Analogy between a hydraulic circuit (left) and an electronic circuit (right).

The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. Electricity (as well as heat) was originally understood to be a kind of fluid, 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 paradigms (electronics and hydraulics).

There is no unique paradigm for establishing this analogy. Two paradigms can be used to introduce the concept to students:

• 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 head 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 electric potential with gravitational potential.
• Completely enclosed version with pumps providing pressure only; 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 hydraulic ohm 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 phasors in this definition, so possess a phase as well as magnitude.
• A slightly different paradigm is used in acoustics, where acoustic impedance 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 magnetic mirror 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).

## Hydraulic analogy: Hydraulic analogy with horizontal water flow

### Hydraulic analogy: Voltage, current, and charge

Hydraulic pressure difference

## Hydraulic analogy: Equation examples

Some examples of analogous electrical and hydraulic equations:

type hydraulic electric thermal mechanical
quantity volume ${\displaystyle V}$ [m] charge ${\displaystyle q}$ [C] heat ${\displaystyle Q}$ [J] momentum ${\displaystyle P}$ [Ns]
potential pressure ${\displaystyle p}$ [Pa=J/m=N/m] potential ${\displaystyle \phi }$ [V=J/C=W/A] temperature ${\displaystyle T}$ [K] velocity ${\displaystyle v}$ [m/s]
flux Volumetric flow rate ${\displaystyle \Phi _{V}}$ [m/s] current ${\displaystyle I}$ [A=C/s] heat transfer rate ${\displaystyle {\dot {Q}}}$ [J/s] force ${\displaystyle F}$ [N]
flux density velocity ${\displaystyle v}$ [m/s] current density ${\displaystyle j}$ [C/(m·s) = A/m²] heat flux ${\displaystyle {\dot {Q}}''}$ [W/m] stress ${\displaystyle \sigma }$ [N/m = Pa]
linear model Poiseuille's law ${\displaystyle \Phi _{V}={\frac {\pi r^{4}}{8\eta }}{\frac {\Delta p^{\star }}{\ell }}}$ Ohm's law ${\displaystyle j=-\sigma \nabla \phi }$ Fourier's law ${\displaystyle {\dot {Q}}''=\kappa \nabla T}$ Dashpot ${\displaystyle \sigma =c\Delta v}$

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

## Hydraulic analogy: 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.

Usefulness requires that the reader or student has 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 pump cavitation 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.

• Fluidics
• Hydraulic circuit
• Mechanical-electrical analogies

## Hydraulic analogy: Notes

1. Paul J. Nahin, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age, JHU Press, 2002 Buy book ISBN 0801869099 page 59
2. A. Akers, M. Gassman, & R. Smith, Hydraulic Power System Analysis. Taylor & Francis, New York, 2006, Chapter 13, Buy book ISBN 0-8247-9956-9.
3. A. Esposito, "A Simplified Method for Analyzing Circuits by Analogy". Machine Design, October 1969, pp. 173-177.
4. Brian J. Kirby, Micro- and Nanoscale Fluid Mechanics, p. 69, Cambridge University Press, 2010 Buy book ISBN 1139489836.
5. Schelleng, John C. "The violin as a circuit." The Journal of the Acoustical Society of America 35.3 (2005): 326-338. http://www.maestronet.com/forum/index.php?app=core&module=attach&section=attach&attach_id=13435
6. "Axial feedback stabilization of a flute mode in a simple mirror reactor, by M. A. Lieberman and S. L. Wong, Plasma Physics, Vol. 19, pp. 745-55 (1977). The article contains an L-C circuit that is unstable because the "capacitance" is negative: http://iopscience.iop.org/0032-1028/19/8/005/pdf/0032-1028_19_8_005.pdf
7. http://amasci.com/emotor/cap1.html
• Hyperphysics
• Hyperphysics 2 - Doesn't have the reservoir.
• With animations! - The paddle behaves like a resistor if the paddle itself is massless and has friction, and like an inductor if it is frictionless and has mass. Also has equivalent equations.
• Ohm's law
• Capacitor analogy
• Capacitor circuit analogy
• Newsgroup thread with lots of good analogies and a few bad ones
• Understanding Electricity - an analogy with water
• Water analogy to transistors Slightly better version would be something like an exhaust gas recirculation (EGR) valve
• Measuring electricity
• Plumbing analogy
• Animation
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