Hydraulic analogy

The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by ) is the most widely used analogy for "electron fluid" in a metal conductor. Since is invisible and the processes at play in  are often difficult to demonstrate, the various s are represented by  equivalents. Electricity (as well as ) was originally understood to be a kind of, and the names of certain electric quantities (such as current) are derived from hydraulic equivalents. As with all analogies, it demands an intuitive and competent understanding of the baseline s (electronics and hydraulics).

Paradigms
There is no unique paradigm for establishing this analogy. Two paradigms can be used to introduce the concept to students using pressure induced by gravity or by pumps.

In the version with pressure induced by gravity, large tanks of water are held up high, or are filled to differing water levels, and the potential energy of the water is the pressure source. This is reminiscent of electrical diagrams with an up arrow pointing to +V, grounded pins that otherwise are not shown connecting to anything, and so on. This has the advantage of associating with.

A second paradigm is a completely enclosed version with pumps providing pressure only and no gravity. This is reminiscent of a circuit diagram with a voltage source shown and the wires actually completing a circuit. This paradigm is further discussed below.

Other paradigms highlight the similarities between equations governing the flow of fluid and the flow of charge. Flow and pressure variables can be calculated in both steady and transient fluid flow situations with the use of the analogy. Hydraulic ohms are the units of hydraulic impedance, which is defined as the ratio of pressure to volume flow rate. The pressure and volume flow variables are treated as s in this definition, so possess a phase as well as magnitude.

A slightly different paradigm is used in acoustics, where is defined as a relationship between pressure and air speed. In this paradigm, a large cavity with a hole is analogous to a capacitor that stores compressional energy when the time-dependent pressure deviates from atmospheric pressure. A hole (or long tube) is analogous to an inductor that stores kinetic energy associated with the flow of air.

A circuit was used to model feedback stabilization of a hydrodynamic plasma instability in a   In this application, the effort was to keep the plasma column centered by applying voltages to the plates, and except for the presence of turbulence and non-linear effects, the plasma was an actual electric circuit element (not really an analog).

Voltage, current, and charge
In general, is equivalent to. This model assumes that the water is flowing horizontally, so that the force of gravity can be ignored. In this case electric potential is equivalent to. The (or  or potential difference) is a difference in pressure between two points. Electric potential and voltage are usually measured in. is equivalent to a hydraulic ; that is, the volumetric quantity of flowing water over time. Usually measured in.

is equivalent to a quantity of water.

Basic circuit elements
A is equivalent to a tank with one connection at each end and a rubber sheet dividing the tank in two lengthwise (a ). When water is forced into one pipe, equal water is simultaneously forced out of the other pipe, yet no water can penetrate the rubber diaphragm. Energy is stored by the stretching of the rubber. As more current flows "through" the capacitor, the back-pressure (voltage) becomes greater, thus current "leads" voltage in a capacitor. As the back-pressure from the stretched rubber approaches the applied pressure, the current becomes less and less. Thus capacitors "filter out" constant pressure differences and slowly varying, low-frequency pressure differences, while allowing rapid changes in pressure to pass through.

An is equivalent to a heavy paddle wheel placed in the current. The of the wheel and the size of the blades restrict the water's ability to rapidly change its rate of flow (current) through the wheel due to the effects of, but, given time, a constant flowing stream will pass mostly unimpeded through the wheel, as it turns at the same speed as the water flow. The mass and surface area of the wheel and its blades are analogous to inductance, and friction between its axle and the axle bearings corresponds to the resistance that accompanies any non-superconducting inductor. An alternative inductor model is simply a long pipe, perhaps coiled into a spiral for convenience. This fluid-inertia device is used in real life as an essential component of a. The of the water flowing through the pipe produces the inductance effect; inductors "filter out" rapid changes in flow, while allowing slow variations in current to be passed through. The drag imposed by the walls of the pipe is somewhat analogous to parasitic resistance. In either model, the pressure difference (voltage) across the device must be present before the current will start moving, thus in inductors voltage "leads" current. As the current increases, approaching the limits imposed by its own internal friction and of the current that the rest of the circuit can provide, the pressure drop across the device becomes lower and lower.

An ideal (ideal ) or ideal  is a  with feedback control. A pressure meter on both sides shows that regardless of the current being produced, this kind of pump produces constant pressure difference. If one terminal is kept fixed at ground, another analogy is a large body of water at a high elevation, sufficiently large that the drawn water does not affect the water level. To create the analog of an ideal, use a : A current meter (little ) shows that when this kind of pump is driven at a constant speed, it maintains a constant speed of the little paddle wheel.

