# Transmission Line model [closed]

Can some one explain me why exactly classical circuit theory fails to account for the wave nature of EM fields and how the transmission line model accounts for it better

• Is there a definition of classical circuit theory that constrains it in such a way as to preclude transmission line theory? When I was taught, I don't remember such a distinction being made. Commented Jul 8, 2022 at 10:55
• By "classical circuit theory" do you mean "lumped elements modelisation" (as opposed to "distributed elements modelisation") ? Commented Jul 8, 2022 at 15:18

Really short answer¹: Because the speed of light is finite. Your circuit model works well for as long every cause and effect are instantly linked. That approximation works well for short (in terms of wavelengths) distances.

The model I was taught in the first semester of uni came with a pretty clear warning that, hey, this is a linearized, wavelengths >> circuit and component sizes model.

¹ to answer the question "why does that approximate transmission lines better", one would honestly have to answer "because the whole of transmission line theory. I think "teach me transmission lines" might be a bit broad for this platform. Other than that, this really seems to be your homework assignment...

Classical circuit theory is essentially a limited approximation of the general electrodynamics presented by J.C Maxwell about 160 years ago. Circuit theory assumes that if there's a voltage at the one end of a wire, the same voltage is at the same time also at the other end of the wire. The existence of electromagnetic waves in 3D space around the circuit and the propagation time of the electromagnetic wave are simply omitted to make practical applications simpler.

Actually the whole fact that practical circuits exist in the 3D space is omitted. There are only nodes and wires and other parts between them as a purely logical structure with no geometry. The omission doesn't harm if the dimensions of the circuit is small enough when related to the used signal frequencies or said better when related to the wavelengths at the operating frequencies. No strict limit exists, but generally if the longest dimension in the circuit is shorter than 10% of the wavelength the classic circuit theory is acceptably accurate.

As said, circuit theory omits the existence of waves in the space around the circuit. In practice circuits generally work also as antennas which radiate waves to the space, but that radiation is neglible if the dimensions of the circuit are small enough or they are specially designed to have such geometry (=transmission line) that the electromagnetic wave propagates along the wires, not outwards spreading further from the wires like it happens around antennas.

In circuit theory phenomenons in the circuits are assumed to happen as voltages between wire joints and currents through wires and other parts. People who understand electrodynamics know that actually everything in the circuits happens as varying 3D space vector fields (=electric, magnetic) which occur as well inside the materials as in the space around them. These fields cause such thing as current in conductors and they have also inter-dependencies such as induction (=changing magnetic field cause an electric field).

In circuit theory we generally centralize the time dependent field phenomenons to happen inside special parts like inductors, transformers and capacitors. They are presented with equations which bind together currents, voltages and time. An example: The voltage u between the ends of an ideal inductor L and the current i through the inductor obey the next equation:

u = L(di/dt) ; L is the inductance

If that equation is not true for a part, it isn't an inductor, but something else.

The law for capacitors is as simple, only change the places of current and voltage. Transformers are trickier, they need a couple of equations for one transformer. Transmission lines are even more trickier.

You obviously have seen how transmission lines are modeled as LC-ladder circuits. The losses in the materials are modeled by inserting also resistors. When a math limit equations are searched by letting the number of the ladder elements increase infinitely one gets practically applicable differential equations for transmission lines. They have been known since 1876 as Telegrapher's Equations. They present current and voltage waves through the circuit and really give usefully accurate results how transmission lines behave.

In physics that's nonsense, which happens to give useful results because the currents and voltages calculated properly in a transmission line by using Maxwell's 3D vector field equations happen to be quite the same, but the differences between the results of proper electrodynamics and results given by circuit theory (=Telegrapher's Equations) can be calculated and measured. Fortunately the differences are small in low loss transmission lines which have 2 parallel conductors. Making calculations in physically sound way with 3D vector fields would be far too complex for practical engineering work. But such calculations are needed to design accurate microwave parts and antennas.

Conclusion: Circuit theory rejects waves in the very start of the theory but the results are still usable in small enough circuits in frequencies low enough. To make the circuit theory usable with transmission lines a fake wave theory - Telegrapher's Equations - which uses voltage and current waves in a circuit is developed. It totally omits the actual 3D space wave around the wires, which for ex. in a coaxial cable propagates in the insulation layer between the conductors, but it still fortunately gives usable results in simple enough geometries

• >>> It totally omits the actual 3D space wave around the wires, which for ex. in a coaxial cable propagates in the insulation layer between the conductors, but it still, fortunately, gives usable results in simple enough geometries <<< ... These losses could be "included" in the distributed "resistance" and "conductance" of the line ... ? Commented Jul 8, 2022 at 15:24
• The wave in a coaxial cable is mostly in the insulator between the conductors. It's not loss, it's the transferred electric power. It doesn't flow inside the metal, no matter it can be calculated as P = U x I . Metal is needed to direct the wave along the cable without radiating to the space. The wires in a pair cable or twisted pair have the same role.
– user136077
Commented Jul 8, 2022 at 16:13