Transmission Line model [closed]

Question

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

Answer 1

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.

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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...

Answer 2

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.

Microwave circuits apply the latter approach. Low frequency circuits like an audio amplifier operates in so low frequencies that the radiation is neglible due the small dimensions when compared to the wavelength. But the radiation can often be measured easily to exist if the circuit has wires say 1% of the wavelength. For ex. an usual 88-108 MHz FM radio receiver can be often detected by using another usual radio receiver. It is sensitive enough to hear the stray radiation of another radio receiver at short distances, say 5 or 10 meters. In addition computer equipment radiate often so much noise that an usual radio receiver can be useless in the same room.

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