The Electronic-Hydraulic Analogy

Why bother?

I studied Chemical and Biological engineering instead of electrical, so I spent much more time learning about fluid flow and pressure drops than I did electronic circuits. Though our curriculum did include electronics as part of our control systems class, our exposure to circuits was far more limited than the other engineering disciplines.

I've always been fascinated by electronics though: playing with simple circuits and microcontroller projects in high school and making guitar effect pedals and audio circuits now.

Electricity is invisible. You can't see electrons moving through a wire, and the concepts can feel frustratingly abstract when you're first learning. Water, on the other hand, is intuitive—I already understand how it behaves and know the equations that govern it.

As I dug deeper into electronics, I noticed that many formulae had structural equivalents in domains I'd already encountered. The same governing equations—Fourier's Law from heat transfer, Poiseuille's law from fluid mechanics—create equivalent relationships between quantities, just as Ohm's law does for electricity.

This analogy holds up so well that it's been given its own name: the electronic-hydraulic analogy. It maps electrical concepts onto a system of pipes and water tanks. As I was learning analog electronics to make guitar pedals, I found returning to these conceptual comparisons extremely useful for building circuit intuition.

Charge → Volume

The most fundamental mapping: electrical charge (\(Q\), measured in coulombs [\(\text{C}\)]) corresponds to a volume of water (\(V\), measured in cubic meters [\(\text{m}^3\)]). Just as you can have a certain amount of water stored in a tank, you can have a certain amount of charge accumulated somewhere in a circuit.

A single electron carries a tiny, fixed charge, about \(1.6 \times 10^{-19}\) coulombs. Think of this as a single water molecule: a discrete, indivisible (if you break H2O into atoms, it's not really "water" anymore) unit of "stuff" that makes up what flows through the system. In practice you deal with bulk quantities and equations for emergent behavior, you rarely think about individual electrons or individual water molecules.

Wires → Pipes

Pipes are how we move our volumes of water around, and a conductor is simply a pipe for electrons. An ideal wire with zero resistance is like a perfectly smooth, frictionless pipe—water (charge) flows through with no pressure (voltage) drop. Real wires have some resistance, just as real pipes have friction losses.

The analogy extends to wire gauge: thicker wires have lower resistance, just as wider pipes allow more flow for the same pressure difference. This is why high-current applications require thick cables.

Current → Flow Rate

Current (\(I\), measured in amperes [\(\text{A}\)]) is the rate at which charge flows past a point. One amp equals one coulomb per second:

\[ I = \frac{dQ}{dt} \]

The hydraulic equivalent is volumetric flow rate, cubic meters per second passing through a cross-section of pipe. A fat pipe with water rushing through carries a high flow rate; a thin trickle has a low rate. Analogously, a thick copper wire carrying lots of electrons per second has high current; a thin wire with sparse electron flow has low current.

Similar conservation laws also apply: in a closed hydraulic system, mass is conserved. Water doesn't appear or disappear, what flows in (and isn't generated, consumed, or accumulated) must flow out. The same applies to charge in a circuit, and this is the basis of Kirchhoff's Loop Laws (the current law, namely).

Voltage → Pressure

Voltage (\(V\), measured in volts [\(\text{V}\)]) represents electrical potential difference between two points. This maps to pressure difference in the hydraulic world.

More precisely, voltage corresponds to hydraulic head—which accounts for both pressure and elevation. If you lay your hydraulic circuit down so that it's horizontal and all pipes are at the same height, you can simply think of it as pressure difference.

A battery is like a pump: it does work to maintain a pressure difference between its terminals. Water flows spontaneously from high pressure to low pressure; conventional current flows from high voltage to low voltage. Without a pressure difference, water sits still and without a voltage difference, no current flows.

The units reinforce this: a volt is a joule per coulomb (energy per unit charge), while pressure is energy per unit volume (joules per cubic metre). Both describe how much "push" is available to move their respective substances.

Resistance → Pipe Friction

Some pipes are easier to push water through than others, defined by its frictional resistance (viscous shear forces). A short, wide pipe with smooth walls will see water flow through easily. A long, narrow pipe with rough walls? You need a lot more pressure to get the same flow.

Electrical resistance (\(R\), measured in ohms [\(\Omega\)]) works the same way. A resistor is like a constriction in a pipe, it makes it harder for current to flow, and you need more voltage to push the same amount of charge through.

This is the satisfying part of the analogy: when you write out the governing equations in their fundamental form, they have identical structure. All transport phenomena follow the same pattern for flux (the rate of flow of a physical quantity through a given surface/area)

\[ \text{Flux} = \text{Conductance} \times \text{Gradient of Potential} \]

For electricity, this is Ohm's law in its microscopic form:

\[ \mathbf{J} = \sigma \mathbf{E} \]

where \(\mathbf{J}\) is current density [\(\text{A/m}^2\)], \(\sigma\) is electrical conductivity, and \(\mathbf{E}\) is the electric field [\(\text{V/m}\)]. Current density is just another flux (charge flowing through a cross-sectional area per unit time). The electric field is the gradient of voltage.

For laminar flow in pipes, Poiseuille's law has the same shape:

\[ Q = \frac{\pi r^4}{8 \mu L} \Delta P \]

If we rearrange this to match the macroscopic form of Ohm's law (\(\Delta V = IR\)):

\[ \Delta P = \underbrace{\frac{8 \mu L}{\pi r^4}}_{R_{\text{hydraulic}}} Q \]

That term in the middle is hydraulic resistance. It depends on viscosity \(\mu\), pipe length \(L\), and radius \(r\). (Note the a brutal fourth-power dependency on radius: halve the pipe diameter and resistance increases sixteenfold, plus pipe clogging tends to build upon itself leading to a runaway blockup).

A resistor in a circuit behaves exactly like a narrow section of pipe: it restricts flow and causes a pressure (voltage) drop proportional to the flow rate (current) passing through.

Limitations

All analogies eventually break down. The electronic-hydraulic analogy helps build basic intuition, but it fails when pushed to more advanced analyses:

• Compressibility. Real fluids compress slightly under pressure; charge doesn't accumulate or compress in wires. In circuits, current is the same everywhere in a stretch of wire.
• Fields and action at a distance. Electric fields can influence charge behavior without physical contact. There's no hydraulic equivalent to electromagnetic induction or the field that drives current.
• Propagation speed. Individual electrons drift at millimetres per second, but the electrical signal propagates near the speed of light. This doesn't happen in water.
• More complex components. The analogy starts to get strained for capacitors and inductors, and more complex components past that. Workable analogies exist, but the intuition starts to break down.

For understanding resistors, basic circuit behaviour, and building intuition about voltage and current, the analogy serves well. I'll explore component-specific analogies in future posts.

Reference

Electrical	Hydraulic
Charge \(Q\) [C]	Volume \(V\) [m³]
Current \(I\) [A = C/s]	Flow rate \(\dot{V}\) [m³/s]
Voltage \(V\) [V = J/C]	Pressure \(P\) [Pa = J/m³]
Resistance \(R\) [Ω = V/A]	Flow resistance [Pa·s/m³]
Wire	Pipe
Battery	Pump
Resistor	Constriction / Valve