# Why do fundamental circuit laws break down at high frequency AC?

Actually, it is all about the waves. Even when dealing with DC, it is all managed by the electrical and magnetic fields and waves.

The "fundamental laws" aren't breaking down. The rules you have learned are simplifications that deliver accurate answers under certain conditions - you haven't yet learned the fundamental laws. You are about to learn the fundamental laws after having used simplifcations.

Part of the assumed conditions for the simplified rules is that the circuit is much smaller than the wave length of signal(s) involved. In those conditions, you can assume that a signal is in the same state across the circuit. That leads to a lot of simplifications in the equations describing the circuit.

As the frequencies get higher (or the circuits larger) so that the circuit is an appreciable fraction of the wavelength, that assumption is no longer valid.

The effects of wavelength on the operation of electrical circuits first became obvious at low frequencies but with very large circuits - telegraph lines.

When you start working with RF, you reach wavelengths such that the size of a circuit that sits on your desk is an appreciable fraction of the wavelength of the signals used.

So, you start having to pay attention to things you could conveniently ignore before.

The rules and equations you are now learning also apply to simpler, lower frequency circuits. You can use the new things to solve the simpler circuits- you just have to have more information and solve more complicated equations.

The fundamental laws of EM are Maxwell's Equations: $$\nabla \cdot \mathbf{E} = 4\pi\rho$$ $$\nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}$$ $$\nabla \times \mathbf{B} = \frac{1}{c}\left( 4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t}\right)$$

They have always been the fundamental laws of EM, but at lower frequencies, we find solving those multidimensional differential equations to be rather hard, and not all that beneficial to support our understanding of the circuit. You don't want to have to invoke symmetry to properly solve an an equation for propagation along a wire if the net difference between a short 18ga wire and a long 0000 wire is 0.0000001% with respect to the behaviors you are interested in.

Accordingly, people have already integrated these equations for simple cases, like wires at low frequencies, and found the equations you were given in earlier classes. Well, more precicely, we found these equations first, then found Maxwell's equations as we pushed deeper into EM, and then eventually showed that the original equations were consistent with Maxwell's.

Personally, I find it best to explore this by example. I'd like to take an example from the famous tome: The Art of High Speed Digital Design (subtitle: A Handbook of Black Magic). In their introduction, they point out how important capacitor type choices are. They make the extraordinary claim that at high speeds, a capacitor can look like an inductor because its leads are two parallel wires. Parallel wires have an inductance.

If we use the concept of impedance, we can calculate the effects of parasitic inductance on our capacitor. The impedance of a capacitor is \$\frac{-1}{\omega C}\$, and the impedance of an inductor is \$\omega L\$. We'll ignore parasitic resistance for now, though it's an important detail too in many cases. Put them in series and you see the impedence of the circuit \$\frac{-1}{\omega C} + \omega L = \frac{\omega^2 CL - 1}{\omega C}\$. As you can see, at high frequencies, that CL term starts to dominate, making the whole circuit look more like an inductor. At lower frequencies, where \$\omega^2CL \ll 1\$, you can ignore this. At high frequencies, you can't.

Likewise, at high frequencies, it gets harder to ignore the fact that wires emit EM radiation. At low frequencies, this effect is trivial, but at high frequencies, a large amount of power can be dissipated in the wire itself.

Because the assumptions required by the lumped element model are violated. The lumped element model is what allows you to analyze devices like resistors connected by nodes, without considering the physical layout of devices and the circuit.

The lumped element model assumes:

1. The change of the magnetic flux in time outside a conductor is zero.

$${\frac {\partial \phi _{B}}{\partial t}}=0$$

1. The change of the charge in time inside conducting elements is zero.

$${\frac {\partial q}{\partial t}}=0$$

1. The characteristic length (the ‘size’ of the nodes and devices) is much less than the wavelength of the signal of interest.

$$L_c << \lambda$$