Other circuit elements
A is equivalent to a one-way  with a slightly leaky valve seat. As with a diode, a small pressure difference is needed before the valve opens. And like a diode, too much  can damage or destroy the valve assembly.

A is a valve in which a diaphragm, controlled by a low-current signal (either constant current for a  or constant pressure for a ), moves a plunger which affects the current through another section of pipe.

is a combination of two transistors. As the input pressure changes, the pistons allow the output to connect to either zero or positive pressure.

A is a  operated by a flow meter. As water flows through in the forward direction, the needle valve restricts flow more; as water flows the other direction, the needle valve opens further providing less resistance.

Principal equivalents
EM wave speed is equivalent to the  in water. When a light switch is flipped, the electric wave travels very quickly through the wires.

Charge flow speed is equivalent to the particle speed of water. The moving charges themselves move rather slowly.

is equivalent to the a constant flow of water in a circuit of pipes.

is equivalent to water oscillating back and forth in a pipe

AC and s is somewhat equivalent to being transmitted through the water pipes, though this does not properly mirror the cyclical reversal of alternating electric current. As described, the fluid flow conveys pressure fluctuations, but fluids do not reverse at high rates in hydraulic systems, which the above "low frequency" entry does accurately describe. A better concept (if sound waves are to be the phenomenon) is that of direct current with high-frequency "ripple" superimposed.

Inductive spark used in s is similar to, caused by the inertia of water

Equation examples
Some examples of analogous electrical and hydraulic equations:

If the differential equations have the same form, the response will be similar.

Limits to the analogy
If taken too far, the water analogy can create misconceptions. For it to be useful, one must remain aware of the regions where electricity and water behave very differently.

 : Electrons can push or pull other distant electrons via their fields, while water molecules experience forces only from direct contact with other molecules. For this reason, waves in water travel at the speed of sound, but waves in a sea of charge will travel much faster as the forces from one electron are applied to many distant electrons and not to only the neighbors in direct contact. In a hydraulic transmission line, the energy flows as mechanical waves through the water, but in an electric transmission line the energy flows as fields in the space surrounding the wires, and does not flow inside the metal. Also, an accelerating electron will drag its neighbors along while attracting them, both because of magnetic forces.

Charge: Unlike water, movable charge carriers can be positive or negative, and conductors can exhibit an overall positive or negative net charge. The mobile carriers in electric currents are usually electrons, but sometimes they are charged positively, such as the positive ions in an, the in s or  in s and some (very rare) conductors.

Leaking pipes: The  of an electrical circuit and its elements is usually almost equal to zero, hence it is (almost) constant. This is formalized in, which does not have an analogy to hydraulic systems, where the amount of the liquid is not usually constant. Even with liquid the system may contain such elements as s and open pools, so the volume of liquid contained in a part of the system can change. For this reason, continuing electric currents require closed loops rather than hydraulics' open source/sink resembling spigots and buckets.

Fluid velocity and resistance of metals: As with water hoses, the carrier drift velocity in conductors is directly proportional to current. However, water only experiences drag via the pipes' inner surface, while charges are slowed at all points within a metal, as with water forced through a filter. Also, typical velocity of charge carriers within a conductor is less than centimeters per minute, and the "electrical friction" is extremely high. If charges ever flowed as fast as water can flow in pipes, the electric current would be immense, and the conductors would become incandescently hot and perhaps vaporize. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe. Resistance in most electrical conductors is a linear function: as current increases, voltage drop increases proportionally (Ohm's Law). Liquid resistance in pipes is not linear with volume, varying as the square of volumetric flow (see ).

For this reason there is no hydraulic explanation for such things as a 's charge pumping ability, a 's and voltage drop,  functions,, etc., however equivalent devices can be designed which exhibit similar responses, although some of the mechanisms would only serve to regulate the flow curves rather than to contribute to the component's primary function.

In order for the model to be useful, the reader or student must have a substantial understanding of the model (hydraulic) system's principles. It also requires that the principles can be transferred to the target (electrical) system. Hydraulic systems are deceptively simple: the phenomenon of is a known, complex problem that few people outside of the fluid power or irrigation industries would understand. For those who do, the hydraulic analogy is amusing, as no "cavitation" equivalent exists in electrical engineering. The hydraulic analogy can give a mistaken sense of understanding that will be exposed once a detailed description of electrical circuit theory is required.

One must also consider the difficulties in trying to make an analogy match reality completely. The above "electrical friction" example, where the hydraulic analog is a pipe filled with sponge material, illustrates the problem: the model must be increased in complexity beyond any realistic scenario.