shaft speed, which is in a different part of the circuit (it does not appear explicitly in Fig. 12).

Unfortunately, hydraulic circuit symbols, unlike electrical symbols, do not provide for a different symbol to indicate dependency. Dependency is explicit only when applied to Equation 5. Electrically dependent current generators are indicated by replacing the circular generator of Fig. 13 with a rotated square *(Fig. 14)*. The symbol indicates it is dependent, but not what it is dependent on. More information is needed.

Refer again to Fig. 11, which depicts the pressure-flow characteristics of the pressure-compensated pump. As the pressure increases, the *Knee* is reached so that the internal feedback regulator takes over the control of the pump’s displacement. At low pressures, when operating below the knee, a spring biases the displacement to its maximum value.

Further attempts to increase pressure to the right of the knee are met with a commensurate reduction in displacement and a limitation in the pressure’s rise rate. Called the *Positive pressure region*—where the machine behaves more like a positive pressure source—it’s the analogy to the positive voltage source. Though the slope of the curve in this region is shown with less steepness than what’s actually achieved with a real pump, it is intended to illustrate the principle of pressure compensation. The real pump’s steeper slope is never vertical, however, without electronic control of displacement.

The maximum pressure of the pressure-compensated pump in Fig. 11 is labeled *Blocked port pressure*. Two points can be made about this notation. First, the fluid-power industry does not normally use that wording, preferring instead to call this the *deadhead* pressure. Obviously, deadhead means blocked outlet port. Second, the knee and deadhead pressures are adjustable, if not at field-application time, then at the factory final test time.