EC-3.4 The Wave and the Diffusion Equation

What this is and why it exists

So far the unknown has depended on one variable, usually time. A great deal of electronics involves quantities that depend on position as well. Voltage along a line, a field in space, carrier concentration through a doped region.

Two partial differential equations account for most of it. They look almost identical on the page and behave completely differently, and seeing that contrast once is worth more than either derivation alone.

Both equations are derived twice in a typical degree, once in electromagnetics and once in device physics. Neither time is it done as mathematics, which is why the topic sits here instead. Meeting them together makes the later derivations short.

The vocabulary

• Partial derivative — a derivative with respect to one variable while the others are held fixed.
• Partial differential equation — an equation relating partial derivatives of an unknown that depends on more than one variable.
• Wave equation — the second derivative in time equals a constant times the second derivative in position.
• Diffusion equation — the first derivative in time equals a constant times the second derivative in position.
• Boundary condition — a statement about the unknown at the edges of the region.
• Separation of variables — assuming the answer is a function of position multiplied by a function of time.
• Propagation speed — the constant in the wave equation, the speed at which a shape travels.
• Diffusion length — the distance over which an injected excess falls to a small fraction of its starting value.

The mental model

The first thing that changes with two independent variables is what an arbitrary constant becomes. Solving an ordinary equation leaves you with constants. Solving a partial one leaves you with arbitrary functions, which is a much larger family. That is why boundary conditions carry so much more weight here.

Now the two equations, and the single derivative that separates them.

The wave equation has a second derivative in time. Its solutions are any shape moving to the left and any shape moving to the right, both at one fixed speed set by the constant. The shape is preserved as it travels. That preservation is the defining property of a wave. It is why a pulse launched down a good transmission line arrives at the far end looking like the pulse you sent.

The diffusion equation has a first derivative in time. It describes something flowing from where there is more of it toward where there is less. Nothing travels at a fixed speed. Instead sharp features smooth out fast, and an injected excess spreads and fades. Heat through a metal bar does this. So do minority carriers injected across a junction, which is why this equation appears in every treatment of semiconductor devices.

One derivative apart, opposite behaviour. It is worth pausing on why. A second derivative in time means the system has inertia, so it can overshoot and come back, which is oscillation. A first derivative means it has no inertia at all and can only relax.

The method to learn here is separation of variables. Assume the answer is a function of position multiplied by a function of time. Substitute it in and divide through by the product. You are left with an equation where one side depends only on position and the other only on time. Since they must be equal for every position and every time, both sides must equal the same constant. One partial equation has become two ordinary ones, each solvable by what you already know.

That gives you an infinite family of separated solutions, one for each allowed value of the constant. The boundary conditions then decide which combination of them fits, and the combination is a sum with coefficients. Working out those coefficients is exactly the Fourier series problem, which is the next place this course goes.

So the sequence has a reason. The Fourier series is not introduced as an ornament. It is introduced because separation of variables produces an infinite sum. That sum has to be matched to a boundary, and matching it is what the series is for.

What you should now be able to explain or do

• Tell a wave equation from a diffusion equation by inspection, and say what each does.
• Explain physically why one preserves a shape and the other smooths it away.
• Carry out separation of variables and say why both sides must equal a constant.
• Say why boundary conditions matter more here than for an ordinary equation.
• Name one place in electronics where each equation appears.

Check yourself