# Hydraulic-Electric Analogies: Transmission Comparisons, Part 1

The hydrostatic transmission and Ward Leonard generator-motor drive are so perfectly analogous to one another that they must be included in any complete review of analogies.

There are, literally, thousands of electronic gadgets and devices for which no effective hydraulic analogy exists. On the other hand, some electric-hydraulic analogies are so perfectly analogous to one another that they have to be included in any reasonably complete review of analogies. Those two devices are the Ward Leonard generator-motor drive and the hydrostatic transmission. But before delving into the details of those two machines, let’s revisit some of the earlier concepts, namely the use of analytical schematics to characterize pumps, motor and generators, plus pressure-compensated pumps.

### Analytical Schematics of Pumps and Motors

The concepts on the ideal variable-displacement pump covered previously are depicted in Fig. 1. Analytical schematics of the ideal pump are so named because they assist in writing the defining equations used for calculating machine sizes and performance. Recall that an ideal machine is 100% efficient. Although unrealistic, it can still be useful, especially as a starting point to more practical pumps.

The schematics are helpful in understanding the inner workings of pumps and motors without having to know countless internal details. Analytical schematics are merely extensions and variations on two-port theory, which has found wide application in the many input-output devices found in electrical and electronic systems. The schematics arise because of scrutiny of the pump construction or motor as well as studying the performance data from objective lab tests.

Figure 1 shows the ideal torque and flow-generation portions (designated as I within the circular envelopes) of a variable-displacement pump. The circle on the left is the counter-torque generator and the one on the right is the flow generator. The input port consists of the pair of lines on the left (labeled as TiP), an arrow, and the two small circles.

Two-port parlance always has two connecting points—one for flow to enter and one for flow to exit. Physically, the input port is the pump’s input shaft to which input torque, speed, and power are applied. In electrical two-port theory, the two small, white-filled circular connecting points at the left boundary, taken together, comprise the “input port.”

Torque is analogous to pressure and shaft speed is analogous to flow. Thus, the shaft speed “flows” through the counter-torque generator. The input shaft and ideal torque generator also comprise the mechanical side of the circuit and are also referred to as the “input side of the circuit.” It follows, then, that the hydraulic ports comprise the “output side of the circuit.”

The output side of the circuit is the flow generator and its two output connecting points, labeled A-port, pump and B-port, pump. As in the case of the input port, the output port consists of the two small white connecting points on the right side of the pump envelope (dash-dot line). They provide the connecting points where external plumbing is connected to carry transmit flow to the load and return it to the pump inlet.

### Practical Hydraulic Pump Analytical Schematic

The analytical schematic for the practical pump can be conceived by first looking at the machine construction. When the pump is tested in standard fashion—say, in compliance with ISO 4409 [21]—the output data show a measurable drop in output flow as output load pressure is increased with the input shaft regulated at a constant speed. The cause is simple: Internal leakage is directly proportional to outlet pressure. That is, the pumping elements, gears, vanes, or pistons displace a fixed amount of fluid each and every revolution, but some of that displaced fluid does not find its way out the outlet port to do useful work. Instead, it slips from the high-pressure regions within the pump and back to the inlet port. There are three identifiable and separate internal leakage paths. They can be visualized by reviewing the construction of a pump, such as the vane pump cutaway in Fig. 2.

Referring to Fig. 2, when the rotor is spinning clockwise and “dragging” its vanes along with it, the blue port is the outlet, high-pressure port, and the yellow is the inlet, low-pressure port. That means all the internal regions to the right of the two vanes (one at top dead center and the other at bottom dead center) are separating the low-pressure regions from the high pressure. Huge pressure differences will be imperfectly sealed by the vanes, which are in sliding contact with the inner surface of the cam ring, the rotor-vane slots, and the two end-plate surfaces. The sliding contact means that gaps, tiny though they may be, are present at all the sliding surfaces. The parts are riding on a thin film of fluid, usually oil, providing lubrication for the sliding parts. Some of the leakage fluid passes directly from the high-pressure zone to the inlet kidney port to be immediately recirculated to the outlet. This is referred to as port-to-port leakage and is carried through the port-to-port leakage path, RLPP, in Fig. 3.

There is another identifiable leakage path. It carries fluid from the high-pressure port—mostly along the ends of the vanes and the rotor—into the low-pressure cavity inside the pump housing. Because the shaft must penetrate from the outside environment to the interior of the pump, the shaft must have a sliding seal. The seal prevents the hydraulic fluid from seeping from the pump interior to its exterior.

This means the pump housing can experience relatively high pressure, as much as 50% of the outlet pressure. Both the housing and the shaft seal must accommodate this high pressure, which raises the cost of the shaft seal and the housing. To relieve the housing (case) pressure, many pumps and motors have an external drain port, commonly referred to as the case drain. It connects from the pump housing to the fluid reservoir. The high pressure-to-case-drain leakage path is identified with the RLACD algebraic symbol, meaning the leakage goes from Port A to the case drain.

At this point symmetry takes over in the assessment of internal leakage. Figure 3 shows a third leakage path symbolized with RLBCD. An examination of the pump’s innards reveals that if a leakage path exists from the high-pressure region to the case, then a similar path must exist between the low-pressure inlet port and the case. Such a path does exist, and its effects can be measured, for instance, when both the outlet port and inlet port are at high or relatively high pressure, as can be the case in the hydrostatic transmission. The leakage resistance loss coefficients are symbolized with RL, followed by an identifier extension on the subscript. The higher the leakage resistance, the lower the internal leakage at a given pressure, and the more efficient the pump.

### Differentiating Between Ports

Clearly, the hydraulic pump is an input-output device: mechanical power in and hydraulic power out. But the two-port concept brings up a contradiction between electrical and hydraulic nomenclature. With electrical systems, the port in two-port concept actually consists of two separate points at which two separate wires are connected. Yet, it is the connection pair that is called a port. Similarly, the high- and low-pressure points at which hydraulic plumbing is connected comprise two separate hydraulic ports (external connection points), but only one port in the electrical two-port context.

The term port used in this discussion is a single point at which an external connection is to be made. That is, the hydraulic context will be used. The subject is brought up to clarify the situation where readers are interested in understanding more about two-port theory in the electrical context. It’s intended to reconcile the two different interpretations of what a port is.